Wednesday, December 9, 2015

Let's be realistic about math zombies, II

The most obvious reason for “ferreting out the math zombies” is to identify who needs help. The zombies we should worry about are those whose understanding of the material is insufficient for advancing to the next level of math.

In the Edworld, unfortunately, much of the testing and scoring seems to be an end in itself. These assessments, rather than giving teachers feedback about what sorts of help different students need, often simply serve to brand certain students as “unready” or deficient, and to block them from taking certain classes or enrolling in certain schools or programs.

Education Realist takes this even further. First, s/he identifies a sort of math zombie that s/he feels is valued by a certain sort of teacher: the sort of teacher who, s/he feels, agrees with what Barry and I wrote in our article. Such teachers:

rip through the material, explaining it both procedurally and conceptually but focus on procedural competence. They assign difficult math problems in class with lots of homework. Their tests are difficult but predictable.
For the record, Barry and I do underscore the virtues of procedural competence. But, in terms of focus, what our article was doing was weighing the virtues of having students practice procedures with having them explain answers in words and diagrams. We haven’t made any blanket statements about the ideal balance between procedural practice and mathematical concepts. Nor have we advocated that teachers “rip” through the material: we’ve simply proposed that it’s more efficient to have students spend more time doing problems that challenge them mathematically and less time explaining answers to problems that don’t challenge them mathematically.

But moving back to the sort of teacher who rips through material and focuses on procedural competence and whose tests are difficult but predictable, this sort of teacher, apparently, values students who:
diligently memorize the cues and procedures, and obediently regurgitate the procedures, aping understanding without having a clue.
Leaving off the “aping understanding without having a clue” qualifier, I'm guessing that the post-1960s textbooks I mention in my previous post favor this sort of student. But I lack Education Realist’s certainty that such students are valued by material-ripping, procedure-focusing teachers. Nor am I so sure that students who memorize and regurgitate procedures have, ipso facto, no clue. In fact, one way to develop an understanding of what underpins procedures is to see them in action, and one of the best ways to see them in action is to enact them yourself. I can think of many examples where people, myself included, started out by applying procedures that we didn’t truly understand, and only in the course of doing so developed an understanding of the underlying meanings or concepts (whether in math, computer programming, or some sort of mechanical project).

But back to these memorizing and regurgitating students. Education Realist feels that such students are completely lacking, not just in conceptual understanding, but also in intellectual curiosity and academic ethics:
The students don’t care in the slightest. They are there for the A… [They] simply go through the motions, stuffing procedures into episodic memory with nothing making it to semantic, all to be forgotten as soon as the test is over.
And not only are teachers who emphasize procedures valuing and rewarding such students; so, apparently, is the AP, because, apparently, “AP Calc tests reward zombie math.”

At this point, one might expect Education Realist to cite studies showing the percentages of students who (1) earn 4s and 5s on AP calculus tests and (2) can’t handle the college level math classes they place into. But such statistics matter only if your prime concern is helping students who are unready for the next level of math.

Education Realist’s concern, it seems, is a bit different. S/he feels that we’re living in a time of math zombie-favoritism, and that this, plus the increased drive for calculus for all and AP testing, means that the zombies are (somehow, despite all the downsides of being a zombie) undeservedly winning, and that:
smart, intellectually curious non-zombies bow out of the game, decide they’ll go to a state school or community college.
I, too, am concerned about how today’s schools marginalize the smart and the intellectually curious. But I say the most likely culprit isn’t the teacher who focuses on procedural competence over conceptual understanding: as I noted earlier, the first (via “discovery learning,” in the best sense of the term) can lead to the second. Nor is the culprit the AP. No, the most likely culprits are (1) a dumbed down, post-1960s math curriculum that favors formulaic breadth over conceptual depth; (2) NCTM and Common Core-inspired requirements that students spend significant amounts of time working in groups and explaining their answers in words and diagrams to mathematically under-challenging problems. And even if (2) is still relatively rare in high schools, it's increasingly common in K8, particularly in those schools that are otherwise relatively functional, and this only further entrenches (1), which is all too common throughout K12.


lgm said...

