Hidden Treasure in Math Class

A course design can facilitate learning, and a course design can hinder learning.  I suspect that we get so focused on the details of our math courses that we may not notice whether our course is facilitating or hindering.

In our Math Literacy class, we have been working on algebraic reasoning.  On the surface, the class looks like we are not ‘covering’ very much because we don’t include some typical algebraic (developmental) topics.  We found some hidden treasure this week in class.

As we often do, part of class is based on groups figuring out problems with some guidance and reflection.  Today this meant that we had each group do an equation ‘tag-team’ style — each student could either do the next step, or erase the last step.  Students had a little trouble playing by the rules, and wanted to switch to ‘their’ method to solve the equation.  The payoff came when we talked about the different choices, as more students figured out that they have options for linear equations.

The hidden treasure came next, not that students saw it as totally good.  We looked at how we could solve equations of a type never seen before, starting with a simple rational equation (namely, 5 = 200/x).  Students could see the solution (40) though not always obtained formally, so we talked about doing ‘opposite’ operations to solve.  We followed this with a radical equation (the pendulum model), which is not normally seen in this level of math course.  To solve for the length inside the radical, we listed the calculation steps if we knew the length and wanted to calculate the period.  Then, we reversed — the opposite operations in the reverse order.

To me, the hidden treasure in this is that students get to think about both types of skills that we use in mathematics — we have routine procedures (often based on properties) and we have reasoning about statements (often based on relationships unique to the problem).  Wouldn’t it be wonderful if students developed both strategies, instead of just using routine procedures (often memorized)?

It’s clear that my hidden treasure was not perfectly clear to students; after this discussion, we had a worksheet which included an equation of related design.  They generally understood the reverse order idea, but thought they should do them in a different order — a choice which requires applying properties of expressions.  Our conversation was more satisfying than normal because we had used the reasoning approach, and talked about choices.

Students may still ‘want’ a recipe for solving equations and simplifying expressions.  Giving students a recipe hides the math treasure; emphasizing choices and reasoning allows for the possibility of students finding our hidden treasure.

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Are Math Courses Worthwhile?

Colleges usually require students to take math courses, whether for an associates degree or a bachelor degree.  The common approach is to identify a level on the “developmental math to calculus” ladder that seems like the best fit.  I believe that this approach is bound to failure, partially because these math courses are normally not worth taking … from the students’ point of view.

Much of the current conversation focuses on developmental mathematics, and whether those courses are worth the investment of institutional resources.  This approach hides the assumption that the gateway college courses are worthwhile for the institution and its students.

During a meeting about general education last week, a comment was made that we all know what college algebra is; to be fair, the speaker meant college algebra as opposed to intermediate algebra.  It’s true that math faculty can tell when they see a college algebra course — because it matches a generic description of college algebra.  The suggestion was also made that a college algebra course includes more demanding problem solving than intermediate algebra.

A student perspective on courses is naturally simpler than ours, but perhaps we need to attend to that perspective to solve our deep-rooted curricular problems.  A course is worthwhile for students when one or more of these conditions is met:

• The content of the course is naturally appealing to a curious mind.
• The abilities developed in the course enable success in other courses (easily seen as such).
• The process of learning in the course is stimulating and/or rewarding (innately).

I’m not describing students who think a course is worthwhile because it was easy, nor those who see primarily value in the social relationships.  I am thinking of students who are looking for an academic reason for taking a course.

A course such as college algebra is doomed to fail all student criteria, at least for most students.  It seems like we, as mathematicians, want students to take these courses so that we can spot the unusual students for whom such an artificial set of content appeals to students via the third condition (the learning is innately stimulating or rewarding).  We seem to take pride in the tidy logic and coherence of the traditional content, forgetting that students might need something different for their needs.

In other disciplines, a gateway course is often seen as an attractor for students — show students how wonderful the discipline is so that they want to see more.  Sociology and french inspire students, sometimes, because wise faculty design such courses to be potentially inspiring to a broad cross-section of students.  When was the last time your college algebra course inspired somebody who was not already STEM-bound?

We would like to have more math and STEM students, but we put courses in front of students that have a strong track record of discouraging student interest in our discipline.  Whether we call it college algebra or pre-calculus, a central goal of the course should be student inspiration.  I do not think our typical courses serve students sufficiently well to be worthwhile.

My concern also applies to other courses besides college algebra or pre-calculus.  We often use statistics as an easier path for students, a sort of “we can’t win anyway, so let’s make it easier” approach.  It’s time for us to re-build our gateway math courses so that they are appropriate introductions into the science called mathematics.  The emerging models of developmental mathematics — AMATYC New Life, Dana Center Mathways, Carnegie Foundations Pathways — can form a foundation for these new gateway math courses.

