Deborah Schifter is a principal research scientist at the Education Development Center in Newton, Massachusetts. Schifter says that while students are capable of powerful mathematical reasoning, traditional math education focuses on the memorization of facts and computational procedures which do very little to broaden and deepen students’ number and operations sense. As a result, in the process of schooling, students tend to lose contact with their own mathematical ideas and, unable to keep hold of the multitude of procedures needing to be remembered, come to rely on faulty "rules of thumb". Schifter argues for the need to transform mathematics pedagogy to create classrooms in which making sense of mathematics is both the means and the goal of instruction. In Reasoning About Operations: Early Algebraic Thinking in Grades K-6 (1998), Schifter stresses the importance of teaching mathematics contextually (versus skill in isolation) to better develop and deepen students’ ability to effectively problem solve.
So what does teaching math in context mean to us in the classroom and for our students? It means that instead of putting numbers on the board or under a document camera and asking students to perform a calculation, we are going to put those numbers into a story or word problem that students can relate to and understand. And not just the first, or second time, but every time you want your students to perform a calculation, it should be in context. The story problems should have meaning to the students and be relevant to their lives, not necessarily to yours or their parents’ lives. This is where, “...making sense of mathematics is both the means and the goal of instruction.” (Schifter, 1998). Let’s look at a couple of examples.
The traditional division algorithm is very difficult for many students because computation is usually isolated from any context and when the divisor is not an obvious (basic) factor of the dividend many students don’t know where to begin. Consider the following:
Straight division computation example: 375 ÷ 15 = ? (out of context and meaningless)
In an adult context: Your family car can go 375 miles on a tank of gas. If the tank holds 15 gallons, how many miles to the gallon does your car get? This story problem holds little to no meaning for children.
In a relevant student context: The field day t-shirts were delivered to the school in 15 boxes. If 375 t-shirts were ordered, how many t-shirts are in each box?
Do you see the difference? Students wouldn’t necessarily have to know the traditional process and steps for division to solve a problem that was put in a relevant context. Students could create a model to help them the solve problem and then work towards understanding the steps to the more efficient algorithm. This blog topic is not about teaching division and the do’s and don’ts of this very abstract operation, that will be a whole other blog topic on its own! These examples are to highlight how important it is to teach any and all math operations in a context relevant to children.
The idea of teaching all operations in relevant context is supported by all respectable math researchers. John Van de Walle says, “...contextual problems and models (counters, drawings, number lines), are the main teaching tools that you have to help students construct a rich understanding of (math) operations.” Doesn’t it make more sense to teach math this way? It has been my experience that students have a much better understanding and display more confidence when we attach some meaning to the numbers. This holds true for kindergarten students as well as fifth grade students. This is really an easy change that doesn’t require a multitude of resources.
Lesson idea: give students several pairs or sets of numbers and have them write the context for the numbers but not solve the problems. You can set whatever parameters you want to accommodate student needs and further your curricular goals. For example you might give the numbers 25, 67, and 11 and tell students they need to create a multi-step story problem. Students will then exchange their work and solve each other’s problems. If/when students have any difficulty with the problems they are solving, they go to the creator of the problem for help. A student might create something like the following with 25, 67, and 11:
Jody, Keiffer and Amarion were best friends and liked to share baseball cards. Together, the three boys had 67 cards. If Jody has 25 cards, and Keiffer has 11 cards, how many cards does Amarion have?
You might start with this type of closed-ended problems (only one correct answer) and then go on to teach students how to use those numbers to create open-ended questions (problem has more than one correct answer). This will probably require you to model (possibly several times) the thought process on this, but this whole idea is very doable for students. There are multiple benefits to having students write their own problems and none more important than writing the action that creates the need for a particular math operation. I have even, with my students, created a list of “key” words (those words that some teachers mistakenly teach as indicators for operations) and then tell them they may not use any key words in their story problems. This increases vocabulary while promoting critical thinking! It’s a win, win!