Fitzsimmons, C. J. & Thompson, C. A. (2022). Developmental differences in monitoring accuracy and cue use when estimating whole-number and fraction magnitudes. Cognitive Development, 61.
Have you ever felt totally sure that you answered a question correctly, only to be wrong? When taking a test, have you ever felt your confidence wash away as you realize you don’t know the material well enough to answer exam questions? Or, on the flip side, have you ever doubted your ability, then realized you were right all along?
These questions illustrate meta-cognitive monitoring: knowing when you’re right or wrong. Knowing what you know is pretty important – if you’re sure that you know something, you likely stop studying and avoid help seeking or error checking.
In fact, when children or adults solve math problems and then rate how confident they are that they are right, their confidence is better than their actual performance at predicting whether they ask for help (1, 2, 3). Unfortunately, people are not great at judging when they’re right and when they’re wrong. But why is that?
Why are people bad at judging when they are right and wrong?
My colleague Clarissa Thompson and I wondered whether people mis-judge their performance because they become familiar with features of the math problems, like the numbers involved. For example, some people might feel more comfortable working with numbers like 1, 2, or 1/2, than with numbers like 12, 29, or 15/30. We expected that people would feel more confident that their answers were correct when they solved problems with more familiar numbers, even if they were just as good at solving problems with numbers they were less familiar with.
We tested our hypothesis about the relation between people’s familiarity and confidence by having people solve math problems with fractions. Fractions are important for many things in life like math, science, and cooking. And, fractions can look different but mean the same thing. For example, 1/2 and 15/30 are equivalent fractions – they both represent 0.5 – but they have different digits for numerators (top number) and denominators (bottom number).
We expected people to say they were more familiar with fractions with smaller digits, like 1/2, compared to larger digits, like 15/30. And, we expected people to be more confident when they solved problems with fractions that were simpler-looking and more familiar, like 1/2, compared to those that looked more complicated, like 15/30.
Investigating the role of familiarity in fraction estimation and confidence
In our studies, children and adult participants estimated where fractions went on 0 to 1 number lines, like the one in the figure below. Immediately after they estimated each number, they rated how confident they were that their estimate was in the correct location. Critically, we had them estimate equivalent fractions that mean the same thing and go at the same place on the number line, like 1/2 and 15/30, but that we expected people to be more and less familiar with.
After the participants estimated all fractions on number lines and judged their confidence in their answers, they rated how familiar they were with each fraction.
As expected, children and adults were more familiar with simpler-looking fractions, like 1/2, than more complicated looking, but equivalent fractions, like 15/30. And, consistent with our hypothesis, participants were more confident in their estimates of simpler-looking, familiar fractions than more complex looking, unfamiliar fractions. Participants were more confident in their estimates of simpler fractions even when they performed equally well on the two kinds of problems – that is, when their number-line estimates were just as accurate for simpler fractions and more complicated fractions. In other words, when people solved problems, they were more sure that they were right when they were familiar with features of the problems, regardless of their performance.
This finding made a lot of sense: In education and daily life, people are more likely to encounter smaller numbers like 1, 2, and 1/2, than numbers like 15, 30, and 15/30. We reasoned that it is people’s experiences with these numbers that leads them to be more familiar and confident in their estimates.
Does experience make people more familiar and confident in solving problems?
In our follow-up experiment, we investigated whether we could increase children’s and adults’ familiarity and confidence by having them interact with fractions. At the very beginning of the study, we randomly assigned half of the participants to get some extra experience with a subset of fractions. They named them (e.g., two-ninths), identified the top and bottom numbers, and estimated them on number lines without any feedback.
But, that brief experience didn’t work – children and adults were equally familiar with the subset of fractions regardless of whether they interacted with them at the beginning of the study or not. And, given that this experience didn’t affect their familiarity, it’s no surprise that participants who saw fractions at the beginning of the study were just as confident as participants who did not see these fractions. In hindsight, this makes sense: how could a brief 5-minute experience overcome the years of experience people have with numbers in their day-to-day life?
However, we still suspected that experiences with numbers were important for familiarity and confidence, so we decided to check some existing data from our lab where children played fraction games over multiple sessions. After children played fraction board games intended to improve their understanding, they became more confident in their ability to compare, order, and estimate fractions even though they didn’t get any better at solving the problems. This finding supports our hypothesis that experiences matter: When you get lots of experience with the same set of numbers, you become more confident in your ability to solve problems that contain those numbers – even if you’re not actually any better at solving them.
Up to this point, I’ve discussed research focused on fractions. But we also tested people’s metacognitive monitoring as they estimated whole numbers on number lines. The results were a bit more complicated for this task, and we’re continuing to think about and research why people struggle to judge when they are accurate or not as they estimate whole numbers.
Concluding remarks
Children and adults are often poor judges of when they’re accurate or not. It seems that a feeling of familiarity with the problem can bias people’s metacognitive monitoring, at least for fraction problems.
So, the next time you skim over content and think, hey this feels familiar, I know it, it’s possible that your confidence is biased. The reverse could also be true: When you start solving some unfamiliar problems, you may be less confident because they’re novel, then waste valuable time asking for help when you already know the answer. One good way to know how much you know or not is by practice testing yourself – a method that has benefits for improving your actual knowledge and your awareness of your own knowledge (4).
Additional References:
- Fitzsimmons, C. J., & Thompson, C. A. (2023). Why is monitoring accuracy so poor in number line estimation? The importance of valid cues and systematic variability for US college students. Metacognition and Learning, 1-32.
- Fyfe, E. R., Byers, C., & Nelson, L. J. (2022). The benefits of a metacognitive lesson on children’s understanding of mathematical equivalence, arithmetic, and place value. Journal of Educational Psychology, 114(6), 1292.
- Nelson, L. J., & Fyfe, E. R. (2019). Metacognitive monitoring and help-seeking decisions on mathematical equivalence problems. Metacognition and Learning, 14, 167-187.
- Rivers, M. L. (2021). Metacognition about practice testing: A review of learners’ beliefs, monitoring, and control of test-enhanced learning. Educational Psychology Review, 33(3), 823-862.
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