Popular in Professional

Culture and Society

The City as a Growth Machine: Toward a Political Economy of Place

New York University Harvey Moltoch

Education

The Role of Education in Reducing Racial Inequality

University of Southern California Pedro A. Noguera

Law

Diversity and Social Mobility in the UK legal profession

Law Society I. Stephanie Boyce

Internet Law

The Law of AI for Good

University of San Diego Orly Lobel

Pharmacy

Cytotoxic Activity and Anti-Migratory Effect of Berberine-Phytantriol Liquid Crystalline on Non-Small-Cell Lung Cancer

University of Technology Sydney Kamal Dua

Human Rights Law

Transforming Restorative Justice for Transitional Settings

University of Nottingham Kerry Clamp

Education

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

View more insights in Education - Pedagogy - STEM Education

View more figures from University of Kentucky

View the page of Pooja Sidney

Read the Study

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

Generating Workspace

Export as PDF

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.

Language:

Export as PDF

X-TIMESTAMP-MAP:LOCAL=00:00:00.000,MPEGTS=900000

00:00.195 --> 00:03.445
When teachers are teaching something that is new

00:03.625 --> 00:07.445
and difficult, they want to leverage some prior knowledge

00:07.445 --> 00:08.845
that children already have.

00:09.145 --> 00:12.325
So my particular work is in the, in mathematics.

00:12.865 --> 00:14.445
And so when we think about, say,

00:14.605 --> 00:16.085
learning about fractions in fourth

00:16.085 --> 00:17.925
or fifth grade, kids have a wealth

00:17.925 --> 00:21.725
of knowledge about whole numbers that teachers want them

00:21.865 --> 00:25.245
to bring in or transfer in to what they're learning.

00:25.505 --> 00:29.005
So an instructional analogy is a way

00:29.025 --> 00:32.565
of making a connection between something that kids,

00:32.565 --> 00:35.285
something that's very familiar, to something that's new

00:35.285 --> 00:36.645
and perhaps more challenging.

00:37.225 --> 00:39.525
Uh, when we think about an explicit analogy,

00:39.945 --> 00:42.885
we think about an analogy in which the teacher is

00:43.485 --> 00:46.085
directly bringing children's attention to the fact

00:46.085 --> 00:47.205
that there is a connection.

00:47.465 --> 00:50.165
So that might be like in science, for example,

00:50.505 --> 00:53.245
one very common explicit analogy is saying that,

00:53.245 --> 00:55.245
the structure of an atom is like the structure

00:55.245 --> 00:57.245
of a solar system with something in the middle,

00:57.625 --> 00:59.565
and then some elements organized around it.

00:59.985 --> 01:03.165
Um, an implicit analogy that's a newer idea

01:03.165 --> 01:05.085
that my colleagues and I have been working

01:05.155 --> 01:06.605
through and thinking about.

01:07.065 --> 01:09.645
Um, the idea with an implicit analogy is

01:09.645 --> 01:11.325
that you set up the analogy

01:11.425 --> 01:13.685
for the student without the student's direct

01:13.685 --> 01:14.845
attention to it.

01:15.225 --> 01:17.205
So the way that we operationalize

01:17.205 --> 01:20.365
that is we use a warmup activity to activate

01:21.205 --> 01:22.925
relevant prior knowledge in the mind first,

01:23.425 --> 01:25.605
and then go into direct instruction so

01:25.605 --> 01:27.605
that the analogy is clear to the instructor,

01:28.145 --> 01:30.245
but you're not necessarily drawing attention

01:30.245 --> 01:32.045
to it for the child.

01:32.465 --> 01:35.365
So that kind of gets into the, the sort

01:35.365 --> 01:37.405
of prior literature about analogies.

01:37.665 --> 01:38.725
And, you know, in,

01:38.785 --> 01:41.685
in many instances when we're thinking about the best ways

01:41.745 --> 01:44.005
to teach kids, it's good to be direct.

