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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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## Essays About Math: Top 10 Examples and Writing Prompts

Love it or hate it, an understanding of math is said to be crucial to success. So, if you are writing essays about math, read our top essay examples.

Mathematics is the study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them . It can be used for a variety of purposes, from calculating a business’s profit to estimating the mass of a black hole. However, it can be considered “controversial” to an extent.

Most students adore math or regard it as their least favorite. No other core subject has the same infamy as math for generating passionate reactions both for and against it. It has applications in every field, whether basic operations or complex calculus problems. Knowing the basics of math is necessary to do any work properly.

If you are writing essays about Math, we have compiled some essay examples for you to get started.

## 1. Mathematics: Problem Solving and Ideal Math Classroom by Darlene Gregory

2. math essay by prasanna, 3. short essay on the importance of mathematics by jay prakash.

- 4. Math Anxiety by Elias Wong

## 5. Why Math Isn’t as Useless as We Think by Murtaza Ali

1. mathematics – do you love or hate it, 2. why do many people despise math, 3. how does math prepare you for the future, 4. is mathematics an essential skill, 5. mathematics in the modern world.

“The trait of the teacher that is being strict is we know that will really help the students to change. But it will give a stress and pressure to students and that is one of the causes why students begin to dislike math. As a student I want a teacher that is not so much strict and giving considerations to his students. A teacher that is not giving loads of things to do and must know how to understand the reasons of his students.”

Gregory discusses the reasons for most students’ hatred of math and how teachers handle the subject in class. She says that math teachers do not explain the topics well, give too much work, and demand nothing less than perfection. To her, the ideal math class would involve teachers being more considerate and giving less work.

You might also be interested in our ordinal number explainer.

“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.”

In her essay, Prasanna gives readers a basic idea of what math is and its importance. She additionally lists down some of the many uses of mathematics in different career paths, namely managing finances, cooking, home modeling and construction, and traveling. Math may seem “useless” and “annoying” to many, but the essay gives readers a clear message: we need math to succeed.

“In this modern age of Science and Technology, emphasis is given on Science such as Physics, Chemistry, Biology, Medicine and Engineering. Mathematics, which is a Science by any criterion, also is an efficient and necessary tool being employed by all these Sciences. As a matter of fact, all these Sciences progress only with the aid of Mathematics. So it is aptly remarked, ‘Mathematics is a Science of all Sciences and art of all arts.’”

As its title suggests, Prakash’s essay briefly explains why math is vital to human nature. As the world continues to advance and modernize, society emphasizes sciences such as medicine, chemistry, and physics. All sciences employ math; it cannot be studied without math. It also helps us better our reasoning skills and maximizes the human mind. It is not only necessary but beneficial to our everyday lives.

## 4. Math Anxiety by Elias Wong

“Math anxiety affects different not only students but also people in different ways. It’s important to be familiar with the thoughts you have about yourself and the situation when you encounter math. If you are aware of unrealistic or irrational thoughts you can work to replace those thoughts with more positive and realistic ones.”

Wong writes about the phenomenon known as “math anxiety.” This term is used to describe many people’s hatred or fear of math- they feel that they are incapable of doing it. This anxiety is caused mainly by students’ negative experiences in math class, which makes them believe they cannot do well. Wong explains that some people have brains geared towards math and others do not, but this should not stop people from trying to overcome their math anxiety. Through review and practice of basic mathematical skills, students can overcome them and even excel at math.

“We see that math is not an obscure subject reserved for some pretentious intellectual nobility. Though we may not be aware of it, mathematics is embedded into many different aspects of our lives and our world — and by understanding it deeply, we may just gain a greater understanding of ourselves.”

Similar to some of the previous essays, Ali’s essay explains the importance of math. Interestingly, he tells a story of the life of a person name Kyle. He goes through the typical stages of life and enjoys typical human hobbies, including Rubik’s cube solving. Throughout this “Kyle’s” entire life, he performed the role of a mathematician in various ways. Ali explains that math is much more prevalent in our lives than we think, and by understanding it, we can better understand ourselves.

## Writing Prompts on Essays about Math

Math is a controversial subject that many people either passionately adore or despise. In this essay, reflect on your feelings towards math, and state your position on the topic. Then, give insights and reasons as to why you feel this way. Perhaps this subject comes easily to you, or perhaps it’s a subject that you find pretty challenging. For an insightful and compelling essay, you can include personal anecdotes to relate to your argument.

It is well-known that many people despise math. In this essay, discuss why so many people do not enjoy maths and struggle with this subject in school. For a compelling essay, gather interview data and statistics to support your arguments. You could include different sections correlating to why people do not enjoy this subject.

