Monday, December 7, 2015

Let's be educationally realistic about math zombies

Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers—from multi-digit arithmetic through to multi-variable calculus—doesn’t understand the underlying math? Or that the mathematician with the Asperger’s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?
One of the sections of Barry Garelick and my recent piece on that has turned out to be most controversial is this one. In follow-up posts/comments on two blogs (Dan Meyer's and Education Realist's), people have argued that such "math zombies" do exist, citing kids they've worked with who make it all the way through high school calculus and the AP exams without understanding the underlying math, and (Education Realist):
the ever-growing complaints of college math professors about students with strong math transcripts but limited math knowledge.
I am closely associated with a number of college math professors, and am both familiar with and sympathetic to these complaints. Of particular concern are the many post-1960s high school texts featuring what some mathematicians have called “cookbook calculus." This is a curriculum that favors breadth over depth, and superficial applications of ready-made formulas over in-depth discussions of where these formulas come from and math problems that are conceptually challenging.

It is precisely for this reason that the careful reader will notice the phrase “multi-variable” modifying the word “calculus” in the above excerpt from our article. Multi-variable calculus is rarely taught in high school; readiness for multi-variable calculus implies what simply getting As in high school math doesn't imply: readiness of college-level math. To further clarify that we are talking about post-high school math skills, we extended the hypothetical zombie to someone operating on the frontiers of math.

OK, perhaps there are students who succeed in college-level math and even function quite well at the frontiers of math who are, nonetheless, zombies. When it comes to zombies, it gets philosophical: how can we know? But here’s another, more practical, question: who cares? As long as someone finishes a given math class ready for the next level of math, who cares whether they’re a partial or total math zombie?

Indeed, given the limitations of working memory, being a partial math zombie is probably a good thing. I'll go even further: it's probably a prerequisite to mathematical success--just as it is, mutatis mutandis, for success in everything from prose writing to piano performing.

As for ferreting out those math zombies who aren’t ready for the next level of math, I’m with Education Realist. As I wrote earlier here, you do this by assigning more conceptually challenging math problems—of the sort that simply can’t be solved if you lack the requisite depth of understanding. As Education Realist noted on DanMeyer’s blog, one can “ask test questions that ferret out zombies.”

Education Realist, though, isn’t OK with simply ferreting out those zombies who aren’t ready for the next level of math. For Education Realist, as we’ll see in my next post, the problem of math Zombiedom is much bigger and deeper than poor preparation for higher-level math.


SteveH said...

Zombie Math is an attempt at using a new term to give life to the old and failed use of "rote math." Some even go so far as to claim that getting a 4 or 5 in AP Calculus AB or BC is not a guarantee of any understanding - that there is some other kind of real understanding (not just due to a slower coverage of material) that is not taught in a "traditional" approach to math education pedagogy. However, use of "rote" and "zombie" are never defined let alone calibrated. This is just an attempt to bolster vague and ill-defined top-down, hands-on, engagement-driven, student-centered educational pedagogy. It's one more attempt to claim that educational turf is somehow more important than math content and skills.

Everyone knows that those majoring in math or other heavy STEM degrees might require re-taking calculus in a more rigorous college form. Many colleges don't automatically accept good AP scores. For many, however, AP Calculus is a great way to fulfill college math requirements. Is it better to go more slowly in high school and leave off at pre-calc or is it better to retake calculus at a more rigorous college level? In either case, there are no specific alternatives suggested as a solution to so-called zombie math. It's just a way to push some vague "understanding" educational pedagogy. It's not just a way to say that slower coverage means better understanding. The implication is that one ALSO has to change how things are taught - top-down. It's not just a matter of spending extra time for context and depth added on top.

My son got a 5 in AP Calculus BC but still had to take a college math placement test because he was going to major in math. If anyone were to complain and ask for changes to the AP Calculus curriculum, I would listen to these people. They are not complaining about a lack of educational "understanding" pedagogy. They are concerned about long-term mastery content and skills. I would like to see more formal proofs taught in high school, but even that is no guarantee of understanding. However, I saw no understanding problems with my son due to how the material was covered in his math classes. This doesn't mean that he could not have had more understanding, but those complaining about zombies provide no details or calibration.

SteveH said...

"... you do this by assigning more conceptually challenging math problems—of the sort that simply can’t be solved if you lack the requisite depth of understanding."

