Our Security Blanket is a Problem for Our Students

Acceleration is both a buzz-word and a set of solutions in developmental mathematics. In a basic way, the New Life model is based on acceleration to college mathematics for most of our students. The courses in the New Life model — Mathematical Literacy and Algebraic Literacy — are being well received; dozens of colleges have implemented one or both courses.

However, we are resisting a simple change that promises significant improvement at little risk — eliminating any college course prior to the level of beginning algebra or mathematical literacy. I’m talking about courses called pre-algebra, basic math, and or arithmetic. I believe that these courses have insignificant benefits while presenting risks to students.

The vast majority of these courses focus on procedural skills in a few content domains (decimals, fractions, percents, very basic geometry, and perhaps extremely limited algebraic skills). Historically, these courses are a relatively recent development from a remedial point of view:

The myth that we must fill all student deficiencies before they can take a college-level math course.

We all have deficiencies; human beings have a capacity to function in spite of them. We tend to accept without question the surface logic that says a student needs to master arithmetic before they can master algebra. [The New Life courses do not de-emphasize algebra; our focus is on diverse mathematics and understanding, including algebra.] A course like beginning algebra or Math Lit continues to be one of the key gatekeepers to college success.

At the global level, I have never seen any study reporting a large correlation between pre-algebra (or arithmetic) skills and success in beginning algebra; sure, there are a few studies (including my own) that show a significant correlation … due primarily to large sample sizes. Significance does not show a meaningful relationship in all cases. A correlation of 0.2 to 0.3 is only connected with 5% to 10% of the variation in outcomes; other student factors (like high school GPA) have larger correlations.

At the micro level, we often justify a pre-algebra course by justifying the components. Fractions are needed before algebra, because the algebra course covers rational expressions. Other content areas have similar rationales. This justification has two major problems:

The need in the target course is artificially imposed in many cases (‘needed for calculus, so we do this in beginning algebra’). [This is a pre-calculus course has the responsibility for this need.]
The pre-algebra content is almost always a procedurally bound, right answer obsessed quick tour with no known transfer to an algebraic setting.

When the New Life model was developed, we did not assume any particular content connections. We looked at the content of Mathematical Literacy, and determined that nature of the knowledge needed before students would have a reasonable chance of success. The list of prerequisites to Math Lit is quite short:

Understand various meanings for basic operations, including relating each to diverse contextual situations

Use arithmetic operations to solve stated problems (with and without the aid of technology)

Order real numbers across types (decimal, fractional, and percent), including correct placement on a number line

Use number sense and estimation to determine the reasonableness of an answer

Apply understandings of signed-numbers (integers in particular)

For the vast majority of students, any gaps in these areas can be handled by just-in-time remediation. This list certainly does not justify a prerequisite course. A similar analysis from a beginning algebra reference would yield a similar list, I believe.

In spite of what we know, we continue to offer courses before beginning algebra or Math Lit, and continue to require students to pass them before progressing in the sequence.

This has been a long-debated topic in AMATYC — why does an arithmetic-based course need to be a prerequisite to algebra? Essentially, I think this is our problem — these courses are security blankets for us. We feel like we are doing the safe thing and helping our students by giving them this ‘chance to be successful’; we believe that these courses offer real benefits for students, even though the data is pretty clear that they do not (in general).

It is uncomfortable, perhaps even scary, for us to consider the possibility that all students be placed into beginning algebra or Mathematical Literacy. We worry about the risk. We seem unconvinced that another math course in a sequence is creating known risks and problems for our students.

We can easily see the problem by a simulation. Let’s assume that 70% of the students pass pre-algebra, that 80% of those continue to beginning algebra (or Math Lit), and 60% of these pass.

Enter pre-algebra, pass beginning algebra … about 34%

Compare this to these same students starting out in beginning algebra. There is no sequence; the percent who pass beginning algebra is simply the pass rate for a group with somewhat higher risk.

Skip pre-algebra, pass beginning algebra … about 40% to 50%

The real world is not as rosy as the first scenario. At my college, less than 50% of our pre-algebra passers complete beginning algebra (and a fourth of these barely pass, having little chance at the next level).

Related to this issue is the body of research on the connection between placement into developmental mathematics and completion of college. One such study is by Peter Bahr (http://www.airweb.org/GrantsAndScholarships/Documents/Bahr%202012%20Aftermath%20of%20Remedial%20Math.pdf) A consistent finding in these studies is that completion is inversely proportional to the ‘levels below college’ that students are placed at — even if they pass the math courses.

We should be very upset by the situation. Few researchers talk about this, but we know. Pre-algebra (and arithmetic) courses tend to have a higher (sometimes much higher) proportion of minority students, as well as people with employment and economic problems. Community colleges are supposed to be about upward mobility; instead, we’ve created a system which has been shown to keep certain groups from advancing.

Let go of that security blanket called pre-algebra (or arithmetic). Take the very small risk of helping a lot more students get though their mathematics and their program. Completion leads to economic opportunity. Let’s get out of the way, as much as possible!

Developmental Mathematics Revival!

Bringing a new vitality to college mathematics 2011 to 2019