We Don’t Know How To Teach Math

And we barely know why we teach it, too.

We just don’t. Not to and for everyone.

If we didn’t, it wouldn’t be o.k. for so many people to say, “Oh, I hate math.” And yet, you can be in a room full of really smart, successful people, and you can guarantee that a not insignificant subsection of the people there will say that they hate math.

That means we are not doing a good job somewhere. Somehow in the “How,” “Why” and “What,” we are falling down on the job. And because we don’t do it well, math remains the third rail of progressive education. Seymour Papert (source – the Papert scholar, Gary Stager) said that math represents the failure of progressive education because the way we teach math always reintroduces coercion back into education.

And yes, there are those who fall in love with the pure beauty of higher level mathematics. For those of us who would have moments of epiphany when you could just “see” the math unfold in a way that seemed to explain the universe, math class could be amazing. But the fact that there are some of those folks seem to be justifying not changing the fundamental pedagogy and structure of math, thereby ignoring all the kids who never understood or cared about a sine wave.

Conrad Wolfram’s TED Talk about the need to understand the difference between computation and real mathematics is a good start to try to figure out how to do it better.

Some school – like the iSchool in NYC – just punt. They teach the stuff of math using computer software, and then try to use their project-based instructional time to have kids apply the math in their projects. Does it work? It probably beats sitting and listening to a math teacher write and explain the math on the board.

But it begs a fundamental question for me – why do we just teach math? Problem sets and the occasional poorly worded word problem seems just wrong to me when mathematical thinking is problem-finding and problem-solving of the highest order. Of all the subjects that seem to have been most damaged by siloing and a lack of true interdisciplinarity, math seems to suffer the most.

After all, math is the language of the physical world. There’s more real math in the arc of a frisbee in flight than in all the word problems in a textbook.

Math is the language of probability. Any poker player who has to consider the pot odds to know if they are making a good bet is doing applied mathematics.

Math is problem-finding and problem-solving as anyone who has ever tried to figure out why they have no money at the end of the month can tell you.

And let’s not forget that the same mathematical problem solving algorithms are involving in every computer programming challenge we could possibly undertake.

What if we completely rethought the way we taught math so that everything was structured around using math to seek out and solve problems? What mathematical concepts would become paramount? What pedagogies would come to the forefront? And how much could we finally get so many people to be willing to say that they simply “hate math?”

– Posted using BlogPress from my iPad

29 thoughts on “We Don’t Know How To Teach Math

  1. Alleluia!!!! I am a high school Math teacher and would LOVE to teach Math the way you have described. However, we have an antiquated curriculum AND those Standardized tests that assess the antiquated curriculum. We NEED to teach Mathematics as Problem Solving. That skill will be useful to ALL of our students. Chris, thanks for putting into words what I have been feeling for years.

  2. Love your post, Chris. I wouldn’t argue against all math, but I would start with Algebra – the gatekeeper to broken dreams. Why does that have to be paramount in every students life? Make it optional, for goodness sakes. As you pointed out there are so many more interesting things for kids to do.

    • Algebra–not as taught, but as a subject–is clearly something that not only do students need to learn, but in fact most people *do* learn more than they realize. If you’ve ever figured out how much a bulk purchase was costing per unit, that’s algebra. If you’ve ever built something to fit into something else and had to figure out measurements, that’s algebra. That’s why it shouldn’t be optional.

      • I respectively disagree. You certainly can use algebra to solve both problems and that is cool, but do you really need to use traditional algebra to solve them? All you need is everyday math which is what most people would use to solve them.

      • I also disagree. While the examples you point out are indeed and can be solved with Algebra, if that is all Algebra was then most people would breeze through.

        Even now, I am studying pre-algebra in college. I have made the Deans list but I have not taken any real math all last year. But now I must learn how to contend with Parabolas and Functions. I don’t ever see any human being not in a STEM field having to ever use a Parabola or figure out how to find the X and Y intercepts of a curve. So no, Algebra as a whole is not used in everyday life. Picking from the most basic of examples is a bad example.

