The Bad Part of Dev Math

This past weekend, I was at our state affiliate conference.  MichMATYC has a long history (relatively), and we have had a number of AMATYC leaders from our state (including three AMATYC presidents).  We’ve been heavily involved with the AMATYC standards (all 3 of them).  However, you can still see some bad stuff among our practitioners.

One of the sessions I attended focused on lower levels of dev math — pre-algebra and beginning algebra.  The presenter shared some strategies which had resulted in improved results for students; those improved results were (1) correct answers and (2) understanding.  That sounded good.

However, the algebra portion was pretty bad.  The context was solving simple linear equations, and the presenter showed this sequence:

• one step equations (adding/subtracting; dividing)
• two step equations (two terms on one side, one on other)
• equations with parentheses, resulting in equations already seen

All equations were designed to have integer answers; the presenter’s rationale was that students (and instructor) would know that a messy answer meant there had been a mistake.  All equations were solved with one series of steps (simplify, move terms, divide) — even if there was an easier solution in a different order.

When asked about the prescriptive nature of the work, the presenter responded that students understood that it was reversing PEMDAS (which, of course, makes it even worse for me).

The BAD PART of dev math is:

1. Locking down procedures to one sequence
2. Building on memorized incomplete information (like PEMDAS)

As soon as students move from linear equations taught in this way to any other type (quadratic, exponential, rational) they have no way to connect prior knowledge to new situations.  In other words, the student will seem to ‘not know anything’ in a subsequent class.

To the extent that this type of teaching is common practice, developmental mathematics DESERVES to be eliminated.  Causing damage is worse than not having the opportunity to help students.  When we offer a class on arithmetic (even pre-algebra), the course is very likely to suffer from the BAD PART; offering Math Literacy to meet the needs in ‘pre-algebra’ and ‘basic algebra’ will tend to avoid the problem — but is no guarantee.

All of us have course syllabi with learning outcomes.  Those outcomes need to focus on learning that helps students, not learning that harms students.  Reasoning and applying need to be emphasized, so that students seldom experience the BAD PART.

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For students heading in a “STEM-ward” direction, understanding functions will become critical.  Unfortunately, a combination of a prior procedural emphasis and some innate cognitive challenges tends to result in a condition where students lack some basic understandings.

For example, in my intermediate algebra class, we provide problems such as:

For f(x) shown in the graph below, (A) find the value of f(0), (B) find the value of f(1), and
(C) find x so that f(x)=0.

Since there is no equation stating how to calculate function values, students need to use the information in the graph.  The vast majority of students make 2 novice errors:

• Error of x-y equivalence:  providing the same answer for (A) and (C)
• Error of symmetry: Since the answer for (A) is x=1, stating the answer for (C) as x=1

To improve this understanding, I use the longest (time measured) group activity in the course.  This is definitely a situation where “Telling” does not correct the errors [I’ve tried that 🙁  ], and the small group process helps dismantle some of the errors.  Clearly, the correct understanding for reading function graphs is critical for success in pre-calculus and eventually in calculus.

Another function concept we dealt with this week is ‘domain’.  Now, once students have found a domain, there is a tendency for some students to think they should find the domain of any and all functions, regardless of the directions for the situation.  This “inertia error” (what was started … continues) is not a long-term problem.  Here is a typical problem for the long-term problem:

Find the domain for the function graphed below:

In this particular class, I provide a fair amount of scaffolding … in a small group project, we explored the behavior of rational functions (without using that label) including what the “undefined” x-value means on the graph.  We don’t use the word asymptote; rather, we talk about the fact that some x-value results in division by zero, and the graph of the function can not show any ‘point’ for such inputs.  This leads to the graphing of the function, including the behavior around the ‘gap’.

Students struggle quite a bit with this type of problem.  Sometimes, they continue the ‘function values from graph’ thinking, and latch on to x=0 or y=0 to make some statement about a ‘domain’.  Many students will correctly identify the x-values for the gaps (yay) but make illogical statements about the domain.  The typical student error is:

• (-infinity, -2) ∪ (-2, infinity)  … or even just one interval (-2, infinity)

This type of error usually follows from a process-focus, detached from the underlying meaning.  I am trying to get them to see:

• gap on graph equates to excluded values in the domain

The process focus looks at the first part of  this.  Like the function value errors, the effective treatment of this problem requires time and individual conversations.

This type of function work is not typical for an intermediate algebra course.  However, it would be typical for an algebraic literacy course.  As we transition from traditional content to modern content in our courses, I am expecting that our intermediate algebra courses will fade away … to be replaced by variations of the algebraic literacy course.

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The Power of Understanding Math

One of the most rewarding experiences we can have is when a student exceeds our expectations.

This is a story of a student who initially struggled with everything and is now being successful within an individualized course structure.  In this class, I never ‘lecture’ to a group of students.  Class time is used for studying, help, consultation, and testing; we call it the “Math Lab” though it’s not what most people mean by that phase.  The common meaning of “Math Lab” is a drop-in help center open to a variety of students in a set of math classes.  Our Math Lab is a way to take a few math courses … our math help for other classes is separate from the Math Lab.