In this essay, begin by reading articles and essays about the importance of studying math. Then, write about the different ways that having proficient math skills can help you later in life. Next, use real-life examples of where maths is necessary, such as banking, shopping, planning holidays, and more! For an engaging essay, use some anecdotes from your experiences of using math in your daily life.

Many people have said that math is essential for the future and that you shouldn’t take a math class for granted. However, many also say that only a basic understanding of math is essential; the rest depends on one’s career. Is it essential to learn calculus and trigonometry? Choose your position and back up your claim with evidence.

Prasanna’s essay lists down just a few applications math has in our daily lives. For this essay, you can choose any activity, whether running, painting, or playing video games, and explain how math is used there. Then, write about mathematical concepts related to your chosen activity and explain how they are used. Finally, be sure to link it back to the importance of math, as this is essentially the topic around which your essay is based.

If you are interested in learning more, check out our essay writing tips !

For help with your essays, check out our round-up of the best essay checkers

Martin is an avid writer specializing in editing and proofreading. He also enjoys literary analysis and writing about food and travel.

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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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## 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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## How mathematical practices can improve your writing

Writing is similar to three specific mathematical practices: modelling, problem-solving and proving, writes Caroline Yoon. Here, she gives some tips on how to use these to improve academic writing

## Caroline Yoon

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## .css-1txxx8u{overflow:hidden;max-height:81px;text-indent:0px;} A framework to teach library research skills

The trouble with bloom’s taxonomy in an age of ai, emotions and learning: what role do emotions play in how and why students learn, playing the promotion game: how to navigate upshifting, using the snowflake method to build belonging on campus.

I feel for my students when I hand them their first essay assignment. Many are mathematicians, students and teachers who chose to study mathematics partly to avoid writing. But in my mathematics education courses, and in the discipline more generally, academic writing is part of our routine practice.

Mathematicians face some challenging stereotypes when it comes to writing. Writing is seen as ephemeral, subjective and context-dependent, whereas mathematics is seen as enduring, universal and context-free. Writing reflects self, but mathematics transcends it: they are distinct from each other.

This is a false dichotomy that can discourage mathematicians from writing. It suggests writing is outside the natural skill set of the mathematician, and that one’s mathematics training not only neglects one’s development as a writer but actively prevents it. Rather than capitulate to this false dichotomy, I propose we turn it around to examine how writing is similar to three specific mathematical practices: modelling, problem-solving and proving.

## Three mathematical practices that can improve your writing

Mathematical modelling.

Let us consider a hypothetical mathematics education student who has spent weeks thinking, reading and talking about her essay topic, but only starts writing it the night before it is due. She writes one draft only – the one she hands in – and is disappointed with the low grade her essay receives.

She wishes she had started earlier but she was still trying to figure out what she wanted to say up until the moment she started writing. It was only the pressure of the deadline that forced her to start; without it, she would have spent even more time thinking and reading to develop her ideas. After all, she reasons, there is no point writing when you do not know what to write about!

This “think first, write after” approach, sometimes known as the “writing up” model is a dangerous trap many students fall into, and is at odds with the way writing works. The approach allows no room for imperfect drafts that are a necessary part of the writing process . Writing experts trade on the generative power of imperfect writing; they encourage writers to turn off their internal critics and allow themselves to write badly as a way of overcoming writing inertia and discovering new ideas. The “shitty first draft” is an ideal (and achievable) first goal in the writing process. Anyone can produce a sketchy first draft that generates material that can be worked on, improved and eventually rewritten into a more sharable form.

Mathematical modelling offers a compelling metaphor for the generative power of imperfect writing. Like polished writing, polished mathematical models are seldom produced in the first attempt. A modeller typically begins with some understanding of the real situation to be modelled. The modeller considers variables and relationships from his or her understanding of the real situation and writes them into an initial mathematical model.

The model is his or her mathematical description of the situation, written in mathematical notation, and the modeller who publishes a mathematical model has typically created and discarded multiple drafts along the way, just as the writer who publishes a piece of writing has typically written and discarded multiple drafts along the way.

- Resource collection: Skills every early career academic needs
- Top tips to improve the teaching of mathematics in universities
- One write way to student success in mathematics

## Problem-solving

Writing an original essay is like trying to solve a mathematics problem. There is no script to follow; it must be created by simultaneously determining one’s goals and figuring out how to achieve them. In both essay writing and mathematical problem-solving, getting stuck is natural and expected. It is even a special kind of thrill.

This observation might come as a surprise to mathematicians who do not think of their problem-solving activity as writing. But doing mathematics, the ordinary everyday act of manipulating mathematical relationships and objects to notice new levels of structure and pattern, involves scratching out symbols and marks, and moving ideas around the page or board.

