When children learn about fractions, their prior knowledge of whole numbers often interferes, resulting in a whole number bias. However, many fraction concepts are generalizations of analogous whole number concepts; for example, fraction division and whole number division share a similar conceptual structure. Drawing on past studies of analogical transfer, we hypothesize that children’s whole number division knowledge will support their understanding of fraction division when their relevant prior knowledge is activated immediately before engaging with fraction division. Children in 5th and 6th grade modeled fraction division with physical objects after modeling a series of addition, subtraction, multiplication, and division problems with whole number operands and fraction operands. In one condition, problems were blocked by operation, such that children modeled fraction problems immediately after analogous whole number problems (e.g., fraction division problems followed whole number division problems). In another condition, problems were blocked by number type, such that children modeled all four arithmetic operations with whole numbers in the first block, and then operations with fractions in the second block. Children who solved whole number division problems immediately before fraction division problems were significantly better at modeling the conceptual structure of fraction division than those who solved all of the fraction problems together. Thus, implicit analogies across shared concepts can affect children’s mathematical thinking. Moreover, specific analogies between whole number and fraction concepts can yield a positive, rather than a negative, whole number bias.
In learning mathematics, children’s new knowledge often builds on what they already know. Many studies demonstrate positive effects of prior knowledge on later mathematics learning (e.g.,
These questions are particularly relevant in the domain of fraction learning. A large body of work demonstrates that children’s prior knowledge of whole numbers negatively biases their understanding of fraction symbols (e.g.,
Taken together, these lines of work on fraction understanding paint a mixed picture concerning the effects of children’s prior whole number knowledge. One possible explanation for these mixed findings is that some aspects of children’s prior knowledge of whole numbers support fraction understanding, whereas other aspects do not. Thus, children’s whole number knowledge may not generally support fraction learning, though specific, prior whole number concepts may be leveraged to support specific, novel fraction concepts. However, the specific pathways through which children’s whole number knowledge may support (or bias) fraction understanding remain poorly understood.
In this work, we apply an analogical transfer framework (e.g.,
For example, in one classic study of analogical problem solving among adults,
This framework can be used to make predictions about when prior knowledge should support or hinder children’s mathematics learning. Specifically, children’s prior knowledge of similar mathematics problems should support new learning to the extent that the problems share the same underlying relationships among elements, or
In the current study, we investigate
Research on implicit analogies has largely been conducted with adults, and the timing of the analogies varies across studies. In some studies, the initial problem is solved immediately before the target problem (e.g.,
The timing of analogies during instruction is likely to matter for children’s ability to effectively use implicit analogies. Children may be especially likely to rely on very recently activated concepts or procedures (e.g.,
The current study is focused on conceptual understanding of fraction division—a topic that poses great challenges for many children (
The analogical transfer framework makes predictions regarding what kinds of prior knowledge will be most useful for understanding fraction division, as it implies
In contrast, other operations on fractions (i.e., fraction addition, subtraction, and multiplication) should be less useful analogues for fraction division, because they differ from fraction division in conceptual structure, despite surface similarities. Fraction addition and fraction subtraction involve additive relationships, which are distinct from the multiplicative relationship inherent in fraction division. Fraction multiplication can be construed as iterating a group, whose size is specified by one operand, the number of times specified by the second operand. Although multiplication and division are fundamentally related, as both can be grounded in a multiplicative grouping structure, the relationship between the operands is not the same across operations. Despite these differences in conceptual structure, other operations on fractions are perceptually similar to fraction division, as they share the representation of quantity as a fraction symbol. Fraction division is also similar to other operations on fractions in terms of timing within curricula; fraction arithmetic problems of various sorts often occur within the same instructional unit.
Of course, the conceptual similarity between whole number operations and fraction operations is true for each operation, not only for fraction division. In this research, we focus on fraction division because children’s conceptual understanding of fraction division has been shown to be weak (
Two previous studies have shown that analogies between whole number division and fraction division
Building on this prior work, the present study was designed to address two primary aims. First, we aim to contribute to a resolution of the whole number bias debate. Specifically, we aim to replicate and extend previous findings (e.g.,
To address these aims, we measured children’s conceptual understanding of fraction division under two experimental conditions, one in which children’s knowledge of a structurally similar domain (whole number division) was activated immediately before they were asked to demonstrate fraction division and another in which children’s knowledge of a perceptually similar, but structurally dissimilar, domain (fraction multiplication) was activated immediately before they were asked to demonstrate fraction division. Importantly, we activated children’s knowledge of both whole number and fraction addition, subtraction, multiplication, and division concepts in both conditions. The critical differences between the two experimental conditions were in the
