# Confirmation bias in math education

Confirmation bias is a quality of human mental processes which makes us tend to think in terms of positive examples and tests that would confirm our working hypothesis, rather than negative examples or tests that would falsify the conjecture.

The term was coined by P. C. Wason after his 2-4-6 task experiment, where he asked the participants to find a rule which governed whether a triples of numbers would fit or not. First, they were told that $(2,4,6)$ fits. Then the subjects would ask about their own triples of numbers and the experimenter would answer if that triple fits or not. Only a small percent of people found the true rule, which was "any increasing sequence", instead generating more specific rules like "the middle is the average". The problem was that participants offered triples that would align with their hypothesis, rarely trying falsifying it. In fact some test subjects didn't get any "negative results", that is, the rule could have been "any triple of numbers".

Confirmation bias in math education:

Confirmation bias happens in math education in two flavors: student's and teacher's.

The student's version is when the pupils try to solve some problem, but aren't critic about their own opinions. Perhaps this is most important to science education, but in math pattern recognition plays a huge role and fallacious reasoning hurts progress.

However, I'm more interested in the teacher's flavor, which is of different kind. It regards how the teacher thinks of his/her students, for example,

• if the teacher suspects that the student has some problem, he will try to help before checking if it really is that problem;
• if the teacher thinks that the child does not understand some concept, he/she will try to prove that it indeed is the case;
• the teacher my want to stress why his/her solution is better than some other solution presented by the student (perhaps not even accepting it).

Question:

• Is the teacher's confirmation bias a serious problem?
• How could we make the teachers aware of it (not as frequently, but even colleagues from university fall for this)?
• How could I make some particular teacher aware of it (as a parent, or as a student)?
• I notice you have a section entitled Confirmation bias in math education, but I can't really see how this is specific to math ed. Could you elaborate? Mar 28 '14 at 17:54
• @BenjaminDickman The student's confirmation bias is more about science than math, however, it happens that it is in the math component of these sciences (e.g. pattern recognition, model testing, etc.), so the most accurate description might be "confirmation bias in math part of science education" which is still relevant to math education. Mar 28 '14 at 18:21
• But how are your "examples" different from, say, those for an English instructor who is teaching a student how to write a convincing paragraph in an essay? Why is this "more about science than math"? I'm not properly understanding the question, which looks to me like it can be approximated with: Is it bad if (math...) teachers refuse to change their minds about the unhelpful pedagogical strategies they use? Should we (if so - how do we) make teachers aware of this tendency? Mar 28 '14 at 18:29
• @BenjaminDickman The teacher's confirmation bias is more general, and would fit academia.se, however, I think that it is more a problem of math education than other domains. In particular the handling (and so the answers) would be different, e.g. in history class you wouldn't argue which year World War II started (i.e. there is only one solution); in art class you might have a different view than the teacher, but are no real arguments, e.g. you could be wrong, because you lack experience. Mar 28 '14 at 18:39
• Your link seems wrong. I tracked it down and while it is on the same general topic of logic, it is some sort of card experiment, not the triples experiment. Your link: tandfonline.com/doi/abs/10.1080/14640746808400161 A pdf copy: web.mit.edu/curhan/www/docs/Articles/biases/… Dec 16 '17 at 16:20

Is the teacher's confirmation bias a serious problem?

Yes and this is not specific to math class.

Example of Confirmation Bias in Assessment

On Monday, the teacher thinks: "If my students have mastered addition and subtraction of of fractions, then they will be able to calculate, say $\frac{2}{5}+\frac{7}{10}=\_\_\_\_\_\_\_$ and $\frac{1}{5}-\frac{2}{11}=\_\_\_\_\_\_\_$ using an efficient algorithm. So, if they can get the answer and use the LCDs of 10 and 55, they've mastered adding fractions." Students then do very well on the assessment scoring an average of 90%. Everyone is pleased and the teacher feels the notion of "My class has mastered the addition and subtraction of fractions" to be confirmed.

On Tuesday, the teacher asks the class to compute $7-\frac{1}{2}=\_\_\_\_\_\_$.

• Student A says it can't be done because you can't take pizzas away from numbers.
• Student B says the answer is $3\frac{1}{2}$ because you're taking half of $7$ away from $7$.
• Student C says the question is impossible because you have to have common denominators to add or subtract fractions and $7$ doesn't even have a denominator.
• Student D says it depends what you're taking half of, so there are many correct answers.
• Most of the class stares silently with an obviously confused look on their faces.

What happened?