But I don't know what these are. Do they show a lack of understanding of the material covered in the textbook or AP curriculum? Is this understanding only obtained using whatever vague pedagogy is being pushed by educators? Is this argument supposed to provide any level of support for the understanding silliness going on in K-6 math? "Understanding" is too often thrown out there without any real explanation.

All we get are strawmen like this:

"Math answers aren’t math understanding any more than the destination of your car trip indicates the route you took."

AP Calculus math track classes are nothing like this. If you can do all of the problems in a homework set, can you be a zombie? If so, what, exactly, is this lack of understanding? With 7 1/2 years of math and engineering courses in college, 40 years of writing over a million lines of computer-aided engineering software, 6 years of full-time teaching of college math and computer science classes, and 15 years listening to this educational understanding math vagueness, I don't know what this lack of understanding is. Explain it better or go away. Explain why it's the pedagogy and not slower coverage.

What this really comes down to a turf battle. Educational pedagogues desperately want to claim some sort of primacy over content and skills. The only way to do that is to devalue content and skills as having little connection with some sort of real understanding.

Too often the hidden agenda is not discussed. When math coverage is slowed down, do educators even care to take some of the extra time to enforce mastery of problem sets? Or, are they using up that time with hands-on projects with the hope that engagement will get the rest done? Are they trusting the process or are they pushing and actually ensuring mastery? When one claims a poor connection between mastery and understanding, I know what their answer is. To me, it's called failure and educational conceit. It's amazing that colleges produce any proper mathematicians and engineers given the understanding complaints of K-12 educators and the unwillingness of colleges to use their pedagogy.

FedUpMom said...

Ugh. This reminds me of a similar strawman who rears his ugly head in the reading wars. There we hear about kids who can correctly read every word on the page, but who don't show deep comprehension. It's because of those evil phonics!

Really? Anyway, I'd prefer having a kid who can read but can't show comprehension to a kid who can't read at all at the end of first grade.

Karen W said...

Math zombie is genius. How can one ever demonstrate to their satisfaction that a person isn't a math zombie? It's like being roped into arguing against balance. You can't win and they don't have to talk about why whole classes of students aren't fluent in math facts (or standard algorithms) even though they should be quite capable of learning them.

Katharine Beals said...

Well put--it is genius! There's a basic problem with falsifiability here, particularly since "math zombie" theorists say that you can follow procedures perfectly and get correct answers and still be a math zombie.

Barry Garelick said...

In all these debates about "rote" versus "understanding" there is never a definition of understanding. The depth of knowledge about a subject varies, but one can have some grasp of mechanical fundamentals to allow for reasoning with those mechanics. SteveH makes the point that one cannot progress in mathematics with just rote memorization. I agree.

For example, the student who knows "by rote" that if three pencils cost 15 cents, to find the cost of 1, you divide 15 by 3. But a good math course will give students variants of the initial problem rather than the same problem over and over. A variant might be to then ask what five pencils cost. Or that the cost of a pencil was reduced by 2 cents, and you previously paid 15 cents for 3 pencils, how much change would you get back if you bought 5 pencils at the new price and paid for it with a 1 dollar bill?

A student may work the problem mechanically--some students may be able to explain what they're doing if asked and some may not. Fewer still may be able to provide a written explanation. But students who can consistently get problems and their variants correct are operating on something beyond rote--they are reasoning.

At beginning stages of learning, understanding is partial. That is, students may master the procedural aspect of a problem but be unable to explain the underpinnings--even when taught. But later they may be able to. Students may not be able to fully understand what 4/5 of 2 represents even though they can calculate the answer. But over time the student who was unable to provide an example can do so. I recall being surprised that things I found puzzling about arithmetic when I was in the lower grades were now very clear when I took algebra. I now had a new (symbolic) method for representing those ideas and could rely on algebra to express what I had a difficult time expressing in words with only an arithmetic understanding.

Barry Garelick said...

In all the discussions and debates such as those at the dy/Dan blog, about high schoolers possessing only a mechanical (or rote) understanding of math there is rarely a discussion about what happens at K-6 or K-8. With the disdain for giving students many problems (i.e., it is termed "drill and kill" and is deemed to be something bad) students do not have as much experience working problem sets in which the problems are ramped up and scaffolded, so that they are forced to think beyond the initial worked example to solve variants of the problem(s).

