# First-Year Mathematics and Its Application to Science: Evidence of Transfer of Learning to Physics and Engineering

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Transfer in Science/Engineering Educational Research

#### 2.2. Quantitative Measures of Transfer

- (i)
- A student gave the correct answers in both sections. His or her transfer score was 2, and it is assumed that transfer of learning occurred;
- (ii)
- A student gave the wrong answer in a mathematics question; however, he or she answered correctly on its corresponding non-mathematics question. A score of 1 was given, as it was considered that to some extent, transfer of learning had occurred;
- (iii)
- If students gave a right answer in a mathematics section, but did not get the corresponding question in non-mathematics section, a score of 0 was awarded;
- (iv)
- If students answered incorrectly in both questions, a score of 0 was given.

- Can transfer of mathematics learning be observed in the biology, molecular bioscience, engineering, and physics exam performances?
- How is transfer related to overall attainment in mathematics and science/engineering courses?
- What are the relationships between general educational attainment (university entrance rank), mathematics attainment, science/engineering attainment, and the transfer of learning between mathematics and science/engineering?

## 3. Materials and Methods

#### 3.1. The Mathematics and Science Courses Examined and Their Assessment

#### 3.2. Operationalisation of the Transfer of Mathematics

#### 3.3. Demonstration of Calculation of the Transfer Scores and the Transfer Index

#### 3.4. ATAR-Adjusted Transfer Index

#### 3.5. Modeling the Relationships between Transfer and Attainment in Mathematics and Science/Engineering

## 4. Results

#### 4.1. Can Transfer of Mathematics Learning be Observed in Biology, Molecular Bioscience, Engineering and Physics Exam Performance?

#### 4.2. How Is Transfer Related to Overall Attainment in Mathematics and Physics/Engineering Courses?

#### 4.3. What Are the Relationships between General Educational Attainment (University Entrance Rank), Mathematics Attainment, Physics/Engineering Attainment, and the Transfer of Learning between Mathematics and Physics/Engineering?

