Lots of our incoming community-college student literally don't now what the equals relation means. It seems pretty amazing, but there it is. Over and over again every day, no matter how many times it was explained. "True or false? 0 = 100" "True" "No, it's false" "I don't get it". Last semester I had at least one or two students who were so enormously challenged that we could repeat this every day all semester and they'd never get it right. But it seems like fully half of my students answer the second one incorrectly. So far, everyone, confidently answers "true" to the first. But the new discovery for me was how extremely simple a question it takes to make this visible.Īll I've done is start asking, "True or false? (a) 6 = 6, (b) 3 = 5". Ginsburg, "The effects of instruction on children's understanding of the 'equals' sign." The Elementary School Journal 84.2 (1983): 199-212. Now, this isn't a tremendously novel observation, e.g., see: Baroody, Arthur J., and Herbert P. A surprising proportion of my students are very confused about what the equality (=) sign means. I discovered something last semester that made me insert a new little thing in the first day of my basic-level (remedial, liberal arts) community college math courses. Get the whole article by Willingham (including citations for all the claims above) at the AFT website. Evidence on this point is positive but limited, perhaps because automatizing factual knowledge poses a more persistent problem than difficulties related to learning mathematics procedures. One would expect that interventions to improve automatic recall of math facts would also improve proficiency in more complex mathematics. Fourth, when children have difficulty learning arithmetic, it is often due, in part, to difficulty in learning or retrieving basic math facts. Third, knowledge of math facts is associated with better performance on more complex math tasks. Second, students who do not have math facts committed to memory must instead calculate the answers, and calculation is more subject to error than memory retrieval. With enough practice, however, the answers can be pulled from memory (rather than calculated), thereby incurring virtually no cost to working memory. First, it is clear that before they are learned to automaticity, calculating simple arithmetic facts does indeed require working memory. This interpretation of the importance of memorizing math facts is supported by several sources of evidence. The less working memory a student must devote to the subtraction subproblems, the more likely that student is to solve the long division problem. Students who automatically retrieve the answers to the simple subtraction problems keep their working memory (i.e., the mental “space” in which thought occurs) free to focus on the bigger long division problem. For example, long division problems have simpler subtraction problems embedded in them. This automatic retrieval of basic math facts is critical to solving complex problems because complex problems have simpler problems embedded in them. Moreover, retrieval must be automatic (i.e., rapid and virtually attention free). The answers must be well learned so that when a simple arithmetic problem is encountered (e.g., 2 + 2), the answer is not calculated but simply retrieved from memory. Let’s take a close look at each.įactual knowledge refers to having ready in memory the answers to a relatively small set of problems of addition, subtraction, multiplication, and division. In its recent report, the National Mathematics Advisory Panel argued that learning mathematics requires three types of knowledge: factual, procedural, and conceptual. Some solid thoughts from Daniel Willingham on the need for automaticity in basic mathematics skills (re: ) from his article "Is It True That Some People Just Can't Do Math?" ( American Educator, Winter 2009-2010):
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