A common complaint about math text books is that the books give the same problem over and over, but with different numbers. This still persists. In addition, another solution is to continually give "real world" problems for which the students have little or no prior experience, which is the opposite of scaffolding, and supposedly builds a problem-solving "schema" without benefit of necessary intermediate steps. Instead, students faced with a new type of problem which requires acquisition of a skill or procedure they haven't have, are supposedly motivated to learn what is needed to solve the problem on a "just in time" basis. This works well if the problem is scaffolded so that what they learn is part of a well-thought out sequence of topics, procedures and skills. But in a world dominated by philosophies of "struggle is good", and "scaffolding eclipses real reasoning and is just more rote", the math students must master is fragmented.

Comparisons with students in previous eras who came to high school proficient in operations with fractions, decimals and percents, fall on deaf ears. We are told "Oh, they just had a rote understanding". Or: "The demographics have changed" which goes to a theory of "racial dilution".

Rarely examined is how the students who are succeeding in STEM subjects are acquiring their knowledge. Often, parents supplement or supplant what they are getting at school, or students are tutored or attend learning centers such as Sylvan and Huntington. In such situations, students are given many problems, are pushed to memorize and in short do the things that are held in disdain under the prevailing educational dogma. When this is pointed out, the fall back is "Well those students are the exception" as in "traditional methods work for them". If so, there are many exceptional students given the ever increasing numbers of learning centers across the U.S.

SteveH said...

"There we hear about kids who can correctly read every word on the page, but who don't show deep comprehension."

That's EXACTLY what I was told by my son's Kindergarten teacher when I asked about a reading test they gave the kids - which they wanted to hide from parents. "Some kids can read encyclopedias but not comprehend a word." I wanted to ask her if they checked his comprehension.

"Math zombie is genius. How can one ever demonstrate to their satisfaction that a person isn't a math zombie? It's like being roped into arguing against balance."

EXACTLY! It plays well and puts parents on the defensive. I called them preemptive parental strikes. I never argued with my son's K-6 teachers. It would not be productive to basically tell them that they are completely wrong. I was told by a teacher to my face that my son had "superficial knowledge." It didn't even sink in to them when bright kids in fifth grade still did not know the times table after years of "trust the spiral" Everyday Math. If you wait long enough, even the kids will think they're stupid. I had kids tell me that when I taught an after-school SSAT prep class for 7th and 8th graders. Some make it through the nonlinear change in education in middle school, but many never recover. I've tutored high school kids in math who are bright, but borderline failing because they never learned how to properly do nightly math homework. This is not a skill that magically goes away when they get to college.

"..."math zombie" theorists say that you can follow procedures perfectly and get correct answers and still be a math zombie."

This is the big lie, but somehow everyone is supposed to take it for granted. However, we never know the details of the pedagogy that is supposed to fix it and how they can tell. If students can pass any math class with just rote understanding, then the teachers are incompetent at giving tests.

So, are they talking about an AOPS or AMC level of ability? No, but I would even disagree that those questions define full understanding. That's overkill on a limited range of material just to find difficult questions for a competition. Understanding is also not the ability of a student to really tear apart one interesting (fractals?) area of math without showing mastery of other material in the curriculum. I would not do so well on an AOPS or AMC test, but what effect does speed have for complicated problems and understanding? I have the skills to tediously and carefully analyze a technical journal article. Is that understanding? Yes, but it's more than that. That skill was developed by years and years of doing homework sets on my own. That skill and understanding were not engagement driven. They were individual nightly homework driven.

SteveH said...

"Rarely examined is how the students who are succeeding in STEM subjects are acquiring their knowledge."

With full inclusion and officially NO-STEM top-down math in K-6, parents have to ensure practice and mastery of more than what they get at school. When my son was in K-6 I never worried about "understanding." I worried about basic math skills and understanding only at a simple level of fractions and percentages. Once he got into Pre-Algebra (with a proper textbook), it was all about making sure he did the entire homework problem set correctly. If any test result was not good, I knew that the lack of understanding(!) would build up. Understanding was defined by successfully doing the entire homework problem set. To call that ability zombie math is incredibly stupid or self-serving.

Magically, all of that stupid sort of educational pedagogy disappeared in high school. In most high schools, integrated math has lost and everyone now focuses on proper textbooks and the AP Calculus track. Life goes on successfully for many who survived K-6 in spite of Meyer, etal. Talk of zombie math is a feeble attempt to resurrect old and fuzzy educational math pedagogy in high school where it has completely lost the battle. K-6 is another case. It's a land of dream-world educational pedagogy that thinks that they can pull off full inclusion and differentiated instruction with "trust the spiral" curricula. It's a land where teachers feel comfortable admonishing parents and telling them that their child has superficial knowledge of "mere facts." This doesn't stop them from sending home notes to parents telling them to practice "math facts" and complaining that they need to get parents to not want just what they had when they were growing up, because, well, we parents must be superficial or stupid.

