Algebra is a problem; in schools, in colleges, in the society. One of the most recent proposals to solve that problem by eliminating elementary algebra from the school curriculum altogether came from our colleague, Andrew Hacker, a retired political scientist from the Queens College (NYT July 28,2012). Is algebra necessary? – he asks. The underlying implicit questions are: What algebra? For whom?
In order to answer these fundamental questions it’s worthwhile to ask why our students fail algebra in troves. The study led by Prof Akst from BMCC in 2005 (Geoffrey Akst, David Crook, Virginia Moreno, Cheryl Littman, and Angela Grima, Performance in selected mathematics courses at the City University of New York: Implications for retention. CUNY February 2006)., also a retiree at present, had shown that only around 37% of our students who take remedial arithmetic at CUNY manage to pass the subsequent course of elementary algebra.
It turns out that the problem doesn’t start in the 9th grade as Hacker suggests. It starts much earlier, in the 3rd grade, because one of the central obstacles student encounter in mastering algebra is fractions of arithmetic. And that difficulty stays with them into algebra and later into college math. An excellent study PROM/SE of 2006 (http://promse.msu.edu/research_results/PROMSE_research_report.asp) conducted by the Michigan State University investigating the ability of students in grades 3 through 12 to understand and to master fractions reveals that the lowest percentage of passing students was seen in the 3rd grade, only 24% passing; only 56% in the 6th grade and 73% in the 8th grade passing the PROM/SE exam used in the study.
Following the logic of our colleague, we should abandon teaching of fractions because students have serious problems with them, which persist into college. In fact, recent reorganization of CUNY remedial programs moves exactly in this direction by eliminating arithmetic exam from the exit from remediation. Then why don’t we abandon teaching of fractions? Because fractions, together with ratios and proportions underlie the proportional reasoning – the basic component of mathematical thinking. They are necessary not just for engineering and scientific work but also for our nurses, whose major mathematical job is to be able to do many unit conversions at once for application of different medicines. Thus the logic of our colleague would directly impact the well-being of citizens, in particular of retirees who have to extensively rely on the expertise of nurses.
Why do fractions constitute such a stumbling block for young students? Because fractions represent the first conscious effort at abstract mathematical thinking. They can’t be taught by mere memorization of rules or algorithms but involve the process of understanding mathematical concepts. In fact proportional reasoning involving fractions in the fundamental way is recognized as the gateway to algebra and that brings us to the stated questions (Berk et al, 2009; Lo and Watanabe, 1997).
The major part of algebra involves the process of generalization of arithmetic and the experience of CUNY instructors of remedial algebra and high school teachers of mathematics indicates that one of the reasons of our students’ failing algebraic exams is, indeed, their weakness in arithmetic numerical skills. It is this weakness in numerical skills that doesn’t allow students to develop a clear notion of variable as generalization of a number.
Now, that we understand a bit better sources and reasons for the difficulties in algebra, we have two options:
One can either eliminate algebra or eliminate the difficulties with algebra.
Andrew Hacker chooses the first option and provides a series of arguments in its support. However, he doesn’t mention the possibility of the second option of eliminating difficulties—the possibility embraced by the new Common Core Standards in mathematics, which aim precisely at eliminating those difficulties in algebra, among others.
I think Hacker and I would both agree that the primary purpose of public education is to give every young person the intellectual tools he or she may need in order to productively develop and follow his or her talents and inclinations. Hacker argues that algebra (as opposed, say, to arithmetic or quantitative reasoning) is not an intellectual tool necessary for a person’s productive development; in addition, when it is included in the required curriculum, it proves instead (because of its difficulty) to be an obstacle to such development.
The way algebra is generally taught in the US seems to be conceived as a collection of rote recipes (“algorithms”, in Hacker’s terms) for manipulating mathematical expressions consisting of letters and arithmetic operations, and for translating various types of artificial “word problems” into such expressions. If this is what is meant by “algebra”, then I think Hacker is right: there is no reason on earth why students should be required to undergo instruction in “algebra”.
However, if algebra is conceived as the general science of quantitative relationships, the concepts underlying such relationships and the tools that have been developed for utilizing them, then algebra so conceived is an inseparable component of modern “quantitative reasoning” – for example, of an actual understanding of arithmetic or of the ability to understand, and critically reason about, statistical relationships (how can you understand a statistical graph without having the concept of a function? or of a Gaussian distribution whose simplest representation is the composition of the exponential function with the “dreaded” quadratic function). In fact, I would thus argue that algebra conceived is not naturally separable from any serious conception of “quantitative skills”, if the notion of “skill” includes “understanding”. Consequently, I think that any seriously conceived teaching of the tools and skills that Hacker thinks should be part of the curriculum would actually and naturally include algebraic thinking.
Where Hacker misses the point, I think, is that just changing the topics included in the math curriculum would not solve the problem as the difficulties with fractions and proportional reasoning in early grades demonstrate. What needs change is the very conception of, and attitude towards, math and its teaching, on the part of policy makers, curriculum builders, teacher educators and teachers. But this, as we know, involves a kind of vicious circle, since all of these people have been educated in the very same system that needs changing. Unfortunately, most recent directives from CUNY central concerning remedial mathematics reinforce that vicious circle, as “understanding” is conspicuously from the mandated Student Learning Outcomes for mathematics.
Taking into account that 70%+ of future teachers of mathematics, accordingly to AMATYC (Beyond Crossroads 2005), progress through community colleges and 75%-80% of freshmen have to take remedial mathematics, we get 50%+ of those future teachers for whom “understanding” had been eliminated from their mathematics education. Good bye fractions and proportional reasoning! That is how the vicious circle which led us to the present situation is vigorously maintained by our own university.
What is the solution? We have to cut the vicious circle, the famous Gordian knot.
How to do it?
My own predilection is mathematics teaching-research conducted by ourselves in our own classrooms aimed at improvement of learning in those very same classrooms, and beyond. Only through our objective look upon our (individual and collective) practice, standing face to face with the reality of the mathematics classroom situation yet with the desire to improve it, we creatively break the vicious circle, the Gordian knot. Because then we start learning from ourselves, not from the teachers, who of course gave us what they could. And we learn it with the help of best research tools, which give answers to the questions we need to ask to improve it. What is needed, is the critical mass of mathematics teachers and instructors, who via their classroom mathematics teaching-research can break through the boundaries of the vicious circle, re-educating themselves as well as students and thus opening the depth, the beauty and usefulness of algebra for the generations to come.