## 5. Discussion

#### 5.1. Transfer from Mathematics to Science

#### 5.2. Relationships between Transfer and Educational Attainment

#### 5.3. Strengths, Limitations, and Insights for Future Transfer Research

#### 5.4. Implications for Teaching and Learning Science

## Author Contributions

## Conflicts of Interest

## References

- Australian Industry Group. Lifting Our Science, Technology, Engineering and Maths (STEM) Skills. 2013. Available online: http://www.utas.edu.au/__data/assets/pdf_file/0005/368546/lifting_our_stem_skills_13.pdf (accessed on 28 November 2017).
- Office of the Chief Scientist. Mathematics, Engineering & Science in the National Interest. 2012. Available online: http://www.chiefscientist.gov.au/2012/05/mes-report/ (accessed on 28 November 2008).
- Stokke, A. What to Do about Canada’s Declining Math Scores. 2015. Available online: https://www.cdhowe.org/sites/default/files/attachments/research_papers/mixed/commentary_427.pdf (accessed on 28 November 2017).
- Government of India. Annual Report 2013–14; Department of Science and Technology: New Delhi, India, 2014. Available online: http://www.dst.gov.in/annual-report-2013-14-0 (accessed on 28 November 2017).
- Science Council of Japan. Nihon no Tenbou—Rigaku Kougaku Karano Teigen (Prospects of Japan: Suggestions from science and engineering). 2010. Available online: http://www.scj.go.jp/ja/info/kohyo/pdf/kohyo-21-tsoukai-3.pdf (accessed on 28 November 2017).
- The Royal Society. Vision for Science and Mathematics Education. 2014. Available online: https://royalsociety.org/education/policy/vision/ (accessed on 28 November 2017).
- National Research Council. The Mathematical Sciences in 2025; National Academies Press: Washington, DC, USA, 2013. [Google Scholar]
- U.S. Congress Joint Economic Committee. STEM Education: Preparing for the Jobs of the Future. 2012. Available online: https://www.jec.senate.gov/public/index.cfm/democrats/2012/4/stem-education-preparing-jobs-of-the-future (accessed on 28 November 2017).
- Massachusetts Institute of Technology. Strategic Plan 2011–2016: From Imagination to Impact. 2010. Available online: http://odge.mit.edu/wp-content/uploads/2012/09/ODGE-Strategic-Plan-web.pdf (accessed on 28 November 2017).
- U.S. Department of Education. STEM 2026: A Vision for Innovation in STEM Education. 2016. Available online: https://innovation.ed.gov/files/2016/09/AIR-STEM2026_Report_2016.pdf (accessed on 28 November 2017).
- Bransford, J.D.; Brown, A.L.; Cocking, R.R. How People Learn: Brain, Mind, Experience, and School; National Academy Press: Washington, DC, USA, 1999. [Google Scholar]
- Mayer, R.E.; Wittrock, M.C. Problem-solving transfer. In Handbook of Educational Psychology; Berliner, D.C., Calfee, R.C., Eds.; Simon & Schuster Macmillan: New York, NY, USA, 1996. [Google Scholar]
- Thorndike, E.L.; Woodworth, R.S. The influence of improvement in one mental function upon the efficiency of other functions. Psychol. Rev.
**1901**, 8, 247–261. [Google Scholar] - Karakok, G. Students’ Transfer of Learning of Eigenvalues and Eigenvectors: Implementation of Actor-Oriented Transfer Framework; ProQuest LLC: Ann Arbor, MI, USA, 2009. [Google Scholar]
- Lobato, J.; Rhodehamel, B.; Hohensee, C. “Noticing” as an alternative transfer of learning process. J. Learn. Sci.
**2012**, 21, 433–482. [Google Scholar] [CrossRef] - Bock, D.D.; Deprez, J.; Dooren, W.V.; Roelens, M.; Verschaffel, L. Abstract or concrete examples in learning mathematics? A replication and elaboration of Kaminski, Sloutsky, and Heckler’s study. J. Res. Math. Educ.
**2011**, 42, 109–126. [Google Scholar] [CrossRef] - Fyfe, E.R.; McNeil, N.M.; Borjas, S. Benefits of “concreteness fading” for children’s mathematics understanding. Learn. Instr.
**2015**, 35, 104–120. [Google Scholar] [CrossRef] - McNeil, N.; Fyfe, E. “Concreteness fading” promotes transfer of mathematical knowledge. Learn. Instr.
**2012**, 22, 440–448. [Google Scholar] [CrossRef] - Siler, S.; Willow, K. Individual differences in the effect of relevant concreteness on learning and transfer of a mathematical concept. Learn. Instr.
**2014**, 33, 170–181. [Google Scholar] [CrossRef] - Barnett, S.M.; Ceci, S.J. When and where do we apply what we learn? A taxonomy for far transfer. Psychol. Bull.
**2002**, 128, 612–637. [Google Scholar] [CrossRef] [PubMed] - Nakakoji, Y.; Wilson, R. Transfer of mathematical learning to science: An integrative review. Manuscript in preparation.
- Jackson, D.; Johnson, E. A hybrid model of mathematics support for science students emphasizing basic skills and discipline relevance. Int. J. Math. Educ. Sci. Technol.
