what i have learned in general mathematics essay

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Math Essay | Essay on Math for Students and Children in English

February 13, 2024 by Prasanna

Math Essay: Mathematics is generally defined as the science that deals with numbers. It involves operations among numbers, and it also helps you to calculate the product price, how many discounted prizes here, and If you good in maths so you can calculate very fast. Mathematicians and scientists rely on mathematics principles in their real-life to experiments with new things every day. Many students say that ” I hate mathematics ” and maths is a useless subject, but it is wrong because without mathematics your life is tough to survive. Math has its applications in every field.

You can also find more  Essay Writing  articles on events, persons, sports, technology and many more.

Long and Short Essays on Math for Students and Kids in English

We are presenting students with essay samples on an extended essay of 500 words and a short of 150 words on the topic of math for reference.

Long Essay on Math 500 Words in English

Long Essay on Math is usually given to classes 7, 8, 9, and 10.

Mathematics is one of the common subjects that we study since our childhood. It is generally used in our daily life. Every person needs to learn some basics of it. Even counting money also includes math. Every work is linked with math in some way or the other. A person who does math is called a Mathematician.

Mathematics can be divided into two parts. The first is Pure mathematics, and the second is Applied mathematics. In Pure mathematics, we need to study the basic concept and structures of mathematics. But, on the other side, Applied mathematics involves the application of mathematics to solve problems that arise in various areas,(e.g.), science, engineering, and so on.

One couldn’t imagine the world without math. Math makes our life systematic, and every invention involves math. No matter what action a person is doing, he should know some basic maths. Every profession involves maths. Our present-day world runs on computers, and even computer runs with the help of maths. Every development that happens requires math.

Mathematics has a wide range of applications in our daily life. Maths generally deals with numbers. There are various topics in math, such as trigonometry; integration; differentiation, etc. All the subjects such as physics; chemistry; economy; commerce involve maths in some way or the other. Math is also used to find the relation between two numbers, and math is considered to be one of the most challenging subjects to learn. Math includes various numbers, and many symbols are used to show the relation between two different numbers.

Math is complicated to learn, and one needs to focus and concentrate more. Math is logical sometimes, and the logic needs to be derived out. Maths make our life easier and more straightforward. Math is considered to be challenging because it consists of many formulas that have to be learned, and many symbols and each symbol generally has its significance.

Some of the advantages of Math in our daily life

• Managing Money: Counting money and calculating simple interest, compound interest includes the usage of mathematics. Profit and loss are also computed using maths. Anything related to maths contains maths.
• Cooking: Maths is even used in cooking as estimating the number of ingredients that have to be used is calculated in numbers. Proportions also include maths.
• Home modelling: Calculating the area is essential in the construction of the home or home modelling. The size is also measured using maths. Even heights are also measured using maths.
• Travelling: Distance between two places and time taken to travel also includes maths. The amount of time taken revolves around maths. Almost every work is related to maths in some way or another. Maths contains some conditions that need to be followed, and maths has several formulas that have to be learned to become a mathematician.

Short Essay on Math 150 Words in English

Short Essay on Math is usually given to classes 1, 2, 3, 4, 5, and 6.

Maths is generally defined as the science of numbers and the operations performed among them. It deals with both alphabets along with numbers and involves addition, subtraction, multiplication, division, comparison, etc. It is used in every field. Maths consists of finding a relation between numbers, calculating the distance between two places, counting money, calculating profit and loss.

It is of two types pure and applied. Pure math deals with the basic structure and concept of maths, whereas applied mathematics deals with how maths is used it involves the application of maths in our daily life. All the subjects include maths, and hence maths is considered to be one of the primary and joint issues which need to be learned by everyone. One couldn’t imagine their life using maths. It has made our experience easy and straightforward. It has prevented chaos in our daily life. Hence learning maths is mandatory for everyone.

10 Lines on Math in English

• Father of Mathematics was Archimedes.
• Hypatia is the first woman know to know to have taught mathematics.
• From 0-1000 ,letter “A” only appears first in 1,000 ( “one thousand “).
• Zero (0) is the only number that can not be represented by Roman numerals.
• The Sign plus (+) and Minus(-) were discovered in 1489 A.D.
• Do you know that a Baseball field is of the perfect shape of a Rhombus.
• Jiffy is considered to be a unit of time for 1/100th of a second.
• 14th March International Day of Mathematics.
• Most mathematics symbols weren’t invented until the 16th century.
• The symbols for the division is called an Obelus.

FAQ’s on Math Essay

Question 1. What is Mathematics in simple words?

Answer: Mathematics is the study of shapes, patterns, numbers, and more. It involves a comparison between two numbers and calculating the distance between two places.

Question 2. Do we need mathematics every day?

Answer: Yes, we need mathematics every day, from buying a product to sell anything you want. Maths is present in our daily life, and no matter what work we do, maths is involved, and the application of maths is current in our everyday life.

Question 3. Who was the No.1 Mathematicians in the world?

Answer: Isaac Newton, who was a profound mathematician, is considered to be one of the best mathematicians in the world.

Question 4. What are the applications of maths?

Answer: Maths have various applications in our daily life. Maths is present everywhere from counting money to the calculating distance between two places. We could find math applications around.

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“What I learned from Studying Mathematics for 15 Years”

Mathematics has been an integral part of my life for the past 15 years, shaping my thinking, problem-solving skills, and overall approach to learning. Throughout my academic journey, from elementary school to high school and beyond, the subject of mathematics has provided me with invaluable lessons and experiences that have had a profound impact on my personal and intellectual growth. In this essay, I will reflect on the multitude of ways in which I have learned and benefited from mathematics over 15 years.

One of the fundamental lessons I have learned from mathematics is the importance of a strong foundation. Mathematics builds upon itself, with each concept and skill acting as a building block for the next. In my early years of learning mathematics, I focused on understanding numbers, basic operations, and simple equations. This laid the groundwork for more advanced topics such as algebra, geometry, and calculus. By emphasizing the importance of mastering foundational concepts, mathematics taught me the value of patience, perseverance, and the ability to build knowledge incrementally.

Moreover, mathematics has honed my critical thinking and problem-solving abilities. Through solving mathematical problems, I have developed a logical and systematic approach to tackling complex challenges. Mathematics has taught me to break down problems into smaller, more manageable parts and to apply various problem-solving strategies to find solutions. It has instilled in me the importance of precision, attention to detail, and the ability to think both analytically and creatively. These skills have proven to be immensely useful not only in mathematical contexts but also in other areas of life, such as decision-making, planning, and problem-solving in general.

In addition to analytical skills, mathematics has enhanced my ability to think abstractly. Concepts such as algebraic expressions, geometric shapes, and mathematical proofs have pushed me to think beyond the concrete and engage with abstract ideas. This has expanded my capacity for imagination and creativity, enabling me to approach challenges from different perspectives. Mathematics has shown me that there can be multiple pathways to arrive at a solution and has encouraged me to think outside the box. This flexibility of thinking has been invaluable in various academic disciplines and real-world scenarios.

Furthermore, mathematics has fostered in me a strong sense of discipline and perseverance. Solving complex mathematical problems requires patience, persistence, and the willingness to put in the necessary effort. Mathematics has taught me that success often comes after multiple attempts, failures, and revisions. It has shown me that setbacks and mistakes are opportunities for growth and learning. This mindset has been a guiding principle in my life, empowering me to embrace challenges, push through obstacles, and persist in the face of adversity.

Moreover, mathematics has deepened my appreciation for precision and accuracy. In mathematics, there is no room for ambiguity or approximation. The discipline demands precise calculations, clear explanations, and rigorous proof. This emphasis on accuracy has permeated other aspects of my life, such as writing, research, and problem-solving in general. Mathematics has taught me the importance of attention to detail and the need to justify and validate my assertions with sound reasoning and evidence. These skills have been instrumental in my academic pursuits and have helped me develop a meticulous approach to any task I undertake.

Beyond the cognitive and academic benefits, mathematics has also provided me with a sense of wonder and beauty. The elegance and symmetry inherent in mathematical principles and patterns have sparked my curiosity and ignited a lifelong passion for the subject. Mathematics has shown me that there is an inherent beauty in the way numbers, shapes, and equations interconnect and interact with each other. This aesthetic appreciation has enriched my worldview and fostered a deeper appreciation for the beauty and orderliness of the natural world.

In conclusion, my journey with mathematics over the past 15 years has been transformative and enriching. The subject has taught me far more than just numbers and formulas. It has imparted valuable life lessons, nurtured critical thinking and problem-solving skills cultivated abstract thinking and creativity, instilled discipline and perseverance, fostered precision and accuracy, and awakened a sense of wonder and beauty. Mathematics has shaped the person I am today and has equipped me with a strong foundation for lifelong learning and growth. As I continue to embark on new academic and professional endeavors, I carry with me the invaluable lessons and experiences gained from 15 years of learning mathematics.

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Active Learning in Mathematics, Part IV: Personal Reflections

By Benjamin Braun, Editor-in-Chief , University of Kentucky; Priscilla Bremser, Contributing Editor , Middlebury College; Art Duval,  Contributing Editor , University of Texas at El Paso; Elise Lockwood,  Contributing Editor , Oregon State University; and Diana White,  Contributing Editor , University of Colorado Denver.

Editor’s note: This is the fourth article in a series devoted to active learning in mathematics courses.  The other articles in the series can be found here .

In contrast to our first three articles in this series on active learning, in this article we take a more personal approach to the subject.  Below, the contributing editors for this blog share aspects of our journeys into active learning, including the fundamental reasons we began using active learning methods, why we have persisted in using them, and some of our most visceral responses to our own experiences with these methods, both positive and negative.  As is clear from these reflections, mathematicians begin using active learning techniques for many different reasons, from personal experiences as students (both good and bad) to the influence of colleagues, conferences, and workshops.  The path to active learning is not always a smooth one, and is almost always a winding road.

Because of this, we believe it is important for mathematics teachers to share their own experiences, both positive and negative, in the search for more meaningful student engagement and learning.   We invite all our readers to share their own stories in the comments at the end of this post.  We also recognize that many other mathematicians have shared their experiences in other venues, so at the end of this article we provide a collection of links to essays, blog posts, and book chapters that we have found inspirational.

There is one more implicit message contained in the reflections below that we want to highlight.  All mathematics teachers, even those using the most ambitious student-centered methods, use a range of teaching techniques combined in different ways.  In our next post, we will dig deeper into the idea of instructor “telling” to gain a better understanding of how an effective balance can be found between the process of student discovery and the act of faculty sharing their expertise and experience.

Priscilla Bremser:

I began using active learning methods for several reasons, but two interconnected ones come to mind.  First, Middlebury College requires all departments to contribute to the First-Year Seminar program, which places every incoming student into a small writing-intensive class. The topic is chosen by the instructor, while guidelines for writing instruction apply to all seminars.  As I have developed and taught my seminars over the years, I’ve become convinced that students learn better when they are required to express themselves clearly and precisely, rather than simply listening or reading.  At some point it became obvious that the same principle applies in my other courses as well, and hence I was ready to try some of the active learning approaches I’d been hearing about at American Mathematical Society meetings and reading about in journals .

Second, I got a few student comments on course evaluations, especially for Calculus courses, that suggested I was more helpful in office hours than in lecture.  Thinking it through, I realized that in office hours, I routinely and repeatedly ask students about their own thinking, whereas in lecture, I was constantly making assumptions about student thinking, and relying on their responses to “Any questions?” for guidance, which didn’t elicit enough information to address the misunderstandings around the room. One way to make class more like office hours is to put students into small groups. I then set ground rules for participation and ask for a single set of problem solutions from each group. This encourages everyone to speak some mathematics in each class session, and to ask for clarity and precision from classmates.  Because I’m joining each conversation for a while, I get a more accurate perception of students’ comprehension levels.

This semester I’m teaching Mathematics for Teachers, using an IBL textbook by Matthew Jones . I’ve already seen several students throw fists up in the air, saying “I get it now!  That’s so cool!” How well I remember having that response to my first Number Theory course; it’s why I went into teaching at this level in the first place.  On the other hand, a Linear Algebra student who insists that  “I learn better from reading a traditional textbook” leaves me feeling rather deflated. It seems that I’ve failed to convey why I direct the course the way that I do, or at least I haven’t yet succeeded.  The truth is, though, that I used to feel the same way.  I regarded mathematics as a solitary pursuit, in which checking in with classmates was a sign of weakness.  Had I been required to discuss my thinking regularly during class and encouraged to do so between sessions, I would have developed a more solid foundation for my later learning. Remembering this inspires me to be intentional with students, and explain repeatedly why I direct my courses the way that I do.  Most of them come around eventually.

Elise Lockwood:

I have a strong memory of being an undergraduate in a discrete mathematics course, trying desperately to understand the formulas for permutations, combinations, and the differences between the two. The instructor had presented the material, perhaps providing an example or two, but she had not provided an opportunity for us to actively explore and understand why the formulas might make sense. By the time I was working on homework, I simply tried (and often failed) to apply the formulas I had been given. I strongly disliked and feared counting problems for years after that experience. It wasn’t until much later that I took a combinatorics course as a master’s student. Here, the counting material was brought to life as we were given opportunities to work through problems during class, to unpack formulas, and to come to understand the subtlety and wonder of counting. The teacher did not simply present a formula and move on, assuming we understood it. Rather, he persisted by challenging us to make sense of what was going on in the problems we solved.

For example, we once were discussing a counting problem in class (I can’t recall if it was an in-class problem or a problem that had been assigned for homework). During this discussion, it became clear that students had answered the problem in two different ways — both of them seemed to make sense logically, but they did not yield the same numerical result. The instructor did not just tell us which answer was right, but he used the opportunity to have us consider both answers, facilitating a (friendly) debate among the class about which approach was correct. We had to defend whichever answer we thought was correct and critique the one we thought was incorrect. This had the effect not only of engaging us and piquing our curiosity about a correct solution, but it made us think more carefully and deeply about the subtleties of the problem.

Now, studying how students solve counting problems is the primary focus of my research in mathematics education. My passion for the teaching and learning of counting was probably in large part formed by the frustrations I felt as an undergraduate and the elation I later experienced when I actually understood some of the fundamental ideas.

When I have been given the opportunity to teach counting over the years (in discrete mathematics or combinatorics classes, or in courses for pre-service teachers), I have tried my hardest to facilitate my students’ active engagement with the material during class. This has not taken an inordinate amount of time or effort: instead of just giving students the formulas off the bat, I give them a series of counting problems that both introduce counting as a problem solving activity and motivate (and build up to) some key counting formulas. For example, students are given problems in which they list some outcomes and appreciate the difference between permutations and combinations firsthand. I have found that a number of important issues and ideas (concerns about order, errors of overcounting, key binomial identities) can emerge on their own through the students’ activity, making any subsequent discussion or lecture much more meaningful for students. When I incorporate these kinds of activities for my students, I am consistently impressed at the meaning they are able to make of complex and notoriously tricky ideas.

More broadly, these pedagogical decisions I make are also based on my belief about the nature of mathematics and the nature of what it means to learn mathematics. Through my own experiences as a student, a teacher, and a researcher, I have become convinced that providing students with opportunities to actively engage with and think about mathematical concepts — during class, and not just on their own time — is a beneficial practice. My experience with the topic of counting (something near and dear to my heart) is but one example of the powerful ways in which student engagement can be leverage for deep and meaningful mathematical understanding.

Diana White:

What stands out most to me as I reflect upon my journey into active learning is not so much how or why I got involved, but the struggles that I faced during my first few years as a tenure-track faculty member as I tried to switch from being a good “lecturer” to all out inquiry-based learning.  I was enthusiastic and ambitious, but lacking in the skills to genuinely teach in the manner in which I wanted.

As a junior faculty member, I was already sold on the value of inquiry-based learning and student-centered teaching.  I had worked in various ways with teachers as a graduate student at the University of Nebraska and as a post-doc at the University of South Carolina, including teaching math content courses for elementary teachers and assisting with summer professional development courses for teachers.  Then, the summer before I started my current position, I attended both the annual Legacy of R.L. Moore conference and a weeklong workshop on teaching number theory with IBL through the MAA PREP program.  The enthusiasm and passion at both of these was contagious.  

However, upon starting my tenure track position, I jumped straight in, with extremely ambitious goals for my courses and my students, ones for which I did not have the skills to implement yet.  In hindsight, it was too much for me to try to both switch from being a good “lecturer” to doing full out IBL and running an intensely student centered classroom, all while teaching new courses in a new place.  I tried to do way too much too soon, and in many ways that was not healthy for either me or the students, as evidenced by low student evaluations and frustrations on both sides.

