I recently finished reading Jo Boaler’s book What’s Math Got To Do With It? and I need a way to reflect on what I had learned and consider ways to implement her research in my own teaching. This book was fantastic. In every chapter Jo made at least one point that just blew my mind. I was highlighting things in the PREFACE, people. I don’t usually even read the preface.
I’m going to use quotes from the book and expand on them with my own thoughts. I’m going to cover the introduction and first two chapters in this post, and I’m planning follow-up posts to cover the rest of the book.
Introduction: Understanding the Urgency
…we do not and cannot know what mathematics students will need in the future, but the best preparation we can give them is to teach them to be quantitatively literate, think flexibly and creatively, problem solve, and use intuition as they develop mathematical ideas. (p. 8)
Huh. That list kinda sounds like skills that student actually need to be successful in life, not just math. This is always the problem with asking the question, “When will I use this in real life?” The answer is usually, “You’re not going to.” Most people don’t need to know how to solve a quadratic equation, graph a sine curve, or integrate a function in “real life”. They do need to know how to think flexibly and creatively, problem solve, and use intuition as they develop ideas. If this was the focus of math class instead of computation, maybe we could finally stop asking that question.
Chapter 1 – What is Math? And Why Do We All Need It?
What is math? “…ask mathematicians what math is and they are more likely to describe it as the study of patterns.” (p. 19) Notice this doesn’t say computation. It doesn’t say memorizing rules or steps. It doesn’t say any of the things that we typically associate with math class. Mathematicians describe true mathematics as looking for and analyzing patterns in the world around them (not always related to numbers).
Imre Lakatos, mathematician and philosopher, describes mathematical work as “a process of ‘conscious guessing’ about relationships among quantities and shapes.” Those who have sat in traditional math classrooms are probably surprised to read that mathematicians highlight the role of guessing. … [W]hen children who have experienced traditional math classes are asked to estimate, they are often completely flummoxed…. This is because they have not developed a good feel for numbers, which would allow them to estimate instead of calculate, and also because they have learned, wrongly, that mathematics is all about precision, not about making estimates or guesses. Yet both are at the heart of mathematical problem solving. [emphasis mine] (p. 25)
This is the hardest part of any conversation about how math class should be taught. It’s hard to explain to people that their beliefs about what math truly is are wrong, usually because they never got far enough in math (usually graduate-level work) to see how creative and exploratory math really is.
True mathematics is: exploratory and creative, collaborative, and all about asking questions.
People commonly think of mathematicians as solving problems, but as Peter Hilton, an algebraic topologist, has said, “Computation involves going from a question to an answer. Mathematics involves going from an answer to a question.” … Bringing mathematics back to life for schoolchildren involves giving them a sense of living mathematics. When students are given opportunities to ask their own questions and to extend problems into new directions, they know mathematics is still alive, not something that has already been decided and just needs to be memorized. [emphasis mine] (p. 27)
This is what I need to be focusing on in my Algebra 1 Support classes this year (where I have the freedom to explore without the constraints of a curriculum calendar) – “pos[ing] and extend[ing] problems of interest to students” (p. 27-28). I need to give them the chance to explore, not just memorize and regurgitate.
Now, this analogy is my favorite, right here. I’ve seen it before, I think in A Mathematician’s Lament by Paul Lockhart. (Incidentally, that book is the one that first began to teach me that math could be something more than memorizing and regurgitating.) I love the comparison of the way we teach math to a hypothetical way we could teach music. It really helps to underscore the way we suck all the interestingness out of math.
Imagine music lessons in which students worked through hundreds of hours of sheet music, adjusting the notes on the page, receiving checks and crosses from the teachers, but never playing the music. Students would not continue with the subject because they would never experience what music is. Yet this is the situation that continues, seemingly unabated, in mathematics classes.