Participants were recruited from a public school district in a midsized Midwestern city through invitations distributed to students in their 5^{th} and 6^{th} grade classes. Participants were recruited at these grade levels because fraction operations are often covered in 5^{th} and 6^{th} grade within this district. We expected students to be proficient in modeling all of the whole number problems and in modeling addition, subtraction, and multiplication with fractions. A total of 63 children participated in the study in the summer after their 5^{th} or 6^{th} grade year. One child was excluded due to not finishing the session in the available time. We asked parents or guardians to report children’s gender and race/ethnicity. The final sample was made up of 28 children who had completed grade 5, and 34 children who had completed grade 6. The sample included 24 girls and 38 boys, and race and ethnicity were distributed as follows: 79% White, 3% Hispanic, 6% Asian, 2% AfricanAmerican and 10% multiple races or ethnicities.
We activated and measured children’s conceptual understanding of arithmetic operations using a modeling task in which children were asked to represent equations using physical manipulatives (unlabeled fraction bars). We chose this task rather than the picture generation and story generation tasks used in prior work (e.g.,
During the study session, participants were asked to use unlabeled fraction bars to model a series of arithmetic expressions. The bars were made up of interlocking blocks of various sizes. Such blocks are sometimes called “fraction blocks” or “fraction towers”, because the size of the fractional pieces is proportional to the size of the whole (e.g., two halves is the same size as three thirds or one whole), and because the pieces can be snapped together and stacked. Children were provided with six kinds of bars: wholes, halves, thirds, fourths, fifths, and sixths. Bars with different sized fractional pieces were different colors, and they were presented in separate containers and connected to make wholes (e.g., the sixths pieces were all in bars of 6/6ths; see
To model the expressions, children were provided with unlabeled fraction bars. The pink bars (a) could not be further divided. The yellow bars (b) could be divided into halves. The light blue bars (c) could be divided into thirds. The dark blue bars (d) could be divided into fourths. The purple bars (e) could be divided into fifths. The black bars (f) could be divided into sixths.
During the study session, children were shown a set of problems, one at a time. The set contained 16 whole number arithmetic problems and 16 fraction arithmetic problems (see
Each session took place in a quiet room on a university campus, and lasted approximately 45 minutes. First, the experimenter introduced the task of using the bars to model the equations, and asked children to say aloud what they were thinking throughout the experiment. Next, the experimenter introduced the unlabeled fraction bars and demonstrated that the differently colored bars could be used to represent different fraction magnitudes. After giving children the opportunity to ask questions about the fraction bars, the experimenter asked the child to model each problem using the bars on the table. Children were asked to start by modeling the first number in the given problem, and then “show what it looks like to
Children were randomly assigned to one of two conditions: blocked by
During the experimental session, the experimenter (the first author) coded the strategies children used to model each problem, based on children’s speech and actions. Children’s strategies were classified as correct or incorrect based on whether they accurately represented the
Addition strategies were coded as correct if children represented the joining of two sets, one as big as the first operand and one as big as the second operand. For example, for 9 + 3/4, children often took out nine whole bars, either nine whole bars or nine conjoined bars that were each equivalent to a whole bar (e.g., nine conjoined bars each made up of two half bars), from the container and put them in one group on the table, then made a second group of three onefourths bars on the table, and finally pushed these groups together to represent addition of the two operands. Subtraction strategies were coded as correct if children represented taking away or cancelling the second operand from the first operand. For example, for 5 – 2/3, children often made a group of five conjoined wholes from the container of thirds bars, and then disconnected two thirds bars from one of these wholes, separating the two thirds bars from the remaining four and onethird bars. Multiplication strategies were coded as correct if children represented one operand as the size of a group and the other operand as the number of groups. For example, for 6 * 2/3, some children made six groups with two thirds bars in each group. Other children counted out six conjoined wholes using the thirds bars, disconnected two thirds from each whole bar, and put all the twothirds bars together in one group as their answer.
Strategies on division problems were coded as correct if they reflected either a
Children who used a correct, quotative strategy modeled the divisor as the size of the groups. For example, for the problem, 15 ÷ 5, a quotative strategy might involve separating five bars at a time from the initial group, yielding three equal groups of five bars each (see
Children who used a correct, partitive strategy modeled the divisor as the number of groups. For example, for the problem 15 ÷ 5, a partitive strategy might involve making a group of 15 whole bars and then dividing them into five groups (see
Importantly, although US learners often use both partitive and quotative models to represent
Sample end states for the modeling task for partitive division of 15 ÷ 5 (Panel A), quotative division of 15 ÷ 5 (Panel B), and quotative division of 8 ÷ 3/4 (Panel C).
No child in our dataset used a partitive strategy for modeling fraction division. In this regard, it is worth noting that partitive division is
Strategy  Whole Number Division Examples (15 ÷ 5)  Fraction Division Examples (8 ÷ 3/4) 