Figuring out specifically what was going wrong was a challenge, though.  Those who came to observe, both from my department and our Center for Faculty Development, did not find anything specific that was major, and student comments were somewhat generic – frustration that they felt the class was disorganized and that they were having to teach themselves the material.  

I thus backtracked to more in the center of the spectrum, using an interactive lecture  Things smoothed out and students became happier.  What I am not at all convinced of, though, is that this decision was best for student learning.  Despite the unhappiness on both our ends when I was at the far end of the active learning spectrum, I had ample evidence (both from assessments and from direct observation of their thought processes in class) that students were both learning how to think mathematically and building a sense of community outside the classroom.  To this day, I feel torn, like I made a decision that was best for student satisfaction, as well as for how my colleagues within my department perceive me.  Yet I remain convinced that my students are now learning less, and that there are students who are not passing my classes who would have passed had I taught using more active learning. (It was impossible to “hide” with my earlier classes, due to the natural accountability built into the process, so struggling students had to confront their weaknesses much sooner.)

It is hard for me to look back with regrets, as the lessons learned have been quite powerful and no doubt shaped who I am today.  However, I would offer some thoughts, aimed primarily at junior faculty.  

Don’t be afraid to start slow.  Even if it’s not where you want to end up, just getting started is still an important first step.  Negative perceptions from students and colleagues are incredibly hard to overcome.

Don’t underestimate the importance of student buy-in, or of faculty buy-in.  I found many faculty feel like coverage and exposure are essential, and believe strongly that performance on traditional exams is an indicator of depth of knowledge or ability to think mathematically.

Don’t be afraid to politely request to decline teaching assignments.  When I was asked to teach the history of mathematics, a course for which I had no knowledge of or background in, I wasn’t comfortable asking to teach something else instead.  While it has proved really beneficial to my career (I’m now part of an NSF grant related to the use of primary source projects in the undergraduate mathematics classroom), I was in no way qualified to take that on as a first course at a new university.

I have personally gained a tremendous amount from my participation in the IBL community, perhaps most importantly a sense of community with others who believe strongly in active learning.  

My first experience with active learning in mathematics was as a student at the Hampshire College Summer Studies in Mathematics program during high school.  Although I’d had good math teachers in junior high and high school, this was nothing like I’d seen before: The first day of class, we spent several hours discussing one problem (the number of regions formed in 3-dimensional space by drawing \(n\) planes), drawing pictures and making conjectures; the rest of the summer was similar.  The six-week experience made such an impression on me, that (as I realized some years later) most of the educational innovations I have tried as a teacher have been an attempt to recreate that experience in some way for my own students.

When I was an undergraduate, I noticed that classes where all I did was furiously take notes to try to keep up with the instructor were not nearly as successful for me as those where I had to do something.  Early in my teaching career, I got a big push towards using active learning course structures from teaching “ reform calculus ” and courses for future elementary school teachers.  In each case, this was greatly facilitated by my sitting in on another instructor’s section that already incorporated these structures.  Later I learned, through my participation in a K-16 mathematics alignment initiative , the importance of conceptual understanding among the levels of cognitive demand , and this helped me find the language to describe what I was trying to achieve.

Over time, I noticed that students in my courses with more active learning seemed to stay after class more often to discuss mathematics with me or with their peers, and to provide me with more feedback about the course.  This sort of engagement, in addition to being good for the students, is very addictive to me.  My end-of-semester course ratings didn’t seem to be noticeably different, but the written comments students submitted were more in-depth, and indicated the course was more rewarding in fundamental ways.  As with many habits, after I’d done this for a while, it became hard not to incorporate at least little bits of interactivity (think-pair-share, student presentation of homework problems), even in courses where external forces keep me from incorporating more radical active learning structures.

Of course, there are always challenges to overcome.  The biggest difficulty I face with including any sort of active learning is how much more time it takes to get students to realize something than it takes to simply tell them.  I also still find it hard to figure out the right sort of scaffolding to help students see their way to a new concept or the solution to a problem.  Still, I keep including as much active learning as I can in each course.  The parts of classes I took as a student (going back to junior high school) that I remember most vividly, and the lessons I learned most thoroughly, whether in mathematics or in other subjects, were the activities, not the lectures.  Along the same lines, I occasionally run into former students who took my courses many years ago, and it’s the students who took the courses with extensive active learning, much more than those who took more traditional courses, who still remember all these years later details of the course and how much they learned from it.

Other Essays and Reflections:

Benjamin Braun, The Secret Question (Are We Actually Good at Math?), http://blogs.ams.org/matheducation/2015/09/01/the-secret-question-are-we-actually-good-at-math/

David Bressoud, Personal Thoughts on Mature Teaching, in How to Teach Mathematics, 2nd Edition , by Steven Krantz, American Mathematical Society, 1999.   Google books preview

Jerry Dwyer, Transformation of a Math Professor’s Teaching, http://blogs.ams.org/matheducation/2014/06/01/transformation-of-a-math-professors-teaching/

Oscar E. Fernandez, Helping All Students Experience the Magic of Mathematics, http://blogs.ams.org/matheducation/2014/10/10/helping-all-students-experience-the-magic-of-mathematics/

Ellie Kennedy, A First-timer’s Experience With IBL, http://maamathedmatters.blogspot.com/2014/09/a-first-timers-experience-with-ibl.html

Bob Klein, Knowing What to Do is not Doing, http://maamathedmatters.blogspot.com/2015/07/knowing-what-to-do-is-not-doing.html

Evelyn Lamb, Blogs for an IBL Novice, http://blogs.ams.org/blogonmathblogs/2015/09/21/blogs-for-an-ibl-novice/

Carl Lee, The Place of Mathematics and the Mathematics of Place, http://blogs.ams.org/matheducation/2014/10/01/the-place-of-mathematics-and-the-mathematics-of-place/

Steven Strogatz, Teaching Through Inquiry: A Beginner’s Perspectives, Parts I and II,  http://www.artofmathematics.org/blogs/cvonrenesse/steven-strogatz-reflection-part-1,  http://www.artofmathematics.org/blogs/cvonrenesse/steven-strogatz-reflection-part-2

Francis Su, The Lesson of Grace in Teaching, http://mathyawp.blogspot.com/2013/01/the-lesson-of-grace-in-teaching.html

2 Responses to Active Learning in Mathematics, Part IV: Personal Reflections

In response to Priscilla Bremser, I feel as though it is almost elementary that students who are able to precisely express themselves are better to understand the information conceptually. What I mean by this is that the students who are able to interact with the information will get a better idea of what that information means conceptually rather than the students who simply listen to lecturing.

In regards to your second point, I also find this point to be important, even though it may seem obvious. Similarly to your first point, students who get more personal interaction with the instructor will probably be more likely to understand the information that is being presented. Since I am still in school, we have been discussing the best ways to prompt questions from students. Asking “are there any questions” is not a good way to do this. Breaking up into groups is a good way to see where the students are at conceptually.

However, this may prove to be tricky at the college level because of class size. One way to battle this is to ask for thumbs (either up, down, or in the middle) as to whether they understand the information being presented. This practice will give you a good idea at where the class is as a whole in a quick snapshot and students will be less likely to feel as though they are being singled out.

A few points in this post resonated with me particularly well. First, when Priscilla said that she was more helpful in office hours than in lecture because she asked students about their own thinking in the former, I agreed with it from a student’s perspective. Making class feel more like office hours, with more one-on-one time, helps students feel more like individual learners in the classroom. By suggesting small group work in order to facilitate more participation and allow for more analysis of each student’s performance, I feel that Bremser is acknowledging the ineffectiveness of using the phrase “Any questions”, which is something I try not to use, and hate to hear in my college classes. I also can relate to what Diana White says about trying to switch teaching styles as you would flip a switch. Not having the skills necessary to be at the level you want will be frustrating, and I know that as a future teacher, I will want to be successful right out of the gate. I know that this is unreasonable, and largely impossible, but this is more of a personality flaw that I will have to suppress. When it comes to being evaluated by others, I will have to recognize that many of my evaluators were once young teachers themselves, with the same aspirations, the same experience, and probably the same results as me. I will have to be patient, and use their feedback (and my own) to improve my teaching over time, rather than overnight. I wonder if this is a good assessment of what I should expect of myself when I begin teaching.

Comments are closed.

Opinions expressed on these pages were the views of the writers and did not necessarily reflect the views and opinions of the American Mathematical Society.

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25 Interesting Math Topics: How to Write a Good Math Essay

writing good math essay

Mathematics is a fascinating world of numbers, shapes, and patterns. 

Whether you are a student looking to grasp math concepts or someone who finds math intriguing, these topics will spark your curiosity and help you discover the beauty of mathematics straightforwardly and engagingly.

In this article, I will explore interesting math topics that make this subject not only understandable but also enjoyable.

Why Write About Mathematics

First, it helps demystify a subject that many find intimidating. By breaking down complex mathematical concepts into simple, understandable language, we can make math accessible to a wider audience, fostering greater understanding and appreciation.

Second, writing about mathematics allows us to showcase the practical applications of math in everyday life, from managing personal finances to solving real-world problems.

This helps readers recognize the relevance of math and its role in various fields and industries.

Additionally, writing about mathematics can inspire curiosity and a love for learning.

It encourages critical thinking and problem-solving skills, promoting intellectual growth and academic success.

Finally, mathematics is a universal language that transcends cultural and linguistic barriers.

After discussing math topics, we can connect with a global audience, fostering a sense of unity and collaboration in the pursuit of knowledge

 25 Interesting Math Topics to Write On

 Mathematics is a vast and intriguing field, offering a multitude of interesting topics to explore and write about.

Here are 25 such topics that promise to engage both math enthusiasts and those seeking a deeper understanding of this fascinating subject.

1. Fibonacci Sequence: Delve into the mesmerizing world of numbers with this sequence, where each number is the sum of the two preceding ones.

2. Golden Ratio: Explore the ubiquity of the golden ratio in art, architecture, and nature.

3. Prime Numbers: Investigate the mysterious properties of prime numbers and their role in cryptography.

4. Chaos Theory: Understand the unpredictability of chaotic systems and how small changes can lead to drastically different outcomes.

5. Game Theory: Examine the strategies and decision-making processes behind games and real-world situations.

6. Cryptography: Uncover the mathematical principles behind secure communication and encryption.

7. Fractals: Discover the self-replicating geometric patterns that occur in nature and mathematics.

8. Probability Theory: Dive into the world of uncertainty and randomness, where math helps us make informed predictions.

9. Number Theory: Explore the properties and relationships of integers, including divisibility and congruence.

10. Geometry of Art: Analyze how geometry and math principles influence art and design.

11. Topology: Study the properties of space that remain unchanged under continuous transformations, leading to the concept of “rubber-sheet geometry.”

12. Knot Theory: Investigate the mathematical study of knots and their applications in various fields.

13. Number Systems: Learn about different number bases, such as binary and hexadecimal, and their significance in computer science.

14. Graph Theory: Explore networks, relationships, and the mathematics of connections.

15. The Monty Hall Problem: Delight in this famous probability puzzle based on a game show scenario.

16. Calculus: Examine the principles of differentiation and integration that underlie a wide range of scientific and engineering applications.

17. The Riemann Hypothesis: Consider one of the most famous unsolved problems in mathematics involving the distribution of prime numbers.

18. Euler’s Identity: Marvel at the beauty of Euler’s equation, often described as the most elegant mathematical formula.

19. The Four-Color Theorem: Uncover the fascinating problem of coloring maps with only four colors without adjacent regions sharing the same color.

20. P vs. NP Problem: Delve into one of the most critical unsolved problems in computer science, addressing the efficiency of algorithms.

21. The Bridges of Konigsberg: Explore a classic problem in graph theory that inspired the development of topology.

22. The Birthday Paradox: Understand the surprising likelihood of shared birthdays in a group.

23. Non-Euclidean Geometry: Step into the world of geometries where Euclid’s parallel postulate doesn’t hold, leading to intriguing alternatives like hyperbolic and elliptic geometry.

24. Perfect Numbers: Learn about the properties of numbers that are the sum of their proper divisors.

25. Zero: The History of Nothing: Trace the historical and mathematical significance of the number zero and its role in the development of mathematics.

How to Write a Good Math Essay

Mathematics essays , though often perceived as daunting, can be a rewarding way to delve into the world of mathematical concepts, problem-solving, and critical thinking.

Whether you are a student assigned to write a math essay or someone who wants to explore math topics in-depth, this guide will provide you with the key steps to write a good math essay that is clear, concise, and engaging.

1. Understanding the Essay Prompt

Before you begin writing, it’s crucial to understand the essay prompt or question.

Analyze the specific topic, the scope of the essay, and any guidelines or requirements provided by your instructor.

Mostly, this initial step sets the direction for your essay and ensures you stay on topic.

2. Research and Gather Information

You need to gather relevant information and resources to write a strong math essay. This includes textbooks, academic papers, and reputable websites.

Make sure to cite your sources properly using a recognized citation style such as APA, MLA, or Chicago.

3. Structuring Your Math Essay

Start with a clear introduction that provides an overview of the topic and the main thesis or argument of your essay. This section should capture the reader’s attention and present a roadmap for what to expect.

The body of your essay is where you present your arguments, explanations, and evidence. Use clear subheadings to organize your ideas. Ensure that your arguments are logical and well-structured.

Begin by defining any important mathematical concepts or terms necessary to understand your topic.

Clearly state your main arguments or theorems. Please support them with evidence, equations, diagrams, or examples.

Explain the logical steps or mathematical reasoning behind your arguments. This can include proofs, derivations, or calculations.

Ensure your writing is clear and free from jargon that might confuse the reader. Explain complex ideas in a way that’s accessible to a broader audience.

Whenever applicable, include diagrams, graphs, or visual aids to illustrate your points. Visual representations can enhance the clarity of your essay.

Summarize your main arguments, restate your thesis, and offer a concise conclusion. Address the significance of your findings and the implications of your research or discussion.

4. Proofreading and Editing

Once you’ve written your math essay, take the time to proofread and edit it. Pay attention to grammar, spelling, punctuation, and the overall flow of your writing.

Ensure that your essay is well-organized and free from errors.

Consider seeking feedback from peers or an instructor to gain a fresh perspective.

5. Presentation and Formatting

A well-presented essay is more likely to engage the reader. Follow these formatting guidelines:

• Use a legible font (e.g., Times New Roman or Arial) in a standard size (12-point).
• Double-space your essay and include page numbers if required.
• Create a title page with your name, essay title, course information, and date.
• Use section headings and subheadings for clarity.
• Include a reference page to cite your sources appropriately.

6. Mathematical Notation and Symbols

Mathematics relies heavily on notation and symbols. Ensure that you use mathematical notation correctly and consistently.

If you introduce new symbols or terminology, define them clearly for the reader’s understanding.

7. Seek Clarification

If you encounter difficulties or ambiguities in your math essay, don’t hesitate to seek clarification from your instructor or peers.

Discussing complex mathematical concepts with others can help you refine your understanding and improve your essay.

8. Practice and Feedback

Writing math essays, like any skill, improves with practice. The more you write and receive feedback, the better you’ll become.

Take your time with initial challenges. Instead, view them as opportunities for growth and learning.

With dedication and attention to detail, you can craft a math essay that not only conveys your mathematical knowledge but also engages and informs your readers.

Josh Jasen or JJ as we fondly call him, is a senior academic editor at Grade Bees in charge of the writing department. When not managing complex essays and academic writing tasks, Josh is busy advising students on how to pass assignments. In his spare time, he loves playing football or walking with his dog around the park.

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My Reflection in Mathematics in the Modern World

Related Papers

Cheryl Praeger

as the wonderings about the status of school mathematics are becoming louder and louder, the need for a revision of our reasons can no longer be ignored. In what follows, I respond to this need by taking a critical look at some of the most popular arguments for the currently popular slogan, “Mathematics for all.” This analysis is preceded by a proposal of how to think about mathematics so as to loosen the grip of clichés and to shed off hidden prejudice. It is followed by my own take on the question of what mathematics to teach, to whom, and how.

Ten pages paper, will be presented at '5th International …

Mette Andresen

As the time enters the 21st century, sciences such as those of theoretical physics, complex system and network, cytology, biology and economy developments change rapidly, and meanwhile, a few global questions constantly emerge, such as those of local war, food safety, epidemic spreading network, environmental protection, multilateral trade dispute, more and more questions accompanied with the overdevelopment and applying the internet, · · · , etc. In this case, how to keep up mathematics with the developments of other sciences? Clearly, today's mathematics is no longer adequate for the needs of other sciences. New mathematical theory or techniques should be established by mathematicians. Certainly, solving problem is the main objective of mathematics, proof or calculation is the basic skill of a mathematician. When it develops in problem-oriented, a mathematician should makes more attentions on the reality of things in mathematics because it is the main topic of human beings.

