# Learning Multiplication by Translating across Microworlds

^{*}

## Abstract

**:**

## 1. Introduction

## 2. On Learning Multiplication with Multiple Models and Representations

## 3. Theoretical Framing

## 4. Methodological Approach

#### 4.1. TouchTimes—Two Microworlds of Multiplication

#### 4.2. Research Context

- A bunch of buttons fell on the floor. Nick gathered them in heaps of 5 buttons. He made 8 heaps. How many buttons are there?
- Aria planted some tomatoes in her mini-garden. She planted them in rows of 5 tomatoes and made 9 rows. How many tomatoes did she plant?

- 3.
- Colin has 4 hoodies and 5 sweatpants. How many ways can he match the hoodie and the sweatpants?

## 5. Coordinating between Grasplify and Zaplify

#### 5.1. Three Cases of Translation with Ian and Fabian

#### 5.2. Patterns of Translations

#### 5.2.1. Button Problem

#### 5.2.2. Tomato Problem

#### 5.2.3. Combinatorics Problem

#### 5.3. Follow-Up Questions

#### 5.4. Untranslatables

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**(

**a**) Positing the hand horizontally instead of along the left edge; (

**b**) making 5 pips in Grasplify; (

**c**) making 8 pips and 5 pods in Grasplify.

Grasplify | Zaplify | |
---|---|---|

Placement of fingers | Right and left | Side and bottom |

Screen objects | Pips and pods | Horizontal and vertical lines making intersection points |

Multiplication objects | Multiplicand and multiplier; expression (i.e., 3 × 4) | Factors; two numbers (i.e., 3 on the top and 4 on the side) |

Orientation/view | Landscape | Portrait or landscape |

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**MDPI and ACS Style**

Tan, S.; Chorney, S.; Sinclair, N.
Learning Multiplication by Translating across Microworlds. *Educ. Sci.* **2024**, *14*, 423.
https://doi.org/10.3390/educsci14040423

**AMA Style**

Tan S, Chorney S, Sinclair N.
Learning Multiplication by Translating across Microworlds. *Education Sciences*. 2024; 14(4):423.
https://doi.org/10.3390/educsci14040423

**Chicago/Turabian Style**

Tan, Sheena, Sean Chorney, and Nathalie Sinclair.
2024. "Learning Multiplication by Translating across Microworlds" *Education Sciences* 14, no. 4: 423.
https://doi.org/10.3390/educsci14040423