01:44.265 --> 01:45.685
You know, if we want them to know something,

01:46.185 --> 01:47.205
the most efficient way to,

01:47.345 --> 01:48.965
to have them know it is to tell them.

01:49.385 --> 01:53.165
Um, and, and certainly as adults, if you

01:53.165 --> 01:56.445
or I wanted to make an analogy, it would make sense for,

01:56.865 --> 01:59.085
you know, if you're explaining something to me for you

01:59.085 --> 02:01.645
to really explain the thing I know that's familiar,

02:01.865 --> 02:04.045
the new thing and all the connections between them.

02:04.505 --> 02:07.605
Um, and in fact, prior research suggested that,

02:07.995 --> 02:11.725
analogies were really difficult for students to learn from

02:11.785 --> 02:15.005
and they needed a lot of, of that support, explicit support

02:15.545 --> 02:18.725
for how, what prior knowledge do I have that's relevant

02:18.945 --> 02:21.205
and how exactly does it map to the new thing?

02:21.585 --> 02:24.205
Um, but as you can imagine, that takes a lot

02:24.205 --> 02:25.485
of attentional capacity as well.

02:25.885 --> 02:28.525
'cause as the learner, I have to pay attention to elements

02:28.525 --> 02:30.885
of the old idea, elements of the new idea

02:31.185 --> 02:32.765
and all the correspondences,

02:32.825 --> 02:35.565
and then adapt to what I know to something new.

02:35.585 --> 02:37.725
So that takes a lot of cognitive capacity,

02:38.055 --> 02:39.685
works really well for adults.

02:40.105 --> 02:42.565
Um, but I sort of wondered whether that would work just

02:42.565 --> 02:43.645
as well for kids.

02:43.665 --> 02:46.445
And so that's what led us to thinking about,

02:46.635 --> 02:49.365
analogy more as analogical priming

02:49.555 --> 02:50.805
with this implicit analogy

02:51.025 --> 02:54.645
and less of that really explicit direct instruction.

02:54.705 --> 02:56.005
On the analogy itself,

02:56.155 --> 02:58.765
there's this great prior literature on analog

02:59.405 --> 03:00.685
Transfer and analog problem solving.

03:00.745 --> 03:04.165
So that again, is, using something more familiar,

03:04.225 --> 03:05.445
to sort of bootstrap

03:05.905 --> 03:07.605
and, understand the structure

03:07.625 --> 03:10.005
of something more complicated or something newer.

03:10.725 --> 03:13.095
A lot of this great not, sorry.

03:13.165 --> 03:15.935
There's a lot of prior literature that suggests

03:15.935 --> 03:19.775
that students need hints to know what prior knowledge

03:19.775 --> 03:22.655
to retrieve, that they need support for that mapping.

03:23.115 --> 03:25.935
Uh, and a lot of the prior literature also suggested

03:25.935 --> 03:28.295
that this is a difficult process, that it doesn't happen.

03:28.355 --> 03:30.655
And in fact, we see this in the educational context all the

03:30.655 --> 03:32.055
time when, say, kids learn something

03:32.115 --> 03:34.535
or students will learn something in calc one

03:34.755 --> 03:37.055
and then fail to transfer into their engineering classes.

03:37.435 --> 03:40.655
So we see this failure to transfer in the research

03:40.915 --> 03:42.855
and we see a failure to transfer in real life.

03:42.915 --> 03:46.575
And so a lot of researchers sort of were thinking,

03:46.765 --> 03:48.055
that you'd have to make

03:48.055 --> 03:49.575
that connection really explicit to students.

03:50.035 --> 03:52.575
Now, when we looked at the developmental literature,

03:53.075 --> 03:55.655
you can see that even really young infants

03:56.135 --> 03:57.255
transfer spontaneously.

03:57.325 --> 03:59.455
There's this great study by, Chen

03:59.455 --> 04:01.975
and his colleagues, that I love with toddlers.

04:02.115 --> 04:04.575
So they're, I think they're 18 months to 2-year-old.

04:04.575 --> 04:06.575
They're, they're little toddlers

04:06.955 --> 04:10.055
and, they, they're a couple of different conditions.

