Posted March 26, 2017
CORE’s Excellence in Education Blog post this month is on the importance of clearing out math misconceptions in order to make room for new knowledge. See what happened when students were certain about the wrong right answer.
(By Dean Ballard, CORE’s Director of Mathematics)
How many times have you tried to explain something to someone and just can’t break through because he or she seems to be holding on viciously to some misinformation or misconception? Oddly enough this puts me in mind of a recent experience at a school. The principal was late to our meeting and came to the meeting agitated because he could not park in his reserved spot. He had agreed to let a teacher at the school use the spot two days earlier while he was away all day at district meetings. Unfortunately, the teacher’s car broke down and could not be moved from the parking spot. Two days later the teacher still had not arranged a tow truck to remove the car. So often I find with students that a wrong idea is a lot like this broken down car. It occupies a space reserved for the right knowledge and unfortunately the right knowledge cannot be parked in the spot until the wrong knowledge is towed away.
I remember reading a case study about a teacher’s experience re-teaching fraction multiplication to her class. The case study is in a great old book called, Fractions, Decimals, Ratios, and Percents, Hard to Teach and Hard to Learn? [i]. One student in this class thought that the way to multiply mixed numbers such as 6 ^{3}⁄_{4} x 5 ^{1}⁄_{3} was to multiply the whole numbers (6 x 5), then multiply the fractions (^{3}⁄_{4} x ^{1}⁄_{3}), and then add the two together for the final answer. This procedure made sense to him because it followed a procedure similar to adding fractions. Using this student’s method, 6 ^{3}⁄_{4} x 5 ^{1}⁄_{3} = (6 x 5) + (^{3}⁄_{4} x ^{1}⁄_{3}) = 30 + ^{3}⁄_{12} = 30 ^{1}⁄_{4} rather than the correct answer of 36. The teacher went over again the right way to multiply mixed numbers and explained why the distributive property applies in this situation. The student understood the procedure and could repeat the correct procedure. However, in the end he said as he walked away, “I still don’t see why my way doesn’t work.”
Earlier this year when teaching a class about how to find the percent of a number (see my previous blog), the biggest hurdle I encountered was that students thought they already knew the procedure for finding the percent of a number. They “knew” to multiply the percent times the number. So in their minds 200% of 40 = 200 x 40 = 8,000. I realized they must have already been taught a correct procedure such as, “multiply the percent times the number and move the decimal to the left two places in the final answer.” However, they hung onto the part of the procedure that made sense to them (multiply the two numbers together) and forgot the part that did not makes sense to them (move the decimal point two places to the left in the final answer). I saw two needs, to dislodge the misconception that just multiplying the two numbers would give you the answer and to reteach the right method but with some sense making and understanding about why the decimal point is moved over in the final answer. Without understanding it was unlikely the correct method would be remembered accurately, especially in light of their current misconceptions about the procedure. Without dislodging their misconception, it was unlikely the students would accept the “new” idea. So the lesson about finding the percent of a number changed. I first focused on towing away the wrong knowledge in order to make room for new correct knowledge. In this case I used known percent facts that the students were very confident with and that made sense to them, such as 50% of 40 = 20 and 100% of 40 = 40. With these they could see that the procedure they were using did not work, and they knew something was wrong. From this point they were ready to replace the misconception with new knowledge.
Sometimes the only way to deal with misconceptions is to first debunk them. Otherwise students can hold onto them no matter what we give them in terms of new correct knowledge. This experience reminded me of the importance of checking prior knowledge for my lessons in order to dig out any well rooted misconceptions that occupy needed brain space for new knowledge.
[i] Barnett, Goldenstein, and Jackson (1994) Fractions, Decimals, Ratios, & Percents Hard to Teach and Hard to Learn? Heinemann, Portsmouth, NH
To receive future editions of this blog via e-mail, add your name to the mailing list.
I wonder if this approach works for all misconceptions. First, we have to understand it, then debunk it, and then we can begin to build.
Nice article!
Using this student’s method, 6 3⁄4 x 5 1⁄3 = (6 x 5) + (3⁄4 x 1⁄3) = 30 + 3⁄8 = 30 3⁄8 rather than the correct answer of 28. Sorry but 28 is not the answer either. 6 3/4 = 27/4 and 5 1/3 = 16/3. 27/4 x 16/3 = 432/12 = 36.
For the student’s method, (6 X 5) + (3/4 X 1/3) would be 30 + 1/4, not 30 + 3/8. Unless the student had multiple misconceptions….
Also, I believe the correct answer is 36, not 28.
But I really love the idea of needing to clear misconceptions instead of trying to stuff the “correct” methods into that occupied space. It makes sense why students never “get it” …we aren’t helping them understand WHY what they thought was incorrect.
Just subscribed to the blog. Keep the great ideas coming!
I applaud those who noticed significant errors in the computation within this article. Check your work before publishing! The answer you gave for the student is not correct either. 3/4 x 1/3 is NOT 3/8, it is 3/12 which reduces to 1/4. My goodness! Talk about misconceptions!
Thanks to those who took the time to comment on the blog and also for catching and bringing to our attention the math errors. It is truly ironic that in a blog on math errors I made two very obvious errors. Still can’t figure out how that happened. My apologies. We’ve fixed the blog. I also want to clarify, that the book by Barnett, Goldenstein, and Jackson from which I pulled this fraction incident did NOT have those errors in it.