One of the anxieties that often come up in schooling in general and homeschooling in particular is math. At every school I’ve been to there has always been endlessly handwringing over what kind of math instruction is most effective, and once I’ve started homeschooling, I’ve noticed that there is constant, persistent discussion about the pros and cons of various math curricula.
I touched upon the Common Core last week, and some of the most frequent and strident complaints about these standards are the proposed changes they make to math education. Families can’t seem to make heads or tails about the “new new math” and there’s a lot of grousing—some of it understandable, some of it histrionic—about how it seems both overly complicated and infantile. Turns out that if you hate math as a parent, you’re probably better off not helping your kid with his or her homework. Little wonder you have frustrated parents posting their kids’ “ridiculous” homework on Facebook as a way to tell Common Core to suck it.
I love math. I’ve had some very, very good teachers in middle school and high school, starting with Ms. Counihan in 6th grade, who coached my first math team. I was always a little too stolid to ever be a starter on the teams, or a professional mathematician, but I’ve always been fascinated and appreciative of mathematical concepts and puzzles. I’m, therefore, fairly at ease about handling math instruction for my kids.
Nevertheless, I’ve been reading What’s Math Got to Do With It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (2015 revised and updated) by Jo Boaler, and I couldn’t help bookmarking pages and taking notes.
As for Common Core, she says this:
The Common Core math curriculum is not the curriculum I would have designed if I had had the chance. It still has far too much content that is not relevant for the modern world and that turns students off mathematics, particularly in the high school years, but it is a step in the right direction—for a number of reasons. The most important improvement is the inclusion of a set of standards called the “mathematical practices.” The practice standards do not set out knowledge to be learned, as the other standards do, but ways of being mathematical. They describe aspects of mathematics such as problem solving, sense making, persevering, and reasoning. (xxvi)
NOTE: Much of what follows are just excerpts from the book, as I find it well-written, and I’m too lazy to synthesize it all into my own thoughts.
She starts with some of the common misconceptions of mathematics as a discipline and its true nature as practiced by mathematicians.
…mathematics as a learning subject, not a performance subject. Most students when asked what they think their role is in math classrooms say it is to answer questions correctly. They don’t think they are in math classrooms to appreciate the beauty of mathematics, to explore the rich set of connections that make up the subject, or even to learn about the applicability of the subject. They think they are in math classrooms to perform. (xviii)
Ask most school students what math is and they will tell you it is a list of rules and procedures that need to be remembered. Their descriptions are frequently focused on calculations. Yet as Keith Devlin, mathematician and writer of several books about math, points out, mathematicians are often not even very good at calculations as they do not feature centrally in their work. As I had mentioned before, ask mathematicians what math is and they are more likely to describe it as the study of patterns. (19)
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, as I doubt whether they have ever experienced any encouragement to guess in their math classes. When an official report in the UK was commissioned to examine the mathematics needed in the workplace, the investigator found that estimation was the most useful mathematical activity. Yet when children who have experienced traditional math classes are asked to estimate, they are often completely flummoxed and try to work out exact answers, then round them off to look like an estimate. 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. (25)
After making a guess, mathematicians engage in a zigzagging process of conjecturing, refining with counterexamples, and then proving. Such work is exploratory and creative, and many writers draw parallels between mathematical work and art or music. (25)
Another interesting feature of the work of mathematicians is its collaboratory nature. Many people think of mathematicians as people who work in isolation, but this is far from the truth…. The mathematicians interviewed gave many reasons for collaboration, including the advantage of learning from one another’s work, increasing the quality of ideas, and sharing the “euphoria” of problem solving. (26)
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.”… All the mathematical methods and relationships that are now known and taught to schoolchildren started as questions, yet students do not see the questions. Instead, they are taught content that often appears as a long list of answers to questions that nobody has ever asked. (27)
Another important part of the work of mathematicians that enables successful problem solving is the use of a range of representations such as symbols, words, pictures, tables, and diagrams, all used with precision. The precision required in mathematics has become something of a hallmark for the subject and it is an aspect of mathematics that both attracts and repels. For some schoolchildren it is comforting to be working in an area where there are clear rules for ways of writing and communicating. But for others it is just too hard to separate the precision of mathematical language with the uninspiring drill-and-kill methods that they experience in their math classrooms. There is no reason that precision and drilled teaching methods need to go together, and the need for precision with terms and notation does not mean that mathematical work precludes open and creative exploration. On the contrary, it is the fact that mathematicians can rely on the precise use of language, symbols, and diagrams that allows them to freely explore the ideas that such communicative tools produce…. As Keith Devlin reflects: “Mathematical notation no more is mathematics than musical notation is music.” (29)
So why is it so many don’t understand math as this “music of the spheres”? Passive learning.