04:10.195 --> 04:13.175
And some of the toddlers are shown a problem

04:13.175 --> 04:14.655
solving solution from their parents.

04:15.005 --> 04:17.735
It's sort of, series of motor behaviors they have

04:17.735 --> 04:18.855
to do to get to a toy.

04:19.515 --> 04:21.135
And then they're shown a brand new problem.

04:21.675 --> 04:25.215
It the, all the objects are different, all the barriers

04:25.245 --> 04:26.415
between them and the toy,

04:26.435 --> 04:27.695
but the motor actions are the same.

04:27.995 --> 04:30.895
And the kids whose parents showed them how

04:30.895 --> 04:32.495
to solve the different problem

04:33.705 --> 04:35.335
we're really good at learning very

04:35.335 --> 04:36.495
quickly about the new problem.

04:37.125 --> 04:39.375
They, they sort of applied those same motor actions

04:39.375 --> 04:40.735
that were already activated

04:41.235 --> 04:43.015
by watching the parent to this new problem.

04:43.475 --> 04:45.015
So on one side,

04:45.035 --> 04:47.415
the adult literature tells us theological transfer hard.

04:48.035 --> 04:51.815
On the other side we see that infants, young children

04:52.965 --> 04:55.855
very spontaneously and easily transfer

04:55.855 --> 04:58.015
and apply their knowledge from one context to the next.

04:58.315 --> 05:01.215
So then the question that I had is in this, early

05:01.235 --> 05:04.375
to middle childhood period, is there a way of supporting

05:04.375 --> 05:08.695
that spontaneous implicit transfer that we see in young kids

05:09.355 --> 05:12.455
rather than reaching for the cognitively effortful stuff

05:12.805 --> 05:15.055
that we observe in adult,

05:15.075 --> 05:18.655
in the study we had sixth grade students learning about,

05:18.795 --> 05:20.215
the structure of fraction division.

05:20.555 --> 05:22.055
So not just how to solve the problem,

05:22.395 --> 05:24.575
but what does it mean to divide by a fraction.

05:25.235 --> 05:27.615
And we had students learn in three conditions.

05:27.915 --> 05:30.335
So in one condition they got the best explicit

05:30.405 --> 05:32.255
analogy that we could design.

05:32.395 --> 05:33.735
So that included all

05:33.835 --> 05:36.895
of the recommendations from prior research on how

05:36.895 --> 05:39.415
to identify the prior knowledge, how

05:39.415 --> 05:42.175
to represent prior knowledge, how to support connections

05:42.175 --> 05:46.775
for students between the old familiar idea and the new idea.

05:47.595 --> 05:50.135
So we had two whole number division problems,

05:50.135 --> 05:51.375
which are really helpful for

05:51.375 --> 05:52.415
thinking about fraction division.

05:52.615 --> 05:54.495
'cause it turns out they're almost the same.

05:54.835 --> 05:56.775
You can think about whole number division as how many

05:56.775 --> 05:58.605
Times one thing goes into another

05:58.705 --> 06:00.365
and you can think about fraction division as

06:00.365 --> 06:02.285
how many times one thing goes into another.

06:02.865 --> 06:05.085
So in one condition we have two whole number division

06:05.085 --> 06:07.845
problems and then two fraction division problems.

06:07.845 --> 06:10.685
And the instructor does this really nice beautiful explicit

06:10.685 --> 06:11.965
mapping between the two of them.

06:12.505 --> 06:16.045
Um, in another condition, our implicit analogy condition,

06:16.395 --> 06:18.205
they warm up with whole number division

06:18.705 --> 06:20.805
and then we never talk about them again

06:20.875 --> 06:22.085
when we get to fraction division.

06:22.105 --> 06:23.325
So that's the implicit part.

06:23.385 --> 06:26.725
The student's attention is not drawn to the connection,

06:27.345 --> 06:30.485
but importantly, there are a lot of implicit cues.

