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Children’s learning from implicit analogies during instruction: Evidence from fraction division

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Children learn better when their prior knowledge is activated through implicit analogies rather than explicitly guided. This study by Pooja Sidney shows the effectiveness of this approach in teaching fraction division.

Image courtesy of interviewee. December 15, 2023

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This video video discusses how teachers can effectively introduce new and challenging concepts, particularly in mathematics, by leveraging students' prior knowledge. It contrasts two types of teaching strategies: explicit and implicit analogies. An explicit analogy directly connects familiar ideas to new concepts, whereas an implicit analogy activates relevant prior knowledge through warm-up activities without directly highlighting the connection.

Research indicates that students, particularly younger ones, may find implicit analogies more beneficial for learning alignment between concepts than explicit mappings. For example, a study with sixth graders learning about fraction division showed that those engaged with implicit analogies (which involved a warm-up with similar whole number division problems without direct prompts to make connections) performed better than those who received explicit instruction focusing on these connections.

Additionally, the text highlights that explicit analogy conditions sometimes lead to misconceptions, suggesting that drawing students’ attention to the connections may overwhelm their cognitive capacities. It suggests practical recommendations for educators, such as carefully designing warm-up activities that activate relevant prior knowledge and maintaining consistency in visual representations and language used in instruction to facilitate implicit learning connections. The overall implication is that understanding how students think and using strategic teaching methods can improve their learning outcomes significantly.

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When teachers are teaching something that is new

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and difficult, they want to leverage some prior knowledge

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that children already have.

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So my particular work is in the, in mathematics.

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And so when we think about, say,

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learning about fractions in fourth

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or fifth grade, kids have a wealth

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of knowledge about whole numbers that teachers want them

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to bring in or transfer in to what they're learning.

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So an instructional analogy is a way

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of making a connection between something that kids,

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something that's very familiar, to something that's new

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and perhaps more challenging.

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Uh, when we think about an explicit analogy,

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we think about an analogy in which the teacher is

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directly bringing children's attention to the fact

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that there is a connection.

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So that might be like in science, for example,

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one very common explicit analogy is saying that,

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the structure of an atom is like the structure

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of a solar system with something in the middle,

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and then some elements organized around it.

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Um, an implicit analogy that's a newer idea

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that my colleagues and I have been working

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through and thinking about.

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Um, the idea with an implicit analogy is

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that you set up the analogy

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for the student without the student's direct

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attention to it.

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So the way that we operationalize

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that is we use a warmup activity to activate

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relevant prior knowledge in the mind first,

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and then go into direct instruction so

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that the analogy is clear to the instructor,

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but you're not necessarily drawing attention

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to it for the child.

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So that kind of gets into the, the sort

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of prior literature about analogies.

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And, you know, in,

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in many instances when we're thinking about the best ways

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to teach kids, it's good to be direct.

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You know, if we want them to know something,

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the most efficient way to,

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to have them know it is to tell them.

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Um, and, and certainly as adults, if you

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or I wanted to make an analogy, it would make sense for,

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you know, if you're explaining something to me for you

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to really explain the thing I know that's familiar,

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the new thing and all the connections between them.

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Um, and in fact, prior research suggested that,

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analogies were really difficult for students to learn from

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and they needed a lot of, of that support, explicit support

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for how, what prior knowledge do I have that's relevant

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and how exactly does it map to the new thing?

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Um, but as you can imagine, that takes a lot

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of attentional capacity as well.

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'cause as the learner, I have to pay attention to elements

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of the old idea, elements of the new idea

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and all the correspondences,

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and then adapt to what I know to something new.

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So that takes a lot of cognitive capacity,

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works really well for adults.

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Um, but I sort of wondered whether that would work just

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as well for kids.

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And so that's what led us to thinking about,

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analogy more as analogical priming

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with this implicit analogy

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and less of that really explicit direct instruction.

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On the analogy itself,

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there's this great prior literature on analog

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Transfer and analog problem solving.

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So that again, is, using something more familiar,

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to sort of bootstrap

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and, understand the structure

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of something more complicated or something newer.

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A lot of this great not, sorry.

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There's a lot of prior literature that suggests

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that students need hints to know what prior knowledge

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to retrieve, that they need support for that mapping.

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Uh, and a lot of the prior literature also suggested

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that this is a difficult process, that it doesn't happen.

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And in fact, we see this in the educational context all the

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time when, say, kids learn something

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or students will learn something in calc one

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and then fail to transfer into their engineering classes.

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So we see this failure to transfer in the research

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and we see a failure to transfer in real life.

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And so a lot of researchers sort of were thinking,

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that you'd have to make

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that connection really explicit to students.

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Now, when we looked at the developmental literature,

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you can see that even really young infants

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transfer spontaneously.

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There's this great study by, Chen

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and his colleagues, that I love with toddlers.

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So they're, I think they're 18 months to 2-year-old.

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They're, they're little toddlers

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and, they, they're a couple of different conditions.

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And some of the toddlers are shown a problem

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solving solution from their parents.

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It's sort of, series of motor behaviors they have

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to do to get to a toy.

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And then they're shown a brand new problem.

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It the, all the objects are different, all the barriers

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between them and the toy,

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but the motor actions are the same.

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And the kids whose parents showed them how

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to solve the different problem

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we're really good at learning very

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quickly about the new problem.

