Ojose: Common Misconceptions in Mathematics

This is a wonderful little book, not just for what it contains, but for what it represents. In it, Bobby Ojose catalogs 43 things that grade school students frequently misunderstand about basic arithmetic and high school algebra. Each section includes an example of the misconception, a brief explanation of what the student has misunderstood, a few hundred words outlining what teachers can do about it, and then another few hundred words summarizing the research literature related to that particular misconception. Here’s a much-edited excerpt:

Misconception 6: Comparing Decimals

Grades: 4 to 6

Question: Which is greater, 3.215 or 3.7?

Likely Student Answer: 3.215

Explanation of Misconception: …they would…likely state that 3.215 is greater than 3.7 because it has more digits. …the student…did not possess knowledge related to number density…

What Teachers Can Do: …emphasize the importance of lining up decimals points… …start with decimals that have the same number of decimal places… …progress to compare numbers like 7.4 and 7.04… The importance of adding zeroes to make the two numbers the same length [before comparison] should be emphasized…

Research Node: A study by Steinle (2004) identifier the “larger-is-larger” behavior… The study…suggested that teachers need to be aware that always rounding the result of a calculation to two decimal places can reinforce the belief that decimals form a discrete system…

This single insight isn’t going to change a student’s world, but putting a dozen of them together will have the same effect as giving a novice skier a dozen tips about their form, or a novice musician a dozen tips about their breathing. It’s a wonderful example of pedagogical content knowledge (PCK), and if the magic thumb drive I sometimes dream about could bring me just one book, I would ask for a catalog like this describing common misconceptions about programming and how to fix them.