Explanation

A teacher will frequently ask themselves: "If my students have mastered concept X, then they will be able to do task Y." They then infer that "If my students can do task Y, then they have mastered concept X."

Mastering the addition and subtraction of fractions implies you'll do well on a test of $\frac{2}{5}+\frac{7}{10}=\_\_\_\_\_\_\_$.

But doing well on a test of $\frac{2}{5}+\frac{7}{10}=\_\_\_\_\_\_\_$ emphatically does not imply mastery of addition and subtraction of fractions. It does not imply that students have, at any point, incorporated any aspect of fractional number knowledge with prior whole number knowledge.

Many people are highly confused by the notion that fractions can be greater than one whole, yet they passed all their fraction tests in school.

Valid reasoning and intuitive reasoning are miles apart... even if they feel like they should overlap.

How could I make some particular teacher aware of it (as a parent, or as a student)?

Teachers and students need to ask themselves:

1. Can students excel on my assessments and have no idea what's going on? One way to test this is to create math assessments that involve no calculations, only estimates, drawings, discussion, comparing and contrasting, etc.

2. Can students excel on my assessments and then forget everything they've learned the next day? Will they be able to build on it in a year with no further review?

3. Have they fully integrated all the new content (fractional arithmetic) with old content (whole number arithmetic)?

4. When doing math, can I explain why every step is true? Why it's useful? Or am I just following instructions mindlessly? i.e. Am I making sense of this or just answer-getting?

Ultimately, we are all biased and we all intuitively seek evidence of what we want, expect, and believe. To counter that bias, we must deliberately seek evidence that students have not mastered what they appear to know.

For those ready for formal study of the counterintuitive nature of valid inferences from assessments, they should study sensitivity and specificity and related topics. This is mainstream in medicine. Education desperately needs it to become mainstream as well.

How could we make the teachers aware of it (not as frequently, but even colleagues from university fall for this)?

I've linked to this many times but here it is again... Many high school graduates struggle with $0.03=0.030$. They've been passing assessments like the one above for years and thus consider their notions of being "good enough" at math as decisively confirmed.

Who should have disconfirmed such notions? When? And how? Probably not with traditional calculation-based assessments...

In first-year calculus, many students "confirm" their mastery by getting a high grade. But if you ask them to explain how we know the fundamental theorem of calculus is true, you get a blank stare. You'd get a similar blank stare if you asked them why the power rule works. You might even get a blank stare if you asked them what you can do with calculus that you can't do with prior math.

• P.S. This is just a teacher's confirmation bias with respect to assessment. There is confirmation bias in a million other places, not because teachers are dumb but because they're human and all humans suffer from confirmation bias. Dec 1 '17 at 18:42

Is the teacher's confirmation bias a serious problem?

It's one of a list of assessment errors. It's specially noteworthy, because it hits the tension field instruction vs. construction. As such, it is a problem, but not much more serious than other errors in this tension field.

How could we make the teachers aware of it?

You can't help teachers not accepting or devalueing alternative valid solutions. If they genuinely think them wrong, they are at the wrong place. If they argue not to confuse other students, they are principally right. In that case one needs to start on a more basic level of instruction vs. construction.

You can help a teacher projecting non-understanding onto other students by letting thus students take tests from other teachers/educators. They probably won't succeed fully, but they will show some competence that the biased teacher didn't test for or recognize. You can then show him these results or even show him how thus student managed some other problems with his competence shown.

You can help a teacher overeagerly helping students by giving him time for the students. If a teacher is under time pressure (like "Students must be able to do this calculation/solve that class of problems after $n$ weeks.“) then he won't risk a student trying out his own ways and thus possibly constructing an alternative solution but also possibly failing at all. The call for Construction need not only be, how teachers should make students learn, but also how teachers' success in letting students construct is evaluated.

How could you make some particular teacher aware?

Tell him of the student's success with his alternate solution or his method of learning.

• First: It seems like you are using your answer to support something you call construction; the move from confirmation bias to this seems contrived to me, but perhaps you could provide some references about the former and clarify how these concepts relate. Second: I disagree about the impossibility of helping teachers who aren't valuing alternative solutions. I also don't think the practice of pulling students from one teacher to take tests with another is very practical: Probably this will lead to deeper departmental/administrative problems. As for your last suggestion, ...(cont'd) Mar 28 '14 at 18:03
• ...I'm still confused as to how providing an overly eager teacher with more time relates to a "call for Construction." Mar 28 '14 at 18:05
• I have never heard about "tension field instruction vs. construction", could you elaborate a bit on this topic? Mar 28 '14 at 19:05
• @BenjaminDickman OK, I'm German and the German terms differ somewhat from the English ones. I'm referring to Constructivism en.wikipedia.org/wiki/… . In German, the base word Konstruktion is used to describe methods propagated by constructivism. The opposite is Instruktion. Mar 29 '14 at 10:13
• So, if you could supply the appropriate English terms, I will gladly edit my post accordingly. Mar 29 '14 at 10:14

Here is a second example of devastatingly terrible and common confirmation bias. [It's probably more important than my first post...]

