A Lexicon of Math Mistakes

Question

Neil Postman wrote an interesting (and freely available) article called "The Educationist as Painkiller." I highly recommend you read the article for your own enjoyment and as a background to this question, especially pp. 4-6. Here is one of his main ideas, from p. 4 (emphasis added):

This, then, is the strategy I propose for educationists—that we abandon our vague, seemingly arrogant, and ultimately futile attempts to make children intelligent, and concentrate our attention on helping them avoid being stupid... The physician knows about sickness and can offer specific advice about how to avoid it. Don't smoke, don’t consume too much salt or saturated fat, take two aspirins, take penicillin every four hours, and so forth. I am proposing that the study and practice of education adopt this paradigm precisely. The educationist should become an expert in stupidity and be able to prescribe specific procedures for avoiding it.

Throughout the rest of the article, Postman addresses a few historical writings about stupidity and summarizes what people have said about stupidity in history.

I grant that, unlike the study of sickness and injustice, the study of stupidity has rarely been pursued in a systematic way. But this does not mean that the subject has no history...

Here's my question: What are the main types of mistakes do we encounter in our classrooms? Of course we can all think of many, many mistakes that our students have made, but perhaps it will be helpful to classify, categorize these mistakes for a "Lexicon of Math Mistakes" which our students should avoid.

Answer 1

Edit 9/5/14: It has recently come to my attention that another helpful paper is:

Confrey, J. (1990). A review of the research on student conceptions in mathematics, science, and programming. Review of research in education, 3-56. Link.

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Though there are sure to be more technology-related errors today, you can find a late 70s article on this subject from the JRME that discusses the topic at hand:

Radatz, H. (1979). Error analysis in mathematics education. Journal for Research in Mathematics Education, 163-172. Link.

Here is a quick list of some error types identified in the paper above:

1. Errors Due to Difficulties in Obtaining Spatial Information

2. Errors Due to Deficient Mastery of Prerequisite Skills, Facts, and Concepts

3. Errors Due to Incorrect Associations or Rigidity of Thinking

4. Errors Due to the Application of Irrelevant Rules or Strategies

Note that for #3 the author cites a German paper of Pippig (1975) that further classifies these errors as:

A. Errors of perseveration

B. Errors of association

C. Errors of interference

D. Errors of assimilation

E. Errors of negative transfer from previous tasks

All topics are given further discussion at the main source.

The Radatz paper is mainly about errors among children, which is why it mentions authors such as Ginsburg (MESE 1); however, you might also check the papers on errors that cited it - found here.

For example, the first paper to show up there is a JRME article co-authored by Zaslavsky (MESE 2):

Movshovitz-Hadar, N., Zaslavsky, O., & Inbar, S. (1987). An empirical classification model for errors in high school mathematics. Journal for Research in Mathematics Education, 3-14. Link.

The authors of this last paper note:

The only basic assumption was that most of the errors high school students commit in mathematics are not accidental and are derived by a quasi-logical process that somehow makes sense to the student. This assumption is an extension to high school mathematics of Ginsburg's (1977, p. 129) principles underlying errors in arithmetic.

Citation Ginsburg, H. (1977). Children's arithmetic: The learning process. New York: Van Nostrand.

The high school categories of errors are:

1. Misused data

2. Misinterpreted language

3. Logically invalid inference

4. Distorted theorem or definition

5. Unverified solution

6. Technical error

Each category is characterized/discussed and has examples provided in the paper.

Answer 2

As a first guess, I'd give the following categories:

Overgeneralization

of known theorems or techniques. Application of these theorems or techniques in unsuited situations.

Overconfidence in intuition

Making intuition into false theorems. Applying intuition where it contradicts known theorems.

Nonapplication

of known theorems or techniques. (Mistakes in problem solving, that leads to not solving the problem.)

Noncorrection

of committed mistakes, although obivously wrong or by testing.

Nonrigorosity

in proofs or arguments. Leaving gaps, unresolved issues or forgetting cases.

Nonconformity

to mathematical standards, thereby writing something false down, although maybe something correct was thought.

Algorithmic mistakes

Mistakes in the ordering or execution of steps of an algorithm.

Answer 3

Through the repetitive drilling of pointless, boring, exercises many people develop a disconnect between the mathematical formalism and semantics. Many people have lost the ability to reason logically about simple mathematical entities since they don't understand them as entities. This is, in my view, a horrible situation and the biggest mistake, as in a sense it is; the misconception of that mathematics is all about computation.

