It’s pretty clear from listening to Philip Gonsalves that if he had his way, we’d throw out most of what we do in math classes and start over.
“We fundamentally have to change the way we teach math,” he told a group of West Contra Costa Unified School District elementary school teachers at a January training session.
Gonsalves is coordinator of Alameda County Collaborative for Learning and Instruction in Mathematics (ACCLAIM), a joint effort of the Alameda County Office of Education, California State University, East Bay and Chabot College. Gonsalves and fellow coordinator Drew Kravin are working with West Contra Costa Unified School District elementary and middle school teachers this school year improve the way we teach and assess students in math.
Gonsalves is a former middle school teacher, worked at Lawrence Hall of Science where he wrote many GEMS (Great Explorations in Math and Science) teacher’s manuals, and has taught at Cal State East Bay for the past 25 years.
Many of the things we say to kids during math lessons only confuse matters and often aren’t true, he said. For example, when we multiply 87 x 465, in one step we say “multiply 8 x 4” when we really mean 80 x 400. Or when we teach students to multiply decimals and start out by telling them to ignore the decimal.
“Who in their right mind tells kids to ignore the decimal point?” Later in the same problem we might tell them to count the decimal places in the two factors to determine how many decimal places to put in the product (answer), he said, but we don’t tell them why this works.
These sorts of instructions don’t help students develop number sense – a solid understanding of the value of numbers. Without good number sense, students can come up with wildly wrong answers and not realize it.
With methods such as the partial-products method and generic rectangle (see photos), students use the actual value of the digits. He also encourages visual representations – for example, building a problem with special blocks that represent different place values or drawing a picture similar to what the blocks would look like. (See photo.)
One of his chief recommendations is that we teach students several methods for attacking the same problem, and give them fewer problems to solve but expect them to show more than one way to solve each one.
“Show multiple methods side by side and let the kids pick the way that works best in a particular situation.”
Showing that the same problem can be done in many ways, he said, “frees them up to play with mathematics. The more you play with it the more you get it.”
The more commonly used approaches rely too much on memorization, rather than understanding, Gonsalves said, which is why students so often forget what they were taught earlier.
Multiple methods can extend even to basic math facts. He recommends having children make their own flash cards of the facts they are having trouble remembering. On one side they write the problem more than one way – for example writing a multiplication problem horizontally and vertically. On the back, they draw more than one picture to represent the same problem. (See photo.)
Gonsalves encourages elementary teachers to be aware of what students will be asked to do when they are older, so that what they teach prepares them for the next steps. Middle and high school teachers need to know what students have been taught in elementary school, so they can refer back to those activities and talk about how they connect to the current work.
Many of the methods Gonsalves and Kravin teach aren’t commonly included in textbooks or other teaching materials, which often only show a single method for solving a particular type of problem. Fortunately, the trainers have posted extensive materials on their program’s website. The catch is teachers (and parents) will need to practice the methods themselves before teaching them.
“You shouldn’t be assigning any problem you haven’t done,” he said.