Learning Math from Errors
By: Megan Sumeracki
When we talk about effective, evidence-based learning strategies, we often note how hard these strategies can feel. This is not a bad thing in itself—challenges can be really good! However, what can be problematic for individual learners is that the difficulty is often misconstrued as “not learning as much,” and can even lead to students liking the strategy less and not choosing it as often (or at all). Students don’t like making lots of mistakes! Framed another way, strategies that feel easy (like rereading) make us think that we are learning a lot, and we tend to like this experience more. If it feels easier and easier, that must be because we are getting better and better, right? Unfortunately, not necessarily.
So, we talk about the difficulty that comes with strategies like retrieval practice, spacing, and interleaving. Perfection does not necessarily mean we’ve mastered the material or learned it well. Embracing errors and challenges can be a positive thing for learning.
In today’s blog post, I want to cover a paper (1) by Deanne Adams and colleagues that demonstrates learning from errors. What I think is neat about this paper is the researchers gave students erroneous math examples (specifically, decimals, which can be hard for middle schoolers) rather than waiting for them to make the errors themselves. The researchers also required the students to identify explanations for the errors (like elaboration through self-explanation) and correct the errors themselves. They compared this learning condition to what appears to be the standard in math: solving practice problems.
The Experiment:
The participants were middle school students (6th and 7th grade in the US, ages 11-13). Students learned decimals in the experiment in one of two different conditions. The experimental condition included erroneous examples; the majority of the problems were presented to students as completed erroneously by another student (in reality these were created by the researchers). This was compared to a “business as usual” control condition (problem solving).
In the erroneous examples condition, students were given two erroneous examples and then solved one problem for a total of 36 problems (i.e., 24 were erroneous examples and 12 were problems to solve). For each problem presented as erroneously completed, students corrected the error. Then, they answered a few multiple-choice questions that required them to identify correct explanations for the problem as well as the errors the fictitious student must have made. They received feedback along the way.
The image above is a screenshot of the paper itself (1), showing what the screen looked like in the erroneous examples condition. The image can be difficult to see, but you can see the image in the actual paper here (scroll to Figure 2). For clarity, I will also describe it: In this problem, the students were asked to find the next two numbers in the sequence: 0.0, 0.4, 0.8, ___, ____. The answers provided by “another student” were .12 and .16. The students then have to fix the error (by typing in 1.2 and 1.6, the correct answers), and pick explanations for the error and the correct answer. For example, the student who made the error thinks that they can treat the numbers after the decimal as whole numbers.
The researchers were intentional in the way they implemented the erroneous examples condition. Specifically:
They presented the erroneous examples as another student’s errors to avoid embarrassment;
they made the examples interactive by prompting students for explanations about the errors made, had them correct the errors, and gave them feedback; and
they worked to minimize/eliminate cognitive overload (see this post).
In the problem-solving condition, students were given 36 problems to solve. For each, they answered a few multiple-choice questions that required them to reflect on the correct answers and explain their solutions. They received feedback along the way.
The image above is a screenshot of the paper itself (1), showing what the screen looked like in the problem-solving condition. The image can be difficult to see, but you can see the image in the actual paper here (scroll to Figure 3). For clarity, I will also describe it: In this problem, the students were asked to find the next two numbers in the sequence: 0.0, 0.4, 0.8, ___, ____. The answers are 1.2 and 1.6. The students then have to pick an explanation of how to solve the problem that is correct (there was a difference of .04 between each number, and they should carry a one from the tenths to the ones place).
Importantly, students engaged with the same problems in each of the two conditions. What differed was whether they practiced solving all of the problems, or practiced solving some problems and engaged with error correction and explanation for the others.
The researchers measured students’ learning through post-tests administered immediately after the two learning conditions were completed and one week later.
There were some other aspects of the procedure as well, and more specific details can be found in the paper. For example, the students completed a pre-test to assess their prior knowledge, answered questions about how they thought the learning activies went (e.g., how much they liked it, ease of interacting on the computer), and answered questions about how accurate they thought they were during the post-test.
The Results:
There were no differences between the two groups on the immediate post-test. However, on the one-week delayed post-test, the students in the erroneous examples condition performed better than the students in the problem-solving condition.
(Note, these results were statistically controlled for prior knowledge as assessed by the researchers, and so they represent increases in learning due to the intervention. If you’re interested or a stats person, check out the paper for more details.)
Importantly, when the researchers looked at students who had lower and higher levels of prior knowledge, the erroneous examples condition consistently led to better performance on the delayed post-test compared to the problem-solving condition. This is good news, because it means this learning method tends to work for students at varying levels of ability, and not just students at one end of the spectrum.
Students in the erroneous examples condition were also more accurate at judging how well they were performing during the post-tests.
However, students in the erroneous examples condition reported liking the lesson less than the students in the problem-solving condition. Unfortunately, another situation where the strategy that students tend to like more is the one that helps them learn less!
References:
(1) Adams, D. M., McLaren, B. M., Durkin, K., Mayer, R. E., Rittle-Johnson, B., Isotani, S., & van Velsen, M. (2014). Using erroneous examples to improve mathematics learning with a web-based tutoring system. Computers in Human Behavior, 36, 401-411. https://doi.org/10.1016/j.chb.2014.03.053
