It could be because the mathematical procedures that are taught in schools will be useful to students later, but I am pretty sure this is false. Almost everyone forgets those procedures as they get older because most people in our society use virtually none of the procedures they learned in school in their day-to-day life. Obviously there are engineers, mathematicians, and scientists who use the mathematics they have learned, possibly on a daily basis, but I think if you dig deeper into the work they do, many of these people use tools to help to do their work (like Mathematica, for example), look up the finer details of mathematical procedures that they do not use often, or who use only a very specialized portion of their mathematical knowledge regularly.

It could be that we want to expose students to different ways of thinking about the world. In this case we would be less concerned with the exact set of mathematical procedures they have learned, and more concerned with learning mathematics as a way of thinking and knowing. I see little evidence that this is an explicit goal of mathematics instruction given that; the students are assessed only on the procedures, teachers are assessed on their students understandings of those procedures, and that the set of mathematical procedures we want students to know is so prescribed such that it is virtually identical around the world.

It could be that we would like students to learn transferable problem solving skills. In this case, we want to teach mathematics in such a way as to promote the likelihood that students will be able to transfer what they learn to other areas. Cross-disciplinary study would be the norm, rather than the exception. It turns out that "teaching skills that transfer" is not as simple as one thinks. In fact, my understanding is that most of the times when people learn skills in one context, they do not end up transfering those skills to other contexts. Instruction that aims for transferable skills has to provide opportunities for students to make connections between different areas, reflect on what they have learned, and develop metacognitive strategies so that students think about their thinking. What evidence is there that these types of activities are a regular part of math classes?

It could be that we would like students to see the beauty and elegance of mathematics. One way to do this could be through exploring mathematical art. Another might be to look at some famous examples of truly elegant uses of mathematics. We could also ask students to talk about mathematics in the abstract and come to a shared understanding of what elegance and beauty in mathematics mean. As far as I know, none of these activities is a common one in math classes. It is depressing to me that this way of thinking which has so much beauty in it is shared in such a way that almost no one in our society ever gets to experience beautiful mathematics.

If one or more of the reasons I suggested above is something you think is a good reason to teach mathematics, how are you ensuring that you meet this goal with what happens for students in your classroom?

What other reasons are there to teach mathematics?

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David is a Formative Assessment Specialist for Mathematics at **New Visions for Public Schools** in NYC. He has been teaching since 2002, and has worked in Brooklyn, London, Bangkok, and Vancouver before moving back to the United States. He has his **Masters degree in Educational Technology from UBC**, and is the **co-author of a mathematics textbook**. He has been published in **ISTE's Leading and Learning**, **Educational Technology Solutions**, **The Software Developers Journal**, **The Bangkok Post** and **Edutopia**. He blogs with the **Cooperative Catalyst**, and is the **Assessment group facilitator for Edutopia**. He has also helped organize the first **Edcamp in Canada**, and **TEDxKIDS@BC**.

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## Comments

## One could also ask "Why teach

One could also ask "Why teach Shakespeare?" With respect to the question at hand, learning mathematics is FAR more than understanding procedures. In fact, procedural knowledge becomes less important BECAUSE of the technology now available. Having said that, a basic understanding of procedures is important as it provides some degree of "cohesion" in problem solving from its inception to its end. It provides the problem solver with an understanding of what goes on "behind the scenes" as technlogy is utilized to do the "grunt work". Some would likely say this is not important but I believe it is. More importantly, if too much knowledge/understanding is compromised through reliance on memorizing algorithms with no supporting knowledge of why they work (which has little to do with mathematics learning IMO), students will have a much more difficult time in moving forward to designing solutions to "real-life" problems that DO call upon mathematics. If high school students are studying "pre-calculus", a reasonable assumption is that they may choose to study calculus at post-secondary; these students should then be given as much knowledge/understanding as possible to adequately prepare them. We should not compromise understanding by giving algorithms to "get by" to improve the appearance of the student's high school transcript. Too often, students are ill-prepared for post-secondary mathematics study and they meet with disaster. This is something that needs to be addressed.

