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.

Bias in Mathematics Education: Did You See an Elephant?

People in the profession of education — including mathematics education — are prone to exhibit some common modes of reasoning.  We tend to value linearity within learning, compliant students, and evidence which supports our current outlook.  Until we overcome this bias in evidence, there is no hope to make real progress for our students.

A concept used in social science research (which is what education is) is ‘confirmation bias’.  Although the image above refers to ‘facts’, for our purposes the word ‘evidence’ might be a better fit (and I also include the phrase “established scientific research”).  We are so cursed by this bias that we seldom are aware that we are extremely biased.

Some examples:

• At a conference, we select sessions dealing with what we are currently working on … ‘what we want to hear’ becomes a guarantee of what we hear.
• In our department, we discuss issues almost exclusively with colleagues who are known to agree with us on problems and solutions.
• When we read professional material, we seek out mathematics or pedagogy that we are already using.

My concern today is not the ‘other’; the concern is us.  Although it is certainly true that Complete College America (CCA) and the organizations bringing us the “Core Principles” of remediation are suffering from severe confirmation bias, their problem would not be able to impact us … unless we are already in a weakened rhetorical state.

Our theories are often as immature as the mythical blind mind finding out what an elephant is like — we experience 1/100th of the entire domain, and conclude that we have a theory for the entirety.  Something like “students have short attention spans, so never try to have a prolonged exploration of a complex topic” or “yes corequisite remediation works after all” or “showing students I care will result in them learning”.

Not only do we have confirmation bias about the learning process, but we have the same type of bias about mathematics itself.  If you don’t rebel at the phrase “mathematics hasn’t really changed”, you have not been paying attention.  If you expect that mathematics remains stagnant, that is exactly what you will see — in spite of overwhelming evidence which conflicts that point of view.

The phrase “growth mindset” is all the rage. Apparently, this only applies to students.

What is My Legacy?

When a person approaches the end of a job or career, whether as a transition to a new position or as a retirement, it is natural to examine and ponder what was accomplished.  Perhaps there were awards received, new products introduced (‘courses’ in the college world), or new processes developed (‘pedagogy’ in the college world). As I approach my ‘inflection point’ (retirement), my thoughts have been on similar matters.  However, I have a great colleague who asked an excellent question:

What is YOUR legacy?

Most of us would like to reflect on a set of positive accomplishments, and might consider that set to be our ‘legacy’.  However, the concept of legacy suggests a carrying forward … an impact that extends past our personal inflection point.  As ‘teachers’, we often view our impact on individual students to be a critical part of our ‘legacy’; certainly, knowing that we played a positive role in somebody’s education and achievement feels good — I just don’t think that ‘feels good’ is a basis for including this in our legacy.

I am also not using the word in the sense often cited in other contexts — business and leadership in particular, where ‘legacy’ refers to a future-oriented plan of creating impact and power.  I am thinking more of legacy in reference to any long-term impacts that might be important to me personally.

To some extent, history establishes a person’s legacy.  In ten years, what evidence or impact will there be?

Professionally, I have had goals involving the local curriculum and national curriculum at the college mathematics level.  Establishing a legacy within these areas is difficult, it seems.  Much of the work turns out to be temporary or unimportant aspects of courses or pedagogy — meaning that the 10 year impact (legacy) is approaching zero (from the positive side, at least).

How can we judge or establish our ‘legacy’ on important matters?  I suspect that trying to estimate our own legacy is to engage in a flawed process, due to a variety of biases.  Other people need to provide information.  Sometimes, this validation comes through an awards process.  On that front, I have received three teaching awards … AMATYC (national), MichMATYC (state), and college; the fact that the awards occurred in that order says something, though I am not sure what.  In spite of those awards, I still have a critical view of my work as a teacher.  Perhaps related to this, my goals for a ‘legacy’ have little to do with my teaching (though I obsess about my performance).

My goals are more about curricular impact.  In developmental mathematics, my work has been directed at modernizing the curriculum while providing effective remediation within a shorter period (one or two courses maximum).  At the college level, I have tried to get people talking — and doing something — about the very obvious problems with the college algebra – pre-calculus – calculus I to III ‘swamp’ of worthless artifacts from a by-gone era.

In some ways, people associate my work with ‘pathways’ in mathematics, and pathways has certainly been a big thing.  This is an alignment concept, in general; in my view, alignment of math with program is an excuse to avoid doing any worthwhile mathematics while pretending that we are helping students by letting them get by with only the math they will encounter for their ‘program’ (as if they have a stable vision of what that is).  In particular, the rush to use statistics as a gen ed math course is often based on questionable inputs concerning occupational needs combined with institutional avoidance of all things algebraic. If people think “pathways are part of your legacy”, I can only hope that they realize I do not support much of that work.

Within developmental mathematics, we have a tri-modal system: updated (like math literacy, quantway, foundations of math reasoning), eliminated (co-requisite), and traditional (same-old remedial courses).  The presence of the updated curriculum, I’d like to think, is part of my legacy.  Will it last 10 years to really be part of my legacy?  Of course, we don’t know … things change, often in directions that can not be anticipated.  I do know that a system with two conflicting designs (updated and traditional) is unstable and unlikely to be sustained in the long term.  I am honestly quite dismayed by the proportion of our remediation which still consists of traditional courses for many students; not only does this harm students, it leaves the system easily attacked by the disruptive influencers who want to eliminate remedial courses.   The fact that some proportion (whether it is 30% or 50%) refuse to update their remedial mathematics is likely to result in the total elimination of remedial mathematics in 5 to 10 years.

Certainly, within the college credit math courses (college algebra to calculus I-III) along with quantitative reasoning we, as a community of professionals, seem more concerned with improving pedagogy than we are with improving mathematics.  Most of us do not include any numerical (computing) methods in the basic courses, and quite a few of us actually ban computing devices from those classes (like calculators).  As AMATYC and MAA work on projects, each group seems to dance around the fundamental flaws in the curriculum in an effort to not ‘upset’ anybody.  Sometimes, the reasoning for this awkward dance is that our colleagues are actively looking for pedagogy so that is what our publications and conferences should be about.  Of course our colleagues are looking for pedagogical help — mostly because that is what we are offering.  If we don’t offer definite curricular guidance, we will never know whether our colleagues are really interested.  In spite of the lack of change, I remain convinced that many of us are ready to start the long process of discussion and dialogue concerning how we should change our broken mathematics curriculum in the first two years.  Our colleagues may need a lot of encouragement and we need a large dose of patience; the process needs appropriate opportunities for meaningful critique and reflection combined with solid design work dealing with the needs of our students and their careers.  The fact that this change process will take years and will be difficult at time is a poor excuse to avoid the problems (at the expense of our students).

As commented, judging one’s own legacy is difficult and perhaps a misguided effort.  In spite of that, I wonder if I have a legacy that extends past my own institution in the time frame approaching the ten year standard.  Perhaps I am looking for feedback on that, though other people will still be limited by the problem of predicting patterns in to the future.  Part of my “legacy discouragement” is that the trends I see in mathematics do not encourage me to think an updated mathematics curriculum will survive a ten year standard.

Do you think the updated developmental mathematics curriculum will expand to encompass the majority of the colleges still offering remedial mathematics courses?  Or, will we face the eventual condition where colleges offer either elimination options (corequisites) or traditional developmental courses (not updated)?

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.