# Interdisciplinary Learning in Mathematics and Science: Transfer of Learning for 21st Century Problem Solving at University

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Transfer and 21st Century Learning

#### 1.2. The Relationship between Mathematics and Science

#### 1.3. University Mathematics Service Courses and Science Courses

- What are the processes of transfer (if any) evident in students’ “think-aloud” accounts while solving physics exam questions requiring knowledge and understanding from their mathematics service courses?
- What are the challenges in transfer of learning reported by students and by academics?
- What teaching and learning factors do students and academics believe to enhance transfer?

## 2. Literature Review

#### 2.1. Transfer of Learning

Students must apply the skills and knowledge they gain in one discipline to another and what they learn in school to other areas of their lives. A common theme is that ordinary instruction doesn’t prepare learners well to transfer what they learn, but explicit attention to the challenges of transfer can cultivate it.

#### 2.2. Cognitive Explanations of Transfer

#### 2.2.1. Higher Order Thinking and Bloom’s Taxonomy

#### 2.2.2. Mathematical Problem-Solving Theories and Transfer

#### 2.3. Socio-Cultural Explanations of Transfer

## 3. Methods

#### 3.1. Research Design

#### 3.2. Student Think-Aloud Study

#### 3.2.1. Sample

#### 3.2.2. Data Collection

#### 3.2.3. Think-Aloud Tasks

#### 3.2.4. Data Analysis

#### 3.3. Interviews with Academics

#### 3.3.1. Sample

#### 3.3.2. Data Collection and Analysis Methods

## 4. Results and Discussion

#### 4.1. Research Question 1: What Are the Processes of Transfer (If Any) Evident in Students “Think-Aloud” Accounts While Solving Physics Exam Questions Requiring Knowledge and Understanding from Their Mathematics Service Courses?

“in the first question with the Planck’s formula, it’s difficult to understand the question to begin with. So reading through the question is a lot to figure out. And you also have to recognise lots of different symbols, maths symbols which I’m sure if I didn’t know what they were I would be very lost, even more than I was.”

#### 4.2. Research Question 2: What Are the Challenges in Transfer of Learning Reported by Students and by Academics?

#### 4.3. Research Question 3: What Teaching and Learning Factors Do Students and Academics Believe to Enhance Transfer?

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Four levels of procedures of mathematical problem solving. (Source: Seo 2010, p. 230 translated by Yoshitaka Nakakoji).

**Figure 3.**(

**a**) Think-aloud reports on transfer processes for question 1. (

**b**) Think-aloud reports on transfer processes for question 2.

Classification | Explanation | Examples |
---|---|---|

1. Knowledge (i) Knowledge of specifics (ii) Knowledge of ways and means of dealing with specifics (iii) Knowledge of the universals and abstractions in a field | Knowledge contains behaviours and tests situations which emphasize the remembering, either by recognition or recall, of ideas, material, or phenomena (p. 62). | (i) To define technical terms in mathematics by giving their attributes, properties, or relations. (ii) Knowledge of the mathematical techniques and methods used by scientists in seeking to answer questions about the world. (iii) The recall of major theorems in mathematics (pp. 64–77). |

2. Comprehension (i) Translation (ii) Interpretation (iii)Extrapolation | The emphasis is on the grasp of the meaning and intent of the material (p. 144). | (i) The ability to translate abstract concepts in mathematics by giving an illustration or example. (ii) The ability to interpret various types of numerical data. (iii) Skill in predicting continuation of trends using mathematical models. (pp. 92–96). |

3. Application | Remembering and bringing to bear upon given material the appropriate generalizations or principles (p. 144) | (i) The ability to apply scientific principles, postulates, theorems in mathematics, or other abstractions to new situations (ii) The ability to apply the laws of trigonometry to practical situations |

4. Analysis (i) Analysis of elements (ii) Analysis of relationships (iii) Analysis of organizational principles | Emphasizes the breakdown of the material into its constituent parts and detection of the relationships of the parts and of the way they are organized. (p. 144) | (i) Ability to distinguish a conclusion from statements which support it. (ii) Ability to detect logical fallacies in arguments and proof in mathematics. (iii) Ability to recognize form and pattern in mathematics as a means of understanding their meaning (pp. 146–48) |

5. Synthesis (i) Production of a unique communication (ii) Production of a plan, or proposed set of operations (iii) Derivation of a set of abstract relations | Definition: the putting together of elements and parts so as to form a whole. In comparison with the subordinate classifications, this classification is less practical and more emphasis on uniqueness and originality. (p. 162) | (i) Skill in writing, using an excellent organization of ideas and statements in mathematics. (ii) Ability to integrate the results of an investigation into an effective plan or solution to solve a problem in mathematics. (iii) Ability to make mathematical discoveries and generalizations. (pp. 169–72) |

6. Evaluation (i) Judgments in terms of internal evidence (ii) Judgments in terms of external evidence | Definition: the making of judgments about the value, for some purpose, of ideas, works, solutions, methods, material, etc. In addition, the process may require behavior classified in other subordinate categories (p. 185) | (i) Judging internal standards, the ability to assess the quality of quantitative data analysis in relation to the experiments conducted. (ii) The comparison of major theories, generalizations and facts in mathematics and science. (Based on pp. 189, 192) |

**Table 2.**Examples of words, phrases, and mathematical expression associated with coding four processes in solving transfer questions 1 and 2.

Processes | Words or Phrases | Mathematical Expression |
---|---|---|

- 1.
- Interpretation
| I know the meaning of I’m not sure about | c = fλ y = exp (−kx), where k > 0 |

- 2.
- Integration
| This question means I can’t understand the question. | Requires rearrangement of I (f, T) Understanding of a graph of y = exp (−kx), k = (√2m (V−E))/ℏ |

- 3.
- Planning
| I know how to do Show No idea about | ∂I/∂f = 0 at f = f_{max}Preparation of drawing the graph with considering how it changes with respect to m, V, and E. |

- 4.
- Execution
| Substitution of into (Partial) derivative of | λ = c/f, dλ/df = −c/f^{2}Actual sketch of the graph |

Factors Enhance Transfer | Factors Hinder Transfer | |
---|---|---|

Academics’ perspective | Interdisciplinary learning Showing relevance of maths Practice Confidence & self-belief Prior leaning | Anxiety Poor feedback Poor explanation Translation Disparity of pedagogy Mismatch of expectation Surface learning approaches Poor maths preparedness |

Students’ perspective | Importance of understanding the question Memory or recall Prior learning Intuition | Anxiety Mismatch of expectation Difficulty in understanding the problem Difficulty in recalling maths knowledge & skills |

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Nakakoji, Y.; Wilson, R.
Interdisciplinary Learning in Mathematics and Science: Transfer of Learning for 21st Century Problem Solving at University. *J. Intell.* **2020**, *8*, 32.
https://doi.org/10.3390/jintelligence8030032

**AMA Style**

Nakakoji Y, Wilson R.
Interdisciplinary Learning in Mathematics and Science: Transfer of Learning for 21st Century Problem Solving at University. *Journal of Intelligence*. 2020; 8(3):32.
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**Chicago/Turabian Style**

Nakakoji, Yoshitaka, and Rachel Wilson.
2020. "Interdisciplinary Learning in Mathematics and Science: Transfer of Learning for 21st Century Problem Solving at University" *Journal of Intelligence* 8, no. 3: 32.
https://doi.org/10.3390/jintelligence8030032