The quote below comes from the latest draft of the California Math Framework (Chapter 4):
There are, however, many situations in which humans are poorly served by such generalizations, especially those that lead to inequities or the unjust treatment of people based on characteristics that call forth internalized stories about expected capacities, motivation, behavior, or background. Such stories are often emotional, based on little evidence, and socially buttressed. Action based on these stories does great harm to school communities and individual students.
This tendency to assume, without adequate justification, that generalizations are valid is reinforced by many poorly constructed math assessment questions—for example, “What is the next term in this sequence: 1, 2, 4, 8, …?” instead of the more informative and reasoning-reinforcing question, “What rule or pattern might generate a sequence that begins 1, 2, 4, 8, ...? According to your rule, what is the next term?” Mathematics education must prepare students to use mathematics to comprehend and respond to their world by deepening their understanding of mathematics and of the issues that affect their lives. The goal is that students learn to “use mathematics to examine…various phenomena both in one’s immediate life and in the broader social world and to identify relationships and make connections between them” (Gutstein, 2003, 45).
The first paragraph above makes the usual play for equity and justice, which has been a narrative for the last few years on how mathematics education should be used to achieve this end. This paragraph then serves as a springboard (as in “Speaking of generalizations and the harm it does to people”) to how mathematics can be used to avoid generalizations. It does this by showing how “next term” problems are teaching students to generalize. I have talked about this tendency of encouraging students to use inductive reasoning, thus giving the impression that they serve as “proofs”. They do not. Deductive reasoning is what is needed for proving conjectures.
In particular, I have written about how this misguided notion plays out in the ever-popular problems about “growing tiles” in which students are to find how many tiles will be in, say, the 5th, 20th, 100th, and nth tile diagram based on the first three or four such diagrams. I bring this up because one of the people promoting this type of problem in the beliefs that math is about the science of patterns and that inductive reasoning constitutes proof is Jo Boaler. Here she talks about such a problem and how to implement it in the classroom.
It might appear that Boaler is asking what rule or pattern might generate the sequence of tiles depicted, thus opening it up to more than one answer. This would be in sync with what is being advised in the new California Math Framework as quoted earlier. A closer look at this exercise (carried out over a five day period, by the way) shows otherwise, however. To wit, she says:
“This task has a low floor and a high ceiling – the low floor means anyone can see how the shape is growing, but it extends to high levels and the function that is represented by the shapes is a quadratic function. Tasks such as this are a really nice way to introduce students to variables.”
Thus, she has a definite function/formula in mind that she would like students to come up with via inductive reasoning. And in fact, at the very end she suggests an “extension”:
“Show an algebraic expression and a visual proof for the number of squares in the nth case.”
The fact that she considers this a “proof” eliminates the possibility that she is looking for other equations that could describe the pattern. This leads to students believing that an inductive inference constitutes proof.
It will be interesting to see if those promoting growing tile problems alter their wording to avoid “generalizations”. Whether this will help bring equity and justice to the world is left as a proof for the student.
Boaler is about to make poor Black and Brown children her lab rats, and 10 years from now when we’re worse off they’ll blame teachers and “poverty”.
We’re doing the same in Australia, thanks in no small part to progressivists having been captured by Boaler and her ilk. It’s a dreadful thing. Worse, teachers who don’t buy into such nonsense have to hide their practices and pretend to be doing problem based learning or face discipline.