        • What you are missing is the fact that math is the tool used to explain how things work. Not to get all nerdy and math crazy here but “Algebra” is not simply used to calculate figures or find x and y intercepts on a graph. Algebra is used to calculate angles (pitch) of rooftops, calculate how far you can go with the gas light on before you are stranded on the side of the road, whether you need one gallon of paint or three gallons of paint when repainting your living room walls (because you can’t take back specially colored paint and two gallons mixed at two different times are two different colors). Yes, Algebra that is taught in middle and high school is concentrated on graphing lines, finding slopes and intercepts, maximizing and minimizing gains/losses, and of course those ridiculous word problems that were made up by some textbook publishing company to come out to integers. That is not what Algebra is in real life though. Algebra is modeling what is going on around you. Figuring out whether it is cost effective to replace your single pane windows with new double paned windows or to just replace the seals around the window. Algebra is balancing your checkbook in the positive each month. Algebra is not having to be “that guy” at the checkout line that has to put things back because they picked up too many items and don’t have enough money to pay for them now that they realize how much they cost.

          Algebra IS all around you all the time. You don’t see it in X’s, Y’s or Z’s but it is there. In my opinion, I think that the traditional Algebra I, Algebra II, Geometry sequence that is generally taken in high school needs to be revisited and revised. Mathematical modeling (taking REAL situations and explaining them through mathematics) should take their place. Then we would not hear so many people have such a negative attitude toward mathematics.

  3. Ihor, I don’t think Algebra is the problem, but rather a poor teaching of it. Algebra in and of itself is simply a way to represent the world algorithmically. If a student can’t use the concepts of Algebra, they can’t solve these problems of life. This not to say that we really teach this fundamental aspect of Algebra over simply memorizing rules for manipulating symbols.

    • Algebra is not a problem! I’m just arguing for it to be an optional choice for students not required by everyone. Most people I talk to say they have never used it in their adult life. Nor do they need it. That it doesn’t mean that its a waste of time for someone who genuinely wants to learn it. Most students (and I speak from experience) go through the motions of learning it without getting much out of it. I think requiring it does more damage than good. Mathphobia is rampant. Now that does not mean math should not be taught or rather not be learned. Its (everyday math) important to learn. We need better curricula and approaches if that is to happen.

      • Most four year colleges will not even think about admitting you if you haven’t taken a level ABOVE ALGEBRA II, let alone Algebra 1….you think maybe they (higher learning institutions) have found out the need to know it???

  4. I suspect part of the answer lies in that “math is the language of the physical world” thing. Which is exactly right: it’s a language. We don’t require all kids to learn languages, and while language instruction can be awesome and fun, like math it’s often seen as dull, mostly memorization and drilling. Kids in a class can practice their new language with each other–which is more than we ever have them do in math classes–but even that is pretty awkward and difficult, because everyone’s vocabulary is so limited and if they actually want to communicate the temptation is to lapse back into their native language.

    I wonder what would happen if we could have kids learning math use the language of math to talk to one another and to fluent speakers: explain to a computer how to build something virtual, or give coordinates and directions to a 3d printer. Have adults who are fluent in math run a math class the way language teachers run language classes: you must speak to me in the new language. Lots of pantomime, of course, but I wonder how that would go.

    • I think that would great. But I don’t think that the stakeholders would have the courage to implement it even if they agreed with us. “yes, but…”

  5. I think that your proposed solutions to math education — in short, to rely on mathematical applications to other fields — fails to capture much of the mathematical experience. Math is about solving practical problems, sure, but it’s also about imagination, play, beauty, and abstract structures. Paul Lockhart’s new book, Measurement does a great job of conveying this side of math.

    But, for a second: why do we teach math? I’m feeling a bit tired of this question.

    There’s no good reason to teach any particular subject to everybody in the world. The world would look largely the same if we swapped Music for Math, Law for History, or taught Psychology to 9th graders instead of Biology. (Evidence? Since we’re doing such a lousy job teaching the subjects we currently teach, the impact of radically changing our curriculum would be negligible.) There is rigor in all these disciplines. There are legitimate reasons why humans would enjoy studying them. There are boring parts. There are ways of making these subjects come alive. There are ways of ruining them. People will always need things that they aren’t taught in school, and people will learn the things that they need to learn.

    In the end, we teach math because we used to teach math. You have something you’d rather teach? Fine with me. Just don’t ruin it.

