I have
written a number of entries regarding “understanding” in math. I have discussed
various misunderstandings about understanding in math. There are two statements I haven’t addressed,
which for me raise many questions.

I have heard
many people express the thought that “Calculation is the price we used to have
to pay to do math. It’s no longer the case. What we need to learn is the mathematical
understanding.”

And often on
the heels of this statement I will be told that they had done well in math all
through elementary school, but when they got to algebra in high school they hit
a wall. Or, similarly, they did great in
high school, and hit a wall with calculus.

There is
much information that we do not have from such statements.

- Was the education they received
really devoid of any kind of understanding and all rote?
- Are there people who get A’s in math
in high school who are really math zombies and cannot progress to the next
level?
- Are these complaints limited to those
who were educated in the era of traditional or conventionally taught math?
- And
of those, how much of what they experienced is due to concepts not explained well,
emphasis on procedures only, and grade inflation?

- Are
there gaps in their math education which compound on themselves?

- And
to what extent are these problems the result of the obsession over
understanding?

Considering
these questions, I have listed some ideas for future studies based on
communication I’ve had with people in math education:

- To
what extent does success in high school math programs correlate with success in
higher level math and science courses in college? (Differentiated by regular
track vs AP/IB/honors track)

- For
successful math students in high school, and college math what did they do
that’s different than those who were successful in high school but did not do
well in college math?

And a
corollary of such a study would be:

- What percent of the student population has had math tutors, or been enrolled in learning centers?
- And for such students what are the basic teaching techniques used for math?

Finally, two more:

- What effect has the emphasis on understanding been on students who have been identified as having a learning disability?
- And a more difficult question, is there evidence that such emphasis has resulted in students being labeled as having learning disabilities?

I of course
am interested in any studies you may know of that would shed light on these
questions.

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Reblogged this on Nonpartisan Education Group.

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Another thought on understanding…ahem…understanding, might also be understood by reviewing old textbooks and curricula. The term itself has become polarized, and misconstrued in an effort to allow for more consultants and edufads being proliferated at ed schools to appear effortlessly in proD sessions for teachers and education conferences.

I’ve more to say on this issue – back in a bit.

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These are excellent questions posed and would generate valuable data.

I heard from the director of our Highly Capable program two years ago, during curriculum adoption and discussing their “new” approach to math in elementary, that they wanted curriculum to focus more on “understanding concepts,” for students to have “deep understanding.” When I asked her what that looked like, she told me they would spend more time in groups discussing open-ended problems and would “go deeper” into each mathematical concept. When I explained my concerns about the slow pace of CC, she reiterated my student would “understand” concepts better and would be able to think “flexibly.”

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Yes, a very typical response.

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Make sure you point out that slower is not equal. Researchers have to analyze the effects of slow separately. And of course, they have to define what constitutes a test of understanding. The rubrics my son had in K-6 were completely worthless as with the state’s annual test results (NECAP and later CCSS), which, in any case, were a year late and many tutoring dollars short. State tests offer a terrible feedback loop. What are teachers in K-6 – potted plant facilitators?

For college, proper math and STEM preparedness is defined by good grades in traditional AP/IB classes, SAT I and II in math scores, the AP test score, and any scores on the AMC/12 (et al.) tests. If research hypotheses are based on something else, then they have to explicitly define them and explain why they are important for student careers. Many seem to conflate better average grades on state test proficiency with “deep” understanding and STEM preparedness. Anyone should be able to get “better” understanding if they slow the pace down, even if they use hand puppets for teaching.

There is also the issue that slow can never catch up in reality. Slow often ends up as repeated partial learning with small incremental steps and the CCSS goal of no remediation for College Algebra – very low statistical expectations. This is when students get algebra as high school freshmen?!? What’s going on in those four years with CCSS?

The College Board is now jumping in with Pre-AP Algebra in 9th grade that finally admits the need for mastery of very many unit skills. They let K-8 off the hook and claim that “social justice” requires Pre-AP math. Then they claim that these slow and deeply prepared (?!?) students can accelerate to a higher slope to get to AP calculus as a senior. Do they have any data whatsoever to show that this can be done on a regular basis? Can they show any (non-extraordinary) cases where slow leads to a STEM career?

Slow is not necessarily bad, and traditional math offers different levels, like college prep, honors, and AP levels. What “they” are offering, however, is something completely different and slower, and they offer no comparable tests for success. They define oranges and then do research to show that they are better than apples. However, they offer no data that shows that their oranges get to STEM careers or even do better at Vo-Tech schools.

In many ways this is not a research issue. Many of these educators seem to see “the” problem as a statistical mean improvement problem and not one of maximizing individual student success. They offer low expectation full-inclusion with blather about understanding to cover up for their complete lack of proper alternate paths until it’s too late. Research is only meaningful if we are on the same page and I don’t expect research will change them from defining oranges.

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Thanks for your comment; cogent as usual.

I see your point about research–given that there’s lots of research out there showing ineffectiveness of various teaching techniques (inquiry-based for example), the education establishment essentially goes on doing what it’s doing. On the other hand, there has been some backing down on “learning styles”–it just took a lot of time. I believe research is necessary for those who DO take it seriously–that would be teachers like me who want to do the right thing by our students.

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