Amarnath Murthy

There is nothing in our lives, in our world, in our universe, that cannot be expressed with mathematical theories, numbers, and formulae. Mathematics is the queen of science and the king of arts; to me it is the backbone of all systems of knowledge. Mathematics is a tool that has been used by man for ages. It is a key that can unlock many doors and show the way to different logical answers to seemingly impossible problems. Not only can it solve equations and problems in everyday life, but it can also express quantities and values precisely with no question or room for other interpretation. There is no room for subjectivity. Though there is a lot of mathematics in politics, there is no room for politics in mathematics. Coming from a powerful leader two + two can not become five it will remain four. Mathematics is not fundamentally empirical —it does not rely on sensory observation or instrumental measurement to determine what is true. Indeed, mathematical objects themselves cannot be observed at all! Mathematics is a logical science, cleanly structured, and well-founded. Mathematics is obviously the most interesting, entertaining, fascinating, exciting, challenging, amazing, enthralling, thrilling, absorbing, involving, fascinating, mesmerizing, satisfying, fulfilling, inspiring, mindboggling, refreshing, systematic, energizing, satisfying, enriching, engaging, absorbing, soothing, impressive, pleasing, stimulating, engrossing, magical, musical, rhythmic, artistic, beautiful, enjoyable, scintillating, gripping, charming, recreational, elegant, unambiguous, analytical, hierarchical, powerful, rewarding, pure, impeccable, useful, optimizing, precise, objective, consistent, logical, perfect, trustworthy, eternal, universal subject in existence full of eye catching patterns.

Journal of Humanistic Mathematics

Gizem Karaali

Katja Lengnink

Mathematics plays a dominant role in today's world. Although not everyone will become a mathematical expert, from an educational point of view, it is key for everyone to acquire a certain level of mathematical literacy, which allows reflecting and assessing mathematical processes important in every day live. Therefore the goal has to be to open perspectives and experiences beyond a mechanical and tight appearance of the subject. In this article a framework for the integration of reflection and assessment in the teaching practice is developed. An illustration through concrete examples is given.

Swapna Mukhopadhyay

Michele Emmer

It is no great surprise that mathematical structures and ideas, conceived by human beings, can be applied extremely effectively to what we call the "real" world. We need only to think of physics, astronomy, meteorology, telecommunications, biology, cryptography, and medicine. But that's not all mathematics has always had strong links with music, literature, architecture, arts, philosophy, and more recently with theatre and cinema

Liliya Samigullina

The article considers mathematics as a way of teaching reasoning in symbolic non-verbal communication. Particular attention is paid to mathematical ways of thinking when studying the nature and its worldview. The nature is studied through the theory of experimental approval of scientific concepts of algorithmic and nonalgorithmic "computing". Various discoveries are analyzed and the role of mathematics in the worldview is substantiated. The greatest value of mathematics is development of knowledge in order to express it in abstract language of mathematics and natural science, i.e., to move to the meta-pedagogical level of understanding of problems

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Reflections: Students in Math Class

Published by patrick honner on June 14, 2012 June 14, 2012

At the end of the term I ask students to write simple reflections on their experiences from the year:  what they learned about math, about the world, about themselves.  It’s one of the many ways I get students writing in math class .

It’s a great way to model reflection as part of the learning process, and it’s also a good way for me to get feedback about the student experience.

Mostly, it’s fun!  I love sharing and discussing the reflections with students, and it always results in great end-of-year conversations.

Here are some of my favorites.

After learning a little more about math, I think math is created rather than discovered.  This makes mathematicians and scientists the creators, not merely the seekers.

I learned a lot of things from my classmates that I wouldn’t have learned if I were to just study on my own.

I have learned that I still have very much to learn about myself.

Mathematics is magical; it can lead you to a dead end, but then it can miraculously open up an exit.

Learning how to think of things in three dimensions completely changed the way I saw math.

By seeing algebraic and geometric interpretations, I learned how to communicate math in more ways.

The process which turns a difficult problem into a relatively easy problem is the beauty of math.

One of the best parts of reflection is how much it gets you thinking about the future.  Plenty of food for thought here.

For more resources, see my Writing in Math Class  page.

Related Posts

• Writing in Math Class
• Writing in Math Class: Peer Review
• Why Write in Math Class?

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patrick honner

Math teacher in Brooklyn, New York

Hilary · August 7, 2012 at 3:39 pm

These are great!

admin · August 8, 2012 at 1:18 am

Yeah, inspiring and thoughtful stuff. It’s a great way to make kids conscious of the role of reflection in learning while getting some practical teaching advice, too.

The key is to get the students writing and reflecting on a regular basis. By the end of the year, the students will have great things to say plus the tools and motivaiton to say them.

Annette · June 17, 2018 at 5:09 pm

I know this is an old post, but this is truly inspiring and I hope you encourage students to continue doing reflections!

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Resources Teaching Writing

Math that connects where we’re going to where we’ve been — quanta magazine.

My latest column for Quanta Magazine is about the power of creative thinking in mathematics, and how understanding problems from different perspectives can lead us to surprising new conclusions. It starts with one of my Read more…

Workshop — The Geometry of Statistics

I’m excited to present The Geometry of Statistics tonight, a new workshop for teachers. This workshop is about one of the coolest things I have learned over the past few years teaching linear algebra and Read more…

People Tell Me My Job is Easy

People tell me my job is easy. You get summers off. You only work nine months of the year. You’re done at 3 pm. You get paid to babysit. Students at that school won’t succeed Read more…

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Home Essay Samples Science

Essay Samples on Mathematics in Everyday Life

Math: the efficient and effective methods to study math.

Math is everywhere; it should be one of the wonders of the world. In a way, Math is a fundamental part of who I am. It’s always been there for me. Yes, a bit strange coming from a high school student. Usually students despise quadratic...

• Mathematics in Everyday Life

Math Discovery and Mathematical Patterns in Standard of Living

Mathematics is literally defined as the study of numbers, quantities, formulas and patterns but in my own understanding, it is the world of numbers and with that it is how the world works. Mathematics is also the study of things, the relationships between things, and...

Problem Solving: Use of Math in Our Everyday Life

What I say about math is that I really don’t like it, but at the end of the day through high school math I have learned how to solve problems and not give up when I don’t fully understand something. I dislike math, but I do need it. The reason why I dislike math is that...

• Problem Solving

Doubt as a Key to Mathematical Knowledge

In my arabic culture, doubt, especially when directed at supperiors, is considered extremely disrespectful. In contrast the proverb, “Doubt is the key to knowledge” indicates that doubt should be looked at in a positive light and specifically as a way of knowing. However doubt is...

The Application of Persistence and Perseverance in Mathematics

All children can benefit from studying and developing strong skills in mathematics. Primarily, learning mathematics improves problem-solving skills, and working through problems can teach persistence and perseverance. Mathematics is essential in daily life for such activities as counting, cooking, managing money, and building things. Beyond...

• Persistence

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The Essence of Mathematics and Its Significance Towards the Behavior of Nature

Mathematics plays an integral part in our daily living, because everything we see, touch, and feel you can’t hide the fact that there is a math involved. Earth doesn’t appear how it looks today if scientists, don’t compute or used mathematics in making our world...

• Mathematical Predictions

The Power of Mathematics: Unveiling its Influence on Nature and Phenomena

Introduction Mathematics plays an integral part in our daily living, because everything we see, touch, and feel you can’t hide the fact that there is a math involved. Earth doesn’t appear how it looks today if scientists, don’t compute or used mathematics in making our...

• Role of Education

The Meaning of Application of Principles in Real Life

The realm of mathematics have a variety of implications on many real word activities that take place in today’s society. From the construction of a buildings to the usage of models in stocks and investment, math has a very effective role in the productivity and...

• Life Without Principle

The Relationship Between Mandala and Mathematic Studies

It is an undeniable fact that numbers have an impact on our lives and cover a very large part of our lives. Although many people think that mathematics consists of only symbols and specific rules, in spite of it seems complex when you look into...

Discovering the Effectiveness of College Algebra

Mathematics education at the college level is facing many challenges. These challenges are occurring at a time when most experts believe that students are going to need stronger mathematical skills than ever before in order to compete in the workforce (National Council of Teachers of...

• College Students

A Report On The Fibonacci Sequence

“Number rules the universe” ~ Pythagoras Numbers are found everywhere in in all aspects of human life. From the start of the day till the moment we fall asleep, we are surrounded with technology of which lots are made possible by numbers. One example are...

• Mathematical Models

The Main Drivers of My Fascination of Mathematics

One of the most striking aspects of mathematics for me is how something so seemingly abstract can have such a major purpose in the inner mechanisms of the universe. For example, which number, when multiplied by itself, is -1? By inspection, you can see that...

• Personal Growth and Development
• Personal Life

The Use Of Probability Theorem In Everyday Life

Throughout daily life probabilty usage is prevalent throughout all hours of life. Barometers can't anticipate precisely how climates manifest, but utilizing apparatuses and special equipment to decide probability for certain types of weather. By example if there's a certain possibility for drizzle, at that point...

Mathematics Is Not Scary, It’s Beautiful

Mathematics is often times seen as dark and scary. People specially students tried to avoid it. There are students are students tries to take courses that doesn’t have math. Well in fact it is unavoidable since math is seen everywhere and it is not scary....

Beauty Is The Creation Of Mathematics

“You are beautiful no matter what they say words can't bring you down, Oh no, You are beautiful in every single way yes words can't bring you down, Oh no, so don't you bring me down today”, sang by Christina Aguilera. I love this song...

The Beautiful Nature Of Mathematics

Beauty, as its definition given by Miriam Dictionary, is the quality or aggregate of qualities in a person or thing that gives pleasure to the senses or pleasurably exalts in the mind or spirit. In addition, as Cambridge Dictionary, is the quality of being pleasing,...

• Golden Mean

Mathematics In Every Aspect Around Us

Mathematics, as complex and absurd it may sound, is literally everywhere. Everywhere in a sense that it is frequently applied in our day-to-day activities, such as cooking (when we make correct measurements of ingredients), planning our daily agenda (how much time we will allocate for...

Mathematics Is Not Just About Numbers, It’s Also About Beauty

It has been said that “Beauty is in the eye of the beholder” for which many may believe is correct, but for some individuals does not accept this quotation. Yes, many may have their differences when it comes to preferences of foods, clothing, gadgets and...

Perfection And Beauty: My Vision Of Mathematics

How we can say Mathematics is perfect? For me Math is perfect because of the many uses or the different uses of mathematics, the benefits of math and the involvement of math in our daily life and how math contributes in our daily life. Mathematics...

The Role Of Mathematics In Creating Beauty

The beauty of mathematics is that it is not a mathematical equation, but rather a concept. I am one of those who believes that mathematics fits not only human life but also in the beauty of nature. It incorporates as part and forms everything in...

• Leonardo Da Vinci
• Natural Environment

Best topics on Mathematics in Everyday Life

1. Math: The Efficient and Effective Methods to Study Math

2. Math Discovery and Mathematical Patterns in Standard of Living

3. Problem Solving: Use of Math in Our Everyday Life

4. Doubt as a Key to Mathematical Knowledge

5. The Application of Persistence and Perseverance in Mathematics

6. The Essence of Mathematics and Its Significance Towards the Behavior of Nature

7. The Power of Mathematics: Unveiling its Influence on Nature and Phenomena

8. The Meaning of Application of Principles in Real Life

9. The Relationship Between Mandala and Mathematic Studies

10. Discovering the Effectiveness of College Algebra

11. A Report On The Fibonacci Sequence

12. The Main Drivers of My Fascination of Mathematics

13. The Use Of Probability Theorem In Everyday Life

14. Mathematics Is Not Scary, It’s Beautiful

15. Beauty Is The Creation Of Mathematics

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Math Essay Ideas for Students: Exploring Mathematical Concepts

Are you a student who's been tasked with writing a math essay? Don't fret! While math may seem like an abstract and daunting subject, it's actually full of fascinating concepts waiting to be explored. In this article, we'll delve into some exciting math essay ideas that will not only pique your interest but also impress your teachers. So grab your pens and calculators, and let's dive into the world of mathematics!

• The Beauty of Fibonacci Sequence

Have you ever wondered why sunflowers, pinecones, and even galaxies exhibit a mesmerizing spiral pattern? It's all thanks to the Fibonacci sequence! Explore the origin, properties, and real-world applications of this remarkable mathematical sequence. Discuss how it manifests in nature, art, and even financial markets. Unveil the hidden beauty behind these numbers and show how they shape the world around us.

• The Mathematics of Music

Did you know that music and mathematics go hand in hand? Dive into the relationship between these two seemingly unrelated fields and develop your writing skills . Explore the connection between harmonics, frequencies, and mathematical ratios. Analyze how musical scales are constructed and why certain combinations of notes create pleasant melodies while others may sound dissonant. Explore the fascinating world where numbers and melodies intertwine.

• The Geometry of Architecture

Architects have been using mathematical principles for centuries to create awe-inspiring structures. Explore the geometric concepts that underpin iconic architectural designs. From the symmetry of the Parthenon to the intricate tessellations in Islamic art, mathematics plays a crucial role in creating visually stunning buildings. Discuss the mathematical principles architects employ and how they enhance the functionality and aesthetics of their designs.

• Fractals: Nature's Infinite Complexity

Step into the mesmerizing world of fractals, where infinite complexity arises from simple patterns. Did you know that the famous Mandelbrot set , a classic example of a fractal, has been studied extensively and generated using computers? In fact, it is estimated that the Mandelbrot set requires billions of calculations to generate just a single image! This showcases the computational power and mathematical precision involved in exploring the beauty of fractal geometry.

Explore the beauty and intricacy of fractal geometry, from the famous Mandelbrot set to the Sierpinski triangle. Discuss the self-similarity and infinite iteration that define fractals and how they can be found in natural phenomena such as coastlines, clouds, and even in the structure of our lungs. Examine how fractal mathematics is applied in computer graphics, art, and the study of chaotic systems. Let the captivating world of fractals unfold before your eyes.

• The Game Theory Revolution

Game theory isn't just about playing games; it's a powerful tool used in various fields, from economics to biology. Dive into the world of strategic decision-making and explore how game theory helps us understand human behavior and predict outcomes. Discuss in your essay classic games like The Prisoner's Dilemma and examine how mathematical models can shed light on complex social interactions. Explore the cutting-edge applications of game theory in diverse fields, such as cybersecurity and evolutionary biology. If you still have difficulties choosing an idea for a math essay, find a reliable expert online. Ask them to write me an essay or provide any other academic assistance with your math assignments.

• Chaos Theory and the Butterfly Effect

While writing an essay, explore the fascinating world of chaos theory and how small changes can lead to big consequences. Discuss the famous Butterfly Effect and how it exemplifies the sensitive dependence on initial conditions. Delve into the mathematical principles behind chaotic systems and their applications in weather forecasting, population dynamics, and cryptography. Unravel the hidden order within apparent randomness and showcase the far-reaching implications of chaos theory.

• The Mathematics Behind Cryptography

In an increasingly digital world, cryptography plays a vital role in ensuring secure communication and data protection. Did you know that the global cybersecurity market is projected to reach a staggering $248.26 billion by 2023? This statistic emphasizes the growing importance of cryptography in safeguarding sensitive information.

Explore the mathematical foundations of cryptography and how it allows for the creation of unbreakable codes and encryption algorithms. Discuss the concepts of prime numbers, modular arithmetic, and public-key cryptography. Delve into the fascinating history of cryptography, from ancient times to modern-day encryption methods. In your essay, highlight the importance of mathematics in safeguarding sensitive information and the ongoing challenges faced by cryptographers.

Writing a math essay doesn't have to be a daunting task. By choosing a captivating topic and exploring the various mathematical concepts, you can turn your essay into a fascinating journey of discovery. Whether you're uncovering the beauty of the Fibonacci sequence, exploring the mathematical underpinnings of music, or delving into the game theory revolution, there's a world of possibilities waiting to be explored. So embrace the power of mathematics and let your creativity shine through your words!

Remember, these are just a few math essay ideas to get you started. Feel free to explore other mathematical concepts that ignite your curiosity. The world of mathematics is vast, and each concept has its own unique story to tell. So go ahead, unleash your inner mathematician, and embark on an exciting journey through the captivating realm of mathematical ideas!

Tobi Columb, a math expert, is a dedicated educator and explorer. He is deeply fascinated by the infinite possibilities of mathematics. Tobi's mission is to equip his students with the tools needed to excel in the realm of numbers. He also advocates for the benefits of a gluten-free lifestyle for students and people of all ages. Join Tobi on his transformative journey of mathematical mastery and holistic well-being.