    • Reply to mpershan
      The second question was not ‘why do we teach math’ – your response to that question is right – i.e., ‘why do we teach anything’?
      The question was “why do we JUST teach math” – Chris suggests using ‘problem sets and the occasional poorly worded word problem seems just wrong’ — and ponders that math should not be siloed – ‘Put away your books, kids- now we are going to do math’ – it needs to be integrated and experienced. We need a different way.

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  7. I think that my poor understanding of math contributes to my inability to teach it well. When I was in University students headed to the elementary teacher training degree program could take a “math-for-teachers” course in place of the more rigorous intro math courses like vector geometry and linear algebra. It was a good methods class for teaching but not a good grounding in mathematical concepts and deep understanding. When given a choice my first instinct in PD as a teacher today is always math sessions or math summer camp courses to brush up and catch up my math instruction. Does anyone else find that not teaching math well goes hand in hand with superficial teacher understanding of the subject?

    • No, I think good teaching is independent of subject matter. I got my degrees in math but that’s not what made me effective with kids. What did was my interest in learning it with my students and being able to say: “I don’t know the answer, let’s work on finding it out together.”

  8. I just got my license in math through a master’s program which was supposed to be competitive. When presenting some micro-teaching among other preservice teachers, the English teachers could not understand enough math to follow my middle-school level lesson and participate in the activity. I can tell you that these teachers, supposedly highly qualified, were not concerned. On the other hand, I felt that I had a good handle on the lessons I saw in Government, English, and Science (disclaimer: I wasn’t a typical student). I’ve been at schools where subject integration is supported, encouraged, mandated, and regular. I’ve been at magnet schools where integration is not practiced. If we work in a system which forces us to hire teachers who barely squeak by on the math Praxis, we can’t expect them to care about math (they got a job teaching some subject without it) or act as a role model for their students who develop the ‘I hate math’ attitude (I didn’t need it; why should you?).

    I don’t think teachers are off the hook here. When English teachers are discussing the writing style of Shakespeare, then either in English or in Math, students could be finding lines of best fit for number of distinct vocabulary words used, frequency of conjunctions or indefinite pronouns, and so on for all of Shakespeare’s plays, similar to the way that researchers use middle-school math to try to figure out if the Bard really wrote some newly-found manuscript (see RadioLab’s “Vanishing Words”, for instance). In Science class when discussing rising sea levels or forest fires, somewhere they could be building simulations and using computer output as the basis for a lesson on data analysis or solving equations. Yes, it takes lots of time both for the teacher and student. Yes, it means that some content might not get covered. But it means that the student gets something more valuable and maybe now they can figure out the rest on their own. I’m too young to understand why a right answer in Chemistry class requires use of significant figures but a word problem in math doesn’t.

    I am blessed to work at a place where I eat lunch with teachers each from a different discipline, and regularly, we ask, “What are your students working on?” “How can I get involved?” Last week, the science teacher passed out his next assignment and asked me, “What don’t you know on this sheet and what could I help explain to you?” I admitted my gaps in knowledge, I asked questions, and I learned something.

    As you could probably tell, I’m not exactly sure what my point is, but I do think that responsibility is shared. Professional development for math teachers should consist of math teachers getting together to discuss and solve math problems. If every other teacher is invited too, they begin to understand what is asked of a student who has to contend with eight separate disciplines. Maybe math starts to matter when its relevance is demonstrated by other teachers.

    • Tim
      A great deal to unpack in your comment here. I, too, teach math and I take pride in being able to discuss books and politics with my English and History colleagues and with my students. However, I recognize that this is an easier task for me than it is for my English and History colleagues because I remain immersed in what they do while they do not remain immersed in what I do in any way that they recognize. Dan Kennedy wrote a brilliant essay about assessment in mathematics ( http://mail.baylorschool.org/~dkennedy/assessment ) and makes an important statement in the essay. He says, and I am paraphrasing, that adults who say that they don’t understand math are saying it because they have walked away from school thinking that understanding math means being able to manipulate symbols in an algebra equation. What those of us who love math recognize is that algebraic thinking is hiding in all sorts of places but most students don’t walk away recognizing is that math is as much a habit of mind – a WAY of thinking – as it is a set of equation solving skills. I think that is the heart of what Chris is trying to get across here. It is on our shoulders to make this change. Our students are pretty astute – they determine what we value in large part based on what/how we assess and evaluate them. If we say we value thought process and problem-solving we better not give them multiple choice tests. If we say we value thinking and problem-solving techniques we better not assess them on carbon copies of problems we have already given them. We need to model what we believe and figure out how to pass that along. Not everyone will walk away loving what we love – that is probably an unreasonable goal – but if we can convince students that math is a tool to help analyze the world and not just a set of algorithms and rules, then our students will be better off than manny of our friends and colleagues who do not teach math.