06:30.665 --> 06:33.205
So the visual representations are the same

06:33.345 --> 06:35.285
for whole number division and fraction division.

06:35.665 --> 06:38.085
The language that we use is almost identical

06:38.225 --> 06:39.965
for whole number division and fraction division.

06:40.305 --> 06:41.925
Um, and then the actions

06:41.955 --> 06:44.725
that they see the instructor do are almost identical.

06:45.145 --> 06:46.925
So we have all these implicit cues

06:47.035 --> 06:49.085
that these things are related without

06:49.645 --> 06:51.805
engaging students in an explicit mapping process.

06:52.185 --> 06:54.765
So we call that the warmup implicit analogy condition.

06:55.025 --> 06:57.325
The third condition is just extra fraction division.

06:57.505 --> 06:59.805
So instead of only getting two problems of instruction,

06:59.805 --> 07:02.245
they got four problems of instruction on fraction division

07:02.465 --> 07:03.605
and no whole number division.

07:03.705 --> 07:06.525
So this would be like just spending additional time teaching

07:06.525 --> 07:08.005
the thing that you wanna be teaching.

07:08.625 --> 07:12.035
So, long story short to what we found was

07:12.035 --> 07:14.155
that children learned about the structure

07:14.175 --> 07:16.715
of fraction division most effectively in the implicit

07:16.715 --> 07:17.755
analogy condition.

07:18.015 --> 07:19.995
And in fact, students in the explicit analogy,

07:19.995 --> 07:23.355
condition did not learn any better than no analogy at all.

07:23.895 --> 07:26.115
Now, the no analogy condition is also a really strong

07:26.115 --> 07:27.995
control 'cause they're spending twice as much time

07:28.575 --> 07:29.915
on the target instruction.

07:30.215 --> 07:32.555
Um, but we didn't see any benefits

07:32.855 --> 07:35.115
of warming up prior knowledge when you're making this

07:35.115 --> 07:36.275
really explicit connection.

07:36.655 --> 07:39.395
Um, one thing that I thought was really interesting is

07:39.395 --> 07:40.675
that we also see

07:40.675 --> 07:43.155
that the explicit analogy leads to misconceptions.

07:43.215 --> 07:45.355
And we've seen that in our prior work as well.

07:46.345 --> 07:48.115
Kids when they're explicitly trying

07:48.115 --> 07:49.155
to make those connections,

07:49.155 --> 07:51.155
sometimes they make the wrong connections.

07:51.695 --> 07:55.555
And so then you see issues pop up that wouldn't have

07:55.555 --> 07:58.955
otherwise popped up if the kids hadn't been trying so hard

07:58.955 --> 08:00.755
to do exactly what the adult told them to do.

08:00.825 --> 08:02.995
Yeah, I think there are some really cool implications.

08:03.295 --> 08:07.675
So one is to un as for instructors to understand that

08:07.675 --> 08:09.795
what children are thinking about, not just children,

08:09.795 --> 08:11.235
we've done this work with adults as well.

08:11.335 --> 08:13.795
So what learners are thinking about when they enter the

08:13.795 --> 08:15.115
classroom shapes

08:15.115 --> 08:16.755
how they think about the first thing they learn.

08:17.455 --> 08:20.835
And so that means that instructors both have to attend to

08:20.835 --> 08:22.835
what students are thinking about at the very beginning

08:22.835 --> 08:25.875
of the lesson, but they also have an opportunity to shape it

08:26.055 --> 08:29.035
by doing a warmup that activates the prior knowledge

08:29.035 --> 08:30.395
that's gonna be most relevant.

08:30.755 --> 08:33.395
I think that's really important because very often we see

08:33.495 --> 08:35.315
warmup activities in math classrooms,

08:35.495 --> 08:38.835
but they're not necessarily chosen with the intent

08:38.835 --> 08:40.115
of activating prior knowledge.

08:40.255 --> 08:41.835
So they might be chosen with the intent

08:41.835 --> 08:45.195
of practicing something you did last week or the week

08:45.195 --> 08:48.435
before, but sometimes the thing you just worked on is not

08:48.455 --> 08:50.795
the prior knowledge that's gonna lead you to success

08:51.375 --> 08:52.675
on the thing you have to do today.