Never mind the fact that full inclusion and NO-STEM in K-6 just hides the tracking at home and increases the academic gap.

Auntie Ann said...

"Demographics have changed" sounds very close to racism to me, and like the old: "soft bigotry of low expectations."

Katharine Beals said...

On Brett Gilland's blog, Education Realist writes "Notice that Beals contradicts herself. In the article, she denied the possibility that zombies exist. Now she’s saying so what if they do?"
The only mention of zombies in the Atlantic article is the paragraph with which this blog post begins. I had thought ER's misreading came from failing to notice the words "multi-variable" and "frontiers of math," but apparently it runs even deeper than that.
As long as someone understands what's going on well enough to advance to the next level (whether in math, or, say, in a debate), then who cares if they're a zombie? But what about people who persistently don't understand what's being said to them? Are they zombies? And should we care?

Katharine Beals said...

Reading comprehension, I should add, involves not just noticing all the qualifiers in a sentence, but also noticing the broader context in which those sentences occur so as to understand the rhetorical point being made. Here's the broader context of the "zombie" paragraph.

Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers—from multi-digit arithmetic through to multi-variable calculus—doesn’t understand the underlying math? Or that the mathematician with the Asperger’s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?

Or is it possible that the ability to explain one’s answers verbally, while sometimes a sufficient criterion for proving understanding, is not, in fact, a necessary one? And, to the extent that it isn’t a necessary criterion, should verbal explanation be the way to gauge comprehension?

My point is that there are people functioning very well in upper level math who wouldn't have met the explain-your-answers verbally-and-diagrammatically at *any* grade level. Let's not turn these kids off to math, or gate-keep them out of classes that challenge them.

Katharine Beals said...

I should have clarified that the middle two paragraphs in my previous comment are taken directly from the Atlantic article. The final paragraph is not.

Some people were confused by comments I made later that went beyond concerns about explain-your-answer to encompass a second concern: the dumbed down, post-1960s middle and high school level texts. There are different problems that arise in different ways at different grade levels. My shift of focus to the high school level, and my clarifications about the zombie quote (about non-linguistically inclined high functioning math types) doesn't mean that I'm suddenly not concerned about the pipeline effects from elementary and middle school, and various problems with the explain-your-answer demands.

SteveH said...

It's a common technique to cherry-pick comments and try to score points while disregarding the overall issues raised. I get that all of the time, and it doesn't clarify the problem.

The problem with "zombie math" is that "they" do not define it. It's used in a way that assumes we all know what it is. It's an arguing trick. However, the claim is that one can be successful on math tests, but still not "understand" important things. We don't know what those important things are.

It could be that teachers (over many years) are not good at writing tests, but I doubt that. One of their teachers would catch them and they will do poorly or fail. If test questions are based on variations of problems in the homework sets, then how would doing well on the tests be a sign of any lack of understanding? What defines understanding in education? There are many levels and are we to presume that some other pedagogical approach (not just based on slowing down the coverage of the material) works better? I see no proof of that. All I ever see is slower coverage with the extra time being used for hands-on engagement. That mixes up two variables.

Is zombie-ness defined by long-term loss of knowledge and skills? Once again, are we to believe that a change in pedagogy is the solution rather than simply slowing down coverage of the material? However, students get mid-term and year-end tests, and many take the AP and SAT II tests long after they have had the material. Are they talking about zombies in all other subjects? Is their top-down, hands-on, engagement approach really the key, or is success (?) just based on slower coverage. How would a traditional pedagogical approach use that extra time? Would they be twiddling their thumbs? Would they not use engagement and "understanding" techniques. Of course, top-down, hands-on, student-centered techniques take a LOT of time and not all important material can get that approach, so what creates this unknown understanding for all of the rest of the material? And how do we know that it even works after separating out the slower coverage variable? Their argument for zombie math is not based on research, but used only to support their own educational philosophy.

I learn complicated new math and really understand it to be able to turn it into tested and validated computer software. However, I find it amazing how much I forget even after 6 months. It's no problem, because I have the background knowledge and skills to figure it out again. These are hard-work (not engagement) skills honed over many, many years of doing nightly homework sets, studying technical journals, and computer software development on work that was anything but fun or engaging. No matter how much "Project Lead the Way" engagement you get in high school, one still has to do the hard work of math problem sets to get into a STEM program in college.