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The New York Times

The learning network | what memorable experiences have you had in learning science or math.

What Memorable Experiences Have You Had in Learning Science or Math?

Questions about issues in the news for students 13 and older.

• See all Student Opinion »

Update: Oct. 16: Winners have been announced! Thank you for participating!

What moments or concepts do you remember best from your education in science, technology, engineering or math, the so-called STEM subjects?

What high or low points come to mind when thinking about classes you’ve taken in school? What do you remember learning informally, outside of school, whether on your own or with friends or relatives?

A special back-to-school edition of the Science Times section is asking what works, and what doesn’t, in STEM education , and we are inviting students to help answer that question by reflecting on their own learning in these areas.

So, if you’re 13 to 19 years old, please tell us below about a memorable moment in your STEM education — whether in school or out of school, whether last week or 10 years ago — and what it taught you.

Note: This is a special edition of our daily Student Opinion question. Posting a comment here by 7 a.m. Eastern on Sept. 27 will enter you in a contest that we will judge in collaboration with The Times’s science desk, and make you eligible to have your writing featured elsewhere on NYTimes.com.

To help answer the question, you might ask yourself:

• What comes to mind when you think back over the best, or worst, moments in the science, technology, engineering or math classes you have taken since you were a child? What lessons, activities or assignments were especially memorable? Why?
• How have your experiences outside of school taught you about scientific, mathematical or technological concepts? For example, you might remember an exhibition at a science museum, or something you made or experimented with in an after-school club .
• Based on your experience, what advice would you give teachers of STEM subjects? Why?

So whether it was the time your third-grade teacher took you outside to see real-world parallel and perpendicular lines; the camping trip you went on with your Girl Scout troop where you learned, firsthand, about how poison ivy spreads; or the summer you spent at an explosives or coding camp, tell us in detail about one important experience and what it taught you — and what advice you might give teachers because of it.

We have a few basic rules for this contest:

• Please keep your responses to 350 words or fewer. ( Here is a word count tool .)
• Anyone who is 13 to 19 years old, from anywhere in the world, is eligible.
• As always for this blog, please omit your last name, but please include your age and hometown.
• Only one comment per student, please.

Teachers: We’ll leave this question open to comments indefinitely, and we invite you to bring all of your classes to answer it, but please remember that if you would like your students to be considered for publication, they must post their responses by Sept. 27. Update: To make it easier to keep track of your students’ submissions, we suggest giving them a class code of some kind to affix to their first names (“JackHCHS”).

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In middle school I came to understand an important distinction that has remained with me ever since. I was attending school in Santa Clara, CA. Arithmetic had been difficult for me through each successive grade. Now in middle school I was faced with algebra and even if I somehow was able to pass the course, an additional hurdle — geometry loomed to further humble me in my parents eyes. Mr. DeVeney taught introduction to algebra, using paperback textbook from SMSG (some math some garbage) was Mr. D’s description. To my complete surprise algebra was not arithmetic more advanced. Algebra was conceptual, imaginative, manipulating symbols. Algebra was mathematics, and I was a much better student of mathematics than I had ever been of arithmetic.

When I was in Grade Seven, I entered a mathematics league with my classmates. After grinding at the competition for one and a half hours, I finally managed to finish and enter my competition. I’ve always been good at math, so I figured I had it in the bag. Three weeks later when the results came, I found out I had 24/25 questions correctly answered. I was ecstatic! I crowed over it to my friends, and to my horror discovered that one of my friends(not the brightest) had also gotten 24/25! I was shocked, because the average score was 5/25, and I was expecting him to get 5-10. To this day it is a reminder to me to never take things for granted.

When I was in tenth grade, I remember going into math class one day somewhat frustrated, I guess, since we had to learn the quadratic formula that day. I know I am a person who forgets things quite easily, so i was like this is going to be a bad day for me. But instead of it being bad it was good for me. My teacher played the quadratic formula song for about half of the 45 minutes we were in class, and on the bus ride home I remember the song completely. To this day i only remember a little bit of it, but in math class i will always remember the quadratic formula now.

AP Chemistry is notorious around the country for being one of the most difficult AP subjects, and at my high school it was no different. However, at my school, things were a little different. My AP Chemistry teacher had made it clear to us from the first day of school that we would be successful in both the course and the challenging exam at the end of the year, and she was right. She had the ability to make us believe in ourselves by being the most committed and confident teacher from which I have ever had the pleasure of learning. I can remember staying after school for hours with my classmates preparing for the endless tests and rigorous labs, with my teacher present. This commitment allowed the class to develop a quasi-competitive, and wholly collaborative dynamic. We all pushed each other throughout the year, and it paid off. We were all successful, just as she had promised on the first day of school. I am now a student studying Chemical Engineering at Northwestern University, and everyday I am reminded of the excellent work of my high school AP Chemistry teacher. Science and engineering is more collaborative than ever before, and the lessons I learned in high school not only benefit my performance, but also enhance my experiences and passion for science. I firmly believe that the commitment and confidence exhibited by my teacher should be put into practice by science and math teachers across the country to ensure that a passion for these subjects fosters the next generation of great scientists and engineers in the United States.

When I was in 8th grade my science teacher really changed my view on Science as a subject. She made me more eager to learn about the different science subjects, like astronomy, earth science and chemistry. That year has changed the way that I study and learn about science even today. It changed my work ethics and even my grades to a degree. Looking back I may not remember everything that I was taught to learn about like songs and movies, but I will always remember the formulas and facts that were in those songs and movies that were drilled into me day after day so that we would remember them. I also will always remember the study skills that I learned in her class like the right ways to approach a lab or test. All and all 8th grade science for me changed my academic life for the better,

As an aspiring politician, I can’t say that STEM subjects are my favorite. By a long shot. Instead of finding the integral of a differential equation, I would much rather be finding how Scott F. Fitzgerald integrates complex literature themes into his novels. I mean, who wouldn’t? ‘Unfortunately’ for me, I’m in the International Baccalaureate Program–a program that puts an emphasis on being “well rounded” (aka, being good at things like math AND science at the SAME time!). In such a program, our teachers understand that while we might not get integration by parts or l’Hôpital’s Rule the first time around, we may treat Dante’s Inferno like child’s play, and, consequently, they’ve derived a solution; they constantly strive to integrate (ha, calculus pun) our math skills with our literacy skills, our foreign language skills with our science skills. This induces an environment where subjects aren’t simply bounded by name, fact, or figure, but rather, application to the “real world”. We are taught to see how each element of our classes connect to the real world, eliminating the ever common, discouraging “I’ll never use this outside of this class” mentality. Rather than a single turning point, this ongoing teaching method and more so understanding of our student body has presented a strong influence to my life. I’m actually, heaven forbid, enjoying my Calc BC class (as evident through cheesy Calculus puns)!

The part of science that I find to be special is the serendipitous inquiry. These are the things that stumble upon and find yourself learning about, either in the classroom or on your own. My eighth grade science teacher called these circumstances “teachable moments”, meaning that they were not part of the curriculum but could be made into a lesson quite easily with just a little bit of excitement and love for science. The best of these “teachable moments” that I encountered was on a common weekday morning, in my first period science class. Not only the twenty some odd students came to class that day, a stray banana managed to find its way into the room as well. This wasn’t just any banana though, it was the didn’t want to eat it at lunch so shoved it back in the locker banana. So in other words it had had time to ferment its inner squishiness for several days. After hearing about the new student (the banana), my teacher picked it up with a smile on his face, and a chuckle that meant he had a crazy idea and we were going to get out of doing the normal work of our 42 minute work period. My teacher pealed the banana into three strips of peel, and placed each on the floor with the inside down, against the tile floor. He then told us that today’s lesson was to test the action of friction and lubrication. Next, our teacher called on three students to ginny pig his experiment, two to stand beside the peel in the event that someone slip and fall, and one to step on the banana peel and test its lubrication. The first of us to step on the banana peel was a kid named Earl, he had heard of the extreme lubrication power of rotten bananas, but never tested it out so he had no idea how slippery it would actually be. When Earl stepped on the banana, it was like his foot became disconnected from the universe for a fraction of a second, and he fell backward and into the two other kids who kept him from hitting the floor. The whole class was in uproar at this point, and ecstatically trying to be selected to be the one to step on the next peel, and get their fix of the lubrication powers of rotten banana. When the period was over, all of the class had had enough laughs to last a weeks worth of class periods, and we left with smiles that screamed “serendipitous science” on our faces.

One memorable moment in my STEM education was when I was hired by the New York Hall Of Science as an Explainer Trainee. Working there has increased my interest in the STEM field, seeing that it is a STEM- centered career field. Because of the experience that I have had at the New York Hall of Science, I have realized that I would like to study mechanical engineering in college. In the process of learning that I wanted to become a mechanical engineer I focused more on my mathematics and science courses. That is my memorable moment in my STEM education.

My high school physics teacher was terrific. You never knew what day the next short snap quiz would be. They counted as part of your final grade. Most everyone everyday got to class way before the bell rang.

Science class has always been an interest of mine. Unlike math, or reading, or writing, science explained to me what was going on around me, not just what happened in some rule book dictating how we are supposed to read or write or solve equations. When I was in 8th grade, I came into my homeroom class and my eccentric teacher Mr. Mullen was holding an impromptu experiment using lubricants. When I walked in I was informed that a gross rotten banana had been kicked into our classroom. Instead of throwing out the squished banana, our teacher had the bright idea to show us how lubricants actually work. We’ve all seen people slipping on banana peels in the movies, or cartoons, and here we were given an opportunity to enjoy the comic relief while still maintaining a scientific standpoint. At the age of 8th grade we all welcomed the idea of watching our classmates slip on banana peels, but for me the appeal was to see if what we had learned about lubricants the week before actually applied in real life. The experiment was set up with three people, one to step on the peal mid-walk, and one holding each of his arms. As he walked forward we were all on the edge of our seats, then as soon as he put weight on the toe that was on the gross mess which was once a banana, he lost his footing and fell, only to be caught by the two guys holding his arms. This experiment was interesting and comical and it gave me a reason to be interested in science. Through this experiment I learned how science is a factor in everything we do in the real world, and even in TV and I’ll never forget it.

In a high school like mine, bright minds are selected and thrust together in a small space. As one would imagine, this creates intense competition. This determination to best those around you makes students reach outside their educational realms and search for something to focus their talents on. With this exploration came, for me, an unexpected love for the intricacies and perplexities of STEM research. Therefore, when I look at my educational experience (as an objective third party), it becomes clear that there is no single moment one can remember that taught him something. He who, upon finding no SINGLE memory, searches for the one moment, never truly experienced the wonders of the STEM fields. Instead, it is he who finds culmination of atmosphere, competition, curiosity, and drive, who has. For me, the value of the STEM fields does not only lie in the material they teach us (mathematics, technology, engineering, and the sciences), but in the life skills they provide and the passion they instill. There is no single MOMENT to look back on that will cause a student to truly appreciate STEM education, but each student who does, as I do, values sitting at a computer and researching for hours on end as much as they cherish hearing their name called in honor of their research.

Science has been an interesting subject for me because its not all about one topic. There’s basically and wide variety to study in science and there is always new discoveries being found. When I was a freshmen I had to come up with a science project. I thought long and hard on what I was going to do. While looking through cereal I noticed the words iron and I thought to myself hmmmmm is there really iron in cereal. So while I googled it I came up with extrodinary answers and soon developed the perfect experiment. I grabbed a bag of frosted flakes and began my process. I filled the bag with ceral and added water then mashed it all up. Then I grabbed a magnet and looked very carefully there were black specks the became attracted to the magnet. as soon as I got the iron out I felt like I made the most remarkable discovery in the science world. This is the story of my most memorable science project that I would cherish and remember for a long time.

Science is an incredible way of knowing whats all around you. Without science you wouldn’t know how the waves in the ocean is created. You wouldn’t know that the planets revolve around the sun because we would still believe that the Earth is in the middle of the solar system. Science is involved in your everyday life. Mathematics and Science come hand in hand with each other. But knowing one extremely well will help you out in the other. Math, you are always using equations to solve for certain things. Don’t you do that in science as well? But don’t they as use the same equations for most of the information in which you are tying to figure out. Both science and math interest me and i will further my education and will have a career in them.

In eighth grade I spent hours after school every week learning chemistry with my science teacher, who knew tons of chemistry and was an awesome teacher. I learned tons that semester– we didn’t have any homework or anything outside of class, but it’s still the best experience I’ve ever had with science in school. I left middle school eventually but held onto my love of chemistry– I’m still learning it like he taught me today (if you’re out there, Mr. Barton, thanks for what you did).

I’ve never thought of myself as the erratic mad scientist, curing cancer or treating lung diseases. Yet somehow, I find myself- a 100 pound scruffy 16 year old girl- doing just that. As a young girl, I never believed that I had the capabilities to do science- a word I reserved for the boy with glasses who always got A’s on math tests and learned long division first. It wasn’t until a UT Austin outreach program that I had an epiphany; just because I failed Algebra 1 the first time around didn’t mean I lacked a natural aptitude for science. The possibilities flooded my body, and I saw a world of possibilities; only a few of which were constrained by ability. I threw myself headfirst into particle physics and mechanics, only coming up for air when I felt like my head was going to explode. A year later, I began devising my own theories. I wanted to discover. I wanted to create something and become a part of this world I had never seen possible for myself. The summer of 2013 was when it really all began. After countless emails to professors, I stepped into my first lab. I half expected to crash and burn- after all, that’s what had happened in 8th grade Algebra and countless other chemistry classes. But boy, was I wrong. Every hour was the best hour of my life. Every new procedure, every technique, every machine lit up my mind. It was almost as though I could feel the neurons firing. Tonight I’ll finish up my Pre-Cal homework, browse Facebook a bit, and go to sleep. Tomorrow I’ll wake up and go to school. I’ll struggle a bit in Physics- I always do. But then I’ll catch an afternoon bus to UT, and spend hours in the lab- doing what I’ve always dreamed of. -Pia, Texas

Much of my life has revolved around STEM (mainly technology). However, I enjoy all STEM subjects and find them interesting. The most memorable moment for me this year was when my science teacher showed our class the differences in gas densities through experimentation. She shook up a flask and poured the invisible gas onto a ramp. She then lit the gas on fire to show how it had traveled down the ramp because it was more dense than normal air. I found this fascinating! It’s intriguing experiments like these that make me love science.

the ‘answer’ is TEACHERS, inspiring brilliant energetic and multifaceted humans known as teachers. this can’t be quantified as merely experience, it’s much more…

One of my most memorable experiences in school had to be taking engineering as a freshman four years ago. I had never taken a course like this before and I was the only female in the class. It was terrifying at first but as the months went by I became more comfortable. I remember designing 3D models of shapes and metal parts by hand and then using the computer software to design them. Even though at first I thought I would not do well in engineering I knew I wanted to take it to become an aerospace engineer or a pilot. I realized that sometimes you have to do something you do not want to do, in order to accomplish your goals or dreams. Now four years later I am taking honors calculus and even though I’m not the only girl in the class, math terrifies me. What I have learned in these classes I can use in life. Which is how to work in a group and be indepenent. These skills are vital not only for becoming an engineer but also being a leader in the real world.

My favorite memory from a science class happened a few years ago. It was my first science expo ever and our project was using different fuels (like backing soda and vinegar, diet coke and mentos, and water and alkali tablets) to fuel a few bottle rockets. The interesting story was in the diet coke and mentos, by accident we brought the wrong type of mentos, fruit instead of mint. If you’re not already aware, the only way foam is produced is with mint mentos due to the lack of candy costing and small crevices. That type of rocket just wouldn’t work and our group had to figure out why. I think I learned more from my failure than from a lot of successes I’ve had. I learned to make sure to check and double check before doing anything and to keep learning even after you make a mistake.

I always want for everything in my life to be concrete. By this, I mean that I want things to always have a precise answer and to never be too broad of a topic. However, throughout my middle school experience, I have learned that science is all around me. If someone asked me to list everything in the world involving science, I would never be able to complete the list. There are an infinite number of science relations in the world, from the way we move, to the way the Earth rotates. I have always loved science, but this has been a difficult concept for me. Though, along with the help of my science teachers, I have managed to accept the idea that science is not tangible. Now, I have learned to embrace and adore this concept, thanks to my teachers.

When I was in 6th grade, my class went on a field trip to an interactive exhibit. It tested us on skill and teamwork. We would simulate being in the International Space Station and the other group would be at mission control next door. We were able to get complete the mission quicker than any other class that’s been there. This exciting experience taught me patience, persistence, teamwork, and focus.