08:53.295 --> 08:54.355
And so there's an opportunity

08:54.735 --> 08:55.930
For, for instructors

08:55.930 --> 08:58.540
to get their kids thinking along the right track

08:58.785 --> 09:00.885
before they even engage in instruction.

09:01.225 --> 09:03.525
So I think that's the nice, you know, practical,

09:04.065 --> 09:05.285
value of this work.

09:05.825 --> 09:09.365
Um, another more broader implication I think is for,

09:09.425 --> 09:12.565
you know, the field of mathematics, learning development

09:12.565 --> 09:14.765
and cognition is to think about

09:14.765 --> 09:18.005
how these implicit learning mechanisms are at play even

09:18.005 --> 09:19.365
during direct instruction.

09:19.785 --> 09:22.765
And I think that our field has focused a lot on,

09:23.385 --> 09:26.405
you know, explicit aspects of explicit instruction

09:26.405 --> 09:28.965
that we can manipulate and much less on

09:28.965 --> 09:31.525
how we can manipulate the environment of learning

09:32.205 --> 09:35.925
to make learning more successful, in ways

09:35.955 --> 09:38.485
that might not be readily obvious to students,

09:39.105 --> 09:40.885
but take a lot of planning for the teacher.

09:41.665 --> 09:44.645
So one example of changing the environment is right to, to

09:44.745 --> 09:46.845
to activate students' prior knowledge in some way.

09:47.145 --> 09:49.125
So that would be, for example,

09:49.125 --> 09:50.885
including those warmup activities.

09:51.225 --> 09:54.085
Um, another way of changing the environment, let me think

09:54.085 --> 09:58.405
of another example that's not exactly the implicit analogy.

09:58.435 --> 10:02.405
Example. Okay, so one other way that I think,

10:02.925 --> 10:05.765
teachers could make a a, a slight change that's not

10:06.275 --> 10:11.125
necessarily a new type of instruction is paying attention to

10:11.145 --> 10:14.125
how they use their language for mathematical topics.

10:14.275 --> 10:16.805
Okay. So that the language for topics

10:17.355 --> 10:20.845
that are similar in structure is also similar to help cue

10:21.395 --> 10:24.085
that what we did before is relevant to what we did now,

10:24.145 --> 10:26.885
or to have some continuity in the types of visual

10:27.405 --> 10:28.445
representations you're using.

10:28.865 --> 10:32.285
So if you're using one type of diagram, say number lines

10:32.585 --> 10:34.005
to learn about whole number division,

10:34.115 --> 10:36.125
that we're using the same type of diagram

10:36.585 --> 10:37.925
to learn about fraction division.

10:38.385 --> 10:39.925
Um, you know, that type

10:39.925 --> 10:43.725
of continuity might not be really explicit for the student.

10:43.925 --> 10:45.485
A student may not notice that.

10:46.385 --> 10:49.725
Um, but it might help the student be more likely

10:49.745 --> 10:52.885
to make a spontaneous connection between those two topics

10:53.595 --> 10:58.205
than using diagrams or representations or words

10:58.345 --> 11:00.045
and language that's really different.

11:00.115 --> 11:03.605
Then there's the additional obstacle of trying to figure out

11:03.605 --> 11:05.845
how these things map,

11:05.945 --> 11:09.085
or just a failure to spontaneously transfer

11:09.085 --> 11:11.965
because there's no queue, there's no implicit queue that

11:11.965 --> 11:13.765
that prior knowledge is relevant.