Quotative (Correct)  Set out 15 bars, divide into three groups of 5  Set out 8 cojoined bars of fourths, divide into ten groups of 3/4, set aside remainder 
Partitive (Correct)  Set out 15 bars, divide into five groups of 3  Not observed 
Quotative and Partitive Hybrid (Correct)  Not applicable  Set out 8 bars, divide into 32 groups of 1/4, divide those groups into three groups of 10, representing (8 ÷ 1/4) ÷ 3, set aside remainder 
InvertandMultiply^{a}  Not observed  Make 8 bars, each made of up 4 onethird pieces, representing 8 * (4/3) 
NonDivision Operation(s)  Not observed  Set out eight 3/4 bars (8 * 3/4) 
Division of Whole Number Components  Not applicable  Set out 8 bars, divide into groups of 4, representing (8 ÷ 4) 
Model Operands Only  Set out one set of 15 bars and one set of 5 bars  Set out one set of 8 bars and one 3/4 bar 
^{a}Although this strategy yields a correct
One child generated a unique, hybrid strategy for division by a nonunit fraction that combined elements of quotative division by a unit fraction and partitive whole number division. For the problem 7 ÷ 3/5, this child first represented 7 ÷ 1/5 using a quotative approach, creating 35 groups of size 1/5. She then further divided these pieces into 3 groups using partitive whole number division, yielding 3 groups of 11 with a remainder of 2. This unexpected “hybrid” strategy was classified in a separate category, and was used only by this particular child on two problems that involved division by a nonunit fraction.
On two whole number division problems and two fraction division problems, the quotient included a remainder. We did not require children to correctly name the remainder as a fraction of a group (e.g., on 8 ÷ 3/4, a remainder of 2/4 as 2/3 of one group) in order to code their behavior and speech as reflecting a partitive or quotative division strategy. Instead, we focused on the nature of the grouping structure. Many children ignored the remainder, some children correctly named the remainder (either as a fraction of a group or as a remainder), and some children incorrectly named the remainder.
Children also sometimes implemented incorrect strategies for modeling fraction division; these strategies were classified into several subcategories (see
One common variant of modeling nondivision fraction operations involved a combination of fraction subtraction and fraction multiplication. For example, for 8 ÷ 3/4, this strategy involved first taking eight whole bars made of fourths bars, disconnecting three fourths bars from each whole bar, separating these eight threefourths bars from the original group, and indicating the remaining set of 8 onefourth bars as the answer. Given the similarity of the first part of this strategy (i.e., disconnecting threefourths from each bar) to children’s fraction multiplication strategies, and the similarity of the second part of the strategy (i.e., separating the disconnected bars from the original group) to children’s subtraction strategies, we considered this strategy to represent 8 – (8 * 3/4), a combination of nondivision fraction operations.
It is important to note that all of the “incorrect” strategies resulted in incorrect solutions for the arithmetic expression, with the exception of the invertandmultiply strategy. However, the task put to the participant was to “show” what the expression means. In this context, the invertandmultiply strategy was considered to be incorrect, because it does not reflect the grouping relationship between the dividend and divisor.
On some trials, a child began to use one strategy, abandoned it, and then used a different strategy. For example, for the problem 5 ÷ 1/4, one child set out five whole bars, and then said, “you have to have 1/4 piles?”. Interestingly, this language mirrored this participant’s language in the preceding whole number division trial (for 17 ÷ 6, “you have to have 6 piles”), and reflects an attempt to apply a partitive model of division to fraction division. Then, he began to divide a whole bar into groups of onefourth, reflecting a quotative model of fraction division. However, after making two groups of onefourth, he decided to start again, and said, “it would be like taking one fourth away from each one”, and then demonstrated 5 – (5 * 1/4) instead. In such cases of strategy switching, the child’s final strategy was coded for analyses of accuracy and strategy use. In our analyses of children’s division strategies, we also report evidence of strategy switching on the critical, fraction division trials.
To evaluate the reliability of the online coding of accuracy, strategy use, and strategy switching carried out by the experimenter during data collection, the second author independently coded the data from the video recordings of the sessions for a subset of the participants. The two coders agreed on whether the child’s modeling of division was correct or incorrect on 100% of 80 trials (4 whole number and 4 fraction division trials from each of 10 participants). Moreover, the two coders agreed on the specific strategy code on 99% (79) of these trials. The two coders agreed on the presence or absence of strategy switching on 99% (71) of 72 trials (4 fraction division trials from each of 18 participants).
The coded data and the corresponding analyses on which the results are based can be found in the
Problem Type, Condition  Addition  Subtraction  Multiplication  Division 

