The Assessment Paradox … Do They Understand?

We often make the assumption that solving ‘more complicated’ problems shows a better understanding than solving ‘simpler’ problems.  This is an assumption … a logical one, with face validity.  I wonder if actual student learning refutes it.

My thoughts on this come from a class I’ve been teaching this summer, “Fast Track Algebra” for the first time.  Fast Track Algebra covers almost all beginning algebra along with all intermediate algebra; those separate courses I’ve taught for 45 years … this was my first time doing the ‘combo’ class.  In case you are wondering how we can manage it, the class meets 50% more — 6 hours per week in fall & spring, and 12 hours per week in the summer like my class did.

Our latest chapter test covered compound inequalities, absolute value equations, and absolute value inequalities.  As for most of the content, none of this is review of what students ‘know’ — any knowledge they had on these concepts is partially or fully faulty, so class time is focused on correcting deeper understanding of concepts and procedures.

The class test, in the case of absolute value inequalities, presented 3 levels of problems:

1. simplest, just need to isolate absolute value and easiest solution … like |x| – 2 < 5
2. typical, absolute value already isolated with a binomial expression … like |3w + 4| >8
3. complex, with a need to isolate absolute value and binomial expression … like 2|2k -1| + 6 < 10

The surprise was that most students did better on the ‘complex’ problems than they did the simplest problems.  On the simplest problems, they would only ‘do positive’ while they would do the correct process on the complex problems (both positive and negative).  This was a little surprising to me.

If a student does not do the simpler problems correctly, it is difficult to accept a judgment that they ‘understand’ a concept — even if they got ‘more complicated’ problems correct.  This paradox has been occupying my thoughts, though I have seen some evidence of its existence previously.

So, here is what I think is happening.  As you know, ‘learning’ is a process of connecting the stimulus (such as a problem) with the stored information about what to do.  Most of the homework, and the latest work in class, deals with the ‘complex’ problem types.  The paradox seems to be caused by responding to the surface features of the problem and retrieving a process memorized in spite of a weak conceptual basis.  In the case of absolute value inequalities, they ‘memorized’ the correct process for complicated problems but failed to connect the concept to the simplest problems because they ‘looked different’.

If valid, this assessment paradox raises fundamental questions about assessment across the entire curriculum.  As you know, the standard complaint in course ‘n+1’ is that students can not apply the content of course ‘n’.  Within course ‘n’, the typical response to this complaint is to emphasize ‘applying’ content to more complicated problems.  Perhaps students can perform the correct procedure on complicated problems without understanding and without being able to apply procedures in simple settings.

I see this paradox in other parts of the algebra curriculum.  Students routinely simplify rational expressions with trinomials correctly, but fail miserably when presented with binomials (or even monomials).

Some of us avoid this paradox by emphasizing applications — known as ‘context’, and focusing on representations of problems more than procedural fluency.  With that contextual focus, we will seldom see the assessment paradox.  The challenge on the STEM path is that we need BOTH context with representation AND procedural fluency.

I’m sure most faculty have been aware of this ‘paradox’, and that this post does not have novel ideas for many of us.  I wonder, though, whether we continue to believe that students ‘understand’ because they correctly solved problems with more complexity.

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2 Comments

• 

  By Dwight, August 1, 2018 @ 12:06 pm

  To the matter of absolute value equations and inequalities, What I think the problem of visualizing using the +,- in solving comes from not having the concept of what absolute value means in the beginning. I always start abs. value lessons with the definition of what abs. value means, then later on reinforce it when working with the equation and inequality solving. It seems to work!
  Dwight

• 

  By Jack Rotman, August 2, 2018 @ 12:35 pm

  Good points … I do similar work in my classes.

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