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The dreaded word problem. In a straightforward equation, students can focus on simply recalling the procedure and solving the problem. But with a word problem, significant effort goes into understanding the scenario even before one gets to the ‘real math’ in the problem.
We are studying student responses to questions from the American Mathematics Competition (AMC) on the Edmodo platform. Almost 3 million students answered these questions over a period of a few months providing invaluable data on common student errors. In earlier articles, we examined student errors in topics like ratios and proportions as well as area and perimeter.
This time, we’ll examine student performance in a few different word problems. We’ll also look at potential limitations of this analysis and how we can check if students are answering seriously, simply guessing, or becoming disengaged.
Let’s start by looking at two similar problems and look for the similarities and differences in student responses.
Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour?
1. 7 (18.2%)
2. 9 (17.6%)
3. 10 (29.8%)
4. 12 (22.3%)
5. 14 (12.1%)
Attempted: 2568 | Skipped: 1
George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first 1/2 mile at a speed of only 2 miles per hour. At how many miles per hour must George run the last 1/2 mile in order to arrive just as school begins today?
1. 4 (16%)
2. 6 (37%)
3. 8 (18%)
4. 10 (16%)
5. 12 (12%)
Attempted: 345300 | Skipped: 300
Before we look into the data carefully, ask yourself: Which of these is the easier question? Clearly the question on Kate’s cycling seems much easier than on George’s walk to school. Firstly, there are no fractional quantities in the time or distance. Secondly, only the overall average speed needs to be calculated. (In the second question, the student must calculate George’s speed for part of the total distance.)
Yet we find that the performance is weaker in the first (easier) question. The most common wrong option is the result of simply averaging out the two speeds mentioned. Thus, we see that the temptation to go with a numerically simpler (or ‘intuitive’) answer can be higher in an easier question. As students engage more with a question, they often better understand it and solve the problem.
When we study the performance on the second question, we find option A confuses students, which is found by working backwards from a simple average speed of 3 mph. This confusion is reflected in the way option A (the red dots in the graph below) climbs parallel with the correct option B till about the halfway score in the paper. Also, the average overall performance of students who choose A is higher than the other wrong options.
Option Selected vs. Average score in full ‘paper’
*George’s walk to school: Students who chose B in this question scored twice as much in the paper as those who chose A.
One of the other clear findings was that students struggled in questions where they had to convert a mathematical relationship mentioned in a word problem to algebraic equations.
The larger of two consecutive odd integers is three times the smaller. What is their sum?
A. 4 (15.5%)
B. 8 (14.1%)
C. 12 (27.1%)
D. 16 (27.7%)
E. 20 (15.6%)
Attempted: 2592 | Skipped: 0
The AMC in their internal archives mark this question as an ‘easy’ one, but students did not seem to find the question easy. Not only do a lot more students choose the wrong answer D, even the second most common answer, C, is far more frequently chosen over the correct answer. Expressing mathematical concepts in words may confuse students more than test writers intend, which could have caused students to give up early.
Perhaps understanding the average performance of students who chose each option could help.
Option Selected vs. Average score in full ‘paper’
As we can see, higher scoring students have answered the question correctly — and the difference in performance between students opting for A and the other options is among the highest we have seen across questions.
This still leaves us with the mystery — why would students get such an easy question wrong, so spectacularly? Either there is a lack of engagement or there is an underlying misconception that we are not aware of.
What can we do as teachers to help students do better in word problems? We need to initially understand where they are struggling. In the previous question for example, is there an underlying misconception that needs to be addressed or are they simply giving up mid-way? A good way to determine that would be to require students to explain in 1–2 lines the approach they took. Once the specific error is correctly identified, it is not difficult to address it with either a simpler question testing the same concept or even re-teaching, if the misconception is widespread.
Analysis of student performance is dependent on student focus and engagement. Student engagement is best when the questions are answered by students in a medium-stakes environment. This means students must not feel overly pressured while answering the questions (as may happen in a high stakes standardized test), nor should they take the exercise so lightly that they may be answering randomly. A medium-stakes environment could have interesting and challenging questions or provide the opportunity to share and read peers’ explanations.
Have you also used questions to determine where students are going wrong and used the analysis of their responses to determine how they are thinking? If yes, do share the questions, analysis and insights as they will help us collectively understand better how students solve problems.
Edmodo acknowledges the inputs of Educational Initiatives, Inc. which provided the question performance analysis used in the article.