I was in seventh grade and we were in the middle of expo ( our science fair). The class was building a renewable and eco-friendly city and my group was in charge of electricity. We had wired all the cords to our three foot tall building and were warming up a solar lamp to see if our lights would work. As my partner, Mamie, was holding the heat lamp it started getting really hot. In a few minutes, the heat lamp had burned a huge hole through her thick cotton sweatshirt! She was very disappointed, but we were all ecstatic when we found out that the lights worked and we had used solar power to power a building! We did this buy attaching solar panels to the top of our building and attaching alligator clips from the wires of the solar panels to the lightbulb. We shone a solar lamp over the panels and to our amazement they worked! I will never forget the day that we used solar power to light up a building!

In 7th grade, I had a science teacher who everyone called Dr. Pete. He was respected and favored as the best teacher in the school by almost all the students because of his teaching methods that were abstract, funny, and a little outlandish . . . and yet extremely efficient and interactive. So we were at the end of a long Chemistry unit, at the beginning of the week, and it was five minutes past the time when class was supposed to start. All the students were in there seats, but everyone had the same question in their head—where’s Dr. Pete? Finally, Dr. Pete comes in, struggling to carry a moderately-sized cooler and placing it on the table at the front of the room. Then he announces: “Class, we’re going to start a lab that should last us until the end of the week.” Opening the cooler, a plume of steam dispersed from it. Turns out the cooler was filled to the brim with dry ice (or solid CO2). I don’t remember exactly what we did, but we tested out a whole bunch of properties and did a whole page-sized list full of experiments that lasted us through that entire week. During the week, we consistently went back to everything we had learned in the unit—and I mean everything. It was extremely fun, and I was never bored for an instant in that class. At the end of the week, I headed home for the weekend, the experiments and knowledge still fresh in my mind. As soon as I got home, I went on my computer and started researching as much of the material as I could remember. I was going in-depth with all the elements, their properties, different compounds, their real-world usage—everything. I spent around three hours doing that. I had always loved chemistry, and loved everything about it. But that was the day that I became fascinated by it.

When I was younger, I used to think that science was learning how plants grew and math was 2+3=5. In seventh Grade I was asked how many points are in this triangle. At first I tried to count the squares on the graph paper. Was I right? Not exactly. There is actually an infinite amount of points. At first it hurt my head to think about it, but then I slowly started t realize that math is so much more complex than even the pythagorean theorem. Some numbers are imaginary, there isn’t an answer to every problem, numbers don’t stop. The same thing applies with science. Science can be anything from the sky to your phone to bugs. Math and Science aren’t confined into a single concrete subject. Just like the number of points there are in a triangle, they are infinitely large.

Through out elementary school I had always thought Science was the most boring subject. However, once I reached Middle School that all changed. In seventh grade I had Mr. Twardowski as my Science teacher and he is the reason why I love Science so much now. He always made the class fun and kept all of the students attention for the full 42 minutes. He opened my eyes to so many things and inspired me to appreciate every little thing that surrounds us. One thing I will never forget about his class is an experiment he let us do which was to make glow sticks. Even though that may not sound like a big deal now, in seventh grade is was pretty cool. He always demonstrated and assigned experiments that were relevant and useful to our age group. He turned what I thought was the most boring subject into my favorite subject. I now envy chemists and physicists because of their talents and dedication for what they do. Science is such an amazing field to look into and it is not restricted into just a few things. When talking about Science, the sky is the limit and any person would be amazed at how thought-provoking it can be and how truly fascinating the world that we live in is. Science is a never ending learning experience and there is still so much out there to be discovered and because of Mr. Twardowski, I am motivated to explore beyond my boundaries and use Science as a guide to exploring the world.

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Essay on Importance of Mathematics in our Daily Life in 100, 200, and 350 words.

• Updated on  
• Dec 22, 2023

Mathematics is one of the core aspects of education. Without mathematics, several subjects would cease to exist. It’s applied in the science fields of physics, chemistry, and even biology as well. In commerce accountancy, business statistics and analytics all revolve around mathematics. But what we fail to see is that not only in the field of education but our lives also revolve around it. There is a major role that mathematics plays in our lives. Regardless of where we are, or what we are doing, mathematics is forever persistent. Let’s see how maths is there in our lives via our blog essay on importance of mathematics in our daily life. 

Table of Contents

• 1 Essay on Importance of Mathematics in our Daily life in 100 words 
• 2 Essay on Importance of Mathematics in our Daily life in 200 words
• 3 Essay on Importance of Mathematics in our Daily Life in 350 words

Essay on Importance of Mathematics in our Daily life in 100 words 

Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Also Read:- Importance of Internet

Essay on Importance of Mathematics in our Daily life in 200 words

Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same. 

From making instalments to dialling basic phone numbers it all revolves around mathematics. 

Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. 

Without mathematics and numbers, none of this would be possible.

Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler. 

Also Read:-   My Aim in Life

Essay on Importance of Mathematics in our Daily Life in 350 words

Mathematics is what we call a backbone, a backbone of science. Without it, human life would be extremely difficult to imagine. We cannot live even a single day without making use of mathematics in our daily lives. Without mathematics, human progress would come to a halt. 

Maths helps us with our finances. It helps us calculate our daily, monthly as well as yearly expenses. It teaches us how to divide and prioritise our expenses. Its knowledge is essential for investing money too. We can only invest money in property, bank schemes, the stock market, mutual funds, etc. only when we calculate the figures. Let’s take an example from the basic routine of a day. Let’s assume we have to make tea for ourselves. Without mathematics, we wouldn’t be able to calculate how many teaspoons of sugar we need, how many cups of milk and water we have to put in, etc. and if these mentioned calculations aren’t made, how would one be able to prepare tea? 

In such a way, mathematics is used to decide the portions of food, ingredients, etc. Mathematics teaches us logical reasoning and helps us develop problem-solving skills. It also improves our analytical thinking and reasoning ability. To stay in shape, mathematics helps by calculating the number of calories and keeping the account of the same. It helps us in deciding the portion of our meals. It will be impossible to think of sports without mathematics. For instance, in cricket, run economy, run rate, strike rate, overs bowled, overs left, number of wickets, bowling average, etc. are calculated. It also helps in predicting the result of the match. When we are on the road and driving, mathetics help us keep account of our speeds, the distance we have travelled, the amount of fuel left, when should we refuel our vehicles, etc. 

We can go on and on about how mathematics is involved in our daily lives. In conclusion, we can say that the universe revolves around mathematics. It encompasses everything and without it, we cannot imagine our lives. 

Also Read:- Essay on Pollution

Ans: Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Ans: Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same.  From making instalments to dialling basic phone numbers it all revolves around mathematics. Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. Without mathematics and numbers, none of this would be possible. Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler.  

Ans: Archimedes is considered the father of mathematics.

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What Students Are Saying About the Value of Math

We asked teenagers: Do you see the point in learning math? The answer from many was “yes.”

By The Learning Network

“Mathematics, I now see, is important because it expands the world,” Alec Wilkinson writes in a recent guest essay . “It is a point of entry into larger concerns. It teaches reverence. It insists one be receptive to wonder. It requires that a person pay close attention.”

In our writing prompt “ Do You See the Point in Learning Math? ” we wanted to know if students agreed. Basic arithmetic, sure, but is there value in learning higher-level math, such as algebra, geometry and calculus? Do we appreciate math enough?

The answer from many students — those who love and those who “detest” the subject alike — was yes. Of course math helps us balance checkbooks and work up budgets, they said, but it also helps us learn how to follow a formula, appreciate music, draw, shoot three-pointers and even skateboard. It gives us different perspectives, helps us organize our chaotic thoughts, makes us more creative, and shows us how to think rationally.

Not all were convinced that young people should have to take higher-level math classes all through high school, but, as one student said, “I can see myself understanding even more how important it is and appreciating it more as I get older.”

Thank you to all the teenagers who joined the conversation on our writing prompts this week, including students from Bentonville West High School in Centerton, Ark, ; Harvard-Westlake School in Los Angeles ; and North High School in North St. Paul, Minn.

Please note: Student comments have been lightly edited for length, but otherwise appear as they were originally submitted.

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“What is Mathematics?” and why we should ask, where one should experience and learn that, and how to teach it

• Conference paper
• Open Access
• First Online: 02 November 2017
• Cite this conference paper

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• Günter M. Ziegler 3 &
• Andreas Loos 4  

Part of the book series: ICME-13 Monographs ((ICME13Mo))

111k Accesses

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“What is Mathematics?” [with a question mark!] is the title of a famous book by Courant and Robbins, first published in 1941, which does not answer the question. The question is, however, essential: The public image of the subject (of the science, and of the profession) is not only relevant for the support and funding it can get, but it is also crucial for the talent it manages to attract—and thus ultimately determines what mathematics can achieve, as a science, as a part of human culture, but also as a substantial component of economy and technology. In this lecture we thus

discuss the image of mathematics (where “image” might be taken literally!),

sketch a multi-facetted answer to the question “What is Mathematics?,”

stress the importance of learning “What is Mathematics” in view of Klein’s “double discontinuity” in mathematics teacher education,

present the “Panorama project” as our response to this challenge,

stress the importance of telling stories in addition to teaching mathematics, and finally,

suggest that the mathematics curricula at schools and at universities should correspondingly have space and time for at least three different subjects called Mathematics.

This paper is a slightly updated reprint of: Günter M. Ziegler and Andreas Loos, Learning and Teaching “ What is Mathematics ”, Proc. International Congress of Mathematicians, Seoul 2014, pp. 1201–1215; reprinted with kind permission by Prof. Hyungju Park, the chairman of ICM 2014 Organizing Committee.

You have full access to this open access chapter,  Download conference paper PDF

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What is mathematics.

Defining mathematics. According to Wikipedia in English, in the March 2014 version, the answer to “What is Mathematics?” is

Mathematics is the abstract study of topics such as quantity (numbers), [2] structure, [3] space, [2] and change. [4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. [7][8] Mathematicians seek out patterns (Highland & Highland, 1961 , 1963 ) and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

None of this is entirely wrong, but it is also not satisfactory. Let us just point out that the fact that there is no agreement about the definition of mathematics, given as part of a definition of mathematics, puts us into logical difficulties that might have made Gödel smile. Footnote 1

The answer given by Wikipedia in the current German version, reads (in our translation):

Mathematics […] is a science that developed from the investigation of geometric figures and the computing with numbers. For mathematics , there is no commonly accepted definition; today it is usually described as a science that investigates abstract structures that it created itself by logical definitions using logic for their properties and patterns.

This is much worse, as it portrays mathematics as a subject without any contact to, or interest from, a real world.

The borders of mathematics. Is mathematics “stand-alone”? Could it be defined without reference to “neighboring” subjects, such as physics (which does appear in the English Wikipedia description)? Indeed, one possibility to characterize mathematics describes the borders/boundaries that separate it from its neighbors. Even humorous versions of such “distinguishing statements” such as

“Mathematics is the part of physics where the experiments are cheap.”

“Mathematics is the part of philosophy where (some) statements are true—without debate or discussion.”

“Mathematics is computer science without electricity.” (So “Computer science is mathematics with electricity.”)

contain a lot of truth and possibly tell us a lot of “characteristics” of our subject. None of these is, of course, completely true or completely false, but they present opportunities for discussion.

What we do in mathematics . We could also try to define mathematics by “what we do in mathematics”: This is much more diverse and much more interesting than the Wikipedia descriptions! Could/should we describe mathematics not only as a research discipline and as a subject taught and learned at school, but also as a playground for pupils, amateurs, and professionals, as a subject that presents challenges (not only for pupils, but also for professionals as well as for amateurs), as an arena for competitions, as a source of problems, small and large, including some of the hardest problems that science has to offer, at all levels from elementary school to the millennium problems (Csicsery, 2008 ; Ziegler, 2011 )?

What we teach in mathematics classes . Education bureaucrats might (and probably should) believe that the question “What is Mathematics?” is answered by high school curricula. But what answers do these give?

This takes us back to the nineteenth century controversies about what mathematics should be taught at school and at the Universities. In the German version this was a fierce debate. On the one side it saw the classical educational ideal as formulated by Wilhelm von Humboldt (who was involved in the concept for and the foundation 1806 of the Berlin University, now named Humboldt Universität, and to a certain amount shaped the modern concept of a university); here mathematics had a central role, but this was the classical “Greek” mathematics, starting from Euclid’s axiomatic development of geometry, the theory of conics, and the algebra of solving polynomial equations, not only as cultural heritage, but also as a training arena for logical thinking and problem solving. On the other side of the fight were the proponents of “Realbildung”: Realgymnasien and the technical universities that were started at that time tried to teach what was needed in commerce and industry: calculation and accounting, as well as the mathematics that could be useful for mechanical and electrical engineering—second rate education in the view of the classical German Gymnasium.

This nineteenth century debate rests on an unnatural separation into the classical, pure mathematics, and the useful, applied mathematics; a division that should have been overcome a long time ago (perhaps since the times of Archimedes), as it is unnatural as a classification tool and it is also a major obstacle to progress both in theory and in practice. Nevertheless the division into “classical” and “current” material might be useful in discussing curriculum contents—and the question for what purpose it should be taught; see our discussion in the Section “ Three Times Mathematics at School? ”.

The Courant–Robbins answer . The title of the present paper is, of course, borrowed from the famous and very successful book by Richard Courant and Herbert Robbins. However, this title is a question—what is Courant and Robbins’ answer? Indeed, the book does not give an explicit definition of “What is Mathematics,” but the reader is supposed to get an idea from the presentation of a diverse collection of mathematical investigations. Mathematics is much bigger and much more diverse than the picture given by the Courant–Robbins exposition. The presentation in this section was also meant to demonstrate that we need a multi-facetted picture of mathematics: One answer is not enough, we need many.

Why Should We Care?

The question “What is Mathematics?” probably does not need to be answered to motivate why mathematics should be taught, as long as we agree that mathematics is important.

However, a one-sided answer to the question leads to one-sided concepts of what mathematics should be taught.

At the same time a one-dimensional picture of “What is Mathematics” will fail to motivate kids at school to do mathematics, it will fail to motivate enough pupils to study mathematics, or even to think about mathematics studies as a possible career choice, and it will fail to motivate the right students to go into mathematics studies, or into mathematics teaching. If the answer to the question “What is Mathematics”, or the implicit answer given by the public/prevailing image of the subject, is not attractive, then it will be very difficult to motivate why mathematics should be learned—and it will lead to the wrong offers and the wrong choices as to what mathematics should be learned.

Indeed, would anyone consider a science that studies “abstract” structures that it created itself (see the German Wikipedia definition quoted above) interesting? Could it be relevant? If this is what mathematics is, why would or should anyone want to study this, get into this for a career? Could it be interesting and meaningful and satisfying to teach this?

Also in view of the diversity of the students’ expectations and talents, we believe that one answer is plainly not enough. Some students might be motivated to learn mathematics because it is beautiful, because it is so logical, because it is sometimes surprising. Or because it is part of our cultural heritage. Others might be motivated, and not deterred, by the fact that mathematics is difficult. Others might be motivated by the fact that mathematics is useful, it is needed—in everyday life, for technology and commerce, etc. But indeed, it is not true that “the same” mathematics is needed in everyday life, for university studies, or in commerce and industry. To other students, the motivation that “it is useful” or “it is needed” will not be sufficient. All these motivations are valid, and good—and it is also totally valid and acceptable that no single one of these possible types of arguments will reach and motivate all these students.

Why do so many pupils and students fail in mathematics, both at school and at universities? There are certainly many reasons, but we believe that motivation is a key factor. Mathematics is hard. It is abstract (that is, most of it is not directly connected to everyday-life experiences). It is not considered worth-while. But a lot of the insufficient motivation comes from the fact that students and their teachers do not know “What is Mathematics.”

Thus a multi-facetted image of mathematics as a coherent subject, all of whose many aspects are well connected, is important for a successful teaching of mathematics to students with diverse (possible) motivations.

This leads, in turn, to two crucial aspects, to be discussed here next: What image do students have of mathematics? And then, what should teachers answer when asked “What is Mathematics”? And where and how and when could they learn that?

The Image of Mathematics

A 2008 study by Mendick, Epstein, and Moreau ( 2008 ), which was based on an extensive survey among British students, was summarized as follows:

Many students and undergraduates seem to think of mathematicians as old, white, middle-class men who are obsessed with their subject, lack social skills and have no personal life outside maths. The student’s views of maths itself included narrow and inaccurate images that are often limited to numbers and basic arithmetic.