As an example, when one of my students mis-read a "find a counterexample or prove the following...." as "find a counterexample and prove the following...." and proceeded to correctly give three counterexamples, showing the claim is false, and then immediately set down to give a formal proof for the validity of the claim, simply going through the motions (assume that, let this, notice that apply this...). Complete disconnect of reason.

So I propose: a-reasonalism.

Answer 4

Perhaps one of the largest mistakes is failing to check your work. This has at least 3 manifestations:

1. Failing to check arithmetic by estimation. If you perform a large multiplication or addition, you should check your answer against a guesstimate for number of digits, etc.
2. Failing to check a solution to an equation by plugging into the original. Sometimes this is hard to do, but on an exam it is easy to plug in to check your work.
3. Failing to test a proof on an example. Many proposed proofs can be seen to be false when to apply them to something simple. For instance, is there a unique subgroup of any given order dividing a group's order? The Klein 4-group answers that trivially.

Answer 5

This is sort of an extended comment, but I find I need space to give what I think is a possibly useful answer that addresses underlying assumptions rather than answering the question as asked.

I'll give you the summary first: Confronting misconceptions may be productive in some cases, and indeed the focus on student misconceptions in math and science was central to much research in mathematics education in the 70's and 80's. However, there are reasons that this approach is problematic for instruction. I have a research reference, and an anecdote.

Smith, diSessa, and Roschelle (1993) discussed some of what had been accomplished with understanding student learning using the model of misconceptions, and then gave some critique.

In the section "Is Confrontation an Appropriate Model of Classroom Learning?" p. 126, they wrote:

There are both strengths and weaknesses in this conceptualization of classroom instruction. We need energetic classroom discussions in which students take positions, make sense of and explain problematic phenomena, and articulate justifications for their ideas. Activities that produce states of cognitive conflict are certainly desirable and conducive to conceptual change. However, as judged by constructivist standards, confrontation suffers from important deficits as either a phase of conceptual change or a model of instruction. As cognitive competition, it cannot explain why expert ideas win out over misconceptions. The rational replacement of one conception with another requires criteria for judgment. As knowledge, those criteria must be constructed by the learner, and neither confrontation nor replacement explains the origins of such principles for choosing concepts, crucial data, or theories. In fact, change in how one decides in favor of one conception over another is a complex part of conceptual development (Kuhn & Phelps, 1982; Schauble, 1990), and confrontation is an implausible mechanism for changing principles that decide, for example, the relevance of data to theory.

Confrontation is also problematic as an instructional model. In contrast to more evenhanded approaches to classroom discussions in which students are encouraged to evaluate their conceptions relative to the complexity of the phenomena or problem, confrontation essentially denies the validity of students' ideas. It communicates to students that their specific conceptions and their general efforts to understand are fundamentally flawed. The metaphor of confrontation is also inconsistent with the pedagogical sensitivity and care required to negotiate new understandings in the classroom (Yaekel, Cobb, & Wood, 1991). Finally, some misconceptions are powerful enough to influence what students actually perceive, thereby decreasing the chances that planned confrontation and competition will be successful (Resnick, 1983).

My anecdote (much abbreviated): A few years ago I was assisting a teacher in a 4th grade class by helping two students who were having difficulty comparing fractions. They thought 3/4 was greater than 4/5. A number of ways of approaching this occurred to me, and to keep the story short, I will just say that I had them construct a physical model that you and I would likely agree showed that 4/5 is greater than 3/4. However, as time ran out on us, they looked at their work and said, "yes, see? 3/4 is more than 4/5."

I'm sure you could find things to critique in the details of my approach. But two things struck me later as I was driving home. The first was that sometimes seeing depends on what you are sure you know. But the second is even more relevant: I still did not know much about why those students thought 3/4 was greater than 4/5. Spending all my time trying to help them reach a productive state of confrontation, I didn't learn about what they knew.

When students know something, it represents knowledge that somehow made some sense to them in some other context. But it's not working appropriately in the desired context.

There is research showing that a focus on student conceptions, rather than mistakes, is not only productive for student learning, it also can be a helpful model in preparing teachers for thinking about student thinking (Fennema et al., 1996).

Of course, these conceptions students have are tied to specific content knowledge. Therefore, a taxonomy of mistakes (de-contextualized from specific content) may not necessarily be useful in the action of classroom instruction. Certainly useful is knowing with some accuracy the types of ways students come into the class making sense of the content in a way that is limiting them (yet it makes some sense to them).