Thank you.

## To decide that it's

To decide that it's acceptable that nobody except engineers should be able to do math in the world is pretty egregious... to teach the procedures that only the engineers would do is also egregious.

## I'm not sure if your comment

I'm not sure if your comment is in response to mine or something else. To clarify, I was using study of post-secondary mathematics as an example.......and just so you know, there are other areas in post-sec requiring higher level mathematics, some of which are far more demanding than engineering. :)

## My primary goal in teaching

My primary goal in teaching mathematics is to sharpen the thinking ability of my students in regards to three areas: conceptual understanding (why do the interior angles of a triangle sum to 180°?), problem solving (I prefer complex problems that take about an hour to solve), and math skill acquisition (factoring polynomials). Ignoring any aspect of this triad results in an unstable math education, and one which misses the opportunity for helping students develop the cognitive skills needed for life.

How to ensure these are happening? I measure conceptual understanding through short essay questions on every quiz and test, after including these types of questions on homework as well. Problem solving skills are taught and applied in group settings as well as on tests (eventually, since most enter my class with little previous experience with problem solving). Math skills are what most math teachers know best, and traditional methods are fine here.

## Independence and Questioning

Hi David,

Great Blog entry - I have, for a long time now, told teachers I work with and my own students, that the main goal of my courses, and teaching math in general, is to help produce independent problem solvers. So I guess, I teach math(the subject) specifically because I enjoy it and it lends itself to solving problems - but this could also be applied to philosophy and even literature if looked at it through a problem-solving lens.

I totally agree that students who get caught up in trying to figure out a certain procedure still don't remember it. In fact, just this past week I gave in (or was actually doing a little experiment) in my PBL geometry class wherein the kids were convinced that if they did enough of one type of problem they'd be able to do it on the test for sure. I gave them about 4 or 5 of the same type of problem - find the point on the coordinate plane that was the "closest" point to a line. They learned the procedure - we wrote the equation of the line perpendicular to the one given through the point off of the line and found the intersection. However, on the test, I put the question in a context where it was a cheetah (on the point off the line) and an antelope running (along the line given) and said where would the cheetah have to run in order to get the antelope where it was closest? I'd say almost half the class did not recognize this as a "closest point on a line to a point not on the line" problem.

It's an interesting dilemma and I believe Polya tried to honor it. Do students "learn" problem solving by doing the same type of problem over and over again? Clearly not - they need to try new problems to learn how to deal with those new situations. That's where the questioning comes into play - learning how to question in the best ways possible. I love your list of questions for problem solving for student - very nice. However, doing novel problems doesn't often seem to teachers like they are 'teaching" does it?

One more thing - this goal also brings up a major question of assessment that many teachers overlook - how to assess problem solving (if that becomes your major goal).

thanks for listening - and continuing your work!

Carmel

## Non-educator perspective

I'm an electronic design engineer and robotic software developer who's been thinking & reading about education recently. I was not "good at math" in school, except for the word problems. I thought they were easy and didn't understand why everyone else thought they were hard. I very much disliked all subjects, except for the reading part of reading. In 5'th grade I had the stunning insight that I could write down any old answers on the worksheet, and get back back to thinking about submarine design a lot sooner. In 7'th grade (1975) I got placed in remedial math rather than algebra because I wasn't fast enough at arithmetic. As well as a lack of interest, I also seemed to have a below-average ability for rote memorization. Instead of memorizing math facts, I made up rules like the "almost ten" rule, where to add 9 or 8 or maybe 7 I would add ten and then take away 1 2 or 3. I also had some system where I counted the four tips of a 4, five points on a 5, etc. I found algebra and beyond to be more interesting, though I still couldn't bring myself to do homework, which brought my grades down. Ironically, I eventually did get a BS in applied math (computer science), but had problems with the "cookbook" engineering ODE's class, which I failed twice. Then I hit on the idea of taking the real math-major ODE class, with proofs and so on, which worked better for me.