The students’ image of what mathematicians are like is very relevant and turns out to be a massive problem, as it defines possible (anti-)role models, which are crucial for any decision in the direction of “I want to be a mathematician.” If the typical mathematician is viewed as an “old, white, male, middle-class nerd,” then why should a gifted 16-year old girl come to think “that’s what I want to be when I grow up”? Mathematics as a science, and as a profession, looses (or fails to attract) a lot of talent this way! However, this is not the topic of this presentation.

On the other hand the first and the second diagnosis of the quote from Mendick et al. ( 2008 ) belong together: The mathematicians are part of “What is Mathematics”!

And indeed, looking at the second diagnosis, if for the key word “mathematics” the images that spring to mind don’t go beyond a per se meaningless “ \( a^{2} + b^{2} = c^{2} \) ” scribbled in chalk on a blackboard—then again, why should mathematics be attractive, as a subject, as a science, or as a profession?

We think that we have to look for, and work on, multi-facetted and attractive representations of mathematics by images. This could be many different, separate images, but this could also be images for “mathematics as a whole.”

Four Images for “What Is Mathematics?”

Striking pictorial representations of mathematics as a whole (as well as of other sciences!) and of their change over time can be seen on the covers of the German “Was ist was” books. The history of these books starts with the series of “How and why” Wonder books published by Grosset and Dunlop, New York, since 1961, which was to present interesting subjects (starting with “Dinosaurs,” “Weather,” and “Electricity”) to children and younger teenagers. The series was published in the US and in Great Britain in the 1960s and 1970s, but it was and is much more successful in Germany, where it was published (first in translation, then in volumes written in German) by Ragnar Tessloff since 1961. Volume 18 in the US/UK version and Volume 12 in the German version treats “Mathematics”, first published in 1963 (Highland & Highland, 1963 ), but then republished with the same title but a new author and contents in 2001 (Blum, 2001 ). While it is worthwhile to study the contents and presentation of mathematics in these volumes, we here focus on the cover illustrations (see Fig.  1 ), which for the German edition exist in four entirely different versions, the first one being an adaption of the original US cover of (Highland & Highland, 1961 ).

The four covers of “Was ist was. Band 12: Mathematik” (Highland & Highland, 1963 ; Blum, 2001 )

All four covers represent a view of “What is Mathematics” in a collage mode, where the first one represents mathematics as a mostly historical discipline (starting with the ancient Egyptians), while the others all contain a historical allusion (such as pyramids, Gauß, etc.) alongside with objects of mathematics (such as prime numbers or \( \pi \) , dices to illustrate probability, geometric shapes). One notable object is the oddly “two-colored” Möbius band on the 1983 cover, which was changed to an entirely green version in a later reprint.

One can discuss these covers with respect to their contents and their styles, and in particular in terms of attractiveness to the intended buyers/readers. What is over-emphasized? What is missing? It seems more important to us to

think of our own images/representations for “What is Mathematics”,

think about how to present a multi-facetted image of “What is Mathematics” when we teach.

Indeed, the topics on the covers of the “Was ist was” volumes of course represent interesting (?) topics and items discussed in the books. But what do they add up to? We should compare this to the image of mathematics as represented by school curricula, or by the university curricula for teacher students.

In the context of mathematics images, let us mention two substantial initiatives to collect and provide images from current mathematics research, and make them available on internet platforms, thus providing fascinating, multi-facetted images of mathematics as a whole discipline:

Guy Métivier et al.: “Image des Maths. La recherche mathématique en mots et en images” [“Images of Maths. Mathematical research in words and images”], CNRS, France, at images.math.cnrs.fr (texts in French)

Andreas D. Matt, Gert-Martin Greuel et al.: “IMAGINARY. open mathematics,” Mathematisches Forschungsinstitut Oberwolfach, at imaginary.org (texts in German, English, and Spanish).

The latter has developed from a very successful travelling exhibition of mathematics images, “IMAGINARY—through the eyes of mathematics,” originally created on occasion of and for the German national science year 2008 “Jahr der Mathematik. Alles was zählt” [“Year of Mathematics 2008. Everything that counts”], see www.jahr-der-mathematik.de , which was highly successful in communicating a current, attractive image of mathematics to the German public—where initiatives such as the IMAGINARY exhibition had a great part in the success.

Teaching “What Is Mathematics” to Teachers

More than 100 years ago, in 1908, Felix Klein analyzed the education of teachers. In the introduction to the first volume of his “Elementary Mathematics from a Higher Standpoint” he wrote (our translation):

At the beginning of his university studies, the young student is confronted with problems that do not remind him at all of what he has dealt with up to then, and of course, he forgets all these things immediately and thoroughly. When after graduation he becomes a teacher, he has to teach exactly this traditional elementary mathematics, and since he can hardly link it with his university mathematics, he soon readopts the former teaching tradition and his studies at the university become a more or less pleasant reminiscence which has no influence on his teaching (Klein, 1908 ).

This phenomenon—which Klein calls the double discontinuity —can still be observed. In effect, the teacher students “tunnel” through university: They study at university in order to get a degree, but nevertheless they afterwards teach the mathematics that they had learned in school, and possibly with the didactics they remember from their own school education. This problem observed and characterized by Klein gets even worse in a situation (which we currently observe in Germany) where there is a grave shortage of Mathematics teachers, so university students are invited to teach at high school long before graduating from university, so they have much less university education to tunnel at the time when they start to teach in school. It may also strengthen their conviction that University Mathematics is not needed in order to teach.

How to avoid the double discontinuity is, of course, a major challenge for the design of university curricula for mathematics teachers. One important aspect however, is tied to the question of “What is Mathematics?”: A very common highschool image/concept of mathematics, as represented by curricula, is that mathematics consists of the subjects presented by highschool curricula, that is, (elementary) geometry, algebra (in the form of arithmetic, and perhaps polynomials), plus perhaps elementary probability, calculus (differentiation and integration) in one variable—that’s the mathematics highschool students get to see, so they might think that this is all of it! Could their teachers present them a broader picture? The teachers after their highschool experience studied at university, where they probably took courses in calculus/analysis, linear algebra, classical algebra, plus some discrete mathematics, stochastics/probability, and/or numerical analysis/differential equations, perhaps a programming or “computer-oriented mathematics” course. Altogether they have seen a scope of university mathematics where no current research becomes visible, and where most of the contents is from the nineteenth century, at best. The ideal is, of course, that every teacher student at university has at least once experienced how “doing research on your own” feels like, but realistically this rarely happens. Indeed, teacher students would have to work and study and struggle a lot to see the fascination of mathematics on their own by doing mathematics; in reality they often do not even seriously start the tour and certainly most of them never see the “glimpse of heaven.” So even if the teacher student seriously immerges into all the mathematics on the university curriculum, he/she will not get any broader image of “What is Mathematics?”. Thus, even if he/she does not tunnel his university studies due to the double discontinuity, he/she will not come back to school with a concept that is much broader than that he/she originally gained from his/her highschool times.

Our experience is that many students (teacher students as well as classical mathematics majors) cannot name a single open problem in mathematics when graduating the university. They have no idea of what “doing mathematics” means—for example, that part of this is a struggle to find and shape the “right” concepts/definitions and in posing/developing the “right” questions and problems.

And, moreover, also the impressions and experiences from university times will get old and outdated some day: a teacher might be active at a school for several decades—while mathematics changes! Whatever is proved in mathematics does stay true, of course, and indeed standards of rigor don’t change any more as much as they did in the nineteenth century, say. However, styles of proof do change (see: computer-assisted proofs, computer-checkable proofs, etc.). Also, it would be good if a teacher could name “current research focus topics”: These do change over ten or twenty years. Moreover, the relevance of mathematics in “real life” has changed dramatically over the last thirty years.

The Panorama Project

For several years, the present authors have been working on developing a course [and eventually a book (Loos & Ziegler, 2017 )] called “Panorama der Mathematik” [“Panorama of Mathematics”]. It primarily addresses mathematics teacher students, and is trying to give them a panoramic view on mathematics: We try to teach an overview of the subject, how mathematics is done, who has been and is doing it, including a sketch of main developments over the last few centuries up to the present—altogether this is supposed to amount to a comprehensive (but not very detailed) outline of “What is Mathematics.” This, of course, turns out to be not an easy task, since it often tends to feel like reading/teaching poetry without mastering the language. However, the approach of Panorama is complementing mathematics education in an orthogonal direction to the classic university courses, as we do not teach mathematics but present (and encourage to explore ); according to the response we get from students they seem to feel themselves that this is valuable.

Our course has many different components and facets, which we here cast into questions about mathematics. All these questions (even the ones that “sound funny”) should and can be taken seriously, and answered as well as possible. For each of them, let us here just provide at most one line with key words for answers:

When did mathematics start?

Numbers and geometric figures start in stone age; the science starts with Euclid?

How large is mathematics? How many Mathematicians are there?

The Mathematics Genealogy Project had 178854 records as of 12 April 2014.

How is mathematics done, what is doing research like?

Collect (auto)biographical evidence! Recent examples: Frenkel ( 2013 ) , Villani ( 2012 ).

What does mathematics research do today? What are the Grand Challenges?

The Clay Millennium problems might serve as a starting point.

What and how many subjects and subdisciplines are there in mathematics?

See the Mathematics Subject Classification for an overview!

Why is there no “Mathematical Industry”, as there is e.g. Chemical Industry?

There is! See e.g. Telecommunications, Financial Industry, etc.

What are the “key concepts” in mathematics? Do they still “drive research”?

Numbers, shapes, dimensions, infinity, change, abstraction, …; they do.

What is mathematics “good for”?

It is a basis for understanding the world, but also for technological progress.

Where do we do mathematics in everyday life?

Not only where we compute, but also where we read maps, plan trips, etc.

Where do we see mathematics in everyday life?

There is more maths in every smart phone than anyone learns in school.

What are the greatest achievements of mathematics through history?

Make your own list!

An additional question is how to make university mathematics more “sticky” for the tunneling teacher students, how to encourage or how to force them to really connect to the subject as a science. Certainly there is no single, simple, answer for this!

Telling Stories About Mathematics

How can mathematics be made more concrete? How can we help students to connect to the subject? How can mathematics be connected to the so-called real world?

Showing applications of mathematics is a good way (and a quite beaten path). Real applications can be very difficult to teach since in most advanced, realistic situation a lot of different mathematical disciplines, theories and types of expertise have to come together. Nevertheless, applications give the opportunity to demonstrate the relevance and importance of mathematics. Here we want to emphasize the difference between teaching a topic and telling about it. To name a few concrete topics, the mathematics behind weather reports and climate modelling is extremely difficult and complex and advanced, but the “basic ideas” and simplified models can profitably be demonstrated in highschool, and made plausible in highschool level mathematical terms. Also success stories like the formula for the Google patent for PageRank (Page, 2001 ), see Langville and Meyer ( 2006 ), the race for the solution of larger and larger instances of the Travelling Salesman Problem (Cook, 2011 ), or the mathematics of chip design lend themselves to “telling the story” and “showing some of the maths” at a highschool level; these are among the topics presented in the first author’s recent book (Ziegler, 2013b ), where he takes 24 images as the starting points for telling stories—and thus developing a broader multi-facetted picture of mathematics.

Another way to bring maths in contact with non-mathematicians is the human level. Telling stories about how maths is done and by whom is a tricky way, as can be seen from the sometimes harsh reactions on www.mathoverflow.net to postings that try to excavate the truth behind anecdotes and legends. Most mathematicians see mathematics as completely independent from the persons who explored it. History of mathematics has the tendency to become gossip , as Gian-Carlo Rota once put it (Rota, 1996 ). The idea seems to be: As mathematics stands for itself, it has also to be taught that way.

This may be true for higher mathematics. However, for pupils (and therefore, also for teachers), transforming mathematicians into humans can make science more tangible, it can make research interesting as a process (and a job?), and it can be a starting/entry point for real mathematics. Therefore, stories can make mathematics more sticky. Stories cannot replace the classical approaches to teaching mathematics. But they can enhance it.

Stories are the way by which knowledge has been transferred between humans for thousands of years. (Even mathematical work can be seen as a very abstract form of storytelling from a structuralist point of view.) Why don’t we try to tell more stories about mathematics, both at university and in school—not legends, not fairy tales, but meta-information on mathematics—in order to transport mathematics itself? See (Ziegler, 2013a ) for an attempt by the first author in this direction.

By stories, we do not only mean something like biographies, but also the way of how mathematics is created or discovered: Jack Edmonds’ account (Edmonds, 1991 ) of how he found the blossom shrink algorithm is a great story about how mathematics is actually done . Think of Thomas Harriot’s problem about stacking cannon balls into a storage space and what Kepler made out of it: the genesis of a mathematical problem. Sometimes scientists even wrap their work into stories by their own: see e.g. Leslie Lamport’s Byzantine Generals (Lamport, Shostak, & Pease, 1982 ).

Telling how research is done opens another issue. At school, mathematics is traditionally taught as a closed science. Even touching open questions from research is out of question, for many good and mainly pedagogical reasons. However, this fosters the image of a perfect science where all results are available and all problems are solved—which is of course completely wrong (and moreover also a source for a faulty image of mathematics among undergraduates).

Of course, working with open questions in school is a difficult task. None of the big open questions can be solved with an elementary mathematical toolbox; many of them are not even accessible as questions. So the big fear of discouraging pupils is well justified. On the other hand, why not explore mathematics by showing how questions often pop up on the way? Posing questions in and about mathematics could lead to interesting answers—in particular to the question of “What is Mathematics, Really?”

Three Times Mathematics at School?

So, what is mathematics? With school education in mind, the first author has argued in Ziegler ( 2012 ) that we are trying cover three aspects the same time, which one should consider separately and to a certain extent also teach separately:

A collection of basic tools, part of everyone’s survival kit for modern-day life—this includes everything, but actually not much more than, what was covered by Adam Ries’ “Rechenbüchlein” [“Little Book on Computing”] first published in 1522, nearly 500 years ago;

A field of knowledge with a long history, which is a part of our culture and an art, but also a very productive basis (indeed a production factor) for all modern key technologies. This is a “story-telling” subject.

An introduction to mathematics as a science—an important, highly developed, active, huge research field.

Looking at current highschool instruction, there is still a huge emphasis on Mathematics I, with a rather mechanical instruction on arithmetic, “how to compute correctly,” and basic problem solving, plus a rather formal way of teaching Mathematics III as a preparation for possible university studies in mathematics, sciences or engineering. Mathematics II, which should provide a major component of teaching “What is Mathematics,” is largely missing. However, this part also could and must provide motivation for studying Mathematics I or III!

What Is Mathematics, Really?

There are many, and many different, valid answers to the Courant-Robbins question “What is Mathematics?”

A more philosophical one is given by Reuben Hersh’s book “What is Mathematics, Really?” Hersh ( 1997 ), and there are more psychological ones, on the working level. Classics include Jacques Hadamard’s “Essay on the Psychology of Invention in the Mathematical Field” and Henri Poincaré’s essays on methodology; a more recent approach is Devlin’s “Introduction to Mathematical Thinking” Devlin ( 2012 ), or Villani’s book ( 2012 ).

And there have been many attempts to describe mathematics in encyclopedic form over the last few centuries. Probably the most recent one is the gargantuan “Princeton Companion to Mathematics”, edited by Gowers et al. ( 2008 ), which indeed is a “Princeton Companion to Pure Mathematics.”

However, at a time where ZBMath counts more than 100,000 papers and books per year, and 29,953 submissions to the math and math-ph sections of arXiv.org in 2016, it is hopeless to give a compact and simple description of what mathematics really is, even if we had only the “current research discipline” in mind. The discussions about the classification of mathematics show how difficult it is to cut the science into slices, and it is even debatable whether there is any meaningful way to separate applied research from pure mathematics.

Probably the most diplomatic way is to acknowledge that there are “many mathematics.” Some years ago Tao ( 2007 ) gave an open list of mathematics that is/are good for different purposes—from “problem-solving mathematics” and “useful mathematics” to “definitive mathematics”, and wrote:

As the above list demonstrates, the concept of mathematical quality is a high-dimensional one, and lacks an obvious canonical total ordering. I believe this is because mathematics is itself complex and high-dimensional, and evolves in unexpected and adaptive ways; each of the above qualities represents a different way in which we as a community improve our understanding and usage of the subject.

In this sense, many answers to “What is Mathematics?” probably show as much about the persons who give the answers as they manage to characterize the subject.

According to Wikipedia , the same version, the answer to “Who is Mathematics” should be:

Mathematics , also known as Allah Mathematics , (born: Ronald Maurice Bean [1] ) is a hip hop producer and DJ for the Wu-Tang Clan and its solo and affiliate projects. This is not the mathematics we deal with here.

