Our beginning algebra class is taking a test on ‘exponents and polynomials’ today; this chapter is about as popular as a math chapter can be for my students. The processes are fairly easy, and with some extra effort in class, most students do well on this test. All is not good, however.
Students tend to have a fragile understanding. For whatever reasons, the symbols in front of them do not have full meaning. Here are two examples of what I am talking about.
Subtraction versus “FOIL”:
Seeing a problem like (5x + 4) – (2x – 3), many students convert it into (5x + 4)(-2x+3) and multiply. They know the rule about ‘changing the signs’ when we subtract, but lose the little ‘plus’ between the groups.
Negative exponents versus polynomials:
Seeing a problem like (6x² – 9x)/(3x²), many students convert (2 – 3/x) into (2- 3)/x to get -1/x.
As teachers, we feel good when students show a process that fits with a good understanding. Showing a process does not depend on a good understanding. The relationship works one way cause and effect (understanding leads to good processes); a good process does not lead to, nor is evidence of, good understanding.
So, we give assessments to students and say “they know exponents” because of the processes and answers. In the extreme form, we have a module on exponents and polynomials and certify “mastery” because of a high score on the module assessment. We do not do enough assessments that do a compare and contrast — opportunities for us to see if a student has a fragile understanding, identify the weakness, and then build up a stronger understanding.
I continue to work on this problem. In the case of ‘subtraction versus FOIL’, I use problems like the one shown on assessments early in the semester, during our first class on ‘FOIL’, and later in the chapter. That helps; no magic, but the opportunity to discuss with an individual student is powerful.
I believe we need to work on two components of our instruction if we have any hope of building a strong understanding in place of fragile understanding.
- Combination of active and direct instruction on the concepts, with a focus on “what choices do we have?”
- Assessments that determine the presence of confusion of concepts (aka ‘fragile understanding’)
Our professional expertise is needed, since we can not assess for the presence of specific confusions unless we know what the common types are. To make this even more challenging, we have no assurance that the confusions are global versus local — do students in beginning algebra courses tend to have the same confusion regardless of locality?
The best resource we have is the students in our classes. Having purposeful conversations (oral assessments) is a critical source of information about both a specific student and zones of confusion. These conversations provide insights, and form a way to validate our more convenient forms of assessment (paper & pencil, or computer test). When I grade today’s test on this chapter, I will be comparing what I thought they understood to what I see being shown on the test; just like my students, there should not be any surprises to me on the test.
Of course, there is a good question … does it matter at all? We have a pride in our work and profession, so we respond with an automatic ‘yes’. We should be able to articulate to other audiences why it does matter. Does a fragile understanding enable or prevent a student from completing a math course? How about a science course? Can we develop quantitative reasoning in the presence of fragile understanding? Does a modular design support sufficiently strong understanding? Do online homework systems provide any benefits for understanding concepts?
The issue of fragile understanding is critical to the first two years of college mathematics, whether in a developmental math class college level. I have heard colleagues suggest that the prerequisite for a certain class be raised to calculus II, not because any calculus is needed but only because students have a stronger understanding after passing (surviving) calculus II. We often cover this problem with a vague label “mathematical maturity”.
In response to a recent post, Herb Gross (AMATYC founding president) wrote a comment, in which he emphasized the “WHY” in the math classes he taught. I totally agree with his comment, in which he said that students want the why — they want to understand. Although a human brain can learn with and without understanding, there is a natural preference to learn with understanding.
A fragile understanding, lacking the ‘why’, leads to both short term and long term problems for students. I think we waste their time in a math class if we accept correct answers for the majority (70%) of problems as a proxy for ‘knowing’. Determining that a student knows mathematics is a complicated challenge, and forms a core purpose for having a strong faculty professional development.
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