  9. I teach algebra. Yes, it should be required. Yes, we use it everyday. Requiring math courses beyond algebra can be debated, I’ve taught those too. But knowledge of algebra is an unquestionable need in our society. If you just consider finances you will have your justification. Budgeting, depreciation cost, investing, simple and compound interest involve the concepts of per cents and exponents found in algebra. Predictions over time involves linear equations found in algebra. If you want to compare two cell phone plans you use a system of equations learned in algebra.
    The logical reasoning used in problem solving is found in algebra. I think we can all agree that we need a nation of critical, creative thinkers. Algebra again. Perhaps our elected officials need a refresher course on the basics of the subject. Maybe then we could avoid having a financial crisis every two months.
    There are too many aspects of our educational system that are archaic. I agree with that. Now, with 46 states agreeing to test on a set of common core standards in math and language arts, teachers will be forced into an even more doldrums type of teaching in order for students to pass the test. If they don’t they will be deemed an ineffective teacher.
    I commend the comments written so far. You are clearly well educated people. I often read comments on sites like this and shake my head to the ignorance being displayed. I don’t see that here.
    Did any of you skip taking algebra? I think not.

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  11. It will appear obvious to anyone who knows me that I would say that we need to teach subjects that require mathematical skills and then teach those skills required as the students discover the need for math to be successful. In other words we should be teaching all subjects in a project based environment. Teach geography and when students need to measure teach them how. When they need to understand area or volume teach those skills then. Understanding mathematical relationships should never be taught in isolation – yet what do we do? For most of the first 6 years of education students go to their “math” time to learn how to count, add/subtract, multiply/divide, and deal with ratios. Why? Because that’s how we’ve always done it. Then these students are given some contrived “word problem” to solve. This needs to change.

  12. I haven’t read all the comments here, and I don’t know if I can manage to say what I really mean, in English, Maybe not in Norwegian either. Anyway, I think I both agree and disagree with you. I am a maths teacher, for from 16 to 20 years old people. Their problem started long ago. The problem is that they wasn’t ever taught mathematics. They wasn’t even taught calculations. They have had a bunch of teachers who have learnt modern pedagogics. The last 20-30 years math-teachers have tried to find fancy ways of making maths interesting, without actually teaching maths. We need to go back. We need to teach children all the basics, we need to teach them how to love to calculate things. Step by step. teach them the logics. Make them practice it. Then the logics. Then practice it. Teach them to think abstract. Not underestimating their ability. Pushing them. Make them love the subject mathematics in itself! Not make a game that contains mathematics. Let the maths become the game instead!

  13. What an awesome and thought provoking post! As a former high school math teacher and mathematics teacher educator, this gives me lots of food for thought. I’m beyond the question of why we teach mathematics. Like one commenter said above, we could debate the merits of teaching many other subjects. I totally agree with your assertion that our methods are antiquated. There’s so much research being done in the fields of learning sciences and equity and diversity that could be useful in reaching students who often cringe at the thought of doing math. Keep fighting the good fight!

    • I agree. It is amazing to see how far some people will go to “make math fun” without realizing that just looking at things that are right in front of your face are as far as you need to go. Real can be fun, it doesn’t have to be some crazy made up game. Giving students a solid base and leading them toward an A-HA moment can be truly amazing. And what’s better is that once that A-HA moment has happened, they want another one and another one and another one. What do you know, students like (maybe appreciate would be a better word here) math when they can use it and see it in their lives. You don’t need tricks or gimmicks, just a solid foundation to build on.

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