## Example: Flight

Imagine it's 1902, a year before the Wright Brothers' first success. A skeptic might say:

If it's impossible for a heavier-than-air machine to fly, then the Wright Brothers will fail to make that machine over and over and over again. And so will everyone else who tries. Since everyone has, in fact, failed over and over and over again, it must be impossible to make a heavier-than-air machine.

This is obviously stupid because we already know the conclusion is wrong empirically.

But it's also stupid because the argument itself is invalid (i.e. the premises of prior failure do not imply the conclusion of impossibility). But it could feel right to the person who says it because intuitive interpretations of evidence frequently suck and we all suffer from confirmation bias.

The idea that someone could find a better way to make these machines that no one has thought of yet? Doesn't cross the skeptic's mind.

## Example: Education

If it's impossible for student Y to learn X, then everything I try to get them to learn it will fail over and over and over again. All of their prior teachers will have failed too. Since those prior teachers have failed and I've failed over and over and over again, it must be impossible for student Y to learn X.

This feels right.

But it's stupid.

It's an objectively invalid argument. If the student doesn't get X, it emphatically does not imply the student's inherently incapable of learning X. Perhaps there is a better teacher somewhere in the world that could teach the student X. Maybe the teacher is overestimating the variety of pedagogy he or she has tried so far. Maybe all the teachers come from a school that has traumatized the student and they've only tried teaching the student in that school. Maybe the student simply needs more time. Maybe the student is totally capable of learning X but needs to learn prior knowledge U, V, and W first.

It's very easy to "confirm" incapability as a teacher, but it's actually impossible to prove. How do you know that nothing could ever help the student learn? That there is no teacher anywhere that could do a better job? How do you prove that no learning environment could make a big difference?

Now... How much human potential do we throw away because of confirmation bias? Probably some tragic amount...

• While what you say is a real issue, I don't think that confirmation bias is the problem here. Just for argument's sake, let me claim that "if you were to throw a fair coin randomly million times, you would never get all heads." How do you distinguish this one from "one cannot travel faster than light"? Our best models predict the first one to be false and the second to be true, but these are only models and we don't know how well they match our reality. Cont. Nov 29 '17 at 16:21
• Cont. Suppose you are an entrepreneur; maybe faster-than-light travel is possible after all, but assuming its availability within next 10 years could cause you to do a number of bad or at least inefficient business decisions. Similarly, I don't see the situation you described as "teachers confirming incapability", but rather as "teachers making efficiency trade-offs". I don't argue about the trade-offs in questions being good, bad or ugly – only saying that confirmation bias does not seem like the main issue in this particular example. Fin Nov 29 '17 at 16:21
• There are countless teachers who conclude permanent, inherent incapability simply because of the student's prior performance. This is bad confirmation-based reasoning. Nov 29 '17 at 18:01
• As for light-speed travel, we know it is reasonable to say that there is no plausible angle of attack in the near future because of what we know deductively from physics. There was, however, a plausible angle of attack in 1902 for flight. So, interpreting failed attempts at flight as decisive proof of flight's impossibility is an illogical interpretation of evidence in favor of one's preconceptions, and thus it exemplifies one aspect of confirmation bias. Nov 29 '17 at 18:02
• But then how do you know if there is any plausible angle of attack? That's the very same problem expressed in a different way. Nov 30 '17 at 14:37

I do not see how any of these examples are confirmation bias.

• if the teacher suspects that the student has some problem, he will try to help before checking if it really is that problem;

Not checking is worse than confirmation bias. You do not even think that your assumptions need confirmation.

• if the teacher thinks that the child does not understand some concept, he/she will try to prove that it indeed is the case;

Confirmation bias might come into it while trying to prove it, but not necessarily so.

• the teacher may want to stress why his/her solution is better than some other solution presented by the student (perhaps not even accepting it).

There is no confirmation bias here. Again, there is the assumption that one's opinion does not need confirmation at all.

That said, of course, confirmation bias is a big problem. The most common one is asking if everything is understood so far and being satisfied with one student answering "yes".