00:00 - When teachers are teaching something that is new and difficult, they want to leverage some prior knowledge that children already have. So my particular work is in the, in mathematics. And so when we think about, say, learning about fractions in fourth

00:16 - or fifth grade, kids have a wealth of knowledge about whole numbers that teachers want them to bring in or transfer in to what they're learning. So an instructional analogy is a way of making a connection between something that kids, something that's very familiar, to something that's new and perhaps more challenging. when we think about an explicit analogy, we think about an analogy in which the teacher is directly bringing children's attention to the fact

00:46 - that there is a connection. So that might be like in science, for example, one very common explicit analogy is saying that, the structure of an atom is like the structure of a solar system with something in the middle, and then some elements organized around it. an implicit analogy that's a newer idea that my colleagues and I have been working through and thinking about. the idea with an implicit analogy is

01:09 - that you set up the analogy for the student without the student's direct attention to it. So the way that we operationalize that is we use a warmup activity to activate relevant prior knowledge in the mind first, and then go into direct instruction so that the analogy is clear to the instructor, but you're not necessarily drawing attention to it for the child.

01:32 - So that kind of gets into the, the sort of prior literature about analogies. And, you know, in, in many instances when we're thinking about the best ways to teach kids, it's good to be direct. You know, if we want them to know something, the most efficient way to, to have them know it is to tell them. and, and certainly as adults, if you or I wanted to make an analogy, it would make sense for,

01:56 - you know, if you're explaining something to me for you to really explain the thing I know that's familiar, the new thing and all the connections between them. and in fact, prior research suggested that, analogies were really difficult for students to learn from and they needed a lot of, of that support, explicit support for how, what prior knowledge do I have that's relevant and how exactly does it map to the new thing? but as you can imagine, that takes a lot of attentional capacity as well.

02:25 - 'cause as the learner, I have to pay attention to elements of the old idea, elements of the new idea and all the correspondences, and then adapt to what I know to something new. So that takes a lot of cognitive capacity, works really well for adults. but I sort of wondered whether that would work just as well for kids. And so that's what led us to thinking about, analogy more as analogical priming

02:49 - with this implicit analogy and less of that really explicit direct instruction. On the analogy itself, there's this great prior literature on analog Transfer and analog problem solving. So that again, is, using something more familiar, to sort of bootstrap and, understand the structure of something more complicated or something newer. A lot of this great not, sorry.

03:13 - There's a lot of prior literature that suggests that students need hints to know what prior knowledge to retrieve, that they need support for that mapping. and a lot of the prior literature also suggested that this is a difficult process, that it doesn't happen. And in fact, we see this in the educational context all the time when, say, kids learn something or students will learn something in calc one and then fail to transfer into their engineering classes. So we see this failure to transfer in the research

03:40 - and we see a failure to transfer in real life. And so a lot of researchers sort of were thinking, that you'd have to make that connection really explicit to students. Now, when we looked at the developmental literature, you can see that even really young infants transfer spontaneously. There's this great study by, Chen and his colleagues, that I love with toddlers. So they're, I think they're 18 months to 2-year-old.

04:04 - They're, they're little toddlers and, they, they're a couple of different conditions. And some of the toddlers are shown a problem solving solution from their parents. It's sort of, series of motor behaviors they have to do to get to a toy. And then they're shown a brand new problem. It the, all the objects are different, all the barriers between them and the toy, but the motor actions are the same.

04:27 - And the kids whose parents showed them how to solve the different problem we're really good at learning very quickly about the new problem. They, they sort of applied those same motor actions that were already activated by watching the parent to this new problem. So on one side, the adult literature tells us theological transfer hard. On the other side we see that infants, young children

04:52 - very spontaneously and easily transfer and apply their knowledge from one context to the next. So then the question that I had is in this, early to middle childhood period, is there a way of supporting that spontaneous implicit transfer that we see in young kids rather than reaching for the cognitively effortful stuff that we observe in adult, in the study we had sixth grade students learning about, the structure of fraction division. So not just how to solve the problem,

05:22 - but what does it mean to divide by a fraction. And we had students learn in three conditions. So in one condition they got the best explicit analogy that we could design. So that included all of the recommendations from prior research on how to identify the prior knowledge, how to represent prior knowledge, how to support connections for students between the old familiar idea and the new idea. So we had two whole number division problems,

05:50 - which are really helpful for thinking about fraction division. 'cause it turns out they're almost the same. You can think about whole number division as how many Times one thing goes into another and you can think about fraction division as how many times one thing goes into another. So in one condition we have two whole number division problems and then two fraction division problems. And the instructor does this really nice beautiful explicit

06:10 - mapping between the two of them. in another condition, our implicit analogy condition, they warm up with whole number division and then we never talk about them again when we get to fraction division. So that's the implicit part. The student's attention is not drawn to the connection, but importantly, there are a lot of implicit cues. So the visual representations are the same for whole number division and fraction division.

