Developmental Mathematics Revival!

Symbols as Window

Like humans in general, our students develop expectations based on experience. Habits form, often without awareness or conscious effort. Behaviors exhibit, which are used to measure knowledge. In assessments, we often confuse correct behavior with correct knowledge.

Symbolic work can be difficult for novices. We (experts) see large amounts of information in short symbolic statements. For a novice, symbols are like a map to a city never visited — yes, we can remember how to get from point A to point B on the map … but without any understanding of what these points mean in the city.

On a recent test in my beginning algebra classes, two mistakes were made by at least 20% of the students (one or both):

-3² + 5² = 2^4 = 16
8^6 divided by 8^2 = 1^6

The first error is a coincidental ‘right answer’ for a very wrong method. The second one, not at all. Both involve over-generalizations of ‘same number’ rules. Obviously, there is a very high probability that the students making one or both of these errors have low study skills or habits (like not doing any practice outside of class).

My concern is not these particular students, nor these particular errors. My concern is our overall approach to mathematics. We tend to take one of these approaches to symbolism in mathematics:

Emphasize symbolic procedures, and measure understanding by correctly completing more complicated problems.
Emphasize context and reasoning, and measure understanding by correctly completing related problems with differences in details.

Some reform models take approach #2 to the extreme — very few symbolic procedures are introduced, and most of what is done is arithmetic; algebraic models are used but carried out with technology more than symbolic procedures. We need to learn how to balance the ‘symbols’ and ‘reasoning’ aspects of mathematics — and be willing to embrace both as critical in all mathematics courses.

Clearly, there is much (perhaps a majority) of our traditional algebra curriculum that involves symbolic work without a purpose now or in the student’s future. I seriously doubt that solving a radical equation by squaring each side twice will ever be a survival skill in a student’s future.

Just as clearly (to me, at least), many of our students will need good understanding of various symbolic structures in mathematics, in future science courses (hard science and soft science). Terms, exponents, coefficients, subscripts, groupings, equations, inequalities … are involved in stating properties in sciences and in using predictive models.

When we assess the mastery of symbolism, we need to deal with much more than ‘correct answers’. In the ideal situation, assessments would be done in a one-on-one verbal interview so the expert can probe into the novice’s understanding based on the individual learner. Lacking that luxury, we will need to use diverse assessment tools that deal with process and connections, as well as answers.

Sadly, I had integrated some of this assessment into the beginning algebra class about two weeks ago — dealing with the adding terms error (first error above). On a worksheet, students were faced with adding like terms (10x^4 + 6x^4) before we had dealt with them formally in class. Something like half the students added the exponents as well as adding the ‘terms’ (coefficients). About 40% of these students apparently maintained this erroneous method up until the test.

Correct answers are only correlated with correct knowledge; students are always seeking the simplest rules for achieving correct answers — which can lead to totally wrong rules. Mathematical symbolism can be a window into the houses where students keep their math knowledge. Too often, however, symbolism is confused with the knowledge and correct answers stop the assessment process.

We need to slow down our courses. Learning mathematics is not a fast or spontaneous activity. Learning mathematics is hard work for both us and our students.

Join Dev Math Revival on Facebook:

Assessment in Math Classrooms, Learning math | Jack Rotman | November 30, 2013 4:37 pm

1 Comment

By schremmer, November 30, 2013 @ 6:04 pm

Re. “approaches to symbolism”
Another approach is to let the reading help the understanding. For instance, reading ax^±n as “a multiplied/divided by n copies of x”.
Regards
–schremmer