Generalizing to Failure: “Cross Products”

The human brain naturally takes the leap between example and generalization.  We encounter one used-car-salesperson who pushes us to buy something we don’t want, and we make a generalization that all used-car-salespersons are pushy.  We encounter a method for correct answers in one fraction situation, and we make a generalization that this method works for all fraction situations.  In fact, some of us teach by taking advantage of this ‘constructive’ process.  Caveat emptor!!

Our Math Literacy course forms the basis for this specific post, though the issues with generalizing are universal.  The specific scenario is this … students have previously encountered operations with two fractions (all 4 basic operations), and now we are solving proportions.  Our proportions involve only one variable term, so students occasionally use proportional reasoning to build up or down, and this ‘works’ for now.

The problem is not that students lack any prior good knowledge about cross products.  Almost every student in my classes ‘knew’ what cross products are (in the fraction world).  The problem was that they generalized an incorrect ‘method’:  cross-products form a fraction.

Like this:

“Solve 14/12 = 126/b”

Student answer:  14/1512 = 1/108

This wrong answer becomes correct if there was a division operation instead of an equation.  The fact that students reach college with such bad knowledge is, of course, a function of the math opportunities they had K-12 … students with a good math background have normally been trained to notice such ‘trivial’ features as the symbol between two fractions.

I’m sure that this specific bad generalization comes from another process — using cross products to test for equivalence of fractions.  Those problems are often presented as a pseud-equation like this:

Test:  15/108 =?  10/76

At the micro-level, my message is “don’t let students use cross-products with fractions unless the object was a proportion complete with the “=” symbol”.  Teaching cross products for anything else causes harm to your students, just as teaching PEMDAS causes harm.

However, my main concern is not really this one situation.  In basic algebra, ‘distributing’ is a key skill.  The false generalizations involve these types of problems (resulting in the ‘answers’ shown):

• (x + 3)² = x²  + 9                <distributing an exponent>
• 3(w – 4)² = (3w – 12)²         <distributing before an exponent>
• 5y² = 25y²                           <distributing an exponent>
• (x + y)/y = x + 1                  <‘distributing’ by cancelling>

The first two types are very resistant to learning to correction.  In psychology, this faulty generalization is sometimes called ‘cognitive distortions’ or ‘hasty generalizations’, though I prefer the direct term ‘false generalizations’.

Keep in mind that “We Are the Problem” (where ‘we’ refers to people teaching mathematics at any level).  We focus on correct answers as measures of correct knowledge (see The Assessment Paradox &#8230; Do They Understand?).  Some of us avoid that paradox by requiring written explanations on assessments; that approach does help if done in moderation — having to explain in multiple situations on one assessment comes with significant overhead for us and our students, as well as the known risks of bias in grading the writing.

We have two other tools to help student correct their generalizations:

1. One-on-one (F2F) feedback
2. Problems designed to confront false generalizations

I have been using both approaches for 20 years or more.  My conclusion (hopefully not a false generalization 🙂 ) is that problems are not as effective as we think they are, in catching bad generalizations. The proportion given earlier came from a student who is very thorough in doing homework, and we had just done this problem in class the day before:

• Solve -3/(y + 4) = 2/(y – 1)

This problem is different from all of the homework, and all problems we had done in class — those binomial expressions were very confusing to students.  With suggestions and sometimes direct statements, students eventually used cross products to solve (complete with the distributing).  That experience does not help, though; the experience is short in duration, and seldom engages an emotional response that might help learning).  Prior learning complete the false generalizations is strong, compared to the experiences we control.

The best impact comes from the one-on-one engagement.  Because there is another level of activity (social or emotional), our work is a bit stronger than just the problems themselves.  Some students I worked with on that unusual problem adjusted their knowledge.

My message today has two components.

1. Teach mathematics in a way that offers some control over false generalizations.

Get students engaged with problems that “don’t work” while including some problems that do work with the idea we are trying to learn.  Keep in mind that, while students helping students supports a good classroom environment, other students will tend to have similar false generalizations.  I had a team this semester where 4 of the 5 students believed the same wrong thing; the other student ‘gave in’ because the other 4 agreed.  YOU are the best resource in the classroom to control generalizations.

2. Assume that a significant proportion of your students have false generalizations about “today’s topic”

Because of the focus on correct answers, students can “go far” without having correct understanding.  Typically, this leads to a ‘crash and burn’ experience in pre-calculus/college algebra or intermediate algebra.  Since we don’t want math courses to be a filter, we need to design instruction so students are not weeded out; opportunities to correct prior learning are critical in our efforts at equity and inclusion.

There is no magic for fixing a false generalization.  Take a look at a study on correcting misinformation in health care (https://psycnet.apa.org/record/2014-41945-002).  The situation is not hopeless, but it is discouraging.  Correcting false generalizations is MUCH more difficult than learning true generalizations in the absence of faulty knowledge.  Thus, the first idea above is the most important — regardless of what you teach, or at what level, structure the learning process so that generalizations are almost always correct.  Five true generalizations with no faulty ones are more valuable than 20 true generalizations with 5 faulty ones.

False generalizations will kill your students dreams.

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