Today I do regularly use basic algebra and also linear agebra for computing 3D geometry, and I read and write equations in academic publications, so I'd imagine that places me somewhere in the top 1% of math users in this country, though my skills and aptitude are by no means outstanding in comparison to some of those I work with. On occasions where I need to do any complicated algebra, I use a computer package, because computers are better at doing that kind of thing without making mistakes. I use other specialized tools for electronic circuit simulation and control system analysis, and do not really understand at any level of detail how they work. When you're using these kinds of simulators you need to maintain a degree of skepticism, which has to be based on an understanding of the behavior of the thing being simulated, but not so much on the workings of the simulator itself. You come to recognize pathologies like a "crunchy" current signal, and play with the simulator options until you get a credible result.

I also find math concepts interesting, and have read quite a few nontechnical books about things like transfinite set theory, Godel incompleteness, and so on, but my eyes do still tend to skip over the equations.

So much for my personal experience and comment on what engineers do. Why teach math? Clearly how we teach math has a great deal to do with how it has always been done, and the "why" was therefore not often questioned. Arithmetic has been part of the school curriculum since ancient times. Land has to be surveyed, taxes assessed, and accounts figured. These were done by the educated elites, while ordinary laborers had little use for numeric skills. With the rise of market economies, almost all daily needs were met through the market, so everyone needed some number sense, and benefited from some arithmetic skills. So far as daily living and workplace needs go, manipulating money and quantities of goods to be bought or sold, profits, interest and so on are the main practical needs.

At one time, pencil and paper arithmetic was clearly a useful skill, while today this is something that is hardly done outside of school. It is still important to be able to understand practical arithmetic problems so that you know which key or spreadsheet symbol to use. Traditional elementary math was moderately effective in conveying this sort of understanding, but I don't think it's really known how important practice with manual arithemtic is for conveying this sort of understanding or for developing "number sense".

I've been studying the history of education because I have an interest in how cultural institutions arise, presumably meeting some cultural need. Education is an interesting case because the overt function of school (teaching of useful subjects) does have obviously have some validity, but doesn't seem sufficient to explain the curriculum. Math education being in excess of obvious need is not a new thing. Since classical times, geometry and logic were part of the curriculum, often with various odd numerological accretions, such as (during the middle ages) extensive classification of special kinds of numbers, going well beyond square, prime, ... .

I think that when you consider the social function of school as a whole you appreciate that it can have value largely independent of the usefulness of the subjects taught:

I realize that my educational difficulties were not entirely typical, and as much as anything reflect my personality defect of not liking to be told what to do. Both of my parents had master's degrees, we had thousands of books around the house, and in my neighborhood we were the kids who used big words. I know that in recent years a strong emphasis of education has been on bringing up the test scores of the worst performers. I don't represent that demographic, but I can relate to their lack of motivation. In my recent reading on education I was unreasonably surprised by the statement that people who choose to go into education often liked school. On one hand, that certainly makes sense, but I guess I found the idea that anyone liked school to be somehow incredible.

@robamacl

http://humancond.org

## This is a very interesting

This is a very interesting post. I have always wanted to know why I had to learn math. I am currently studying Elementary Education and have absolutly no use for algebra or calculus. This is a very interesting way of answering that question. Having math taught in schools so that students could gain a different understanding of the world seems probable. The one point that I can see as extremely important in math is problem-solving skills. This is always helpful and is found in every grade level's math class. A third advantage to students learning math that you did not mention is simply for them to be well-rounded individuals. I'm not sure this is a real reason, but it was something my school always stressed. This was a very interesting blog post and helped me to be able to finally think of an answer to that question.

## From A Pre Service Math Teacher

David,

I’m a pre service high school math teacher, and your post highlighted some of my biggest concerns about the journey that I’m embarking on. As a lifelong lover of math, which I know puts me in the minority, I sometimes find it difficult to put myself in the shoes of those who dislike it, struggle to understand it, and/or fail to see its worth in their lives.

