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”.
traditional math 
“We’re working to reframe the question, “What does it mean to be good at math?””
This is arrogant and obscene in so many ways. I’ve said before that this is an academic turf battle between educators and subject mater experts. It’s a failed argument I’ve seen in different areas – It’s all about me and my turf. They claim that their educational process (“trust the spiral”) creates proper mathematical understanding – not simple year-to-year pushing of mastery of skills – not p-sets and individual nightly homework that focuses on unit skills. Look at the comment above carefully. Educators don’t just want to define a best process. They want to redefine what it means to be good at a subject. That’s obscene.
Am I claiming that process doesn’t matter or that a traditional process is the only one? No, but where are their results? They’ve had control over this for two+ decades. They can’t keep blaming “traditional” math. Where are their opt-in or opt-out choices? They’ve lost the battle for math in high school and many middle schools (like ours) are changing back from curricula like CMP to proper algebra textbooks because parents realize it did not provide a proper curriculum path to AP Calculus. Their ideas are losing to reality, but reality hits a brick wall at the change to 7th grade, where it’s all over for many students.
Educators focus on CCSS, but the highest level in PARCC (“distinguished”) only means no remediation for college algebra. This low slope of education starts in Kindergarten and shifts to a much steeper (non-CCSS) slope starting in 7th grade. This transition now has to be supported by parents and tutors – all of whom use direct, traditional, skill-driven support.
My view is that their talk of “understanding” is cover for the lower expectations required by full academic inclusion, the biggest real change in education since I was in school. Learning has become natural and that is supported by their ideas and process. Meanwhile, many parents hide the tracking and non-natural pushing and expectations at home and with tutors. All of my son’s STEM-prepared friends in high school said the same thing – they were helped at home or with tutors.
So where are their success stories? Where do they show STEM-prepared students who didn’t get help or pushing at home? Where are all of these other students who now claim that they like math when they hit college and vocational math walls that cause them to change career paths? I remember college students who had to drop out of nursing because they couldn’t pass a statistics class.
Where’s the beef?
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“San Francisco Unified is the only large urban school district in California with more than half of its students meeting or exceeding standards on both the math and English language arts portions of the Smarter Balanced assessments. Perhaps more significantly, we have cut from 40 percent to 8 percent the proportion of students who are repeating Algebra 1.”
Education is about individual opportunity and support, not statistical means. Our schools improved their statistics when they changed from MathLand to Everyday Math. Lowering the percentage of students who don’t flunk a required course is not something to brag about. CCSS defines (at best) a no remediation path to college, but educators use it to claim some sort of path of best understanding in math. CCSS now officially defines K-6 as a NO-STEM zone. Most school districts offer a non-linear and non-CCSS curriculum path change to a STEM degree program in college, but you’re on your own. They may crow about statistics based on CCSS, but show no statistics of opportunity at the upper end for those kids not helped by parents or tutors. That’s because they don’t want to know what we STEM parents had to do to support our kids.
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