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## Wednesday, November 7, 2007

### Algebra

For many, algebra was a course on x’s and y’s either in 8th grade or high school. We carried around a fat book and we knew we hit the “big time” in math education when using sophisticated terms like “distributive property” and “quadratic formula”. What we likely never knew was that we had been doing algebra since pre-school and actually enjoying it!
In fact, developing algebraic thinking is a top priority in modern elementary and middle school curriculums. Algebra is now second in importance after arithmetic. The National Council of Teachers of Mathematics (NCTM) states that algebra is a cornerstone of mathematics.

In order to “see” the algebra being learned at The San Francisco School, you need to understand that algebra has various definitions reflecting diverse human perspectives over many centuries. For example, one can define algebra as the “language of mathematics”. This includes variables, number properties and manipulating symbolic expressions and equations (see photo). This algebra is being taught explicitly in the 8th grade at the San Francisco School and is commonly referred to as Algebra I. It is being implicitly taught in the 6th and 7th grade curriculum. For example, my 7th grade has recently been working on plotting numbers on a Coordinate Plane (x and y axes) in a unit on negative numbers. Melissa’s 6th graders have been working on the concept of equivalent fractions, which in turn leads to ratios and proportions, components of the slope in a linear equation.

Another definition of algebra is generalized arithmetic. Arithmetic is at the core of elementary math both here al and most other schools. But arithmetic through an algebraic lens means not only learning how to do something (procedure) but also why we do it (concepts.) For example, in Jana’s 4th grade class, students are currently studying multiplication through arrays. Arrays are grid rectangles where length is one number of squares and width is the other. The area of the rectangle is the product of those two numbers. Students use arrays to identify patterns on the multiplication chart as well as define prime and composite numbers by their factors. This model not only explains multiplication, square numbers, factors, primes and composite, but also translates directly into quadratic equation models in the 8th grade.

Algebra is also a tool to study patterns. Seeking, expressing, and generalizing patterns and rules in real world contexts are evident throughout the SFS classrooms. I recently received a packet of work from my son’s experience in Pamela’s 1st grade class. I was impressed with the breadth of activities he had been doing in mathematics. One sheet, for example, included many rows of five teddy bears he had colored. He was obviously learning about all the combinations of sums that make five, but as he explained it:
“This is a pattern, Daddy. First, I colored all the bears red…that means 5+0. Then I colored all the bears yellow for 0+5. Then I colored one bear red and 4 bears yellow for 1+4… It’s a pattern, Daddy.”
The more formal name for what he did is “combinatorics”: the logical analysis of all possible combinations based on identifying patterns. This is precisely the type of activity first graders need to be doing to internalize addition facts from a “hands-on/minds-on” basis while at the same time exploring the patterns they find as they color. Pamela’s teddy bear coloring sheet is deceptive: you think it is about addition when it really sows the seeds that later blossom in algebra.

Algebraic thinking is sprouting up throughout our school. It is not limited to the 8th grade class. In a progressive school, where the meaning of what we do is as important as the activity itself, it is helpful to remember that math class can be as progressive and challenging as any other subject.