Why do I care that mathematicians acknowledge their natural language of symbols and signs as writing? Quite frankly because they are good at it. They have spent years honing their ability to use writing to restructure their thoughts, to dissect their ideas, identify new arguments. They possess an analytic discipline that most writers struggle with.

Yet few of my mathematics education students take advantage of this in their academic writing. They want their writing to come out in consecutive, polished sentences and become discouraged when it does not. They do not use their writing to analyse and probe their arguments as they do when they are stuck on mathematical problems. By viewing writing only as a medium for communicating perfectly formed thoughts, they deny themselves their own laboratories, their own thinking tools.

I am not suggesting that one’s success in solving mathematical problems automatically translates into successful essay writing. But the metaphor of writing as problem-solving might encourage a mathematics education student not to give up too easily when she finds herself stuck in her writing.

Our hypothetical student now has a good draft that she is happy with. She is satisfied it represents her knowledge of the subject matter and has read extensively to check the accuracy of its content. A friend reads the draft and remarks that it is difficult to understand. Our student is unperturbed. She puts it down to her friend’s limited knowledge of the subject and is confident her more knowledgeable teacher will understand her essay.

But the essay is not an inert record judged on the number of correct facts it contains. It is also a rhetorical act that seeks to engage the public. It addresses an audience, it tries to persuade, to inspire some response or action.

Mathematical proofs are like expository essays in this regard; they must convince an audience. When undergraduate mathematics students learn to construct proofs of their own, a common piece of advice is to test them on different audiences. The phrase “Convince yourself, convince a friend, convince an enemy” becomes relevant in this respect.

Mathematicians do not have to see themselves as starting from nothing when they engage in academic writing. Rather, they can use mathematical principles they have already honed in their training, but which they might not have formerly recognised as tools for improving their academic writing.

## Practical tips for productive writing beliefs and behaviours

- Writing can generate ideas. Free writing is a good way to start. Set a timer and write continuously for 10 minutes without editing. These early drafts will be clumsy, but there will also be some gold that can be mined and developed.
- Writing can be used to analyse and organise ideas. When stuck, try to restructure your ideas. Identify the main point in each paragraph and play around with organising their flow.
- Writing is a dialogue with the public. Seek out readers’ interpretations of your writing and listen to their impressions. Read your writing out loud to yourself: you will hear it differently!

Caroline Yoon is an associate professor of mathematics at the University of Auckland.

This is an edited version of the journal article “The writing mathematician” by Caroline Yoon, published in For the Learning of Mathematics and collected in The Best Writing on Mathematics , edited by Mircea Pitici (Princeton University Press).

If you would like advice and insight from academics and university staff delivered direct to your inbox each week, sign up for the Campus newsletter .

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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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## Engaging Maths

Dr catherine attard, promoting student reflection to improve mathematics learning.

- by cattard2017
- Posted on July 16, 2017

Critical reflection is a skill that doesn’t come naturally for many students, yet it is one of the most important elements of the learning process. As teachers, not only should we practice what we preach by engaging in critical reflection of our practice, we also need to be modelling critical reflection skills to our students so they know what it looks like, sounds like, and feels like (in fact, a Y chart is a great reflection tool).

How often do you provide opportunities for your students to engage in deep reflection of their learning? Consider Carol Dweck’s research on growth mindset. If we want to convince our students that our brains have the capability of growing from making mistakes and learning from those mistakes, then critical reflection must be part of the learning process and must be included in every mathematics lesson.

What does reflection look like within a mathematics lesson, and when should it happen?Reflection can take many forms, and is often dependent on the age and abilities of your students. For example, young students may not be able to write fluently, so verbal reflection is more appropriate and can save time. Verbal reflections, regardless of the age of the student, can be captured on video and used as evidence of learning. Video reflections can also be used to demonstrate learning during parent/teacher conferences. Another reflection strategy for young students could be through the use of drawings. Older students could keep a mathematics journal, which is a great way of promoting non-threatening, teacher and student dialogue. Reflection can also occur amongst pairs or small groups of students.

How do you promote quality reflection? The use of reflection prompts is important. This has two benefits: first, they focus students’ thinking and encourage depth of reflection; and second, they provide information about student misconceptions that can be used to determine the content of the following lessons. Sometimes teachers fall into the trap of having a set of generic reflection prompts. For example, prompts such as “What did you learn today?”, “What was challenging?” and “What did you do well?” do have some value, however if they are over-used, students will tend to provide generic responses. Consider asking prompts that relate directly to the task or mathematical content.

