Mathematical Literacy: Factoring in the Mathematics Curriculum

Some people will be surprised that our Mathematical Literacy course includes some factoring.  Over the years, the topic of factoring has been a focal point of conversations — almost with the assumption that a reform math course would not cover any factoring.  Sometimes, we go to the extreme view of “anything not practical right now … will be omitted”, and factoring is usually not very practical.

In our Mathematical Literacy course we covered factoring  last week — true, this is just the GCF (no trinomial methods nor special formulas).  Since we only include GCF as a method students have an easier time.  However, if we had time, I would not mind if we covered a little more factoring.

For language skills, it is important that people be able to express thoughts concisely (simplify); in some important situations, it is even more important to be able to express thoughts in a more complex way that maintains the equivalent message — persuasive writing and speaking are particular modes in this style.  In a general way, learning (or a process) that can only be used one direction is usually learned only partially.  Deeper learning depends upon a variety of experiences with objects or ideas.

Factoring plays a comparable role in any course emphasizing algebraic reasoning.  A basic issue in algebraic reasoning is “Adding or multiplying?”  Many of our students believe that parentheses always show two things — what to do first (under the curse of PEMDAS) and “this is a product”.  Our work with the GCF puts students right in the middle of this confusion; in other words, the GCF is a great opportunity for students to better understand basic algebraic notation.

Of course, one risk of this work with the GCF is that students get even more confused.  We need to be careful that assessments help students understand better; within the Math Lit class, I need more experience designing the class work so better assessments can be delivered to students.

Of the traditional developmental algebra content, factoring is not my lowest priority — it connects with basic issues of algebra.  I can’t say the same thing for radical expressions, where we deal with procedures only vaguely connected with exponents.  I also place ‘rational expressions’ lower in priority than factoring; outside of the very basic ideas of reducing simple rational expressions, our time on operations and equations with rational expressions list mostly wasted … the emphasis ends up on procedures, not concepts and understanding. Such topics have been included in developmental courses because they are seen as needed in pre-calculus courses … because they are seen as needed in calculus courses.  We should strengthen this flimsy curriculum design based on student needs AND content needs in deliberate ways.

All of us have a role in this process so that mathematics becomes an enabling process rather than a inhibiting process.  Factoring polynomials is not necessarily an evil to be avoided.

 
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Mathematical Literacy: Making Sense of Negative Exponents

One of the unique characteristics of our math literacy class is the organization of content … in a given week, we usually have a blend of numeric and symbolic work mixing in some algebra within a set of concepts that have been building during the semester.

The last class had two primary tasks: Integer exponents in problems, and equations of lines.  In a typical algebra course, integer exponents are ‘covered’ as part of a concentrated sequence; in the Math Lit course, we have already been using basic exponent properties for several weeks (products, quotients, and simple powers).  The idea is to have the work make sense to students, as much as possible.  We had, in fact, done work with scientific notation including small numbers — before we dealt with negative exponents in general.

Since this was our first time doing negative exponents as a general idea, we started with a very basic problem:

Simplify x^2/x^4

We wrote out the powers of x and reduced; then we subtracted exponents.  Very typical stuff.  The difference was the next step:

The two answers are both correct (1/x^2 and x^-2).  What do you think negative exponents mean?

The first student response was ‘do the opposite’, meaning divide.  The second response was ‘turn it over’ meaning reciprocal.  Nonverbal clues indicated that most students understood one of these meanings, and several got both of them.  Only after this interaction did I say anything about what it meant — as we dealt with the idea of writing an expression with positive exponents.

It’s not that this was a magical moment.  Several students in class could not apply what they had just said, and some had a very incomplete understanding of exponents and coefficients.  I’m still looking for the magic path for this ‘sum versus product’ understanding.  However, less of the struggle was about negative exponents than I usually see.

Some progress is encouraging, and evidence of an idea making sense looks like progress.

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Mathematical Literacy: Proportional Reasonng and Dimensional Analysis

Fractions … friend or foe?

Our mathematical literacy course started with a variety of topics, some centered around concepts of proportional reasoning.  That work included some unit conversions, just based on a decision to multiply or divide.  Knowing that we would come back to the topic at a more sophisticated level, I did not go beyond the simpler approach in the book … except to mention that we would have a different method later.

Now, we have the different method.  The basic idea of dimensional analysis makes sense to my students — placing units in fractions to produce the result needed.  Of course, doing problems is not easy for all students all the time.  However, it’s clear that students see these fractions as a good thing (for at least this day).

In this work, we’ve been talking about a ‘path’ — the road from the starting unit(s) to the ending unit(s).  I found it interesting that making this explicit seemed helpful to students; this is the more ‘analysis’ part of the method, and I was not expecting students to like it that much.  For the curious, our work began with simple problems involving just one or two conversion factors; the most complicated involved 5 conversion factors.  [This last problem involved converting a rate from mi/hr to cm/sec.]

We included non-linear units (ie, area and volume) which led to “a foot is not 12 inches”.  This, most likely, did not get understood that well — and I’ll see on the quiz at the start of the next class.

It’s possible that students really do ‘get’ the idea of dimensional analysis; that would be a good outcome!  I also hope that success with this type of fraction work does NOT lead to false generalizations to other fraction work (non-product patterns).  Within our math lit course, dimensional analysis is a step in our proportional reasoning topics — and this seems to be working.

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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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