Mathematics for Teaching

I’m doing some  literature review for my research and I came across this article by L.A Steen in Middle Matters. He was arguing about the Algebra for All standard in the US and part of the article includes description of what is algebra. I thought I should share them in this blog because it is something very important teachers should be aware of when they teach algebra or what they conceive what algebra is and for. Oftentimes, when students ask what algebra is and what they are going to need it for, teachers lazy answer is “Algebra is just like your math in the grades only that this time you work with letters instead of numbers!”

1. Algebra is the language of mathematics, which itself is the language of the information age. The language of algebra is the Rosetta Stone of nature and the passport to advanced mathematics (Usiskin, 1995).
2. It is the logical structure of algebra, not the solutions of its equations, that made algebra a central component of classical education.
3. As a language, algebra is better learned earlier and harder, when learned later.
4. In the Middle Ages, algebra meant calculating by rules (algorithms). During the Renaissance, it came to mean calculation with signs and symbols–using x‘s and y‘s instead of numbers. (Even today, lay persons tend to judge algebra books by the symbols they contain: they believe that more symbols mean more algebra, more words, less.) I think that many algebra classes still promote this view.
5. In subsequent centuries, algebra came to be primarily about solving equations and determining unknowns. School algebra still focuses on these three aspects: employing letters, following procedures, and solving equations. This is still very true. You can tell by the test items and exercises used in classes.
6. In the twentieth century algebra moved rapidly and powerfully beyond its historical roots. First it became what we might call the science of arithmetic–the abstract study of the operations of arithmetic (addition, subtraction, multiplication, etc.). As the power of this “abstract algebra” became evident in such diverse fields as economics and quantum mechanics, algebra evolved into the study of all operations, not just the four found in arithmetic.
7. Algebra is said to be the great gatekeeper because knowledge and understanding of which can let people into rewarding careers.
8. Algebra is the new civil right (Robert Moses). It means access. It means success. It unlocks doors to productive careers and gives everyone access to big ideas.

And I like the education battle cry Algebra for All. Of course not everyone is very happy about this. Steen for example wrote in 1999:

No doubt about it: algebra for all is a wise educational goal. The challenge for educators is to find means of achieving this goal that are equally wise. Algebra for all in eighth grade is clearly not one of them–at least not at this time, in this nation, under these circumstances. The impediments are virtually insurmountable:

1. Relatively few students finish seventh grade prepared to study algebra. At this age students’ readiness for algebra–their maturity, motivation, and preparation–is as varied as their height, weight, and sexual maturity. Premature immersion in the abstraction of algebra is a leading source of math anxiety among adults.
2. Even fewer eighth grade teachers are prepared to teach algebra. Most eighth grade teachers, having migrated upwards from an elementary license, are barely qualified to teach the mix of advanced arithmetic and pre-algebra topics found in traditional eighth grade mathematics. Practically nothing is worse for students’ mathematical growth than instruction by a teacher who is uncomfortable with algebra and insecure about mathematics.
3. Few algebra courses or textbooks offer sufficient immersion in the kind of concrete, authentic problems that many students require as a bridge from numbers to variables and from arithmetic to algebra. Indeed, despite revolutionary changes in technology and in the practice of mathematics, most algebra courses are still filled with mindless exercises in symbol manipulation that require extraordinary motivation to master.
4. Most teachers don’t believe that all students can learn algebra in eighth grade. Many studies show that teachers’ beliefs about children and about mathematics significantly influence student learning. Algebra in eighth grade cannot succeed unless teachers believe that all their students can learn it. (all italics, mine)

I shared these here because in my part of the globe  the state of algebra education is very much like what Steen described. You may also want to read about the expressions and equations that makes algebra a little more complicated to students.

L.A Steen is the editor of the book On the Shoulder of Giants, New Approach to Numeracy, a must read for teachers and curriculum developers. The book is published by Mathematical Sciences Education Board and National Research Council.