An example of powerful reflection prompts is the REAL Framework, from Munns and Woodward (2006). Although not specifically written for mathematics, these reflection prompts can be adapted. One great benefit of the prompts is that they fit into the three dimensions of engagement: operative, affective, and cognitive. The following table represents reflection prompts from one of four dimensions identified by Munns and Woodward: conceptual, relational, multidimensional and unidimensional.

Finally, student reflection can be used to promote and assess the proficiencies (Working Mathematically in NSW) from the Australian Curriculum: Mathematics as well as mathematical concepts. It can be an opportunity for students to communicate mathematically, use reasoning, and show evidence of understanding. It can also help students make generalisations and consider how the mathematics can be applied elsewhere.

How will you incorporate reflection into your mathematics lessons? Reflection can occur at any time throughout the lesson, and can occur more than once per lesson. For example, when students are involved in a task and you notice they are struggling or perhaps not providing appropriate responses, a short, sharp verbal reflection would provide opportunity to change direction and address misconceptions. Reflection at the conclusion of a lesson consolidates learning, and also assists students in recognising the learning that has occurred. They are more likely to remember their learning when they’ve had to articulate it either verbally or in writing.

And to conclude, some reflection prompts for teachers (adapted from the REAL Framework):

- How have you encouraged your students to think differently about their learning of mathematics?
- What changes to your pedagogy are you considering to enhance the way you teach mathematics?
- Explain how your thinking about mathematics teaching and learning is different today from yesterday, and from what it could be tomorrow?

Munns, G., & Woodward, H. (2006). Student engagement and student self-assessment: the REAL framework. Assessment in Education, 13 (2), 193-213.

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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

You have full access to this open access conference paper

- Günter M. Ziegler 3 &
- Andreas Loos 4

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

110k Accesses

9 Citations

3 Altmetric

“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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## The Missing Element for Teachers: Learning What Mathematics Is

## Educational Paths to Mathematics: Which Paths Forward to What Mathematics?

## Mathematics at the Center of Distinct Fields: A Response to Michael and Ted

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.

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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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## 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.

## “Math is a valuable tool and function of the world.”

As a musician, math is intrinsically related to my passion. As a sailor, math is intertwined with the workings of my boat. As a human, math is the building block for all that functions. When I was a child, I could very much relate to wanting a reason behind math. I soon learned that math IS the reason behind all of the world’s workings. Besides the benefits that math provides to one’s intellect, it becomes obvious later in life that math is a valuable tool and function of the world. In music for example, “adolescent mathematics” are used to portray functions of audio engineering. For example, phase shifting a sine wave to better project sound or understanding waves emitted by electricity and how they affect audio signals. To better understand music, math is a recurring pattern of intervals between generating pitches that are all mathematically related. The frets on a guitar are measured precisely to provide intervals based on a tuning system surrounding 440Hz, which is the mathematically calculated middle of the pitches humans can perceive and a string can effectively generate. The difference between intervals in making a chord are not all uniform, so guitar frets are placed in a way where all chords can sound equally consonant and not favor any chord. The power of mathematics! I am fascinated by the way that math creeps its way into all that I do, despite my plentiful efforts to keep it at a safe distance …

— Renan, Miami Country Day School

## “Math isn’t about taking derivatives or solving for x, it’s about having the skills to do so and putting them to use elsewhere in life.”

I believe learning mathematics is both crucial to the learning and development of 21st century students and yet also not to be imposed upon learners too heavily. Aside from the rise in career opportunity in fields centered around mathematics, the skills gained while learning math are able to be translated to many facets of life after a student’s education. Learning mathematics develops problem solving skills which combine logic and reasoning in students as they grow. The average calculus student may complain of learning how to take derivatives, arguing that they will never have to use this after high school, and in that, they may be right. Many students in these math classes will become writers, musicians, or historians and may never take a derivative in their life after high school, and thus deem the skill to do so useless. However, learning mathematics isn’t about taking derivatives or solving for x, it’s about having the skills to do so and putting them to use elsewhere in life. A student who excels at calculus may never use it again, but with the skills of creativity and rational thinking presented by this course, learning mathematics will have had a profound effect on their life.