Whole Number Problems  
Blocked by Number Type  4.00 (0.00)  4.00 (0.00)  3.61 (1.02)  3.77 (0.50) 
Blocked by Operation  4.00 (0.00)  3.87 (0.72)  3.84 (0.73)  3.68 (0.65) 
Fraction Problems  
Blocked by Number Type  3.90 (0.55)  3.60 (1.13)  3.00 (1.67)  1.26 (1.86) 
Blocked by Operation  3.94 (0.36)  3.77 (0.88)  3.29 (1.42)  1.97 (1.96) 
We examined children’s conceptual understanding of each fraction operation under the two experimental conditions. We hypothesized that children whose whole number division concepts were activated immediately before modeling fraction division (i.e., children in the
As our goal was to investigate whether children’s whole number knowledge would support their fraction understanding, for each analysis, we restricted our sample to only those participants who accurately modeled the analogous whole number problems. Thus, the sample size for each analysis varied slightly, depending on the number of participants who were successful with the whole number operation in question. The restricted sample sizes are as follows:
To test whether condition affected children’s ability to model addition, subtraction, or multiplication, we regressed the number of fraction addition, subtraction, and multiplication problems modeled correctly (in three separate regression models) on condition, controlling for child’s grade level. As shown in
To test our main hypothesis, we regressed the number of fraction division equations modeled correctly (0  4) on condition, controlling for child’s grade level. Condition significantly predicted fraction division performance,
The average number of correctly modeled trials for each fraction operation, out of four, by experimental condition, controlling for grade level. Error bars represent the standard errors of each point estimate from each model.
To more closely examine transfer from children’s knowledge of whole number division or other operations on fractions to children’s strategies on fraction division trials, we analyzed the nature of children’s strategy use on fraction division trials in both conditions.
We first examined the consistency of children’s strategy use on whole number and fraction division, and whether the degree of consistency varied across conditions. To address this issue, we examined whether children in the
We coded whether children ever used a quotative strategy on the whole number division trials. Among the 48 children who used either a correct partitive or quotative strategy on all four whole number division trials, 22 children used quotative strategies on all four trials, 7 used a mixture of quotative and partitive strategies across trials, and 19 used partitive strategies on all four trials; see
The distributions of dominant fraction division strategies by whole number division strategy use and by experimental condition are shown in
We also examined whether grouping problems by number type caused children to draw on their recently activated
Whole Number Division Strategies  Quotative^{a}  NonDivision Operations  InvertandMultiply  Division of Whole Number Components  No Model  