Blum, W. (2001). Was ist was. Band 12: Mathematik , Tessloff Verlag, Nürnberg. Revised version, with new cover, 2010.

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Acknowledgment

The authors’ work has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 247029, the DFG Research Center Matheon, and the the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.

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Ziegler, G.M., Loos, A. (2017). “What is Mathematics?” and why we should ask, where one should experience and learn that, and how to teach it. In: Kaiser, G. (eds) Proceedings of the 13th International Congress on Mathematical Education. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-62597-3_5

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Measuring What Counts: A Conceptual Guide for Mathematics Assessment (1993)

Chapter: 4 assessing to support mathematics learning, 4 assessing to support mathematics learning.

High-quality mathematics assessment must focus on the interaction of assessment with learning and teaching. This fundamental concept is embodied in the second educational principle of mathematics assessment.

T HE L EARNING P RINCIPLE

Assessment should enhance mathematics learning and support good instructional practice .

This principle has important implications for the nature of assessment. Primary among them is that assessment should be seen as an integral part of teaching and learning rather than as the culmination of the process. 1 As an integral part, assessment provides an opportunity for teachers and students alike to identify areas of understanding and misunderstanding. With this knowledge, students and teachers can build on the understanding and seek to transform misunderstanding into significant learning. Time spent on assessment will then contribute to the goal of improving the mathematics learning of all students.

The applicability of the learning principle to assessments created and used by teachers and others directly involved in classrooms is relatively straightforward. Less obvious is the applicability of the principle to assessments created and imposed by parties outside the classroom. Tradition has allowed and even encouraged some assessments to serve accountability or monitoring purposes without sufficient regard for their impact on student learning.

A portion of assessment in schools today is mandated by external authorities and is for the general purpose of accountability of the schools. In 1990, 46 states had mandated testing programs, as

compared with 20 in 1980. 2 Such assessments have usually been multiple-choice norm-referenced tests. Several researchers have studied these testing programs and judged them to be inconsistent with the current goals of mathematics education. 3 Making mandated assessments consonant with the content, learning, and equity principles will require much effort.

Studies have documented a further complication as teachers are caught between the conflicting demands of mandated testing programs and instructional practices they consider more appropriate. Some have resorted to "double-entry" lessons in which they supplement regular course instruction with efforts to teach the objectives required by the mandated test. 4 During a period of change there will undoubtedly be awkward and difficult examples of discontinuities between newer and older directions and procedures. Instructional practices may move ahead of assessment practices in some situations, whereas in other situations assessment practices could outpace instruction. Neither situation is desirable although both will almost surely occur. However, still worse than such periods of conflict would be to continue either old instructional forms or old assessment forms in the name of synchrony, thus stalling movement of either toward improving important mathematics learning.

From the perspective of the learning principle, the question of who mandated the assessment and for what purpose is not the primary issue. Instruction and assessment—from whatever source and for whatever purpose—must be integrated so that they support one another.

To satisfy the learning principle, assessment must change in ways consonant with the current changes in teaching, learning, and curriculum. In the past, student learning was often viewed as a passive process whereby students remembered what teachers told them to remember. Consistent with this view, assessment was often thought of as the end of learning. The student was assessed on something taught previously to see if he or she remembered it. Similarly, the mathematics curriculum was seen as a fragmented collection of information given meaning by the teacher.

This view led to assessment that reinforced memorization as a principal learning strategy. As a result, students had scant oppor-

tunity to bring their intuitive knowledge to bear on new concepts and tended to memorize rules rather than understand symbols and procedures. 5 This passive view of learning is not appropriate for the mathematics students need to master today. To develop mathematical competence, students must be involved in a dynamic process of thinking mathematically, creating and exploring methods of solution, solving problems, communicating their understanding—not simply remembering things. Assessment, therefore, must reflect and reinforce this view of the learning process.

This chapter examines three ways of making assessment compatible with the learning principle: ensuring that assessment directly supports student learning; ensuring that assessment is consonant with good instructional practice; and enabling teachers to become better facilitators of student learning.

A SSESSMENT IN S UPPORT OF L EARNING

Assessment can play a key role in exemplifying the new types of mathematics learning students must achieve. Assessments indicate to students what they should learn. They specify and give concrete meaning to valued learning goals. If students need to learn to perform mathematical operations, they should be assessed on mathematical operations. If they should learn to use those mathematical operations along with mathematical reasoning in solving mathematical problems, they must be assessed on using mathematical operations along with reasoning to solve mathematical problems. In this way the nature of the assessments themselves make the goals for mathematics learning real to students, teachers, parents, and the public.

Mathematics assessments can help both students and teachers improve the work the students are doing in mathematics. Students need to learn to monitor and evaluate their progress. When students are encouraged to assess their own learning, they become more aware of what they know, how they learn, and what resources they are using when they do mathematics. "Conscious knowledge about the resources available to them and the ability to engage in self-monitoring and self-regulation are important characteristics of self-assessment that successful learners use to promote ownership of learning and independence of thought." 6

In the emerging view of mathematics education, students make their own mathematics learning individually meaningful. Important mathematics is not limited to specific facts and skills students can be trained to remember but rather involves the intellectual structures and processes students develop as they engage in activities they have endowed with meaning.

The assessment challenge we face is to give up old assessment methods to determine what students know, which are based on behavioral theories of learning and develop authentic assessment procedures that reflect current epistemological beliefs both about what it means to know mathematics and how students come to know. 7

Current research indicates that acquired knowledge is not simply a collection of concepts and procedural skills filed in long-term memory. Rather the knowledge is structured by individuals in meaningful ways, which grow and change over time. 8

A close consideration of recent research on mathematical cognition suggests that in mathematics, as in reading, successful learners understand the task to be one of constructing meaning, of doing interpretive work rather than routine manipulations. In mathematics the problem of imposing meaning takes a special form: making sense of formal symbols and rules that are often taught as if they were arbitrary conventions rather than expressions of fundamental regularities and relationships among quantities and physical entities. 9

L EARNING F ROM A SSESSMENT

Modern learning theory and experience with new forms of assessment suggest several characteristics assessments should have if they are to serve effectively as learning activities. Of particular interest is the need to provide opportunities for students to construct their own mathematical knowledge and the need to determine where students are in their acquisition of mathematical understanding. 10 One focuses more on the content of mathematics, the other on the process of doing mathematics. In both, the assessment must elicit important mathematics.

Constructing Mathematical Knowledge Learning is a process of continually restructuring one's prior knowledge, not just adding to it. Good education provides opportunities for students to connect what is being learned to their prior knowledge. One knows

mathematics best if one has developed the structures and meanings of the content for oneself. 11 For assessment to support learning, it must enable students to construct new knowledge from what they know.

One way to provide opportunities for the construction of mathematical knowledge is through assessment tasks that resemble learning tasks 12 in that they promote strategies such as analyzing data, drawing contrasts, and making connections. It is not enough, however, to expand mathematics assessment to take in a broader spectrum of an individual student's competence. In real-world settings, knowledge is sometimes constructed in group settings rather than in individual exploration. Learning mathematics is frequently optimized in group settings, and assessment of that learning must reflect the value of group interaction. 13

Some mathematics teachers are using group work in instruction to model problem solving in the real world. They are looking for ways to assess what goes on in groups, trying to find out not only what mathematics has been learned, but also how the students have been working together. A critical issue is how to use assessments of group work in the grades they give to individual students. A recent study of a teacher who was using groups in class but not assessing the work done in groups found that her students apparently did not see such work as important. 14 Asked in interviews about mathematics courses in which they had done group work, the students did not mention this teacher's course. Group work, if it is to become an integral and valued part of mathematics instruction, must be assessed in some fashion. A challenge to developers is to construct some high-quality assessment tasks that can be conducted in groups and subsequently scored fairly.

Part of the construction of knowledge depends on the availability of appropriate tools, whether in instruction or assessment. Recent experimental National Assessment of Educational Progress (NAEP) tasks in science use physical materials for a miniexperiment students are asked to perform by themselves. Rulers, calculators, computers, and various manipulatives are examples from mathematics of some instructional tools that should be a part of assessment. If students have been using graphing calculators to explore trigonometric functions, giving them tests on which calculators are banned greatly limits the questions they can be asked and

consequently yields an incomplete picture of their learning. Similarly, asking students to find a function that best fits a set of data by using a computer program can reveal aspects of what they know about functions that cannot be assessed by other means. Using physical materials and technology appropriately and effectively in instruction is a critical part of learning today's mathematics and, therefore, must be part of today's assessment.

Reflecting Development of Competence As students progress through their schooling, it is obvious that the content of their assessments must change to reflect their growing mathematical sophistication. When students encounter new topics in mathematics, they often cannot see how the unfamiliar ideas are connected to anything they have seen before. They resort to primitive strategies of memorization, grasping at isolated and superficial aspects of the topic. As learning proceeds, they begin to see how the new ideas are connected to each other and to what they already know. They see regularities and uncover hidden relationships. Eventually, they learn to monitor their thinking and can choose different ways to tackle a problem or verify a solution. 15 This scenario is repeated throughout schooling as students encounter new mathematics. The example below contains a description of this growth in competence that is derived from research in cognition and that suggests the types of evidence that assessment should seek. 16

A full portrayal of competence in mathematics demands much more than measuring how well students can perform automated skills although that is part of the picture. Assessment should also examine whether students have managed to connect the concepts they have learned, how well they can recognize underlying principles and patterns amid superficial differences, their sense of when to use processes and strategies, their grasp and command of their own understanding, and whether

they can bring these skills and abilities together to produce smooth, proficient performance.

P ROVIDING F EEDBACK AND O PPORTUNITIES TO R EVISE W ORK

An example of how assessment results can be used to support learning comes from the Netherlands. 17 Eleventh-grade students were given regular 45-minute tests containing both short-answer and essay questions. One test for a unit on matrices contained questions about harvesting Christmas trees of various sizes in a forest. The students completed a growth matrix to portray how the sizes changed each year and were asked how the forest could be managed most profitably, given the costs of planting and cutting and the prices at which the trees were to be sold. They also had to answer the questions when the number of sizes changed from three to five and to analyze a situation in which the forester wanted to recapture the original distribution of sizes each year.

After the students handed in their solutions, the teacher scored them, noting the major errors. Given this information, the students retook the test. They had several weeks to work on it at home and were free to answer the questions however they chose, separately or in essays that combined the answers to several questions. The second chance gave students the opportunity not simply to redo the questions on which they were unsuccessful in the first stage but, more importantly, to give greater attention to the essay questions they had little time to address. Such two-stage testing essentially formalizes what many teachers of writing do in their courses, giving students an opportunity to revise their work (often more than once) after the teacher or other students have read it and offered suggestions. The extensive experience that writing teachers have been accumulating in teaching and assessing writing through extended projects can be of considerable assistance to mathematics teachers seeking to do similar work. 18

During the two-stage testing in the Netherlands, students reflected on their work, talked with others about it, and got information from the library. Many students who had not scored well under time pressure—including many of the females—did much better under the more open conditions. The teachers could grade the students on both the objective scores from the first stage and

the subjective scores from the second. The students welcomed the opportunity to show what they knew. As one put it

Usually when a timed written test is returned to us, we just look at our grade and see whether it fits the number of mistakes we made. In the two-stage test, we learn from doing the task. We have to study the first stage carefully in order to do well on the second one. 19

In the Netherlands, such two-stage tasks are not currently part of the national examination given at the end of secondary school, but some teachers use them in their own assessments as part of the final grade each year. In the last year of secondary school, the teacher's assessment is merged with the score on the national examination to yield a grade for each student that is used for graduation, university admission, and job qualification.

L EARNING FROM THE S CORING OF A SSESSMENTS

Assessment tasks that call for complex responses require scoring rubrics. Such rubrics describe what is most important about a response, what distinguishes a stronger response from a weaker one, and often what characteristics distinguish a beginning learner from one with more advanced understanding and performance. Such information, when shared between teacher and student, has critically important implications for the learning process.

Teachers can appropriately communicate the features of scoring rubrics to students as part of the learning process to illustrate the types of performance students are striving for. Students often express mystification about what they did inadequately or what type of change would make their work stronger. Teachers can use rubrics and sample work marked according to the rubric to communicate the goals of improved mathematical explication. When applied to actual student work, rubrics illustrate the next level of learning toward which a student may move. For example, a teacher may use a scoring rubric on a student's work and then give the student an opportunity to improve the work. In such a case, the student may use the rubric directly as a guide in the improvement process.

The example on the following page illustrates how a scoring rubric can be incorporated into the student material in an assess-

ment. 20 The benefits to instruction and learning could be twofold. The student not only can develop a clearer sense of quality mathematics on the task at hand but can develop more facility at self-assessment. It is hoped that students can, over time, develop an inner sense of excellent performance so that they can correct their own work before submitting it to the teacher.

The rubrics can be used to inform the student about the scoring criteria before he or she works on a task. The rubric can also be used to structure a classroom discussion, possibly even asking the students to grade some (perhaps fictional) answers to the questions. In this way, the students can see some examples of how responses are evaluated. Such discussions would be a purely instructional use of an assessment device before the formal administration of the assessment.

S TIMULATING M OTIVATION , I NTEREST, AND A TTENTION

Because assessment has the potential to affect the learning process substantially, it is important that students do their best when being assessed. Students' motivation to perform well on assessments has usually been tied to the stakes involved. Knowing that an assessment has a direct bearing on a semester grade or on placement in the next class—that is, high personal stakes—has encouraged many students to display their best work. Conversely, assessments to judge the effectiveness of an educational program where results are often not reported on an individual basis carry low stakes for the student and may not inspire students to excel. These extrinsic sources of motivation, although real, are not always consonant with the principle that assessment should support good instructional practice and enhance mathematics learning. Intrinsic sources of motivation, such as interest in the task, offer a more fruitful approach.

Students develop an interest in mathematical tasks that they understand, see as relevant to their own concerns, and can manage. Recent studies of students' emotional responses to mathematics suggest that both their positive and their negative responses diminish as tasks become familiar and increase when the tasks are novel. 21 Because facility at problem solving includes facility with unfamiliar tasks, the regular use of nonroutine problems must become a part of instruction and assessment.

In some school districts, educational leaders are experimenting with nonroutine assessment tasks that have instructional value in themselves and that seem to have considerable interest for the students. Such a problem was successfully tried out with fifth-grade students in the San Diego City School District in 1990 and has

subsequently been used by other districts across the country to assess instruction in the fifth, sixth, and seventh grades. The task is to help the owner of an orange grove decide how many trees to plant on each acre of new land to maximize the harvest. 22 The yield of each tree and the number of trees per acre in the existing grove are explained and illustrated. An agronomist consultant explains that increasing the number of trees per acre decreases the yield of each tree and gives data the students can use. The students construct a chart and see that the total yield per acre forms a quadratic pattern. They investigate the properties of the function and answer a variety of questions, including questions about extreme cases.

The assessment can serve to introduce a unit on quadratic functions in which the students explore other task situations. For example, one group of sixth-grade students interviewed an elementary school principal who said that when cafeteria lunch prices went up, fewer students bought their lunches in the cafeteria. The students used a quadratic function to model the data, orally reported to their classmates, and wrote a report for their portfolios.

Sixth-grade students can be successful in investigating and solving interesting, relevant problems that lead to quadratic and other types of functions. They need only be given the opportunity. Do they enjoy and learn from these kinds of assessment activities and their instructional extensions? Below are some of their comments.

It is worth noting that the level of creativity allowable in a response is not necessarily tied to the student's level of enjoyment of the task. In particular, students do not necessarily value assessment tasks in which they have to produce responses over tasks in which they have to choose among alternatives. A survey in Israel of junior high students' attitudes toward different types of tests showed that although they thought essay tests reflected their knowledge of subject matter better than multiple-choice tests did, they preferred the multiple-choice tests. 23 The multiple-choice tests were perceived as being easier and simpler; the students felt more comfortable taking them.

A SSESSMENT IN S UPPORT OF I NSTRUCTION

If mathematics assessment is to help students develop their powers of reasoning, problem solving, communicating, and connecting mathematics to situations in which it can be used, both mathematics assessment and mathematics instruction will need to change in tandem. Mathematics instruction will need to better use assessment activities than is common today.

Too often a sharp line is drawn between assessment and instruction. Teachers teach, then instruction stops and assessment occurs. Results of the assessment may not be available in a timely or useful way to students and teachers. The learning principle implies that "even when certain tasks are used as part of a formal, external assessment, there should be some kind of instructional follow-up. As a routine part of classroom discourse, interesting problems should be revisited, extended, and generalized, whatever their original sources." 24

When the line between assessment and instruction is blurred, students can engage in mathematical tasks that not only are meaningful and contribute to learning, but also yield information the student, the teacher, and perhaps others can use. In fact, an oftstated goal of reform efforts in mathematics education is that visitors to classrooms will be unable to distinguish instructional activities from assessment activities.