— Cam, Glenbard West

## “Just stop and consider your hobbies and pastimes … all of it needs math.”

Math is timing, it’s logic, it’s precision, it’s structure, and it’s the way most of the physical world works. I love math — especially algebra and geometry — as it all follows a formula, and if you set it up just right, you can create almost anything you want in at least two different ways. Just stop and consider your hobbies and pastimes. You could be into skateboarding, basketball, or skiing. You could be like me, and sit at home for hours on end grinding out solves on a Rubik’s cube. Or you could be into sketching. Did you know that a proper drawing of the human face places the eyes exactly halfway down from the top of the head? All of it needs math. Author Alec Wilkinson, when sharing his high school doubting view on mathematics, laments “If I had understood how deeply mathematics is embedded in the world …” You can’t draw a face without proportions. You can’t stop with your skis at just any angle. You can’t get three points without shooting at least 22 feet away from the basket, and get this: you can’t even ride a skateboard if you can’t create four congruent wheels to put on it.

— Marshall, Union High School, Vancouver, WA

## “Math gives us a different perspective on everyday activities.”

Even though the question “why do we even do math?” is asked all the time, there is a deeper meaning to the values it shares. Math gives us a different perspective on everyday activities, even if those activities in our routine have absolutely nothing to do with mathematical concepts itself. Geometry, for instance, allows us to think on a different level than simply achieving accuracy maintains. It trains our mind to look at something from various viewpoints as well as teaching us to think before acting and organizing chaotic thoughts. The build up of learning math can allow someone to mature beyond the point where if they didn’t learn math and thought through everything. It paves a way where we develop certain characteristics and traits that are favorable when assisting someone with difficult tasks in the future.

— Linden, Harvard-Westlake High School, CA

## “Math teaches us how to think.”

As explained in the article, math is all around us. Shapes, numbers, statistics, you can find math in almost anything and everything. But is it important for all students to learn? I would say so. Math in elementary school years is very important because it teaches how to do simple calculations that can be used in your everyday life; however middle and high school math isn’t used as directly. Math teaches us how to think. It’s far different from any other subject in school, and truly understanding it can be very rewarding. There are also many career paths that are based around math, such as engineering, statistics, or computer programming, for example. These careers are all crucial for society to function, and many pay well. Without a solid background in math, these careers wouldn’t be possible. While math is a very important subject, I also feel it should become optional at some point, perhaps part way through high school. Upper level math classes often lose their educational value if the student isn’t genuinely interested in learning it. I would encourage all students to learn math, but not require it.

— Grey, Cary High School

## “Math is a valuable tool for everyone to learn, but students need better influences to show them why it’s useful.”

Although I loved math as a kid, as I got older it felt more like a chore; all the kids would say “when am I ever going to use this in real life?” and even I, who had loved math, couldn’t figure out how it benefits me either. This was until I started asking my dad for help with my homework. He would go on and on about how he used the math I was learning everyday at work and even started giving me examples of when and where I could use it, which changed my perspective completely. Ultimately, I believe that math is a valuable tool for everyone to learn, but students need better influences to show them why it’s useful and where they can use it outside of class.

— Lilly, Union High School

## “At the roots of math, it teaches people how to follow a process.”

I do believe that the math outside of arithmetic, percentages, and fractions are the only math skills truly needed for everyone, with all other concepts being only used for certain careers. However, at the same time, I can’t help but want to still learn it. I believe that at the roots of math, it teaches people how to follow a process. All mathematics is about following a formula and then getting the result of it as accurately as possible. It teaches us that in order to get the results needed, all the work must be put and no shortcuts or guesses can be made. Every equation, number, and symbol in math all interconnect with each other, to create formulas that if followed correctly gives us the answer needed. Everything is essential to getting the results needed, and skipping a step will lead to a wrong answer. Although I do understand why many would see no reason to learn math outside of arithmetic, I also see lessons of work ethics and understanding the process that can be applied to many real world scenarios.

— Takuma, Irvine High School

## “I see now that math not only works through logic but also creativity.”

A story that will never finish resembling the universe constantly expanding, this is what math is. I detest math, but I love a never-ending tale of mystery and suspense. If we were to see math as an adventure it would make it more enjoyable. I have often had a closed mindset on math, however, viewing it from this perspective, I find it much more appealing. Teachers urge students to try on math and though it seems daunting and useless, once you get to higher math it is still important. I see now that math not only works through logic but also creativity and as the author emphasizes, it is “a fundamental part of the world’s design.” This view on math will help students succeed and have a more open mindset toward math. How is this never-ending story of suspense going to affect YOU?

— Audrey, Vancouver, WA union high school

## “In some word problems, I encounter problems that thoroughly interest me.”

I believe math is a crucial thing to learn as you grow up. Math is easily my favorite subject and I wish more people would share my enthusiasm. As Alec Wilkinson writes, “Mathematics, I now see, is important because it expands the world.” I have always enjoyed math, but until the past year, I have not seen a point in higher-level math. In some of the word problems I deal with in these classes, I encounter problems that thoroughly interest me. The problems that I am working on in math involve the speed of a plane being affected by wind. I know this is not riveting to everyone, but I thoroughly wonder about things like this on a daily basis. The type of math used in the plane problems is similar to what Alec is learning — trigonometry. It may not serve the most use to me now, but I believe a thorough understanding of the world is a big part of living a meaningful life.