Whole Number Division: Quotative Only  
Blocked by Number Type  3 (33.3%)  4 (44.4%)  2 (22.2%)  0 (0.0%)  0 (0.0%)  9 
Blocked by Operation  10 (76.9%)  2 (15.4%)  1 (7.7%)  0 (0.0%)  0 (0.0%)  13 
Whole Number: Mixed Quotative & Partitive  
Blocked by Number Type  1 (25.0%)  0 (0.0%)  3 (75.0%)  0 (0.0%)  0 (0.0%)  4 
Blocked by Operation  2 (66.7%)  0 (0.0%)  0 (0.0%)  0 (0.0%)  1 (33.3%)  3 
Whole Number: Partitive Only  
Blocked by Number Type  3 (27.3%)  4 (36.4%)  3 (27.3%)  0 (0.0%)  1 (9.1%)  11 
Blocked by Operation  3 (37.5%)  2 (25.0%)  2 (25.0%)  0 (0.0%)  1 (12.5%)  8 
^{a}One child used quotative division on two fraction division trials and the hybrid strategy on the other two trials. This child’s dominant strategy was coded as quotative division.
Such strategies representing fraction multiplication, fraction subtraction, or a mix of incorrect operations with the given divisor were coded as
We also considered the frequency with which children used nondivision operations as a
Finally, we examined the frequency with which children switched from one strategy to another midtrial on any of the fraction division trials. Given that students in the United States tend to prefer quotative models of division for fraction division problems, we expected that children who had used only partitive strategies on whole number division might have more difficulty modeling fraction division problems, and that this difficulty might manifest in unsuccessful attempts to model fraction division.
The rate of strategy switching was low in the sample as a whole; only five children were coded as using more than one strategy in the course of at least one fraction division trial. However, all five of these children had used only partitive strategies for whole number division. Thus, as expected, children who
Children readily apply their prior knowledge to make sense of problemsolving challenges, and this spontaneous transfer can bias them in either positive or negative ways. In the case of children’s fraction learning, there are many ways in which children’s prior knowledge of whole number arithmetic negatively biases fraction reasoning (e.g.,
We found that children who demonstrated whole number division immediately before modeling fraction division were significantly more successful at modeling fraction division than those who had demonstrated fraction multiplication and other operations on fractions immediately before modeling fraction division. This finding is in line with other research that shows the beneficial effect of analogies between whole number division and fraction division (
Our study leaves open the question of the specific mechanism underlying this recency effect. One possibility is that recency is beneficial in this context because the recently activated knowledge facilitates performance, but it does not remain activated for very long. Another possibility is that recency is beneficial because activation is subject to interference from intervening content. To better understand the mechanism underlying the recency effect, we examined the nature of children’s strategies in each condition.
Among children who used the quotative strategy for whole number division, those who demonstrated fraction division immediately after whole number division were more likely to use quotative strategies on fraction division than children who demonstrated fraction division after other operations on fractions. This finding suggests that the increased success of children in the
We also examined children’s incorrect strategies in each condition for evidence of negative transfer from other operations on fractions. Many children whose knowledge of fraction multiplication (and other fraction operations) was recently activated made errors reflecting negative transfer from other operations on fractions. However, several children whose knowledge of whole number division was recently activated also made such errors. Thus, we did not find a significant difference across conditions in nondivision errors.
One reason why children whose knowledge of whole number division was recently activated may have made other operation errors at similar rates to those children whose fraction multiplication knowledge was recently activated may be that such errors reflect robust misconceptions about fraction division. Other studies have revealed that otheroperations errors are quite common on fraction division problems (e.g.,
We did not include a condition in which children modeled fraction division without any targeted activation of their prior knowledge. Thus, without knowing what strategies are typical in this task, absent any experimental manipulation, we are unable to conclude whether such interference is truly occurring. Future research is needed to establish whether implicit analogies can lead to
In sum, our strategy data suggest that
In many cases, children’s reasoning about fractions appears to be hindered by their strongly activated whole number knowledge. Children’s, and even adults’, reliance on whole number knowledge when reasoning about fractions often appears automatic and difficult to inhibit (e.g.,
Though many previous studies have found evidence for detrimental effects of the whole number bias, we found an advantage for activating children’s knowledge of a specific whole number concept, whole number division, to support their reasoning about an analogous fraction concept, fraction division. These results fit well with the integrated theory of numerical development, posited by
Given the mixed findings on the benefits of drawing on children’s whole number knowledge for understanding fractions, the specificity of the instructional analogy is presumably critical. Had we activated children’s whole number division knowledge more generally, children might have drawn on other aspects of that knowledge that do not apply to fraction division, such as the idea that “
Furthermore, studies that show a negative whole number bias on children’s operation understanding (e.g.,
It remains unclear the extent to which children’s whole number division knowledge supports children’s
In studies of analogies in mathematics instruction, analogies involve explicitly supporting children’s comparisons across structurally similar problems (e.g.,
In contrast, several studies show that adults need not notice an analogy in order for it to affect their understanding of a target concept (e.g.,
Further work is needed to better understand the mechanisms by which implicit analogies support children’s thinking, how long the advantages conferred by implicit analogies last, and how implicit analogies might be capitalized on in instruction. Given previous studies demonstrating that more cognitive support for explicit analogies leads to better learning, future studies are needed to directly compare implicit analogies to explicit, highlycued analogies. It may be the case that implicit analogies and explicit, highlycued analogies have different effects on learning, or that highlycued analogies have longerlasting effects on learning.
It is also unclear whether implicit analogies may be more useful for some types of analogical transfer than for others. In the current study, we considered analogies from earlierlearned to laterlearned problems, and this type of analogy could be construed as
We have argued that children’s recently activated knowledge sets the stage for their understanding of what comes next. This general claim has obvious relevance for the organization of classroom instruction. Teachers often begin lessons by reviewing recently learned concepts or posing “warmup” questions, and it may be useful to think of these activities as implicit analogies. These analogies will support new learning to the extent that the concepts or procedures activated by the analogies are relevant to the new concepts or procedures that follow. Thus, there are likely to be advantages to identifying structurally similar analogues from students’ own prior knowledge before beginning a new lesson, and practicing the most relevant previouslymastered concept to “warm up”.
For fraction learning specifically, instruction on fraction division often follows instruction on other fraction operations in classroom lessons. However, activating students’ knowledge of whole number division may be a better way to start such lessons. This implication is particularly important given the difficulties that children have with fraction division (e.g.,
Finally, our strategy data suggest an unexpected avenue for future research that has potential to influence educational practice: examining the role of the
Though children’s whole number concepts sometimes negatively bias their understanding of fraction concepts, we believe that it is also important to examine the conditions under which children’s whole number knowledge supports fraction learning. In this research, we found that activating children’s most relevant, structurally similar whole number concept (i.e., whole number division) immediately before asking them to model an analogous fraction concept (i.e., fraction division) increased children’s likelihood of demonstrating the fraction concept correctly. Thus, analogies made across number domains that are often thought to cause interference (such as whole numbers and fractions) can be fruitful, as long as targeted links—even implicit ones—are made across specific, analogous concepts.
The analogy between whole number division and fraction division was not made explicit in this study. Instead, the temporal proximity of analogous problems implicitly cued the analogy. These findings add to a growing body of work on the importance of implicit cues in problem solving and learning, and they document that analogical transfer can be cued implicitly as well as explicitly. Moreover, they show that implicit cuing of analogy occurs among children and in mathematics tasks.
Our findings underscore the importance of sequencing in mathematics curricula, and they demonstrate that order matters, even at the level of individual problem types. What children are thinking about immediately before solving difficult problems shapes their problemsolving approaches and affects their likelihood of solving those problems correctly. By applying an analogical transfer perspective to the task of creating contexts for learning, we can gain insight into effective ways to actively draw on students’ prior knowledge for new learning.
Operation  Whole Numbers  Fractions 