The type of traditional teaching that concerns me greatly and that I have identified from decades of research as highly ineffective is a version that encourages passive learning. In many mathematics classrooms across America the same ritual unfolds: teachers stand at the front of class demonstrating methods for twenty to thirty minutes of class time each day while students copy the methods down in their books, then students work through sets of near-identical questions, practicing the methods. Students in such classrooms quickly learn that thought is not required in math class and that the way to be successful is to watch the teachers carefully and copy what they do. … Students who are taught using passive approaches do not engage in sense making, reasoning, or thought (acts that are critical to an effective use of mathematics), and they do not view themselves as active problem solvers. This passive approach, which characterizes math teaching in America, is widespread and ineffective. (40)
One of the major downsides of this kind of instruction is the kind of inequity it tends to engender very early on:
When the researchers looked at ten-year-olds, they found that the below-average group used the same number of known facts as the above-average eight-year-olds, so you could think of them as having learned more facts over the years but, noticeably, they were still not using number sense. Instead, they were counting. What we learn from this, and from other research, is that the high-achieving students don’t just know more, but they work in very different ways—and, critically, they engage in flexible thinking when they work with numbers, decomposing and recomposing numbers.
The researchers drew to important conclusions from their findings. One was that low achievers are often thought of as slow learners, when in fact they are not learning the same things slowly. Rather, they are learning a different mathematics. The second is that the mathematics that low achievers are learning is a more difficult subject. (141, emphasis mine)
Not surprisingly, the researchers found that the lower-achieving students who were not using numbers flexibly were also missing out on other important mathematical activities. For example, one of the important things that people do as they learn mathematics is compress ideas. What this means is, when we are learning a new area of math, such as multiplication, we may initially struggle with the methods and the ideas and have to practice and use it in different ways, but at some point things become clearer, at which time we compress what we know, and move on to harder ideas. At a later stage when we need to use multiplication, we can use it fairly automatically, without thinking about the process in depth. (142)
Nevertheless, people are still very married to the kind of traditional math instruction they are familiar with. Chapter 2 of the book delineates the “math wars” have taken place for and against reforms. If you’re someone who feel conflicted about this controversy, I’d recommend you read this chapter to get a sense of the politics behind the rhetoric.
Anyhoo, what’s the alternative? She gives two examples. One, a “communicative approach” pioneered by Railside High School in California:
The focus of the Railside approach was “multiple representations,” which is why I have described it as communicative—the students learned about the different ways that mathematics could be communicated through words, diagrams, tables, symbols, objects, and graphs. As they worked, the students would frequently be asked to explain work to each other, moving between different representations and communicative forms. When we interviewed students… they told us that math was a form of communication, or a language. (59)
… the teachers enacted an expanded conception of mathematics and “smartness.” The teachers at Railside knew that being good at mathematics involves many different ways of working, as mathematicians’ accounts tell us. It involves asking questions, drawing pictures and graphs, rephrasing problems, justifying methods, and representing ideas, in addition to calculating with procedures. Instead of just rewarding the correct use of procedures, the teachers encouraged and rewarded all of these different ways of being mathematical. (67)
The other, a project-based approach at Phoenix Park School in England:
Instead of teaching procedures that students would practice, the teachers [at Phoenix Park] gave the students projects to work on that needed mathematical methods…. At the start of different projects, the teachers would introduce students to a problem or theme that the students explored, using their own ideas and the mathematical methods that they were learning. The problems were usually very open so that students could take the work in directions that interested them…. Sometimes, before the students started a new project, teachers taught them mathematical content that could be useful to them. More typically though, the teachers would introduce methods to individuals or small groups when they encountered a need for them within the particular project on which they were working. (69-70)
Again, if this interests you at all—or if you have doubts about the efficacy of such an approach—I would recommend you pick up the book and at least read the third chapter, especially the part where the results of the Phoenix Park program is compared to a more traditional approach at the Amber Hill school. There’s a lot to chew on.
Next: Some guidelines for parents