Cited:

Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A Longitudinal Study of Learning to Use Children’s Thinking in Mathematics Instruction. Journal for Research in Mathematics Education, 27(4), 403–434.

Smith, J. P., diSessa, A., & Roschelle, J. (1993). Misconceptions Reconceived: A Constructivist Analysis of Knowledge in Transition. The Journal of the Learning Sciences, 3(2), 115–163.

Answer 6

A few mistakes I observe from time to time while teaching. Note that these are as true of professional mathematics as mathematics education, at least occasionally.

1. Formalism; or, the belief that a rigorous style is necessarily rigorous mathematics, or even correct, or even intuitively valid.
   Of course, style cannot save a bad argument. Furthermore, it might not rescue it from implicit omissions, such as, for example, "A triangle is the union of three lines", despite using a precise concept such as "union". The idea might not even be reasonable, like, say, "Let $x$ be the smallest positive real number".

2. Platonism; or, the belief that an appeal to the intuitive essence of an idea is sufficient replacement for rigor, or even correct, or even universally understood.
   Famously, intuition in calculus cannot always be made rigorous, such as the original definitions of Newton and Leibniz in terms of infinitesimals (they are not to be blamed much, as rigor was different then). At least these could be rescued, unlike "proofs" of the parallel postulate that rejected the consequences of what turned out to be non-Euclidean geometry as being contrary to the concept of a line. Similarly to the first, the definition of continuity "$f(x) - f(a)$ is small" can mean almost anything you want.

3. Luddism; or, the rejection of mathematical "technology", i.e. refined methods.
   Students often insist that what they think they know is more reliable than what they are taught. Even researchers may feel that an elementary technique they can get away with using is preferable to one that is specifically appropriate, but theoretically more difficult.

4. Use of English; or, stringing a complex concept together with words alone, rather than encapsulating its ideas precisely and referring to them with jargon.
   This can be as basic as the antique style "The sum of twice something with an additional quantity of three has the total amount of five, and therefore that thing is itself unity". In the now-rare Euclidean geometry proof class, you see phrases like "corresponding parts of congruent triangles are congruent" used as a mantra.

5. Misuse of symbols; or, stringing a complex deduction together without any words at all.
   Parse this: "$f(x) = x + \sin(x) = f'(x) = 1 + \cos(x) = 0 = \cos(x) = -1 = x = n \pi \text{ ($n$ odd)}$". There are three related computations in there, plus the use of one theorem (on critical points) and a classification of trig function values, which should all be separated. I have heard this described as "knowing only 'equals' as a mathematical verb". Actually, I hope that this is never seen in professional math.

6. Nearsightedness; or, not following your nose; or, getting stuck because the next step looks like it is necessarily hard.
   Any problem where, say, one suddenly has to solve a system of several linear equations. Tedious, yes, but there is a straightforward procedure and it must be done, or if not, finding the alternative is possibly even more work. Numerous exams have been submitted where the solution ends at just such a spot.

7. Farsightedness; or, not simplifying the question because it looks too hard as a whole.
   I knew of a professor who would answer most questions with "Use the definition!". If it weren't for this fallacy, no one would ever have trouble with set-theory exercises like "$f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)$". Tim Gowers and Mohan Ganesalingam have written a program designed to solve this kind of problem, in part, by not making this mistake.

8. Not generalizing; or, doing exercises without getting any better at doing non-exercises.
   Every semester I have students who ask me after exams how they could study better even though they do all the exercises, sometimes even more than I ask. My current advice is that they should be mindful of the implicit lessons to be derived from the experience, but I don't know if this is even helpful.

As the article in the question says, the variety of "stupidity" is infinite. However, I think that people who avoid these eight mistakes are typically among the best students in any class, while those who make even three or so have serious problems. If only I could teach people how not to do them, rather than have them come into my class already knowing...

Answer 7

I know this isn't a full answer, but I wanted to mention John Holt's excellent "How Children Fail" which has many examples on children's failure to learn math.

Most of the issues mentioned in the book are of a more "affective" nature, but are still, very relevant to this discussion. In particular, it seems that many of the examples of "failure" in the book stem from the children not understanding the place of math in the real world and thus fail to develop intuitions about it. It's as if, to some, math is only arbitrary manipulation of symbols devoid of any meaning.