One of the biggest challenges in trying to break the barriers between math teachers and their less-than-enthusiastic students is the age-old question of, “When am I ever going to use this?” I have a Bachelor’s degree in Actuarial Mathematics, and I spent six years in the actuarial profession before taking a leap of faith and deciding to become a teacher. During my time in pension consulting, I considered myself to be the “keeper” of software that did most of the heavy-lifting for me. This experience makes it difficult for me to answer the question stated above, as I rarely used a lot of the years and years of math that I took even though I was in an applied mathematics career. In fact, I have struggled to recall a lot of the Pre-Calculus concepts that I forgot a long time ago in my student teaching, which is difficult to explain to the students who are struggling to master the material.

As a step in working around this issue, I love your comments about teaching mathematics as a way of thinking and knowing. Many important skills, such as analytical reasoning, problem-solving, and abstract thinking, are easily fostered in the math classroom, and these are all highly transferrable across content areas and, in my humble opinion, quite valuable in life. However, I agree that the current standards for assessing students and teachers do not lend themselves to a heavier emphasis on these skills in place of a primary focus on the procedures that will get them the right answers on a final exam or a standardized test. I also feel that the current preoccupation with meeting certain benchmarks takes away from creating opportunities for the students to engage in self-reflection about their own learning, which I am learning is so important.

Finally, what I find to be the most disheartening feature of the typical high school math classroom in America is the oft neglected time to appreciate what you described as the “beauty and elegance of mathematics.” I started out liking math because I was good at it. I grew to love math for the beauty in the theory, the logic, and the expansion of knowledge, even in a general sense, that it affords. I aspire to be the kind of teacher who can illustrate this to her students, even if only a few at a time. Have you designed certain classroom activities with this in mind? I would love to hear about any attempts, successful or otherwise, that you have made in this regard.

Erin

## questions

The questions you ask speak a lot about your philosophy as an educator. I teach middle school math and find I can create much better units if I keep those bigger questions in the front of my mind - the questions about the larger purpose behind teaching this concept or idea or procedure.

I was wondering how you answer those questions in secondary mathematics. I may be teaching algebra next year instead of middle school math, and I will struggle again with the higher purpose of why I'm teaching this. How do you answer that question for yourself and what do you do differently in the classroom as a result?

## You might not be able to

You might not be able to answer that alone. You will end up in a school with a team of other math teachers, and with parents of the students in your school, and with students who have their own goals for their education; as a group, you have to decide what your overall goals are for your math instruction, and then be clear moving forward as a group how the activities and instruction in the classroom matches your goals.

If you end up with more autonomy than is normal in schools, you can do the same thing with your students, and communicate your goals for mathematics education with the parents of your students. One way teachers do this is to invite parents to come in and experience some of the mathematics their students are doing, and work together to decide on the overall goals, even if no one else in your department is thinking about this.

## Algebra

Follow-up questions: In your opinion, why do we teach algebra? And, HOW do we teach algebra to align with that WHY?

I have been looking for an elevator speech. :)

Thank you, by the way, for tweeting out my blog. It is my current project! I am fortunate to have quite a lot of autonomy in my school and a class set of computers, so the world is full of possibilities.

## I think that we teach algebra

I think that we teach algebra because it is a bridge between the more concrete ideas children have developed about numbers, into the more abstract realm of ideas about classes of numbers. It is also reasonable to think about the Powerful Ideas of mathematics, and how many of these ideas are embedded in thinking about algebra. Our goal then is to teach in such a way that a set of over-arching and transferable ideas are accessible to students and to promote critical thinking.

## I like it

I like your elevator speech. The powerful ideas post reminds me of the Standards of Mathematical Practice, a la Common Core, but easier to conceptualize. If I were teaching a class with the goals you describe, I'm not sure it would look much like traditional algebra, but you can use ideas from traditional algebra in it. It will be a challenge.

## I recently used this blog

I recently used this blog post in a workshop, and I thought I would share some of the comments educators wrote about it:

Discipline - we need to learn how to learn so we can be life-long learners.

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