Addition  5 + 17  6 + 
23 + 8  9 + 

7 + 11  4 + 

19 + 4  3 + 

Subtraction  20 – 8  5 – 
14 – 3  2 – 

22 – 15  6 – 

13 – 9  7 – 

Multiplication  7 × 10  3 × 
9 × 8  5 × 

11 × 5  4 × 

3 × 12  6 × 

Division  15 ÷ 5  5 ÷ 
32 ÷ 8  7 ÷ 

18 ÷ 4  4 ÷ 

17 ÷ 6  8 ÷ 
Today, I’m going to ask you to show me what a bunch of math sentences like this one [
This is a study about what kids think about math. So, to know what you’re thinking, I’m going to ask you to talk out loud the whole time. What I mean by talk out loud is that I want you to say out loud everything that you say to yourself silently. Don’t worry about whether or not your thoughts make sense to anyone else, just say whatever you’re thinking.
Before we get started, I’m going to tell you a bit about these blocks. When you use these blocks, consider something that's this big [
For each of these equations, start by showing the first number and then show what it looks like to do the operation using the second number. For example, if I want to model 5 + 17, I should start by showing 5 [
[
Great, one last thing. Try not to solve the problem in your mind, but instead show how you can use the blocks to solve the problem. Some problems will be easy, so you will want to solve it in your mind, but please try to show it with the blocks instead. Some problems will be harder. Even if you’re not sure how to show what the problem means, please try anyways, because even your mistakes can tell us about how kids like you think about math.
If participant asks, “How do I do it?” or “Can I do ____?”
Experimenter
If participant asks, “What if I don’t know the answer?” or “I don’t know the answer.”
Experimenter: “That’s okay, we’re not interested in whether or not you know the right answer. We just want to know what you think.”
If participant only models each number in the equation:
Experimenter: Can you show me what it looks like to (
[
If participant’s action is unclear:
Experimenter: Can you show me which part is the answer?
This research was funded in part by a Marian S. Schwartz Award to the first author from the Department of Psychology at the University of WisconsinMadison.
The authors thank Joana Bielefeld and Samantha Azuma for their help in data collection and coding.
The authors have declared that no competing interests exist.