Smart students were marginalized in nclb when the schools started gaming the tests. Only enough material is taught to just pass the exam, and quite a lot is rote. For example, in my child's included third grade, multiplication was taught with tricks like the 9s finger trick. Learning the concept and using it via the properties to master mental math, as was done prior to nclb, has been re-classified as unnecessary enrichment. Unclassified students are on their own to learn the rest....and the middle class will be afterschooled the following summer to learn the material omitted by the school.
Smart students are marginalized in high school by the decision to not offer honors sections or AP Chem and AP Physics,, and to eliminate Math Club and Science Club. Low expectations arrived during the common core implementations. Students are offered coursework leading to technician degrees, rather than engineering and science degrees. For ex, in my district a 3 period, 3 high school credit course is offered instead of AP physics...they learn how to use hand tools, measurement devices such as calipers and a tire guage, and read a blueprint. Project Lead the Way is more advanced, as is a Boy Scout Merit Badge such as cycling. The principal is still upset with me for putting my kid in virtual AP Physics instead of staying in dumbed down shop. Like I would let a 17 year old drive my vehicle without knowing how to check the air pressure, the brakes, the lights and how and when to change the tires, pads, and headlights.

lgm said...

I will also disagree that AP isnt a culprit. My kid dropped AP Gov because it was rote memorization of the databank of released test questions. No readings, but every Friday wasted on a quiz over the databank section to be memorized as well as landmark court cases. I also note AP Physics has been redesigned recently for less memorization and more thinking.

The with fantastic memories. In first grade, they will have the multiplication tables memorized without any conceptual understanding. The bragging parents will start pressing for a grade skip....and argue that the math reasoning section of the sat10 is unfair to their demographic when junior fails to score above the 80th percentile. They do well in high school, until they are asked to think. Regents Physics and Calc 2 is the washout here...thats where solution books they can memorize with enough simple problems and common test questions to get the 100 become unavailable. If they go into my stem, it will be biology or psych, where their memory serves them well.

SteveH said...

"The with fantastic memories."

This is completely unsupported.

My son is one of those and all throughout his K-8 school years we got the feeling that teachers didn't like him because of that. He made them challenge their beliefs about how kids learn. Some of my son's friends got the same reactions from their teachers. Many want to believe that these kids have something missing. Again, my story is that when my wife and I naively told his first grade teacher that he loved to study geography and could find any country in the world (he learned this on his own), she said "Yes, he has a lot of superficial knowledge." Later that year, when the student teacher was doing (not teaching) a "thematic" unit on "Sands from around the world", he had to show her where Kuwait was on the map. I'm really not making this up.

When our son was in Kindergarten and we asked about a reading test they gave kids (they didn't want to tell parents), his Kindergarten teacher launched into a lecture at us (!) about how some kids can read encyclopedias, but don't know a word they read. I wanted to ask her if she tested him for comprehension, but I knew the answer.

Remembering stuff is ALWAYS good. It's one thing to not like an educational approach that places a top-down emphasis on memorization with no context (like memorizing the list of presidents), but it's quite another to trash "mere facts" and "superficial knowledge." They are not mere or superficial. They form the framework and glue for all understanding. Understanding doesn't attach to nothingness, and Google searches for facts don't automatically create understanding epiphanies. Besides, we haven't had that kind of learning in generations. I can't speak for AP Gov, but some people have spent a long time developing that curriculum. You have to dig into that. I might agree with you, but other AP tests are not like that and you have to develop a better argument than blaming "zombies" just because it appeals to you.

" and argue that the math reasoning section of the sat10 is unfair to their demographic when junior fails to score above the 80th percentile."

I see all sorts of kids who reach some sort of mountaintop in math in high school because their gaps catch up with them. If they do well in class but not on the Regents Test, then the high school classes are bad at preparing them for whatever reason, or the test is bad. My memorization son needed help (emphasis and push) at home with mastery of basic skills. Skills, like facts, form the framework of understanding from the bottom up. "Zombie" is fake in that there is no way to do badly on tests of knowledge and skills, but somehow have more understanding than those who do well - unless the prior courses and teachers were incompetent. Then again, some like to redefine "Degrees of Knowledge" after their own beliefs, like the love of no-one-right-answer questions.

The ability to memorize easily is NOT an indication of an inability to link understanding and usage to those facts. These people are not impervious to the understanding linkages that go on when others memorize slowly. In fact, the linkages happen faster because the framework is already there. Some might have trouble with this, but memorizers as a class do NOT equal a made up "zombie" term just because it sounds good.

lgm said...

Thats the whole problem, isnt it. Some with excellent memories have understanding, some dont. Some never see a problem challenging enough to show the difference. Full inclusion means no challenge for many.