Understanding Math

Written by Alysia Krafel, co-founder 

A major focus of our math program is to help our students reach a deeper level of understanding. Joy and delight in mathematics come from discovery of patterns and the wonderful “ah ha” moments that students get when they see the solution. There is a difference between knowing how to do an arithmetic procedure like this:

2.51 ÷ .03 =

and knowing how and why the trick we all memorized works. At Chrysalis, we work hard to make sure that most students can not only do the problem above, but can explain how and why the procedure works.

For example, imagine we were teaching cooks and we wanted to teach them how to bake a cake. What kind of bakers would we want in the end? If we give them a boxed cake mix and then tell them to follow the directions, they will produce a fine cake quickly with very little mess. They will know there are oil, eggs and water in the cake. But will they know what is in the powder? Could they make a cake from scratch? If the recipe called for three eggs and they only had two, they would probably go to the store or call a neighbor and get more. We call these the Duncan Hines bakers.

If the student followed a cookbook, like Joy of Cooking, they would know that in addition to eggs, oil and water there is flour, sugar, salt, baking soda and baking powder, some flavorings and spices in the cake. They would know the relative amounts of each. They could possibly make a cake from scratch if the cookbook were lost.  If the recipe called for three eggs and they only had two, they would probably adjust the other ingredients and make a smaller cake. They would take longer and make a bigger mess but they would know more in the end. They could possibly create a recipe for a cake. We call these the Joy of Cooking bakers.

If the students were not given a cookbook, but instead were encouraged by their teacher to take what they already knew about making cookies, bread and cupcakes and apply that to creating a cake recipe, the students would learn even more about cooking. Although they would make a considerable mess and might take a few weeks to produce a delicious cake, the students would have a deeper understanding of baking. They would not only know what was in the cake, and the relative amounts of ingredients, but they would know what the eggs were for, what would happen if you left them out and why you use baking powder instead of yeast. If the recipe called for three eggs and they only had two, they would probably increase the oil or add applesauce and make the cake anyway. They would have the knowledge to create totally new recipes. These bakers would understand cakes and create unique solutions to baking problems. We call these the Julia Child's bakers.

Now think about creating mathematicians instead of bakers. Which kind of mathematician do we want our students to grow into?

A Duncan Hines math program is filled with pages that show the student how to do a procedure and then has the student memorize the process with lots of repetitions and drill. A Duncan Hines Mathematician learns 24 x 8 very quickly. But later, in algebra this student will not recognize that A(B + C) is exactly the same 24 x 8. When they encounter a problem, they will simply try to memorize it as a new procedure. That works until you get to word problems or science classes. Many students in this category fail to understand algebra and struggle with the concept of chemical reactions.

The Joy of Cooking and Julia Child's math curricula are often very rare in schools. The pressure of the standards and the tests do not allow time for much exploration in math. The kind of math program which allows exploration and creation of procedures from patterns are called constructivist curriculums. Chrysalis has a constructivist curriculum at all grade levels. It takes more time. It does produce higher quality mathematics students in the end. These student mathematicians will see the connection between 24 x 8 and A(B + C) and will be able to multiply in algebra as easily as they did in fourth grade because the distributive property of multiplication has been deeply developed.

Giftedness in mathematics shows up early and is visible in elementary school students. The Julia Child's mathematicians are developed by teachers who listen while students construct understanding and recognize brilliant thinking. Mathematical conversations occur and more difficult problems are offered. Students are challenged to figure out how to solve new problems using what they already know. These students will not only see the connections between the 24 x 8 and A(B + C) but also extend the pattern to include  (A + B + C) ÷ AB and understand the connections between all three.

While most math curriculums focus on memorization of procedures and churn out Duncan Hines mathematicians, Chrysalis teachers focus on developing Joy of Cooking and Julia Child's mathematicians in all students. Students are allowed to move at their own pace and are given time to construct understanding. As a result, they often become gifted at math because they understand it well enough that they can extend patterns to create their own procedures. We avoid teaching memorized procedures to students, in order to allow the mathematician inside to grow. This is accomplished by a curriculum which uses blocks, cubes and fraction pieces of many sizes and shapes and directs students to construct models of arithmetic problems and then look for patterns. We use objects, discussions, and encourage students to search for multiple solutions at all levels of the math problem.

The curriculum we use for kindergarten through grade 2 is Bridges in Mathematics from the Math Learning Center. At the late elementary level we use Patterns In Arithmetic, developed at the University of California at Irvine, by one of Chrysalis founders, Alysia Krafel, and other teachers, under the direction of Prof. Michael Butler. In the middle school we use College Preparatory Mathematics.