— Rehan, Cary High School

## “Without high school classes, fewer people get that spark of wonder about math.”

I think that math should be required through high school because math is a use-it-or-lose-it subject. If we stop teaching math in high school and just teach it up to middle school, not only will many people lose their ability to do basic math, but we will have fewer and fewer people get that spark of wonder about math that the author had when taking math for a second time; after having that spark myself, I realized that people start getting the spark once they are in harder math classes. At first, I thought that if math stopped being required in high school, and was offered as an elective, then only people with the spark would continue with it, and everything would be okay. After thinking about the consequences of the idea, I realized that technology requires knowing the seemingly unneeded math. There is already a shortage of IT professionals, and stopping math earlier will only worsen that shortage. Math is tricky. If you try your best to understand it, it isn’t too hard. However, the problem is people had bad math teachers when they were younger, which made them hate math. I have learned that the key to learning math is to have an open mind.

— Andrew, Cary High School

## “I think math is a waste of my time because I don’t think I will ever get it.”

In the article Mr. Wilkinson writes, “When I thought about mathematics at all as a boy it was to speculate about why I was being made to learn it, since it seemed plainly obvious that there was no need for it in adult life.” His experience as a boy resonates with my experience now. I feel like math is extremely difficult at some points and it is not my strongest subject. Whenever I am having a hard time with something I get a little upset with myself because I feel like I need to get everything perfect. So therefore, I think it is a waste of my time because I don’t think I will ever get it. At the age of 65 Mr. Wilkinson decided to see if he could learn more/relearn algebra, geometry and calculus and I can’t imagine myself doing this but I can see myself understanding even more how important it is and appreciating it more as I get older. When my dad was young he hated history but, as he got older he learned to appreciate it and see how we can learn from our past mistakes and he now loves learning new things about history.

— Kate, Cary High School

## “Not all children need to learn higher level math.”

The higher levels of math like calculus, algebra, and geometry have shaped the world we live in today. Just designing a house relates to math. To be in many professions you have to know algebra, geometry, and calculus such as being an economist, engineer, and architect. Although higher-level math isn’t useful to some people. If you want to do something that pertains to math, you should be able to do so and learn those high levels of math. Many things children learn in math they will never use again, so learning those skills isn’t very helpful … Children went through so much stress and anxiety to learn these skills that they will never see again in their lives. In school, children are using their time learning calculus when they could be learning something more meaningful that can prepare them for life.

— Julyssa, Hanover Horton High School

## “Once you understand the basics, more math classes should be a choice.”

I believe that once you get to the point where you have a great understanding of the basics of math, you should be able to take more useful classes that will prepare you for the future better, rather than memorizing equations after equations about weird shapes that will be irrelevant to anything in my future. Yes, all math levels can be useful to others’ futures depending on what career path they choose, but for the ones like me who know they are not planning on encountering extremely high level math equations on the daily, we should not have to take math after a certain point.

— Tessa, Glenbard West High School

## “Math could shape the world if it were taught differently.”

If we learned how to balance checkbooks and learn about actual life situations, math could be more helpful. Instead of learning about rare situations that probably won’t come up in our lives, we should be learning how to live on a budget and succeed money-wise. Since it is a required class, learning this would save more people from going into debt and overspending. In schools today, we have to take a specific class that doesn’t sound appealing to the average teenager to learn how to save and spend money responsibly. If it was required in math to learn about that instead of how far Sally has to walk then we would be a more successful nation as a whole. Math could shape the world differently but the way it is taught in schools does not have much impact on everyday life.

— Becca, Bentonville West High School

## “To be honest, I don’t see the point in learning all of the complicated math.”

In a realistic point of view, I need to know how to cut a cake or a piece of pie or know how to divide 25,000 dollars into 10 paychecks. On the other hand, I don’t need to know the arc and angle. I need to throw a piece of paper into a trash can. I say this because, in all reality and I know a lot of people say this but it’s true, when are we actually going to need this in our real world lives? Learning complicated math is a waste of precious learning time unless you desire to have a career that requires these studies like becoming an engineer, or a math professor. I think that the fact that schools are still requiring us to learn these types of mathematics is just ignorance from the past generations. I believe that if we have the technology to complete these problems in a few seconds then we should use this technology, but the past generations are salty because they didn’t have these resources so they want to do the same thing they did when they were learning math. So to be honest, I don’t see the point in learning all of the complicated math but I do think it’s necessary to know the basic math.

