Mathematical Reasoning … Can We Recognize It? Do We Allow It?

My department has been discussing the concept of ‘rigor’, which usually invokes some variant of ‘mathematical reasoning’.  Definitions of either concept often involve communication and flexibility, though our practices may not encourage any of this as much as we would like.

In general, if a learner is simply showing the same behaviors (and mathematical analysis) that have been described … justified … and demonstrated during the class then I do not see much rigor.  Building mathematical reasoning involves exploring something new, and sometimes shows in failed attempts to solve a problem

Of course, the label ‘failed’ is an artificial description based on some arbitrary standard (like a correct ‘answer’).

Recently,  I graded my final batches of final exams.  I want to share two examples of mathematical reasoning which pleased me, in spite of the fact that the solutions submitted by the students were at odds with the expectations on our grading rubric.

Here is the first, in our Math Lit course:

Our rubric called for students to use the area formula once — since that information is shown in the problem.  We actually did very little work in this class with compound shapes.  I was pleased with this student’s analysis, which exceeded anything shown in class.

The second is from our Intermediate Algebra course (which is extinct as of next month):

In this case, our rubric was based on the expectation that students would ‘clear fractions’ (which ain’t the “good thing” it once was).  A handful of students did a reasoning process like that shown above AND recognized that the equation was a contradiction.  [The specific work above is from a student who only got a 2.5 grade in the class.  She has a lot more potential, which I did tell her.]  Although I don’t have any evidence, I can be hopeful that the emphasis on reasoning during my classes contributed to these students seeing a ‘different way’ to solve these problems.

In the algebraic situation, which is more important — to ‘always check solutions’ (awful advice!) or to ‘recognize a contradiction’?  Mathematically, there is no contest; connecting type of statement (contradiction in this case) with the solution set (none) is a fundamental concept in basic algebra … a concept actively discouraged by much of our teaching.

Using the phrase ‘mathematical reasoning’ does not mean that we build any mathematical reasoning in our students.  Our courses cover too much mechanics; “a thousand answers and a cloud of dust’.  Let go of trivial procedures (like extreme factoring of polynomials and simplifying radicals of varying indices with complicated radicands, or memorizing a hundred trig identities).

The easy choice is to emphasize procedures and answers.  The fun choice is to emphasize reasoning and analysis — even in basic courses.

Got (Math) Problem?

We like to believe that taking a mathematics class (or statistics) will improve a student’s ability to solve ‘problems’ with quantitative properties.  A basic flaw with this belief is that most of us (as math educators) do not like to present actual problems to our students — a problem is a situation where the solution is not just to be remembered.  There is a basic element of “have not seen this before” in a problem situation; at one extreme we have exercises (where memory can retrieve exactly what needs to be done) and the other extreme we have non-standard problems (where the presentation is different from experience and the solution involves synthesis).

We generally all hear variations on the phrase “I am not good at word problems”, though repetition has little to do with truth value.  Admittedly, many students are weak at solving problems; I’ve known quite a few colleagues who are not very skilled at this.  If you are interested, there is a wide body of literature of problem solving in mathematics over the past 40 years (or more).

My goal today is to share two ‘problems’ from my Math Literacy class.

As you’d expect, the concept of slope is central in Math Lit.  We begin working with linear and exponential patterns within the first two weeks of class.  Recognizing those patterns involves a first-order analysis of differences and quotients, and this is then presented as the concepts of slope and multiplier.  In the case of slope, students have experience in the homework (and in class) with both calculating and interpreting slope.

A bit later in class, we formalize the linear pattern with y=mx+b with still more work with identifying and interpreting slope.  Students get reasonably good at the routine problems.  However, I put this question on the ‘y=mx+b test’:

The cost to your company to print x paperback sci-fi novels is C=1200+3.50x where C is in dollars and x is the number of books.
A: What is the slope of this line?
B: Producing 298 books would cost $2243.  How much more will it cost to produce 299 books (compared to 298 books)?

Part A was ‘low difficulty’ (about 90% correct answers.  Part B had medium difficulty (~70% correct).  However, half of the students needed to calculate the new value and subtract … even though they had all ‘correctly’ interpreted a similar slope earlier.  Only about a third of the students could see the answer without calculation (with enough confidence to do the problem the ‘easy’ way).

Most ‘steps’ given for teaching problem solving are actually steps for solving exercises involving verbally statements.   The phrase ‘step by step’ is a admission that we are not doing ‘problems’; problem solving improves with more experience solving problems.

