At the AIMS Center, we work with children to learn how they do math, to learn what’s in their thinking process, and to verify the results of research. It may seem like the children are learning, or at least remembering what they are being taught. Standardized test scores, though, seem to paint a bleaker picture. What the test scores don’t tell you is why there is such a discrepancy.
In our contemporary classrooms, many students are learning math as teacher directed and curriculum driven. This process doesn’t allow for whatever skills the student might bring. Also, the curriculum pace is often very fast and practically precludes time for reflection or review. This is what I refer to as classroom math. When we visit classrooms, all too often we observe students responding to math prompts using what we call an algorithmic approach. They get an answer and explain their work in a way that shows they’ve memorized the math facts and procedures for solving certain types of problems. When pressed for deeper meaning to show the connections among the math facts learned, many students’ responses reveal a lack of true understanding.
My colleagues and I have recently worked with some second and third graders who are adept with algorithms and number facts. One girl responded quickly to the question of “what is 8 X 4?” with “eight 4’s is 32”. She was then asked to count to 32 by 4’s, but that part was difficult for her. She had trouble keeping track of how many 4’s she had used as she moved toward 32. Also, she didn’t seem clear on the connection between the two parts of the task, multiplying and counting. As with many children, she had gotten a message that it is not good to use your fingers. We know that fingers are a natural tool for counting, so we told her that fingers were allowed. She was then able to complete the task.
Giving a snap answer to 8 x 4 from a memorized number fact would make it seem like the child is “good at math.” Counting, then, seems slow and inefficient. But counting might be exactly the connection a child needs to make more sense of the number facts. The students who are “good at math” in the early grades are often those who can memorize well, whether they have understanding or not.
Classroom math can crowd out the natural development of math for many children. The math facts/algorithm way is fast and gives the appearance of learning. It is structured around grade level or age. The natural development which leads to real understanding is based on stages, which according to Dr. Les Steffe, are indicated by the types of number sequences in which children can operate.
Most teachers follow traditional educational models with school math. One our goals here is to make teachers aware of the mathematics of children. You have probably observed a child doing a math task in a way that seemed wrong, or didn’t follow adult logic. Would you be able to, instead of intervening with correct school math, ask the child to explain what his/her thinking?