Cognitive Science may Help Learn Pre-Algebra Better


(Photo: Adam Hayes for the New York Times)

Most schools and students have been used to studying math in blocked lessons, that is, the course is divided into several blocks (or topics), with each block explaining a specific math concept. Student study one topic at a time and do practice of the topic repeatedly, and then move on to the next topic. Blocked learning makes students feel that they have learned the topics well by making gradual improvement, but actually this traditional method of learning math does not work that well, since there are still many students seeing algebra, geometry, and other math subjects as their biggest challenge.

US researchers have done an experiment this year on a few of middle school students in pre-algebra classes to test the practicability of a new way of learning Pre-Algebra. The experiment, referred to as the Tampa Experiment, is sponsored by the Institute of Education Sciences in the Department of Education, and involves scholars and psychologists at top colleges like the University of California, Los Angeles. Researchers picked up 8 pre-algebra classes with a total of 140 7th Graders at the Liberty Middle School, and divided the students into two groups.

In an ordinary pre-algebra class, students will learn linear equations, word problems involving proportions, graphs and slopes block by block. However, each group of students in the experiment are asked to study part of the four problem types in a mixed manner: One group got mixed assignments of linear equations and word problems, and regular blocked assignments of the other two types; the other got mixed assignments of graphs and slopes while getting blocked assignments of the first two types. By adding nothing new to learn but simply rearranging the existing study materials, the experiment had students study mixed sets of related things. Such a technique is called "interleaving".

At first, most students complained that the assignments were more difficult than usual and they had to spend more time on the problems. Teachers explained that it was because they had to constantly review what they had learned long (say, one month) ago in order to solve the problems. In nine weeks, each student in the experiment completed 10 assignments with 12 problems each. After another two weeks passed, these students sat a surprise comprehensive exam. Results revealed that students' average score on interleaved material was 72, while that on blocked material was only 38. A student told the New York Times that he found himself spending less time solving the problems in the exam since all the knowledge were "fresh" in his brain. The different is significant, but not surprising, because studies have shown that simultaneously studying distinct but related issues greatly enhances one's ability to identify them quickly and accurately compared with regular learning in blocks.

Although the Tampa Experiment showed that students learned better in the interleaved manner, researchers said it still could not be 100% affirmed that interleaved learning would improve students' Pre-Algebra grades because the sample in the experiment was too small. In the next step, they have planned to launch a larger study which would be including more than a dozen schools with over 80 classes and 1,600 students in total.

The technique of interleaving is actually about cognitive psychology, which studies mental dynamics behind thinking, remembering and problem solving. Studying a mixed set of items (referred to as "cock-tail shaker" by some teachers) actually pushes students to identify the type of a problem before they start to solve it, while in traditional blocked learning, the formula that should be applied to solve the problems is always listed above the exercise for students to follow. Researchers allege that interleaving reinforces the brain's association between specific types of problems and corresponding solution strategies.

The Tampa Experiment is still provoking enough for both teachers and students. It may be more effective if teachers assign students to do pre-algebra practice in an interleaving form rather than in blocks. The final aim is to have students frequently review the knowledge that they have learned a long time ago in the practice so as not to forget it.

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