… Students should not just be memorizing past methods; they need to engage, do, act, perform, and problem solve, for if they don’t use mathematics as they learn it, they will find it very difficult to do so in other situations…. (p. 29)
NO WONDER kids hate math! In its truest sense, math is supposed to be creative and fascinating and wonderful and we make it so boring. “We cannot keep pursuing an educational model that leaves the best and the only real taste of the subject to the end, for the rare few who make it through the grueling years that precede it.” (p. 30)
Chapter 2 – What’s Going Wrong in Classrooms? Identifying the Problems
This chapter was frustrating to read, and I’m going to skip over the section about the math wars because it just made me angry. I just want to say that I truly cannot comprehend the motivations behind a movement that wants to keep math as dry and boring as we currently teach it. In our country, we already have a negative national attitude about school in general and math in particular. Why would anyone fight to keep that? Why would anyone think it’s a good thing to be raising yet another generation who thinks that “I can’t do math, haha” is an acceptable thing to say?
I did have a couple of notable things I pulled out of this chapter, in spite of the feelings of frustration and futility that her discussion of the math wars brought about. First, “It is ironic that math – a subject that should be all about inquiring, thinking, and reasoning – is one that students have come to believe requires no thought.” (p. 42) This statement follows up a discussion about student perceptions of math, in which one student said, “In math you have to remember; in other subjects you can think about it.” (p. 40). This breaks my heart. Students who don’t understand the interconnectedness and logic of mathematics sincerely believe that they have to memorize EVERYTHING in math class. The idea of memorizing every single thing taught in a class for twelve years is enough to give me anxiety. I’ll say it again – NO WONDER kids hate math!
The worst thing about this, though, is that kids don’t start off as number-crunching robots. They learn to be this way because of years of math instruction that teaches them they are supposed to be number-cruching robots. “The fact that many students learn to suppress their thoughts, ideas, and problem-solving abilities in math class is one of the most serious problems in American math education.” (p. 46)
Two other problems with current math instruction that Jo identifies in this chapter are the lack of communication and the lack of reality in math classes. True (graduate-level and beyond) mathematics is very collaborative. One reason for this is the emphasis on reasoning:
Whenever students offer a solution to a math problem, they should know why the solution is appropriate, and they should draw from mathematical rules and principles when they justify the solution rather than just saying that a textbook or a teacher told them it was right. Reasoning and justifying are both critical acts, and it is very difficult to engage in them without talking.
… When students are asked to give their ideas on mathematical problems, they feel that they are using their intellect and that they have responsibility for the direction of their work, which is extremely important for young people. (p. 49)
I already encourage a lot of talking in my classroom, but this year I need to focus on ensuring that those conversations are going in productive directions. I need to make sure that students are reasoning and justifying, and analyzing each other’s justifications.
I really enjoyed Jo’s discussion of ridiculous contexts (think word problems) that have nothing to do with actual real life. My favorite example was this problem, on page 52: A pizza is divided into fifths for 5 friends at a party. Three of the friends eat their slices, but then 4 more friends arrive. What fractions should the remaining 2 slices be divided into? Jo’s point here is that by providing so-called real world applications, students should be able to use their knowledge of the real world to solve the problems. Unfortunately, they actually have to set aside this knowledge in what Jo refers to as Mathland. As she points out, “Everybody knows that … if extra people turn up at a party more pizza is ordered or people go without slices.” Personally, I have even more issues with this problem. Who cuts a pizza into 5 slices? Do you have any idea how difficult it is to cut a circle into an odd number of pieces? How pissed are those two people going to be when they realize they have to share their two slices with 4 more people, so they don’t get as much pizza? And when the 4 new people decided to come to the party, didn’t they text someone and find out if there was any pizza left? They could have picked one up for themselves on the way over! (Then it could have been cut into 4 slices, which is much easier to cut than 5 slices.)
The elimination of ridiculous contexts would be good for many reasons. Most important, students would realize that they are learning an important subject that helps make sense of the world, rather than a subject that is all about mystification and non-sense. (p. 56)
These first two chapters gave me a lot to chew on with regard to how we are currently teaching math and why it’s not a good approach. The next 4 chapters deal with more specifics of how we can improve our math teaching, and will be where the bulk of my reflection needs to happen.