Zombie math is just a new arguing tool to justify their pedagogical approach because "rote" lost it's zing and we haven't seen it in schools for at least 20 years. There may be ways to improve the traditional AP math curriculum in high school, but their pedagogy is not the answer. Where is the definition or "zombie" and where is the proof?

SteveH said...

Brett Gilland is the source of "zombie math". (

He says:

“Here is where the math zombie comes in. It is possible to imagine a student who is fully capable of behaving as a fluent mathematician (at least, fluent enough to get great grades) while still failing to have any internal mathematical states (schema) which coherently map math concepts at any significant level. This student likely has mastered a series of trick to produce consistent procedural results with no conception as to why or how the tricks work. “

This is absolute baloney!

“significant level”?

What is significant? There are many levels and types of understanding, but are all levels driven from the complex (schema) to the easy? We don't expect young students to know about base systems when they learn to count. What, exactly, are math schemas? Are they axioms, corollaries, identities, or proofs? No. They are something that apparently only educators understand.

“mastered a series of trick[sic]”

So, below "significant" its all tricks?

They are bound and determined to denigrate any connection between mastery and understanding. They have NO PROOF of this.

SteveH said...

“…or something like the excellent Level 3 DOK problems by Robert Kaplinski.”

Have either of them looked at good traditional math textbook problem sets? There are many problems at his three "DOK" levels and more. But this is fundamentally flawed. There are not just levels of understanding. There are many different types of understanding. When should students use explicit versus implicit versus parametric equations? What do you do if you don't have enough equations for your variables? How do you come up with extra ones? How do you find an optimum value if you have fewer equations than variables? Traditional math textbooks have lots of higher level problems that require better understanding and application of basic unit skills. These pedagogues just don't like them.

I see their usual educational pedagogical spin on the issue. They get to decide what defines a DOK level, as in this example:

DOK 1 -- 44 + 27 ... Find the sum

DOK 2 -- ?? + 53 = ?? ... Fill in the '?' positions to make a true equation

DOK 3 -- ?? + ?? = ... Fill in the '?' positions to make the largest number

DOK levels 2 and 3 are classic ed-speak problems that supposedly show some higher level application of math "schema." They have no proof of this. The bread and butter of math competitions contain all sorts of odd problem variations where his DOK 3 level examples would provide no "schema" help. These are trivial when kids get to algebra, but it's clearly given as some sort of "schema" for lower grade students. The path to higher degrees of knowledge is not a matter of climbing levels (or going down levels - which is it?), especially ones defined by educational pedagogues. It's also not a top-down process where students start with some magic higher level understanding that make lower level rote and "trick" solutions easy. It doesn't even work like that with proofs. When students are able to do problem sets in traditional math textbooks (working up from simple to complex problems), it is NOT trickery or roe or zombie with no understanding. There is not one magic "significant" level where understanding occurs. I've been over this before with real textbook problems. It's as if they ignore the evidence because it disagrees with their philosophy.

Besides, I see no proof that their pedagogical approaches make students better at doing their high-level DOK types of problems. They slow down coverage and claim better "schema" (?) understanding with absolutely NO PROOF.

When I was in 8th grade taking algebra, we had to write down only one step at a time as we worked through a problem. Each step had to show which rule or identity we used to justify the process. We students were really annoyed because we wanted to do multiple steps at one time. However, this is not what educational pedagogues mean when they want students to explain how they solved a problem. Were we students "zombies" who used only "tricks?" Did we lack "schemas?" How about the schemas of identities?

SteveH said...

Once again, I see this as a turf issue. Education pedagogues desperately want to find something that shows the primacy of their field. It can't be something as simple as pushing, hard work, daily mastery of problem sets, test results, and separation of students by willingness and ability. When my son was in pre-school, I decided that I wanted more for him in math than what I had in my traditional math upbringing. Then I found out that his school used MathLand. They got it completely wrong and now they use Everyday Math and that is almost as bad. They "trust the spiral" and talk about understanding as if they have the "best practice" for everyone. That's used to hide the NON-STEM level and slow coverage of math in K-6. Things changed completely for the better when my son made it to Pre-Algebra using proper textbooks and homework sets, and had non-pedagogues who were certified in math. It got even better in high school when some of his math teachers had backgrounds in industry.