Ed Source has published the latest in a seemingly never-ending series of articles on how best to teach math.
Nearly two decades ago, international math and science tests revealed mathematics instruction in the United States as an inch deep and a mile wide. Since then, we have grappled with how to get depth over breadth in classrooms.
This is confirmed every time I work with teachers or parents, most of whom remember the procedural, answer-based mathematics that they were taught, and the results of that approach. I often hear phrases like: I was really good at math, and then I just didn’t get it anymore; I was never good at math; I was dumb.”
And of course in the world of edu-groupthink, the only reason for this is because the students were taught procedures and nothing else. No other reasons will do. On the other hand, students whose “answer-based mathematics” served them well, are regarded as exceptions; they would have done well in any learning environment because they liked math and were interested in it. The idea that instruction that resulted in success in problem solving served to motivate students to go further is definitely not in the group-think dogma or lexicon.
The parents of students who major in STEM fields understand that well-organized mathematical solutions are their own explanations. Many of the math reformer crowd including the author of these folks seem to regard translating “of” to “multiply” as rote, or mechanical decoding. I, and many others like me, regard it as precisely the kind of “understanding” that is appropriate. The student who goes straight to a mathematical encoding of the problem is the one who likely has the best functional understanding.
The thinking amongst math reformers is that one indication of “understanding” is if a student can solve a problem in multiple ways. Thus, the reformers then insist on having students come up with more than one way to solve a problem. In doing so, they are confusing cause and effect. That is, forcing students to think of multiple ways does not in and of itself cause understanding. They are saying in effect that “If we can just get them to do things that LOOK like what we imagine a mathematician does … then they will be real mathematicians.”
The “answer-based” classroom is now the latest perjorative description along with Phil Daro’s view that math has been taught as “answer getting” with no regard for process or underlying concepts.
Instead, math classrooms become discussion groups. I’ve been told by more than one edu-expert that the content standards of the Common Core math standards are there to serve the eight Standards of Mathematical Practice. Thus, critiquing each others’ work and developing the “habits of mind” outside of the math courses in which instruction would naturally develop such habits is thought to make students look like they’re thinking like mathematicians.
A friend of mine has a son who is majoring in math at MIT. The father had to work with him every night in the lower grades (K-6) to ensure he was mastering the math procedural skills that were not being taught in the son’s classes. When the father was in school, he made it to AP Calculus in high school without any parents’ help. He has remarked that this is not possible today–despite the student’s interest in math. Students don’t just learn it anyway. They need to know how to (dare I say it?) “get answers”. And to use procedures.
“Our PreK-12 math curriculum is taught using principles of “growth mindset,” a concept developed by Carol Dweck, a professor of psychology at Stanford University. Taught with this framework, students learn mathematical reasoning; embrace mistakes as learning opportunities; and work together to build the flexibility and resiliency required for success in math. The goal is to help students stay motivated in the face of challenging work. We’re working to reframe the question, “What does it mean to be good at math?”
Presenting students with open-ended problems with many possible “right answers” is neither necessary nor sufficient to be “good at math”. Getting them to make mistakes by tripping them up with “divergent thinking” type questions is also not necessary in order to obtain the brain growing effect that Jo Boaler has popularized in her writings.
Just teach the students what they need to know, even if it means they are “getting answers”.