This particular student (I’ll call him Philip) was clearly having trouble on the first day.  He did not want to use the online homework system, and that was not a problem for me.  However, he opened the book to the first page of the first section and had lots of questions about the names for types of numbers, about order of numbers on a number line, and (shortly after) about adding signed numbers.  The second day brought questions about the meaning of words in statements of properties, and about the meaning of variables.

It’s not that most students “get” these things, nor that they do not need to work on them.  What was unusual was the level of the struggle (basic) along with the sheer quantity of questions.  I never tell students what my prognosis is for them (I’m sometimes wrong) but I thought this student was going to spend weeks on every chapter.

Philip did, indeed, spend weeks on chapter 1 … a chapter about real numbers in a beginning algebra course.  Following those weeks, Philip then missed several classes due to medical problems related to his PTSD and physical injuries.  With over 6 weeks gone, Philip had only tried that first chapter test.  He was about to encounter the chapter on linear equations and applications, a classic “speed bump” for students struggling to learn algebra.

Somewhere in the month after that, however, Philip began making consistent progress.  In fact, he was getting through the third chapter faster than many students.  That progress has continued, and Philip is very likely to pass the course.

The main point is that something in the way Philip dealt with the struggle made a difference in how he succeeded in the entire course.  Philip works towards understanding everything, including ideas the are relatively minor.  He writes down lists of both vocabulary to learn and problems that he needs help with.  My guess is that his turn-around from struggle to success was caused by his hard work at understanding (and not just knowing what to do).

We all have students in this level of course who interact with the material at a low level; for them, it’s more about remembering what to do than it is about understanding.  I think Philip’s intense effort at understanding provided him with a cumulative positive improvement in the ability to learn new material.

Like most of us, I strive to have all students look for that understanding in learning mathematics regardless of the specific math course.  With other students, I end up trying to pull them someplace they have no intention of going (understanding) while Philip approached the material that way without any influence from me.

As a minor point in this post, I will point out that a struggling student such as Philip will be lost prior to getting any success.  Taking several weeks on one chapter is not an option within a fixed-pace class; instead of accumulating benefits, struggling students accumulate bad grades on assessments.  Our Math Lab, with its focus on individual learning, allows a struggling student to truly become a successful student.

A fixed-pace class has a limited capacity for helping struggling students; they need to be within a relatively small range of struggle in order to succeed.  Our Math Lab expands that range considerably (though there are still limits).

Understanding … a focus on understanding … enables students to obtain power in mathematics by raising their level of functioning to a higher point.

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What’s in that Fraction?

Sometimes students conceptualize math problems in ways that are mysterious to experts, but make sense to them.  On occasion, a bad conceptualization seems to be reinforced by features in the technology they are using.

I was helping a student work with rational expressions in our intermediate algebra course.  This particular student finds the material difficult, and often puts off dealing with the course.  Today, he was starting the first section which includes this problem:

If f(x) = 10/(x+1), find f(½)        [Presented in typical rational expression format.]

I think the student conceptualized fractions as two connected buckets (one for numerator, one for denominator) without seeing any particular meaning for the buckets together.

This student was doing most of their work on an older Casio graphing calculator, which shows fractions like this:

In other words, the calculator has a “a b/c” key used to enter fractions.  The student was trying to type in “10_½+1” so the calculator was showing ’21’ for an answer (which was a mystery answer for this student).  When I suggested using the division symbol instead of the fraction key, there was a resistance … until he discovered that it gave the correct answer for the online homework system.

I think it is pretty common to have students missing concepts in the meaning of fractions.  Frequently, they have trouble connecting a fraction with both one division AND with a combined product and quotient … where this last meaning allows for most of our algebraic work on rational expressions. Our instructional materials frequently emphasize the first concept (a single division), and never make explicit that a fraction also means multiplying and dividing … that “(3x)/(x²+2x)” means multiplying by 3x and dividing by (x²+2x).  Result: memorized rules for how we reduce a fraction.  It’s so much easier to focus on ‘multiplying and dividing by the same factor results in one’ as a concept … rather than ‘cancel common factors’ alone.

We might blame such misconceptions on an over-use of technology, or on a given calculator providing the ‘a b/c’ fraction key.  I think students have the misconception independent of the technology, and that the technology my student was using made it easier for me to identify the issue.

When a person looks for either research on learning fractions, or for suggested instructional sequences, there is agreement that a flexible and more complete set of concepts is critical for the diverse settings where fractions are used.  Our course materials (especially in developmental math, both in pre-requisite and co-requisite models) tend to focus so much on procedures that we never develop any further concepts about fractions.  That is really a shame, since students will forget the procedures; the concepts have a longer shelf life in the human brain.

We should always start with meanings and concepts … especially with fractions.

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