Algebra is often taught poorly, with little connection to other subjects.

People need quantitative reasoning, logic skills, and the ability to apply math skills to real problems.

Most people don’t need all the theory of math.

Almost all I’ve seen of how algebra is taught deserves everything the article says about it, and I would make it optional. Almost all I’ve seen on this blog would provide very good answers to Hacker’s desire to change how math is taught.

Your two questions were “How do we define Algebra?” and “How should we teach it?”. It seems to me that any discussion about the second is predicated on some agreement on the first, and I’m just pointing out that this isn’t trivial.

To be clear, I’m not arguing for getting students to arrive at the door of Algebra 1 with better basic skills. Rather, I’m arguing that they might arrive with a disposition toward thinking relationally about quantities and being flexible in representing relationships. Really understanding equivalence, as well as the relationships between addition and subtraction, and multiplication and division, would go a long way. My basic contention is that Algebra 1 need not be the beginning of algebraic thinking.

My experience in urban L.A. schools is that at least 2/3 of those students have gaping holes in their math foundations, and have been failing mathematics for years before they reach algebra (with the expected decline in self-confidence and a gradual generalizing of poor self-concept to other academic areas). Ask them to add 1/2 + 1/3 and you’ll get anything but 5/6. If they don’t know fractions, they can’t solve proportions, which are the beginning of one-variable algebra. In fact, anyone who has spent more than 5 minutes teaching high school math and has taken the time to investigate the skills of students who fail (looking at math proficiency scores, Math Diagnostic tests, etc) knows this.

These kids are not stupid, and in fact can fill in the holes pretty quickly now that they’re older, IF remediated. When shown how to solve some fractions-related problem, they go: “Really? But that’s so simple!” My experience is that most such students are dying to experience success in mathematics, and once they do so, are grateful and eager for more – and become much less of discipline problems.
Although I thought the film “Waiting for Superman” concerned itself with all the wrong topics, one of the things they did get right was when they pointed out that the charter schools that remediate students as soon as they get them are models of what can be done in this area.

If you put students with grade-level math skills in an algebra class, most students will succeed. And there are even strategies for boosting the performance numbers (SEE Brad Fulton). Algebra (and in general) mathematics is not the problem.

Rather, the question should be: how do you ensure that K-8 students do not fall behind in basic math foundational skills?

Even rephrasing the question as you did still leaves the responses headed for a tar pit that I don’t think will ever be resolved in the national dialogue.

Who said anything about the entire nation agreeing on a definition of Algebra? I’m proposing we don’t start the conversation with the wrong question.

Brian:

@ddmeyer — you have a NorCal neighbor who might disagree with you on the current and future role of some “need” for variable representation and manipulation. @worrydream

Victor wants students to be able to pose and answer questions in dynamic, visual environments … created by someone who understood variable representation and manipulation. If you understand Algebra, you can create those environments (a la Victor). If you don’t, you only get to pose and answer questions in the environments that people like Victor create.

The ability to reason is clearly important in all walks of life and in multiple activities (e.g., analyzing newspaper articles, understanding debates).

Some knowledge of Symbolic Logic can be really helpful in sorting out types of arguments (e.g., transitivity – if A > B and B > C then A > C, true/false arguments and so on), so that you can make better decisions.

Symbolic Logic builds on Boolean Algebra. Learning Algebra in high school was – at least for me — important preparation for learning Symbolic Logic in university, which has been useful preparation for debating, analysis of arguments, and logical reasoning.

The problem, as I see it, is that too FEW individuals learn Symbolic Logic, NOT that too MANY learn Algebra.

My schooling was outside the US, and I used to find it so confusing when people in the school system would refer exclusively to PreAlgebra, Algebra I and Algebra II. ‘Algebra’ in the common parlance is obviously the name of specific courses, not the name of a set of ideas. In order to draw the distinction, I find myself using ‘algebraic thinking’ more and more.

As math teachers, we are obviously failing in communicating our role when the educated public cannot describe the subject we teach half-accurately. People have a good idea of what they learn/learned in History, American History, World History, etc. Same with Physics, Newtonian Physics, Quantum Physics, Nuclear Physics, etc.

I hated history because it seemed to be a boring sequence of dates and conquests. I suppose many people hate algebra for the same reason – a boring sequence of operations and transformations.

What are the big ideas that tie the post-elementary math curriculum together?