Related to slope … We also explore how to find the equation of a line from two points (or two data values).  Later, we learn how to write exponential functions given a starting value and percent change, or from 3 ordered pairs (conveniently with input values of 0, 1 and 2).

This week, I presented the class with a “problem” related to those two types of functions.

Using the data below, find the equation to find the value of a car ($$) based on the age in years.
Age         2             5              7                  10
Value    24400     19000     15400         10000

Coincidentally, students just completed a review problem on identifying linear and exponential patterns where the inputs were consecutive whole numbers.  We had also just calculated slope (again).  In this case, the situation became a problem because the inputs are not consecutive whole numbers.  Very few students could see (working in teams) how to solve the car value problem.  Most students understood with direct guidance (questioning) though one hopes that students will see what we see … that a linear pattern can be established by any consistent pattern of ‘equal slope’ values.

I am sharing these ‘problems’ and my observations in the hope that some people might be interested in exploring real problem solving in their math classes.  Developing problem solving capacity is not tidy, and often frustrating, but this type of work is rewarding to us and (I think) very helpful to students.

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.

The Basis of Basic Algebra: PEMDAS or Order of Operations or ??

My professional work focuses on helping students who have generally completed their K-12 mathematics though they are not able to place in to a college level math course.  Based on doing this for a long time, I share the following conclusions:

• Most students (even those who can place ‘college ready’) have dismal abilities and understanding about arithmetic relationships.  However, this (perhaps surprisingly) has little impact on their success in college.
• The primary issues preventing success in college (in terms of quantitative outcomes) deals with fundamental concepts of basic algebra: expressions & simplifying; equations & solving.  The most fundamental of these issues is order of operations.

So, let’s make this concrete.  We are doing really basic expressions and equations in our Math Lit course; one of the problems for today’s group work was the following:

Solve   15 = -3(y + 2) – 3

Because we are finishing up a unit for a test, we have been doing a lot of distributing in class.  We’ve talked about concepts of order of operations as it relates to expressions like the left-hand side of that equation.  In spite of that, students claim that:

(y + 2) = 3y  (because there is a 1 in front of the y)

Now, it is very easy to tell a student that their work is incorrect; it’s easy to say “you should distribute first” (though we don’t always want to distribute).  I am more interested in diagnosis … WHY is that mistake there?  What understanding needs to change to know what to do with all problems we will see?

It is very disturbing to learn that many “bad things” students do are based on being told in the past to “use PEMDAS”.  In this problem, students honestly think that they have no choice — they MUST combine y and 2; since they know that y=1y, they add 1+2 to get 3y.  Somewhat reasonable … if the requirement to combine were true.

We need to avoid misleading (or incorrect) rules about calculating which lack a sound mathematical basis.  PEMDAS is such a rule; I have written before on this, so I won’t repeat myself (not too much anyway).  See prior posts:  PEMDAS and other lies 🙂 , More on the Evils of PEMDAS! and What does &#8216;sin(2x)&#8217; mean? Or, &#8220;PEMDAS kills intelligence, course 1&#8221;.

Our students would be better served if we focused on the relationships between operations and how that helps with ‘order’ questions — even if we don’t present such complicated (and contrived) problems.    Simple problems are sufficient for much of what we need students to learn:

• -5²  and (-5)²
• 4x²   [does the square apply to the 4?]
• 8+2(x+3)   vs 8+2(6+3)

Algebra is about properties and choices.  Students focus on what they have been told is really important, and PEMDAS is often in this category.  This conflicts with the goals of basic algebra — and with most mathematics our students will work with.  I would rather spend an hour in class exploring the 3 different ways to solve the equation 15 = -3(y + 2) – 3 than in redundant examples drilling “one way” to simplify or “one way” to solve.

Correct answers from PEMDAS are worse than worthless.  Success in basic college math and science classes is based on understanding (thoroughly) a few concepts.  Nobody should be ‘teaching’ PEMDAS, because we should never deliberately harm our students.  Understanding is what enables students to reach their dreams; quick fixes — whether in the form of PEMDAS or ‘co-requisite remediation’ — are more about correct answers than they are about student success or mathematics.

Are you so focused on ‘correct answers’ that you either limit your student’s knowledge or unintentionally cause them harm?  As I tell my students:

Correct answers themselves are almost worthless.  The value comes from our understanding.