Making Algebra “Connect” at the SF School

“Though this be madness, yet there is method in it.”
(William Shakespeare)

Algebra- the word often evokes a strong reaction from people. What do you remember about algebra? What were the "big mathematical ideas" of algebra? What were you really studying?

In case you cannot answer these questions convincingly, try thinking of algebra as the study of patterns and situations in which change occurs. In algebra, we use what we already know to find what we do not yet know. We do this by using patterns and relationships that already exist between numbers. It is a way to see the patterns that are a part of everyday life. These patterns, changes, and relationships can be analyzed and represented in a variety of ways including the use of words, tables, graphs, and symbols.

Traditionally, the goal of algebra instruction has been teaching procedures for manipulating symbols. These procedures are often meaningless to students who try to survive by memorizing and, thus, only retain the ideas for a short time. There is almost no evidence that students develop algebraic and symbolic reasoning from instruction emphasizing teaching procedures for manipulating symbolic expressions. Development of algebraic ideas can and should take place over a long period of time, prior to attempts to deal solely with abstract symbols.

While it is true that SF School 8th graders study what are considered standard algebra topics, they do so within a problem based curriculum. In addition, algebraic concepts such as the use of coordinate graphs, variables, integers and equations are taught in 6th and 7th grades. The ultimate goal is to graduate students with a strong conceptual and procedural understanding of algebra.

At the SF School we are working on the math scope and sequence to support conceptual understanding through concrete and real world examples. This summer the 4th, 5th, 6th and 7th/8th grade teachers will be meeting for three days to examine the math scope and sequence with a particular eye towards solidifying the continuum of concepts and skills that leads towards a successful 8th grade algebra experience. One specific example is how 4th graders work with the concept of fractions by using fraction circles that physically represent the concept of equivalence that later shows up in arithmetic with fractions in 5th and 6th and solving proportions and balancing equations in algebra. SFS students work on Problems of the Week, which are non-routine and often challenging real world applications of math. Evidence of the students’ work blankets the Middle School Math Room’s wall and windows. They do research into the history of math, such as doing Internet research on the life and times of Pythagoras. They have also investigated the mathematical foundations of their cultural surroundings in a project called SCAMP: Story of a Cultural Artifact from a Mathematical Perspective. Student and their families have commented that they had never really perceived all the math that surrounds their everyday world before doing these projects. The combination of mathematical procedural and conceptual knowledge with the historical context allows students to view math not so much as an invention, but rather, as a discovery of pre-existing numerical relationships. We strive to present math as a vibrant area of growth throughout human history and continuing on to the present and beyond.

One advantage of learning algebra in an independent school is that we are not held to unrealistic testing regimens and instead, we can focus on deep understanding as well as procedural fluency. We can base our model on a student-centered approach that moves from concrete materials gradually to algebraic generalizations along these approximate stages:

Build It ... Extend It ... Picture It ... Table It ... Predict It…

… Graph It ... Generalize It ... Formulate a Rule for It.