— Shai, Julia R Masterman, Philadelphia, PA

Learn more about Current Events Conversation here and find all of our posts in this column .

## 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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An Educational Blog

My Teaching Journey

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## Reflecting on My Own Math Experiences

Hi thank you so much for being here..

Welcome! I am so glad you have come across this post! My name is Julia Park and I am a senior at Millersville University! I am an Early Childhood Education major and I have learned so much so far! If you have a moment, feel free to check out my previous blog posts!

In my last post, I shared information about learning centers in math class! In this post, I will be reflecting on my mathematical journey. My experiences in math have really shaped the way I teach my students.

## My Early Math Memories

I believe that early math experiences can really shape a child’s mindset towards mathematics. It has definitely shaped mine. Unfortunately, it has been a long journey of growing my interest in math, and I am still working on it!

When I was in elementary school, even up until my time at Millersville, math has been a huge struggle for me. I have grown up with the incredibly damaging misconception that you have to be a “math person” to excel in math. A lot of my peers had the same mindset, which made it even harder to let go of those limiting thoughts.

I discussed this in my growth mindset blog post , but “math people” do not exist! I have my own reasons as to why I thought there were math people, but children’s experiences often vary. I think my fixed mindset was formed from experiences with not-so-nice teachers, the pressure of time limits and the need for accuracy in class, and a lack of hands-on learning. Those are just a few ideas of why I think I have had a tough time with math and I will be discussing more ideas later in this post!

Although it was hard to get through math class sometimes, I am really grateful that I have had these experiences because I can learn from them and relate to my own students. I want my students to feel comfortable with asking for help and to know that it is possible to learn and grow in many ways!

## What I Have Learned From Past Teachers

Through my time as a student in math class, I have had many different experiences with a variety of teachers. I want to share the good and the bad of what I have gone through because I think it is beneficial for teachers to reflect on all experiences related to learning. We can take what we learn to inform our own teaching practices.

Positive approaches I have learned from teachers:

- Providing assistance outside of class
- Using a hands-on learning approach
- Giving time to practice skills in class
- Utilizing interactive math games
- Facilitating class discussions
- Being kind and encouraging when a student is struggling

Approaches of teachers that were difficult for me:

- Focusing on accuracy only and not effort
- Putting pressure on students to turn in extensive assignments with a limited amount of time
- Teaching new concepts too fast
- Using too many lectures and PowerPoint presentations
- Not having time to reflect on concepts in class
- Being intimidating when a student is struggling

Every student learns differently. These experiences are unique to me and not everyone will be able to relate to what I have taken from my past math classes. However, I think it is important to recognize that although one strategy might work for one student, it might not work for another student. This notion emphasizes the need for differentiation. I will be discussing differentiation more in the next section.

## Strategies I Want to Use to Teach Math

As I finish this semester at Millersville University, I am leaving with so many new ways of teaching math that I was not even aware of previously. I have a new passion for making math class fun and interesting for my students. The following are some examples of strategies I would love to incorporate in my future math class:

- My math instruction will be differentiated based on my students’ needs. I will monitor their progress through various assessments and observations to modify or individualize my instruction when needed.
- Hands-on learning will be included to increase the engagement and participation of my students. I want to make math fun and exciting!!
- Class discussions will be a huge part of my mathematics instruction. Discussions in math class promote a deeper understanding of mathematical concepts in children.
- I would love to try to use interactive notebooks to organize my students’ learning and create engaging experiences. I had not heard of these notebooks until this year and I love them!
- Technology , manipulatives , and children’s literature are just a few tools I plan on using to enhance mathematics instruction for my students.
- Parent involvement is very important for a child’s education and I will consistently keep in contact with families to increase this involvement.
- I am very passionate about modeling a growth mindset for my students. I want my students to believe in themselves and in their ability to grow.
- I will strive to create a safe and welcoming environment for my students. I want them to be comfortable with sharing their ideas and to not fear making mistakes. To do this, I will value effort just as much as accuracy.

## Mistakes Are Learning Opportunities!

One of the biggest lessons I have learned throughout my time at Millersville is that making mistakes is okay. I used to put so much pressure on myself to be perfect and know everything, but that is not healthy. Teachers are not robots made to feed information to students. Instead, we have a purpose to learn alongside our students and to welcome mistakes as learning opportunities.

I am much more comfortable now being honest with my students in moments of uncertainty. I would rather figure something out with them than provide them with the wrong information. It’s really fun to explore ideas with students and work together toward a common goal. These experiences with students are valuable and strengthen the student-teacher relationship. When children trust their teachers, they are more engaged, motivated, and feel an increased amount of comfort when reaching out for help and sharing their thoughts with others.

