Building Understanding in Algebra

Like most of us, I have a tendency to assess student learning with an emphasis on “doing” problems … simplify this, solve that, etc.  We risk missing critical information by this practice — information that would help us build a stronger understanding of algebra.

On a small-group activity this week in our beginning algebra class, I used this question:

Paraphrase the expression  5x² – 4x + 5

This was the first problem of a set of 3, with a heading that included “properties and order of terms & factors”.  Because students have a hard time accepting a math expression as an object (and not always a directive to ‘do something’) many students struggled with the problem.

However, there was one particular error that was quite common, leading to this answer;

21x + 5

Since no work was shown, I was puzzled; I asked each student how they got this result.  Their answer?  The square meant 5 squared, so 25x … then 25x – 4x = 21x.

This is exactly the same issue we deal with when we present “-8²” and “(-8)²”; many of us see those problems as unnecessary.  I don’t agree, as many of my students have struggled mightily with “what does that exponent apply to”. These students can get a majority of correct answers when we say “simplify” because they have memorized the rule about like terms; it’s not that they believe it is wrong to get 21x for the problem — they just know that they are not supposed to do that when the directions are ‘simplify’.

If our students are not clear on “what the exponent applies to”, their understanding is limited to  first degree objects.  Now, we waste a lot of time on polynomial arithmetic that would be better spent on exponential models & numeric methods (to complement symbolic methods).  I have to say, though, that a beginning understanding of our symbolic language is based on the answer to that question “what does the exponent apply to”.

If you teach any algebra (beginning, intermediate, college, or pre-calculus), consider giving your students some open-ended questions about the meaning of our expressions.  Don’t assume that  correct answers is an indication of correct knowledge; the human mind is capable of much memorization and disconnected information.

Helping students build a strong understanding is a labor-intensive process.  Individual and small group dialogues are the most powerful tools to correct bad ideas; just getting feedback like “not correct, it means this” will not be effective.   [This is the reason why about 33% of my class time is spent using those tools.]

Remember that assessments don’t have to involve points or grades.  The best learning in my classes occurs when individuals and small groups struggle through stuff they did not understand correctly.  Every human comes with a drive to understand, and that can be harnessed in our math classes — if we use assessments that create those opportunities for deeper learning.

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