I NTEGRATING I NSTRUCTION AND A SSESSMENT

The new Pacesetter™ mathematics project illustrates how instruction and assessment can be fully integrated by design. 25 Pacesetter is an advanced high school mathematics course being developed by the College Board. The course, which emphasizes mathematical modeling and is meant as a capstone to the mathematics studied in high school, integrates assessment activities with instruction. Teachers help the students undertake case studies of applications of mathematics to problems in fields, such as industrial design, inventions, economics, and demographics. In one activity, for example, students are provided with data on the population of several countries at different times and asked to develop mathematical models to make various predictions. Students answer questions about the models they have devised and tackle more extended tasks that are written up for a portfolio. The activity allows the students to apply their knowledge of linear, quadratic, and exponential functions to real data. Notes for the teacher's guidance help direct attention to opportunities for discussion and the interpretations of the data that students might make under various assumptions.

Portfolios are sometimes used as the method of assessment; a sample of a student's mathematical work is gathered to be graded by the teacher or an outside evaluator.

This form of assessment involves assembling a portfolio containing samples of students' work that have been chosen by the students themselves, perhaps with the help of their teacher, on the basis of certain focused criteria. Among other things, a mathematics portfolio might contain samples of analyses of mathematical problems or investigations, responses to open-ended problems, examples of work chosen to reflect growth in knowledge over time, or self-reports of problem-solving processes learned and employed. In addition to providing good records of individual student work, portfolios might also be useful in providing formative evaluation information for program development. Before they can be used as components of large-scale assessment efforts, however, consistent methods for evaluating portfolios will need to be developed. 26

Of course the quality of student work in a portfolio depends largely on the quality of assignments that were given as well as on

the level of instruction. At a minimum, teachers play a pivotal role in helping students decide what to put into the portfolio and informing them about the evaluation criteria.

The state of Vermont, for example, has been devising a program in which the mathematics portfolios of fourth- and eighth-grade students are assessed; 27 other states and districts are experimenting with similar programs. Some problems have been reported in the portfolio assessment process in Vermont. 28 The program appears to hold sufficient merit, however, to justify efforts under way to determine how information from portfolios can be communicated outside the classroom in authoritative and credible ways. 29

The trend worldwide is to use student work expeditiously on instructional activities directly as assessment. An example from England and Wales is below. 30

Assessment can be integrated with classroom discourse and activity in a variety of other ways as well: through observation, questioning, written discourse, projects, open-ended problems, classroom tests, homework, and other assignments. 31 Teachers need to be alert to techniques they can use to assess their students' mathematical understanding in all settings.

The most effective ways to identify students' methods are to watch students solve problems, to listen to them explain how the problems were solved, or to read their written explanations. Students should regularly be asked to explain their solution to a problem. Each individual cannot be asked each day, but over time the teacher can get a reading on each student's understanding and proficiency. The teacher needs to keep some

record of students' responses. Sunburst/Wings for Learning, 32 for example, recently produced the Learner Profile ™, a hand-held optical scanner with a list of assessment codes that can be defined by the teacher. Useful in informal assessments, a teacher can scan comments about the progress of individual students while walking around the classroom.

Elaborate schemes are not necessary, but some system is needed. A few carefully selected tasks can give a reasonably accurate picture of a student's ability to solve a range of tasks. 33 An example of a task constructed for this purpose appears above. 34

U SING A SSESSMENT R ESULTS FOR I NSTRUCTION

The most typical form of assessment results have for decades been based in rankings of performance, particularly in mandated assessment. Performances have been scored most

typically by counting the number of questions answered correctly and comparing scores for one individual to that for another by virtue of their relative percentile rank. So-called norm referenced scores have concerned educators for many years. Although various criticisms on norm referencing have been advanced, the central educational concern is that such information is not sufficiently helpful to improve instruction and learning and may, in fact, have counterproductive educational implications. In the classroom setting, teachers and students need to know what students understand well, what they understand less well, and what the next learning steps need to be. The relative rankings of students tested may have uses outside the classroom context, but within that context, the need is for forms of results helpful to the teaching and learning process.

To plan their instruction, for example, teachers should know about each student's current understanding of what will be taught. Thus, assessment programs must inform teachers and students about what the students have learned, how they learn, and how they think about mathematics. For that information to be useful to teachers, it will have to include an analysis of specific strengths and weaknesses of the student's understanding and not just scores out of context.

To be effective in instruction, assessment results need to be timely. 35 Students' learning is not promoted by computer printouts sent to teachers once classes have ended for the year and the students have gone, nor by teachers who take an inordinate amount of time to grade assessments. In particular, new ways must be found to give teachers and students alike more immediate knowledge of the students' performance on assessments mandated by outside authorities so that those assessments—as well as the teacher's own assessments—can be used to improve learning. Even when the central purpose of an assessment is to determine the accomplishments of a school, state, or nation, the assessment should provide reports about their performance to the students and teachers involved. School time is precious. When students are not informed of their errors and misconceptions, let alone helped to correct them, the assessment may have both reinforced misunderstandings and wasted valuable instructional time.

When the form of assessment is unfamiliar, teachers have a particular responsibility to their students to tell them in advance

how their responses will be evaluated and what criteria will be used. Students need to see examples of work a priori that does or does not meet the criteria. Teachers should discuss sample responses with their students. When the California Assessment Program first tried out some open-ended questions with 12-grade students in its 1987-1988 Survey of Academic Skills, from half to three-fourths of the students offered either an inadequate response or none at all. The Mathematics Assessment Advisory Committee concluded that the students lacked experience expressing mathematical ideas in writing. 36 Rather than reject the assessment, they concluded that more discussion with students was needed before the administration of the assessment to describe what was expected of them. On the two-stage tests in the Netherlands, there were many fewer problems in scoring the essays when the students knew beforehand what the teacher expected from them. 37 The teacher and students had negotiated a kind of contract that allowed the students to concentrate on the mathematics in the assessment without being distracted by uncertainties about scoring.

A SSESSMENT IN S UPPORT OF T EACHERS

The new visions of mathematics education requires teachers to use strategies in which they function as learning coach and facilitator. Teachers will require support in several ways to adopt these new roles. First, they will need to become better diagnosticians. For this, they will need "… simple, valid procedures that enable [them] to access and use relevant information in making instructional decisions"; "assessment systems [that] take into account the conceptualizations of learning, teaching, and the curriculum that are held by teachers"; and systems that "enable teachers to share assessment data with students and to involve students in making instructional decisions." 38 Materials should be provided with the assessments developed by others that will enable teachers to use assessment tasks productively in their instruction. Help should be given to teachers on using assessment results to encourage students to reflect on their work and the teachers to reflect on theirs.

Teachers will require assistance in using assessments consonant with today's vision of mathematics instruction. The Classroom Assessment in Mathematics (CAM) Network, for example, is an electronic network of middle school teachers in seven urban centers

who are designing assessment tasks and sharing them with one another. 39 They are experimenting with a variety of new techniques and revising tasks to fit their teaching situation. They see that they face some common problems regarding making the new tasks accessible to their students. Collaborations among teachers, whether through networks or other means, can assist mathematics teachers who want to change their assessment practice. These collaborations can start locally or be developed through and sponsored by professional organizations. Publications are beginning to appear that can help teachers assess mathematics learning more thoroughly and productively. 40

There are indications that using assessments in professional development can help teachers improve instruction. As one example, Gerald Kulm and his colleagues recently reported a study of the effects of improved assessment on classroom teaching: 41

We found that when teachers used alternative approaches to assessment, they also changed their teaching. Teachers increased their use of strategies that have been found by research to promote students' higher-order thinking. They did activities that enhanced meaning and understanding, developed student autonomy and independence, and helped students learn problem-solving strategies. 42

This improvement in assessment, however, came through a substantial intervention: the teachers' enrollment in a three-credit graduate course. However, preliminary reports from a number of professional development projects such as CAM suggest that improved teaching practice may also result from more limited interventions.

Scoring rubrics can also be a powerful tool for professional development. In a small agricultural county in Florida, 30 teachers have been meeting on alternate weekends, attempting to improve their assessment practice. 43 The county has a large population of migrant workers, and the students are primarily of Mexican-American descent. The teachers, who teach mathematics at levels from second-grade arithmetic to calculus, are attempting to spend less time preparing the students to pass multiple-choice standardized tests. Working with a consultant, they have tried a variety of new tasks and procedures. They have developed a much greater respect for how assessments may not always tap learning. They found, for

example, that language was the biggest barrier. For students who were just learning English requests such as "discuss" or "explain" often yield little information. The teacher may need, instead, to ask a sequence of questions: "What did you do first?" "Why did you do that?'' "What did you do next?" "Why?" and so on. Working with various tasks, along with the corresponding scoring rubrics, the teachers developed a new appreciation for the quality of their students' mathematical thinking.

Advanced Placement teachers have reported on the value of the training in assessment they get from the sessions conducted by the College Board for scoring Advanced Placement Tests. 44 These tests include open-ended responses that must be scored by judges. Teachers have found that the training for the scoring and the scoring itself are useful for their subsequent teaching of the courses because they focus attention on the most important features and lead to more direct instruction on crucial areas of performance that were perhaps ignored in the past.

Assessment tasks and rubrics can be devices that teachers use to communicate with parents and the larger community to obtain their support for changes in mathematics education. Abridged versions of the rubrics—accompanied by a range of student responses—might accomplish this purpose best. Particularly when fairly complex tasks have been used, the wider audience will benefit more from a few samples of actual student work than they will from detailed descriptions and analyses of anticipated student responses.

Teachers are also playing an active role in creating and using assessment results. In an increasing number of localities, assessments incorporate the teacher as a central component in evaluating results. Teachers are being recognized as rich sources of information about what students know and can do, especially when they have been helped to learn ways to evaluate student performance. Many students' anxiety about mathematics interferes with their test performance; a teacher can assess students informally and unobtrusively during regular instruction. Teachers know, in ways that test constructors in distant offices cannot, whether students have had an opportunity to learn the mathematics being assessed and whether they are taking an assessment seriously. A teacher can talk with students during or after an assessment, to find out how they inter-

preted the mathematics and what strategies they pursued. Developers of external assessment systems should explore ways of taking the information teachers can provide into account as part of the system.

In summary, the learning principle aims to ensure that assessments are constructed and used to help students learn more and better mathematics. The consensus among mathematics educators is that assessments can fulfill this expectation to the extent that tasks provide students opportunities to extend their knowledge, are consonant with good instruction, and provide teachers with an additional tool that can help them to become better facilitators of student learning. These are new requirements for assessment. Some will argue that they are burdensome, particularly the requirement that assessments function as learning tasks. Recent experience—described below and elsewhere in this chapter—indicates this need not be so, even when an assessment must serve an accountability function.

The Pittsburgh schools, for example, recently piloted an auditing process through which portfolios developed for instructional uses provided "publicly acceptable accountability information." 45 Audit teams composing teachers, university-based researchers, content experts, and representatives of the business community evaluated samples of portfolios and sent a letter to the Board of Education that certified, among other things, that the portfolio process was well defined and well implemented and that it aimed at success for all learners, challenged teachers to do a more effective job of supporting student learning, and increased overall system accountability.

There is reason to believe, therefore, that the learning principle can be honored to a satisfactory degree for both internal and external assessments.

To achieve national goals for education, we must measure the things that really count. Measuring What Counts establishes crucial research- based connections between standards and assessment.

Arguing for a better balance between educational and measurement concerns in the development and use of mathematics assessment, this book sets forth three principles—related to content, learning, and equity—that can form the basis for new assessments that support emerging national standards in mathematics education.

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4 things we’ve learned about math success that might surprise parents

Associate Professor of Mathematics Education, York University, Canada

Middle School Teacher. PhD Mathematics Education Student, York University, Canada

Disclosure statement

Tina Rapke received funding from SSHRC: Partnership Engage Grants.

Cristina De Simone does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.

York University provides funding as a member of The Conversation CA.

York University provides funding as a member of The Conversation CA-FR.

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School closures due to conronavirus have put parents in the challenging position of home-schooling their children.

In mathematics education programs for future math teachers, we often discuss the traditional classroom that those studying to become teachers are familiar with. We’re interested in how their own experiences as students can influence their teaching.

Traditional modes of instruction have emphasized that math is best learned through studying and memorizing alone, with the teacher demonstrating procedures and then checking students’ answers .

If parents grew up with this style of instruction, their ideal home-math classroom might look like strict scheduling, workbooks, a child working alone in silence and parents telling children how to solve problems. But if parents enforce this approach, there could be conflicts and maybe even some crying.

But parents, like future educators, can also learn from newer approaches. Here are some practical tips for a different form of home learning.

1. Talking about math

Gone are the days of students sitting quietly while the math teacher does all the talking at the chalkboard. Discussion is important in the mathematics classroom.

Parents should be explicit. Tell your child “we learn by sharing ideas and listening to each other.”

Model active listening skills. Show your child that you are listening by asking questions about what they said to clarify your understanding of their idea. Try saying “tell me more …” or asking “how do you know that?”

Try setting aside your own idea(s) so you can listen and build on their ideas. Instead of saying “yes, but …,” use “ yes, and … ” to help children feel that they’re not being judged and their ideas are important.

2. Attitude

Researchers have identified three underlying interconnected aspects of childrens’ relationships with math that impact how they engage with math: emotional disposition (“I like math”), perceived competence (“I am good at math”) and their vision of math: whether math is about problem solving and understanding or math is about memorization and regurgitation.

Read more: Mathematics is about wonder, creativity and fun, so let's teach it that way

Parents can set a positive attitude for children by being mindful not to say things like “I don’t like math” or “I’m not a math person.” Your child might think they don’t have a chance because you didn’t pass on a math mind .

Academics have debunked common beliefs about the “ math gene ” and explain that there’s lots involved in being good at math . Celebrate the process and not just the final answer. Give high fives for sharing solution strategies, developing a plan to tackle the problem and for not giving up.

Make it clear that making mistakes is OK and can even be a good thing. Many highly successful people see mistakes as learning opportunities and an indication that learning is happening.

3. Working in partnership

A partnership is about working together and can include seeing the teacher as a learner and the student as a teacher . It isn’t about the teacher being “all-knowing” and making all the decisions.

Traditional math teaching, where the teacher assumes an authoritative role, is a major cause of math anxiety . Researchers have found that not all math homework help is beneficial. There is a difference between parents being controlling and being supportive.

With this in mind, wait for your child to ask for help. Try not to control everything. Focus on asking questions about their decisions that will help them figure out possible limitations and benefits of their decisions.

Let children fail. Failure can build confidence . Confidence can come from mastery; mastery can come from practice . Good practice includes analyzing what went wrong and what went right.

Don’t worry about being the expert. Be honest and say “I’m not sure. Let’s figure it out together.”

Start with what children already know . When your child is stuck, ask them to talk through what they are doing.

Take turns doing questions and talking about solution strategies.

Follow your child’s interests and ideas . Let them take the lead, even if you think your approach is better.

4. Basic math skills

If you grew up with traditional math instruction and haven’t thought about math since your school days, it might surprise you to learn that there are multiple ways to solve problems.

You could ask your child to share their way of solving the problem and also share your way.

For instance: What is 24 x 6?

It’s OK if you’re looking for a pencil to do this:

  24 x 6  144

But what are some other ways you might you figure it out?

Multiply 20 x 6 to get 120. Now multiply 4 x 6 to get 24. Add the two figures: 120 + 24 = 144.

Another way would be to focus on 25 x 6 to get 150. Now subtract 6 and you’ve got 144.

Read more: The 'new math': How to support your child in elementary school

In all math problems (including addition or subtraction), your child can use their fingers and you can too.

You can also look for opportunities to highlight math in daily activities.

One fun way is through baking. Arrange three rows of cookie dough with four cookies in each row. Ask how many cookies per batch or how many each family member will get if they share equally.

Being successful at mental math (like the arithmetic you do at the store) happens gradually over time.

Try focusing on basic math skills with your child for 10 minutes or less, every other day.

The takeaway

Think of quality over quantity.

If you want to support math learning at home based on math research: talk with your child, see learning as a partnership and make sure to celebrate their ideas. Your child may teach you something new.

We’d love to hear about how math has provoked families to slow down, have fun, go with the flow and connect.

• Mathematics education
• Coronavirus
• Homeschooling
• Math anxiety
• Math skills
• Math teaching
• Math education
• math teachers

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Cyra Myl Patanyag

Wings to show you what you can become. Roots to remind you where you're from.

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