Check out my blog post about growth mindset to learn more about the importance of making mistakes and the value of having a positive mindset in math class!

## Thank you so much for reading!

I had a blast sharing my mathematical experiences with you all! I have grown so much through the years and I can’t wait to keep growing as I gain more experience. I hope you learned about some ways you can teach mathematics in your own classroom! Thank you for reading. I sincerely appreciate it!

## Published by

Hi! I am Julia Park and I'm a junior at Millersville University. I am currently studying Early Childhood Education. I am so excited to share my journey through my new blog! View all posts by Julia Park

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## COMMENTS

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.

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 ...

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.

Math may seem "useless" and "annoying" to many, but the essay gives readers a clear message: we need math to succeed. 3. Short essay on the importance of Mathematics by Jay Prakash. "In this modern age of Science and Technology, emphasis is given on Science such as Physics, Chemistry, Biology, Medicine and Engineering.

Reflections: Students in Math Class. 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 ...

Patterns are regular ,repeated ,or recurring forms or design and we also learn the difference between the inductive and deductive ,Inductive is the process of reacting a general conclusion by examining specific examples ,Deductive reasoning is the process of reaching a conclusion by applying general assumptions, procedures or principles ...

Free writing is a good way to start. Set a timer and write continuously for 10 minutes without editing. These early drafts will be clumsy, but there will also be some gold that can be mined and developed. Writing can be used to analyse and organise ideas. When stuck, try to restructure your ideas.

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 ...

mathematics is written at university. Writing mathematics involves putting together a coherent argument. I'm not saying you have write an essay, rather that you write mathematics so someone else can understand it. You will get more marks that way and when you go out into the big bad world and get a job you will have a useful skill -

Abstract. This book is a collection of essays on mathematics and the nature of knowledge. We claim that the mathematical sciences, mathematics, statistics and computing, are almost everywhere. In this introductory essay we present in brief our argument why these sciences are essential for human thought and action.

The above statement remains true today, although it was written almost ten years ago in the Mathematical Sciences Education Board's (MSEB) report Everybody Counts (NRC, 1989). In envisioning a future in which all students will be afforded such opportunities, the MSEB acknowledges the crucial role played by formulae and algorithms, and suggests that algorithmic skills are more flexible ...

Using "I" in a reflective paper is generally appropriate, and for other types of writing, "we" may be used occasionally. Writing in mathematics should be careful of tense. When describing facts, use present tense (facts are true). When describing experiments or methods, use past tense (experiments were conducted).

The use of reflection prompts is important. This has two benefits: first, they focus students' thinking and encourage depth of reflection; and second, they provide information about student misconceptions that can be used to determine the content of the following lessons. Sometimes teachers fall into the trap of having a set of generic ...

Stuck on your essay? Browse essays about Mathematics and find inspiration. Learn by example and become a better writer with Kibin's suite of essay help services.

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 ...

A solution to a math problem is an argument. Therefore, it should be phrased and structured in the same way as an argument, with the intent to convince the reader of a certain idea. This is similar to writing an essay, where both the essay and the mathematical solution may be viewed in terms of content and style.

math experience essay. Sort By: Page 1 of 50 - About 500 essays. Decent Essays. My Experience with Math. 739 Words; 3 Pages; My Experience with Math ... Over the past four weeks I have learned a great deal of information and how algebra can be applied in real world applications. The. 722 Words; 3 Pages; Decent Essays. Preview.

Nov. 10, 2022. "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 ...

The intention is to provide teachers a complete resource that addresses all expected learning competencies, as stated in the Department of Education's official curriculum guide. This resource is a teaching guide for General Mathematics. The structure is quite unique, reflective of the wide scope of General Mathematics: functions, business.

In the same manner, the most learned competencies in general mathematics were determined based on the obtained actual test result. Table 4 shows the identified learning competencies in each topic. Six (6) competencies were covered in the area of Functions and their graphs with evaluating a function competency topping on the list.

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 ...

What I Have Learned From Past Teachers. Through my time as a student in math class, I have had many different experiences with a variety of teachers. I want to share the good and the bad of what I have gone through because I think it is beneficial for teachers to reflect on all experiences related to learning. We can take what we learn to ...

essay/reflection - mathematics in modern world for midterms and finals midterms reflection paper studying in the midst of pandemic is quite challenge whole new. ... Both the lessons taught in General Mathematics and Statistics and Probability back in senior high school were encountered once again in this course, but this time with a better and ...