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They, they sort of applied those same motor actions

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that were already activated

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by watching the parent to this new problem.

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So on one side,

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the adult literature tells us theological transfer hard.

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On the other side we see that infants, young children

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very spontaneously and easily transfer

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and apply their knowledge from one context to the next.

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So then the question that I had is in this, early

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to middle childhood period, is there a way of supporting

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that spontaneous implicit transfer that we see in young kids

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rather than reaching for the cognitively effortful stuff

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that we observe in adult,

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in the study we had sixth grade students learning about,

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the structure of fraction division.

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So not just how to solve the problem,

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but what does it mean to divide by a fraction.

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And we had students learn in three conditions.

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So in one condition they got the best explicit

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analogy that we could design.

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So that included all

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of the recommendations from prior research on how

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to identify the prior knowledge, how

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to represent prior knowledge, how to support connections

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for students between the old familiar idea and the new idea.

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So we had two whole number division problems,

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which are really helpful for

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thinking about fraction division.

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'cause it turns out they're almost the same.

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You can think about whole number division as how many

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Times one thing goes into another

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and you can think about fraction division as

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how many times one thing goes into another.

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So in one condition we have two whole number division

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problems and then two fraction division problems.

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And the instructor does this really nice beautiful explicit

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mapping between the two of them.

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Um, in another condition, our implicit analogy condition,

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they warm up with whole number division

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and then we never talk about them again

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when we get to fraction division.

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So that's the implicit part.

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The student's attention is not drawn to the connection,

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but importantly, there are a lot of implicit cues.

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So the visual representations are the same

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for whole number division and fraction division.

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The language that we use is almost identical

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for whole number division and fraction division.

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Um, and then the actions

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that they see the instructor do are almost identical.

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So we have all these implicit cues

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that these things are related without

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engaging students in an explicit mapping process.

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So we call that the warmup implicit analogy condition.

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The third condition is just extra fraction division.

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So instead of only getting two problems of instruction,

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they got four problems of instruction on fraction division

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and no whole number division.

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So this would be like just spending additional time teaching

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the thing that you wanna be teaching.

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So, long story short to what we found was

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that children learned about the structure

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of fraction division most effectively in the implicit

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analogy condition.

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And in fact, students in the explicit analogy,

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condition did not learn any better than no analogy at all.

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Now, the no analogy condition is also a really strong

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control 'cause they're spending twice as much time

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on the target instruction.

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Um, but we didn't see any benefits

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of warming up prior knowledge when you're making this

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really explicit connection.

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Um, one thing that I thought was really interesting is

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that we also see

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that the explicit analogy leads to misconceptions.

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And we've seen that in our prior work as well.

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Kids when they're explicitly trying

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to make those connections,

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sometimes they make the wrong connections.

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And so then you see issues pop up that wouldn't have

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otherwise popped up if the kids hadn't been trying so hard

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to do exactly what the adult told them to do.

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Yeah, I think there are some really cool implications.

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So one is to un as for instructors to understand that

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what children are thinking about, not just children,

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we've done this work with adults as well.

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So what learners are thinking about when they enter the

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classroom shapes

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how they think about the first thing they learn.

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And so that means that instructors both have to attend to

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what students are thinking about at the very beginning

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of the lesson, but they also have an opportunity to shape it

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by doing a warmup that activates the prior knowledge

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that's gonna be most relevant.

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I think that's really important because very often we see

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warmup activities in math classrooms,

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but they're not necessarily chosen with the intent

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of activating prior knowledge.

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So they might be chosen with the intent

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of practicing something you did last week or the week

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before, but sometimes the thing you just worked on is not

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the prior knowledge that's gonna lead you to success

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on the thing you have to do today.

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And so there's an opportunity

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For, for instructors

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to get their kids thinking along the right track

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before they even engage in instruction.

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So I think that's the nice, you know, practical,

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value of this work.

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Um, another more broader implication I think is for,

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you know, the field of mathematics, learning development

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and cognition is to think about

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how these implicit learning mechanisms are at play even

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during direct instruction.

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And I think that our field has focused a lot on,

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you know, explicit aspects of explicit instruction

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that we can manipulate and much less on

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how we can manipulate the environment of learning

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to make learning more successful, in ways

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that might not be readily obvious to students,

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but take a lot of planning for the teacher.

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So one example of changing the environment is right to, to

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to activate students' prior knowledge in some way.

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So that would be, for example,

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including those warmup activities.

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Um, another way of changing the environment, let me think

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of another example that's not exactly the implicit analogy.

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Example. Okay, so one other way that I think,

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teachers could make a a, a slight change that's not

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necessarily a new type of instruction is paying attention to

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how they use their language for mathematical topics.

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Okay. So that the language for topics

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that are similar in structure is also similar to help cue

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that what we did before is relevant to what we did now,

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or to have some continuity in the types of visual

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representations you're using.

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So if you're using one type of diagram, say number lines

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to learn about whole number division,

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that we're using the same type of diagram

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to learn about fraction division.

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Um, you know, that type

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of continuity might not be really explicit for the student.

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A student may not notice that.

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Um, but it might help the student be more likely

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to make a spontaneous connection between those two topics

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than using diagrams or representations or words

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and language that's really different.

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Then there's the additional obstacle of trying to figure out

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how these things map,

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or just a failure to spontaneously transfer

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because there's no queue, there's no implicit queue that

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that prior knowledge is relevant.

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