06:35 - The language that we use is almost identical for whole number division and fraction division. and then the actions that they see the instructor do are almost identical. So we have all these implicit cues that these things are related without engaging students in an explicit mapping process. So we call that the warmup implicit analogy condition. The third condition is just extra fraction division. So instead of only getting two problems of instruction,

06:59 - they got four problems of instruction on fraction division and no whole number division. So this would be like just spending additional time teaching the thing that you wanna be teaching. So, long story short to what we found was that children learned about the structure of fraction division most effectively in the implicit analogy condition. And in fact, students in the explicit analogy, condition did not learn any better than no analogy at all.

07:23 - Now, the no analogy condition is also a really strong control 'cause they're spending twice as much time on the target instruction. but we didn't see any benefits of warming up prior knowledge when you're making this really explicit connection. one thing that I thought was really interesting is that we also see that the explicit analogy leads to misconceptions. And we've seen that in our prior work as well.

07:46 - Kids when they're explicitly trying to make those connections, sometimes they make the wrong connections. And so then you see issues pop up that wouldn't have otherwise popped up if the kids hadn't been trying so hard to do exactly what the adult told them to do. Yeah, I think there are some really cool implications. So one is to un as for instructors to understand that what children are thinking about, not just children, we've done this work with adults as well.

08:11 - So what learners are thinking about when they enter the classroom shapes how they think about the first thing they learn. And so that means that instructors both have to attend to what students are thinking about at the very beginning of the lesson, but they also have an opportunity to shape it by doing a warmup that activates the prior knowledge that's gonna be most relevant. I think that's really important because very often we see warmup activities in math classrooms,

08:35 - but they're not necessarily chosen with the intent of activating prior knowledge. So they might be chosen with the intent of practicing something you did last week or the week before, but sometimes the thing you just worked on is not the prior knowledge that's gonna lead you to success on the thing you have to do today. And so there's an opportunity For, for instructors to get their kids thinking along the right track

08:58 - before they even engage in instruction. So I think that's the nice, you know, practical, value of this work. another more broader implication I think is for, you know, the field of mathematics, learning development and cognition is to think about how these implicit learning mechanisms are at play even during direct instruction. And I think that our field has focused a lot on, you know, explicit aspects of explicit instruction

09:26 - that we can manipulate and much less on how we can manipulate the environment of learning to make learning more successful, in ways that might not be readily obvious to students, but take a lot of planning for the teacher. So one example of changing the environment is right to, to to activate students' prior knowledge in some way. So that would be, for example, including those warmup activities. another way of changing the environment, let me think

09:54 - of another example that's not exactly the implicit analogy. Example. Okay, so one other way that I think, teachers could make a a, a slight change that's not necessarily a new type of instruction is paying attention to how they use their language for mathematical topics. Okay. So that the language for topics that are similar in structure is also similar to help cue that what we did before is relevant to what we did now, or to have some continuity in the types of visual representations you're using.

10:28 - So if you're using one type of diagram, say number lines to learn about whole number division, that we're using the same type of diagram to learn about fraction division. you know, that type of continuity might not be really explicit for the student. A student may not notice that. but it might help the student be more likely to make a spontaneous connection between those two topics than using diagrams or representations or words

10:58 - and language that's really different. Then there's the additional obstacle of trying to figure out how these things map, or just a failure to spontaneously transfer because there's no queue, there's no implicit queue that that prior knowledge is relevant.

Faculti AI can make mistakes