Meaningful Mathematics: That is Worth …

A few years back, my dean informed me that returning adult students wanted to know how their learning would be applied to their lives as opposed to understanding theory.  I was quite surprised by this statement, given all I’ve learned over the years; I had reached the conclusion that the more ‘seasoned’ students wanted to understand the why as well as the how and when.  This cognitive dissonance resulted in a non-discussion as the Dean would not believe my statements.  Of course, these generalizations (hers and mine) are seldom true over a broad range of situations.

This idea of ‘application’ and meaning continues to be a hot-button for me.  Somehow, academia has accepted the strange notion that learning needs to be justified by seeing how knowledge is applied to individual lives.  In the guided pathways movement, mathematics is specifically designated for ‘alignment’ with the student’s program of study.  I’ve written a bit about that; see GPS Part III: Guided Pathways to Success … Informed Choices and Equity and other posts.

There is a need for balance here, as in most things.  Traditionally, college mathematics courses were theory-driven gauntlets designed to ensure that only the fit students reached the point of seeing how mathematics is applied to significant problems and processes.  No ‘meaning’ is permitted until the student has survived entry into calculus with some sanity, and then meaning is only explored within a limited range of classic problems (‘maximize the area of …’).  The absence of meaningful uses of mathematics is only part of the problem with traditional courses.

At the other extreme are some modern courses in quantitative reasoning or statistics.  A note came out this week from Carnegie Math Pathways (the folks doing Statway™ and Quantway™) about how great it was that students could see how to apply the mathematics in their lives (“for the first time …”).  Some of my colleagues emphasize finance work in our QR course for similar reasons.    Yes, that adds some ‘fun’ to the course, and helps with motivation for some students.

A few years back I did an invited talk at a state meeting dealing with general education mathematics.  The talk was apparently well received for the wrong reasons — members of the audience thought I was advocating for a focus on applications and context that students could understand.  I left that meeting dismayed.

Why the dismay?

• Are we only able to motivate student engagement and learning in mathematics if we can convince them with immediate applications?
• Is the value of learning mathematics constrained by specific utilitarian advantages of a constrained set of content?
• Are we so unskilled in teaching mathematics that we see a need to focus on context instead of understanding mathematics?

Some readers might see these statements as disparaging inquiry based learning (and ‘problem based learning’).  My concerns with those pedagogical approaches centers on the balance issue.  As a matter of learning and cognition, context is the classic double-edged sword — yes, context can provide an initial anchor for learning and supports motivation.  However, context also tends to constrain the learning making it difficult for students to transfer their knowledge.

At the heart of my concern is this:

If we focus on utility of mathematics, how are we to inspire the next generation of mathematicians?  Is that inspiration going to be limited to applied mathematics?

For years, I have been saying that every math course should engage students with “useless and beautiful mathematics”.  We should show students how we became inspired to be mathematicians; for most of us, this inspiration combines theory and application — not ‘or’.

Let’s keep mathematics in every math class.

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.

Do we Confuse Good Pedagogy for Good Teaching?

Our professional organizations (both MAA and AMATYC) have published references related to good pedagogy within the last two years.  MAA had the Instructional Practices guide (https://www.maa.org/programs-and-communities/curriculum%20resources/instructional-practices-guide), and AMATYC has IMPACT (http://www.myamatyc.org/).  Lots of good ideas.  References to decent research.  What could be wrong?

Let me use an illustration from the other side of our ‘desk’.  When a student uses procedures without understanding them, we uniformly provide feedback that this is not sufficient.  When a student has some understanding of procedures, but can not understand connections between topics … we tell them that the connections are important.  If a student gets those connections to a reasonable level but can not transfer the knowledge to a slightly different context, we tell them that their learning is not good enough.

However, we tolerate — or even encourage — corresponding misuses of teaching pedagogy.  We see a pedagogy at a conference and ask questions about what to to in the process but rarely a question about WHY does it seem to work.  Very seldom do we even reach the low standard of minimal understanding of the procedures.  Rarely do any of us reach the expert level of knowing how to transfer our understanding to a new situation.

Now, it’s true that ‘good pedagogy’ (like good procedures) create some correct answers … ‘learning’, even if performed without much understanding.  However, the same can be said for some ‘bad pedagogy’; certainly bad stuff has worked reasonably well for me (though I try to not do that stuff anymore).  How do we even identify a method as “good” for teaching?

Sadly, we seem to have only two standards we apply to the process of identifying good pedagogy:

1. It’s good if the method feels right to us.
2. It’s good if somebody has seen good results with it (either better grades or some ‘research’)

Of course, our students do a lot of bad learning by the first standard (such as ideas about fractions or percents).  Students don’t generally use the second standard, and the second standard is actually not a totally bad thing when the results are solid research comparing two or more treatments with somewhat equivalent students.  I don’t expect us to use a gold-standard for research prior to using a pedagogy; I do expect us to do a better job of judging elements of a pedagogy based on understanding the process and the validation of those elements in research over time.

We also fall in to the trap of saying that diverse teaching methods are good.  Now, it is reasonable to assume that a given pedagogy might be well matched to a certain situation; we might even believe that a pedagogy is especially suited to a given mathematical topic (though this is difficult to justify by research).

What should we do differently?   My advice is to keep the classroom pedagogy simple from a student point of view.  I’ve seen teachers use multiple complicated methods over a semester, which requires students to learn our methods which will necessarily have a reduction in their learning of mathematics.  [Students have a finite supply of ‘learning energy’.]

My teaching methods are very simple.  Every day (besides tests) are team based with two activities for learning (start and end of class).  We don’t have assigned roles, and we don’t create artifacts to share with the entire class.  There is only one criteria for measuring the value of our teaching methods:

• Every student learns the most possible mathematics with the highest level of rigor possible every day.

Making this simple method work depends upon my understanding of learning processes as they relate to each topic and concept we explore.  I have studied the learning process for my entire professional career (it’s what my graduate work was in), and what has been shown by research supported by theory is:

The amount and quality of learning are functions of the intellectual interaction of the learner with the material to be learned.

In other words, maximize a quality interaction for each student in order to impact their learning every day.  The learning needs vary with the individual, so the pedagogy must provide a structure for my intervention (based on instant interviews) during class.  My assessment of my methods involves global and individual progress in learning mathematics (including how much rigor is achieved).

One of my students commented last semester: “We could not help but learn.”  I have had more dramatic comments (usually good 🙂 ).  However, this ‘we … learn’ comment is the most valued comment I have received.

My concern involves the frequent copying of teaching methods (often based on the ‘seems right’ criteria).  If you don’t understand how it works … you don’t understand who you will harm with the method.  Although we don’t take a professional oath about this, seeing ourselves as a profession suggests a ‘do no harm’ standard of practice.  Any specific pedagogy has the capacity to harm students; some pedagogies have a decent chance of helping students.

We should not settle for “it works for most students”.  We certainly should not settle for “this generally works, but I do not understand how it works”.  Our lack of understanding will cause harm to students.  Being an expert means that we see simpler solutions that produce broad benefits; using complicated ‘solutions’ means that we don’t understand the problem.

Resist the temptation to copy ‘methods that work’.  Copying methods is not productive for our students; it’s harmful to students if we copy methods without understanding the processes involved.  Your best bet is to keep it simple and interact with students constantly.