Category: math reasoning and applications

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.

 

Pre-Calculus, Rigor and Identities

Our department is working on some curricular projects involving both developmental algebra and pre-calculus.  This work has involved some discussion of what “rigor” means, and has increased the level of conversation about algebra in general.  I’ve posted before about pre-calculus College Algebra is Not Pre-Calculus, and Neither is Pre-calc and College Algebra is Still Not Pre-Calculus 🙁 for example, so this post will not be a repeat of that content. This post will deal with algebraic identities.

So, our faculty offices are in an “open style”; you might call them cubicles.  The walls include white board space, and we have spaces for collaboration and other work.  Next to my office is a separate table, which one of my colleagues uses routinely for grading exams and projects.  Recently, he was grading pre-calculus exams … since he is heavily invested in calculus, he was especially concerned about errors students were making in their algebra.  Whether out of frustration or creative analysis, he wrote on the white board next to the table.  Here is the ‘blog post’ he made:

 

 

 

 

This picture is not very readable, but you can probably see the title “Teach algebraic identities”, followed by “Example:  Which of the following are true for all a, b ∈ ℜ.  In our conversation, my colleague suggested that some (perhaps all) of these identities should be part of a developmental algebra course.  The mathematician part of my brain said “of course!”, and we had a great conversation about the reasons some of the non-identities on the list are so resistant to correction and learning.

Here are images of each column in the post:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

When we use the word “identities” in early college mathematics, most of us expect the qualifier to be “trig” … not “algebraic”.  I think we focus way too much on trig identities in preparation for calculus and not enough on algebraic identities.  The two are, of course, connected to the extent that algebraic identities are sometimes used to prove or derive a trig identity.  We can not develop rigor in our students, including sound mathematical reasoning, without some attention to algebraic identities.

I think this work with algebraic identities begins in developmental algebra.  Within my own classes, I will frequently tell my students:

It is better for you to not do something you could … than to make the mistake of doing something ‘bad’ (erroneous reasoning).

Although I’ve not used the word identities when I say this, I could easily phrase it that way: “Avoid violating algebraic identities.”  Obviously, few students know specifically what I mean at the time I make these statements (though I try to push the conversation in class to uncover ‘bad’, and use that to help them understand what is meant).  The issue I need to deal with is “How formal should I make our work with algebraic identities?” in my class.

I hope you take a few minutes to look at the 10 ‘identities’ in those pictures.  You’ve seen them before — both the ones that are true, and the ones that students tend to use in spite of being false.  They are all forms of distributing one operation over another.  When my colleague and I were discussing this, my analysis was that these identities were related to the precedence of operations, and that students get in to trouble because they depend on “PEMDAS” instead of understanding precedence (see PEMDAS and other lies 🙂 and More on the Evils of PEMDAS!   ).  In cognitive science research on mathematics, the these non-identities are labeled “universal linearity” where the basic distributive identity (linear) is generalized to the universe of situations with two operations of different precedence.

How do we balance the theory (such as identities) with the procedural (computation)?  We certainly don’t want any mathematics course to be exclusively one or the other.  I’m envisioning a two-dimensional space, where the horizontal axis if procedural and the vertical axis is theory.  All math courses should be in quadrant one (both values positive); my worry is that some course are in quadrant IV (negative on theory).  I don’t know how we would quantify the concepts on these axes, so imagine that the ordered pairs are in the form (p, t) where p has domain [-10, 10] and t has range [-10, 10].  Recognizing that we have limited resources in classes, we might even impose a constraint on the sum … say 15.

With that in mind, here are sample ordered pairs for this curricular space:

  • Developmental algebra = (8, 3)           Some rigor, but more emphasis on procedure and computation
  • Pre-calculus = (6, 8)                         More rigor, with almost equal balance … slightly higher on theory
  • Calculus I to III = (5, 10)                   Stronger on rigor and theory, with less emphasis on computation

Here is my assessment of traditional mathematics courses:

  • Developmental algebra … (9, -2)         Exclusively procedure and computation, negative impact on theory and rigor
  • Pre-calculus … (10, 1)                           Procedure and computation, ‘theory’ seen as a way to weed out ‘unprepared’ students
  • Calculus I to III … (10, 3)                      A bit more rigor, often implemented to weed out students who are not yet prepared to be engineers

Don’t misunderstand me … I don’t think we need to “halve” our procedural work in calculus; perhaps this scale is logarithmic … perhaps some other non-linear scale.  I don’t intend to suggest that the measures are “ratio” (in the terminology of statistics; see  https://www.questionpro.com/blog/ratio-scale/ ).  Consider the measurement scales to be ordinal in nature.

I think it is our use of the ‘theory dimension’ that hurts students; we tend to either not help students with theory or to use theory as a way to prevent students from passing mathematics.  The tragedy is that a higher emphasis on theory could enable a larger and more diverse set of students to succeed in mathematics, as ‘rigor’ allows other cognitive strengths to help a student succeed.  The procedural emphasis favors novice students who can remember sequences of steps and appropriate clues for when to use them … a theory emphasis favors students who can think conceptually and have verbal skills; this shift towards higher levels of rigor also serves our own interests in retaining more students in the STEM pipeline.

 

Problem Solving Skills

This is the story of what a student did when confronted with a procedural problem for which she did not ‘remember’ the standard procedure.

One of the in-class assessments I used is a ‘worksheet’; it’s like an open-note quiz over a set of material.  During our last class (intermediate algebra), the worksheet is longer than usual because it ‘covers’ the entire course.  Item 6 on this worksheet is:

Rationalize the denominator 

As I said, she could not remember ‘what to do’.  However, she did a great thing … she recognized that both numbers could be written as a power of 2:

 

 

 

I was very pleased that she did this, but the student was frustrated … she then could not see what to do.  This is pretty typical when novices dive in to the world of ‘non-standard problems’ — problems for which we lack a remembered process.

Of course, it was pretty easy to guide her through the remainder of the work:

=

 

 

Obviously, the expectation (this is our traditional intermediate algebra course) was that students would apply the standard procedure (multiplying top & bottom by the cube root of 4).  Students do not like that procedure, and I tell them that the procedure itself is seldom needed.

The alternate method worked only because there was a common base between numerator and denominator, and I doubt if the student will gain any long-term benefit from this experience.  This was more of a positive thing for me, as a teacher and problem-solver: Noticing a special pattern within a problem is a critical problem solving skill.

I’m sharing this story just because I had fun with it!

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