Our gateway math courses are not usually worthwhile for students, but they can be … and we should make them be worthwhile.

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Mathematical Literacy: One third is less … or is it more than?

Our Math Lit class is dealing with a network of algebraic concepts, including some basic problem solving.  As we often find, students don’t see the point of an algebraic method for simple problems and then have great difficulty using algebra when the problem is a bit more complex.

A simple problem was something like:

The IV is supposed to deliver 50 mL/hr, and the patient is supposed to get 400 mL.  For how many hours will the patient be on the IV?

Most students ‘just divided’, even though they could not explain why that would provide the answer.  When asked for an equation, they could see why ’50n = 400′ would provide a solution; students just did not see the value of the equation.

The next problem was something like:

A house is listed as having an assessed value of $42,000. The assessed value is one-third of the true value of the home.  What is the home actually worth?

Every student started off finding one-third of 42000, with a few then adding this ‘one third’ to the 42000.  Those that added-on were doing a ‘one-third more than’ (a more complicated relationship) rather than a simple factor of 1/3.  In other words, some students thought that the answer should be less than $42000 … and some thought that the answer was 4/3 of $42000.

Students were doing these problems in groups, as they often do in this class.  In this case, however, students did not question each other about their thinking.  Hints and ‘simpler case’ finally got most people to the correct representation.  I suspect that a few students said that this made sense just to be polite.

I suspect that students are being trained to look at “one-third of” as always meaning multiply the numbers — instead of usually meaning that there is a multiplying relationship being stated.  This seemed so strongly held a belief that writing “1/3*n = 42000” did not make sense to them.  Yes, this ‘of’ means multiply — but not ‘multiply by the number stated’.  In addition, I suspect that students are having trouble with the conceptual part of using variables.  This problem is very easy if the ‘one-third of the true value’ is seen as one-third of a variable; this view was difficult for this class of students.

Some similar problems show up in traditional algebra courses, including my intermediate algebra course.  The good thing (or not so good) is that the Math Lit students are not really having that much more trouble with this than students in a ‘higher’ course.  There seems to be a larger baseline ‘desperation’ triggered when a problem involves a fractional relationship, with students reverting to ideas with little or no validity.

This particular relationship (a number is one-third of another)  is not that important within the mathematics of this course.  The more important thing, to me, is students avoiding those bad ideas in a desperate move to answer a question with fractions.  To help with that, I may approach this problem with more scaffolding next time.

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Acceleration in Developmental Mathematics

Whether it is the recent Achieving the Dream conference, reports from Complete College America, or even education blogs, “acceleration” is a hot topic; acceleration is deemed ‘a good thing’.  Is it?

In physical contexts, acceleration is the second derivative of a position function … the rate of change of the rate of change.  To model acceleration, a valid position function has to be established (along with technical requirements such as continuity).  We can estimate acceleration outside of these limitations by use of numeric methods; however, numeric methods can not yield results appropriate for comparisons of different conditions in a scientific manner.

In education, what is ‘acceleration’?  The applied definition is something like “progression to college mathematics done significantly faster” (usually compared to a tedious sequence of traditional developmental mathematics).  Assessment of acceleration models in education is most often done by anecdote — we now get n% of students through their college course in two semesters, and previously we only got t% through.  This corresponds to the numeric methods for physical situations; are the results valid?

Before we can interpret results from acceleration efforts, we need to have a valid model for position.  We do not currently.  The traditional mathematics curriculum in the first two years is primarily a historical artifact continued though social inertia.  We (mathematicians) have not established what mathematics is required and how this mathematics should be ‘packaged’ into steps; without these steps (courses), we lack valid measurements of progress — which is the heart of the acceleration work.

The AMATYC New Life model (Developmental Mathematics Committee) and the Dana Center (University of Texas – Austin) “New Mathways” provide a consistent message about a package (sequence) that offers a scheme to measure valid progress.  Here is a segment of the New Life vision of the curriculum:

With all of the attention on developmental mathematics, there is a tendency to neglect the critical courses which follow:  pre-calculus needs to be a proven preparation for calculus, college algebra needs to be a proven preparation for other STEM courses, and general education mathematics needs to be proven preparation for other quantitative needs.

Until we tackle this large problem area, acceleration may (or may not) be a waste of effort — getting students to the on-ramp faster does not help if the highway is going in the wrong direction (or if the highway is full of unneeded hazards).  Acceleration efforts make the statement that the college mathematics is both a reliable and a valid position (goal); this statement is questionable.    In this way, I see acceleration as sharing a risk with modularization — they both will tend to entrench existing curricular structures at a time when we need to re-build the structure.

Acceleration is done through a variety of methods, and someday we can determine which method is valid for our position function (curriculum).  Until we get closer to that goal, I would not invest significant resources in acceleration.

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