Who needs developmental math? Who needs remediation?

In our beginning algebra class, we started our work in ‘exponents’. I use an activity (guided, discovery) to start this work and talk with students and groups as they answer the questions. The range of questions and confusion is both encouraging and discouraging. Some of the questions showed thoughtfulness and insight; others indicated a naive knowledge of our language system. #placement #tests

A large component of the issues relate to “grouping”, in 3 categories:

The meaning of required, shown grouping
The meaning of optional, shown grouping
The meaning of implied grouping

Many of us have commented on an example of the last group: -5², where the implied grouping exists on the base (5) not the opposite. Students do struggle with this, and … on its own … this type of problem is not worth the trouble. However, many of the same students misinterpret 5x²; when a value of x is provided students will square the 5 as well as the x … if the replacement value is negative, students will either leave off parentheses on that value or write the parentheses but not use them in evaluating. The implied grouping is a key feature of mathematical languages, and it harms students that we are not consistent in the meaning of implied grouping. [Just think about what sin² 3x means … there are two implied groupings in that expression, and both are inconsistent with almost all other implied groupings.]

When a problem had optional grouping shown, as in (5xy)(x²y³), students do not always understand that the meaning has not changed … and often, they think of a different process (like distributing) when they would not if the problem had no grouping at all. Another example would be (5x + 3) – (2x – 5) [required grouping on the 2nd expression] when the student distributes the ‘negative’ and writes (5x + 3) (-2x + 5) … and proceeds to multiply; that’s a case where we would say the grouping is optional but correct with the ‘plus’ between the groups.

So, what do these comments have to do with ‘needing developmental math’ or ‘needing remediation’? These misunderstandings are not gaps in knowledge, nor forgotten information … they are wrong ideas (called ‘baggage’ by some colleagues). Wrong ideas are known to be resistant to instruction; the most common outcome is that the wrong ideas are temporarily covered up by memorized correct information but then re-appear in the behavior after a short period of time.

Much criticism has been leveled at the placement tests we use. The words “evil” and “invalid” often are included in statements about those tests. However, the problem is us not the tests. The tests are constructed to meet ‘market demands’ … we have told the companies that we need to measure skills, so that is what we got. The problem is that skills are a very poor way to identify students needing either a developmental math course or a remedial math course. Missing a skill problem can be caused by either a wrong idea OR a forgotten procedure, resulting in much ambiguity with scores.

Developmental mathematics is not going away. Change is happening … the new courses like Mathematical Literacy and Algebraic Literacy focus first on developing right ideas about the mathematical objects then on procedures. What we need is a new set of specifications for placement tests to determine who needs a course versus those who are either ‘ready now’ or ‘have forgotten some’. I suspect that the ‘entrance tests’ (SAT, ACT) are better measures than the placement tests because the ACT & SAT are not as focused on skills. We need placement tests that identify wrong ideas as well as some fundamental skills.

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placement | Jack Rotman | November 10, 2015 9:52 am

1 Comment

By schremmer, November 11, 2015 @ 10:14 am

Students need to learn the need for a mathematical language so they have to learn it in context, that is in relation with mathematical problems. But, even more important, they must first understand the logic of the mathematical language. In the case of powers, here is the way I proceed:

1. Given the “code” 7*4^+5 (yes, without parentheses and that is exactly what you would enter in a calculator), the students must:

a. Read the code: 7 multiplied by 5 copies of 4
b. “Decode” that is write: 7 x 4 x 4 x 4 x 4 x 4
c. Execute the code: 7168

2. Given the code 1000*5^-2, the students must:

a. Read the code: 1000 divided by 3 copies of 5
b. “Decode” that is write: 1000 / 5 x 5 x 5
c. Execute the code: 8

3. Comment on the code:

a. While the size of the exponent codes the number of copies to be made of the base, it is the sign of the exponent, not *, which codes for whether to multiply or to divide the coefficient by the copies.
b. Thus, the * in the code is only a separator.
c. There is clearly an issue in the case of division (Why / 5 x 5 x 5 instead of / 5 / 5 / 5) but, in retrospect, discussing it at this stage as in

Repeated Multiplication and Division

seems counterproductive.

4. While the exponent must be a signed counting number, the coefficient and the base can be decimal numbers, signed or not. This though, has consequences for the code.

a. ” • ” cannot be used as separator if decimal numbers are involved because of the risk to confuse it with decimal points. In that case, it is better to use ” x ” as separator as in 3.14 x 2.78^-4.
b. ” x ” cannot be used as separator if the base is to be x.
c. Often, teachers prefer using parentheses to separate the coefficient from the base.

5. The purpose of the code:

a. Large size exponents. For instance, -3.14 * 10^±34
b. As long there is no execution, we can compute directly with the code:

ax^±m multiplied by bx^±n = ab*x^±m oplus ±n

ax^±m divided by bx^±n = ab*x^±m ominus ±n

Instances of which need to be thoroughly verified by the students by comparison with the result obtained by first reading and decoding. (While ax^-m divided by bx^-n calls for “division of fractions”, discussing the latter at this stage seems counterproductive.)

c. (Laurent) monomials cannot be added hence (Laurent) polynomials.

The above is dealing only with the syntactics and semantics of the code as I think that involving the context of mathematical problems would overly complicate matters. At least, I don’t see how to do it.

In any case, particularly at the developmental level, I sure do not see the need to spend even one second with code such as “(5xy)(x^2y^2)”.

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