Correcting Misconceptions

by Rivette Pena M.A

Just the word “Math” strikes the fear in the hearts of many. There are an endless amount of misconceptions. It is on the teacher, to foresee these misconceptions and figure out how to address them.

When addressing misconceptions or correcting student mistakes, math professor Dr. Cathy Coker uses an old video clip by Abbot and Costello.

In this clip, Costello tries to convince his landlord that 7 x 13 is 28. He shows him using division, multiplication and then finally addition. All of which are incorrect however, his landlord can’t figure out how he keeps getting the right answer.

Dr. Coker then poses the question to her students, “How would you correct these misconceptions?”

So my question to you is the same, “How would you handle these misconceptions?” A person has a preconceived notion on how to solve problems and you are responsible for tearing down the old ways and introducing the new.

It’s not simply relaying new information. You have to disprove their belief first. Because they don’t believe they are wrong.

Samuel King, a student in Dr. Coker’s college course, reminds us that students can hold on tight to misconceptions.

This Abbott and Costello video reaffirms how confidently someone may feel about how to tackle a math problem, even if they are way off base. While in the extreme, this video demonstrated three altogether different, and definitely incorrect, methods of tackling this problem. Setting aside the multiple arithmetic errors, there are other ways to model out this problem for students.

Most of Dr Coker’s students pointed out that the most important misconceptions were in method and place value. Shaheen Qureshi acknowledged that certain skills must be present in order to be successfull when introducing your students to mulitplying and dividing.

Particularly in this video’s example students [should] have a clear understanding of addition to multiplication.

If a child can’t add then multiplying won’t necessarily be understood fully. I had a student that I was tutoring in math. She had difficulty with her multiplication facts. As I spent more time working with this student, I realized she could not add well enough to skip count. So I realized then that I had to step back and teach her how to skip count. In order for her to excel at multiplication she needed a str4ong addition foundation.

If a teacher is aware of misconceptions, they should be addressed before the lesson. For instance, in elementary school, students may believe that if you multiply two numbers

that the product will be larger than the two factors. However, in middle school this can be a misconception because if one or both of the factors is a fraction then the product will be smaller. This is a misconception we would want to be mindful of when planning our lessons. Our lessons shouls address this misconception in some way in order for the student to move past it and have a deeper understanding of the skill or concept.

If the lesson, has been conducted and misconceptions are now mistakes that need to be clarified. How do you address them in this situation? Stacy Barnwell has this to say :

The numbers in this problem are small enough to use manipulatives to demonstrate that Costello’s answers and methods are incorrect. Students could multiply the numbers on a vertically-lined paper trying Costello’s method, where he lines up each line on the right, and trying the correct method, where each successive line shifts one place left. They could also use manipulatives (paper clips, pieces of paper, etc.) to break 28 into 7 pieces or to multiply 7 piles of 13. Comparing the realworld answer with their various experiments will demonstrate that they must understand and pay attention to place value.

Annette Gomez explains some strategies she would use in this scenario.

In the video, what I agree with is the format the problem is written in, except for the 1st example. All of the misunderstanding comes from the place value and how to correctly do the solve the problems. The way I would solve this is to write out in the follow way (using our place values): 20 13 x 7 91

The strategy here is multiplying the 7 by 3, which is 21. The 1 is written under the ones place and we carry the 2 over to the tens place (on top of the number 13). Then multiply 7 by 1, which is 7, then add the 2 we carried over, which makes is 9. The answer is 91. However, I understand now that there are other newer methods to help solve this such as decomposing, which would break down the number 13, for example, to 10 by 7 and 3 by 7, then add the two numbers 70 and 21 for our answer. I would also use the array strategy, which is using columns or rows to visualize our answer, such as, 7 columns of 13 or 13 columns of 7, etc. When we use these two strategies, we can prove the methods used in the video are wrong. I would show and use decomposing and building arrays to my classroom to help avoid the mistakes made in the video. In my opinion, the decomposing strategy helps to solve faster, and the array is a good visualization method.

Math is something that many kids have trouble with. Just the mention of it makes hearts race and anxiety rise. We need to make this as easy a journey as possible. Empower our youth to excel at math.