A Look Into The Student Mind: Teaching Concepts, Not Formulas (Part 3)

By Guest Author | January 16th, 2018 | No Comments

This article is Part 3 in a five-part series. Find the previous articles here: Part 1. Part 2.

Interested in discussing these findings with other teachers? Follow the Edmodo Topic for this series!

In this series, we’re studying student responses to questions from the American Mathematics Competition (AMC) attempted by hundreds of thousands of students on Edmodo. We saw in the first article that a large data sample like this provides unique opportunities to gain insights into how students think. In the second article, we delved deeper into questions on ratios and proportions and saw how students often use the wrong logic because it worked for them in some other similar-looking question. Being aware of these pitfalls allows teachers to alert students to them and gradually help them overcome them.

This time, we’ll look at a few questions related to area and perimeter and how students answer them.

In our experience, student misconceptions arise when there is an underlying concept that leads to common use of procedures or formulas, but the curriculum emphasizes the formulas more than the concept. Secondly, the concepts rarely have real-world applications to the students themselves..

A fantastic example of these misconceptions is when students study area and perimeter. These concepts are fundamental ideas related to space and measurement, but it appears that most curriculum’s emphasis is more on the formula to calculate them than the idea itself. Students learn formulas for the areas of triangles, various types of quadrilaterals, other polygons, circles, semi-circles and so on. They then move on to surface area and volume — more formulas! Is it surprising that when asked ‘What is area?’ more children answer ‘length times width’ than ‘the space covered by a closed flat shape?’ Is it surprising that many students say that irregular shapes (like a leaf, for example) do not have an area?

The Grade 8 AMC test had this question:

Karl’s rectangular vegetable garden is 20 feet by 45 feet, and Makenna’s is 25 feet by 40 feet. Whose garden is larger in area?

A. Karl’s garden is larger by 100 square feet. (14%)
B. Karl’s garden is larger by 25 square feet. (19%)
C. The gardens are the same size. (25%)
D. Makenna’s garden is larger by 25 square feet. (20%)
E. Makenna’s garden is larger by 100 square feet. (22%)

Attempted: 48046 | Skipped: 16

22% of the students got this question correct. But one of the wrong answers, that the gardens are of equal area ©, is chosen by even more students! This is extremely unusual. The most commonly selected answer is usually the correct one, and when it isn’t, it justifies going deeper into the question and responses.

When we look at the graph which shows how students have performed at different levels of total score in the paper, it appears confused. There is no strongly increasing selection of the correct answer among the better performing students like the data from previous articles. In fact, at no score level (except the perfect score) do more than 60% of the students select the correct answer!

Students who chose E in this question scored almost twice as much in the paper as those who chose any of the other options.

Option vs. Average Score in full ‘paper’

A 32%

B 30%

C 32%

D 31%

E 59%

And yet, we see that those who chose option E have performed significantly better than all the others. Overall this suggests that the question has challenged and stumped many students, but the best ones have answered it correctly.

Where are the children going wrong? Some students may be applying the perimeter formula instead of calculating the area. The perimeter of both Karl and Makenna’s fields would indeed be the same, but not the area.

Still, it is surprising that so many students would get this question wrong — is it possible that students simply guessed or did not answer the question seriously? We’ll have more on that later.

Let us look at one more question related to areas of overlapping shapes. We saw similar patterns in other questions on this topic too.

The shaded region formed by the two intersecting perpendicular rectangles, in square units, is

A. 23 (8%)
B. 38 (41%)
C. 44 (36%)
D. 46 (15%)

Attempted: 2418 | Skipped: 0

While 41% of students have gotten this question from the grade 8 AMC test correct, we can see that a substantial percentage of students have answered C. 44. How do these students get that answer? They have calculated the area of each rectangle (20 sq. units and 24 sq. units) and added them. They have not subtracted the overlap area which is double-counted in that addition.

Is that a careless mistake? If yes, we could hypothesize that if this (or a similar) question were given to the students again, many who made the careless mistake wouldn’t. Experience suggests that careless mistakes occur at a much lower frequency — less than 5% usually. An answer given by more than a third of a student more likely represents an important misconception. This means that if the question is given again, most of these students would give that same answer. Unlike a careless mistake, a misconception needs to be addressed and remediated.

The students are actually associating the area of a rectangle with the formula length into width, and applying it inappropriately. They are not thinking about the physical meaning of area, hence the overlap does not occur to them.

It is also possible that students simply perform some operations using the numbers given in the question, in this case (8 x 3) + (2 x 10). You may be aware of the research around the question “A captain owns 26 sheep and 10 goats. How old is the captain?” (If not, you can look it up on Wikipedia.)

Do you remember the question from the last article? When asked the ratio of the number of girls to boys in a classroom of 28 students, more than a quarter selected 7:4. Since 7 x 4 = 28, sometimes these numbers pop up and take the unsure student towards the wrong answer. Since the number of such students is substantial, it is important for us to understand and try and address this issue.

What are some of the takeaways from today’s examples and discussion?

  • Mechanical application of formula has its pitfalls. As teachers, we need to ensure that students are not losing sight of the underlying concepts while learning procedures, definitions and formulae.
  • We need questions that are designed to waylay students with a weak understanding, both so we can identify those students and also as an example to those students of the pitfalls of weak understanding.

But when we look at large scale data and analyze them, what if students are just guessing? Or not answering questions with seriousness? What if the engagement is missing — possibly due to frustration, boredom or something else? Do share your thoughts and comments on these points. We shall examine these factors in the next article and how we can eliminate some of them from contaminating our results. We will also look at common mistakes students are making while solving word problems with a number of examples from the Edmodo — AMC dataset!

Edmodo acknowledges the inputs of Educational Initiatives, Inc. which provided the question performance analysis used in the article.

About the Author: Guest Author