**2013**, 44, 846–864. [Google Scholar] [CrossRef] - Matthews, K.; Hodgson, Y.; Varsavsky, C. Factors influencing students’ perceptions of their quantitative skills. Int. J. Math. Educ. Sci. Technol.
**2013**, 44, 782–795. [Google Scholar] [CrossRef] - Rylands, L.; Simbag, V.; Matthews, K.; Coady, C.; Belward, S. Scientists and mathematicians collaborating to build quantitative skills in undergraduate science. Int. J. Math. Educ. Sci. Technol.
**2013**, 44, 834–845. [Google Scholar] [CrossRef] - Becker, K.; Park, K. Effects of integrative approaches among science, technology, engineering, and mathematics (STEM) subjects on students’ learning: A preliminary meta-analysis. J. STEM Educ.
**2011**, 12, 23–37. [Google Scholar] - Anderton, R.; Hine, G.; Joyce, C. Secondary school mathematics and science matters: Academic performance for secondary students transitioning into university allied health and science courses. Int. J. Innov. Sci. Math. Educ.
**2017**, 25, 34–47. [Google Scholar] - Sadler, P.M.; Tai, R.H. The two high-school pillars supporting college science. Science
**2007**, 317, 457–458. [Google Scholar] [CrossRef] [PubMed] - Sazhin, S. Teaching mathematics to engineering students. Int. J. Eng. Ed.
**1998**, 14, 145–152. [Google Scholar] - Bransford, J.D.; Schwartz, D.L. Rethinking transfer: A simple proposal with multiple implications. Rev. Res. Educ.
**1999**, 24, 61–100. [Google Scholar] - Detterman, D.K. The case for the prosecution: Transfer as an epiphenomenon. In Transfer on Trial: Intelligence, Cognition, and Instruction; Detterman, D.K., Sternberg, R.J., Eds.; Ablex Publishing Corporartion: Norwood, NJ, USA, 1993. [Google Scholar]
- Gruber, H.; Law, L.; Mandl, H.; Renkl, A. Situated learning and transfer. In Learning in Humans and Machines: Towards an Interdisciplinary Learning Science; Reimann, P., Spada, H., Eds.; Elsevier Science Ltd.: Oxford, UK, 1995. [Google Scholar]
- Hatano, G.; Greeno, J.G. Commentary: Alternative perspectives on transfer and transfer studies. Int. J. Educ. Res.
**1999**, 31, 645–654. [Google Scholar] [CrossRef] - Sternberg, R.J.; Frensch, P.A. Mechanisms of transfer. In Transfer on Trial: Intelligence, Cognition, and Instruction; Detterman, D.K., Sternberg, R.J., Eds.; Ablex Publishing Corporartion: Norwood, NJ, USA, 1993. [Google Scholar]
- Potgieter, M.; Harding, A.; Engelbrecht, J. Transfer of algebraic graphical thinking between mathematics and chemistry. J. Res. Sci. Teach.
**2008**, 45, 197–218. [Google Scholar] [CrossRef] - Britton, S.; New, P.B.; Sharma, M.D.; Yardley, D. A case study of the transfer of mathematics skills by university students. Int. J. Math. Educ. Sci. Technol.
**2005**, 36, 1–13. [Google Scholar] [CrossRef] - Roberts, A.L.; Sharma, M.D.; Britton, S.; New, P.B. An index to measure the ability of first year science students to transfer mathematics. Int. J. Math. Educ. Sci. Technol.
**2007**, 38, 429–448. [Google Scholar] [CrossRef] - Arbuckle, J.L. IBM
^{®}SPSS^{®}Amos™ 22 User’s Guide. 2013. Available online: http://www.uni-paderborn.de/fileadmin/imt/softwarelizenzen/spss/IBM_SPSS_Amos_User_Guide-22.pdf (accessed on 28 November 2017). - Byrne, B.M. Structural Equation Modeling with AMOS: Basic Concepts, Applications, and Programming; Routledge: New York, NY, USA, 2010. [Google Scholar]
- Champagne, A.B.; Klopfer, L.E. A causal model of students’ achievement in a college physics course. J. Res. Sci. Teach.
**1982**, 19, 299–309. [Google Scholar] [CrossRef] - Dehipawala, S.; Shekoyan, V.; Yao, H. Using mathematics review to enhance problem solving skills in general physics classes. In Proceedings of the 2014 Zone 1 Conference of the American Society for Engineering Education, University of Bridgeport, CT, USA, 3–5 April 2014. [Google Scholar]
- Taub, G.E.; Floyd, R.G.; Keith, T.Z.; McGrew, K.S. Effects of general and broad cognitive abilities on mathematical achievement. Sch. Psychol. Q.
**2008**, 23, 187–198. [Google Scholar] [CrossRef] - Deary, I.J.; Strand, S.; Smith, P.; Fernandes, C. Intelligence and educational achievement. Intelligence
**2007**, 35, 13–21. [Google Scholar] [CrossRef] - Australian Government. How much of the Variation in Literacy and Numeracy Can be Explained by School Performance? 2008. Available online: https://archive.treasury.gov.au/documents/1421/HTML/docshell.asp?URL=05%20How%20much%20of%20the%20variation%20in%20Literacy%20and%20Numeracy%20can%20be%20explained%20by%20School%20Performance.htm (accessed on 28 November 2017).
- Haskell, R.E. Transfer of Learning: Cognition, Instruction and Reasoning; Academic Press: San Diego, CA, USA, 2001. [Google Scholar]
- Nakakoji, Y.; Wilson, R.; Poladian, L. Mixed methods research on the nexus between mathematics and science. Int. J. Innov. Sci. Math. Educ.
**2014**, 22, 61–76. [Google Scholar] - Van Someren, M.W.; Barnard, Y.F.; Sandberg, J.A.C. The Think Aloud Method: A Practical Guide to Modelling Cognitive Processes; Academic Press: San Diego, CA, USA, 1994. [Google Scholar]

**Figure 2.**Full path model of the relationships between transfer and ATAR, attainments in mathematics and science/engineering, and demographic variables.

**Figure 3.**Path models showing the relationship among transfer, ATAR, and attainments in mathematics and physics/engineering. (

**a**) normal mathematics and regular physics, (

**b**) advanced mathematics and advanced physics, (

**c**) normal mathematics and engineering

Context: When and Where Transferred from and to | |||||
---|---|---|---|---|---|

Near ←―――――――――――――――――――――――――――――――――――――→ Far | |||||

Knowledge domain | Mouse vs. rat | Biology vs. botany | Biology vs. economics | Science vs. history | Science vs. art |

Physical context | Same room at school | Different room at school | School vs. research lab | School vs. home | School vs. the beach |

Temporal context | Same session | Next day | Weeks later | Months later | Years later |

Functional context | Both clearly academic | Both academic but one nonevaluative | Academic vs. filling in tax forms | Academic vs. informal questionnaire | Academic vs. at play |

Social context | Both individual | Individual vs. pair | Individual vs. small group | Individual vs. large group | Individual vs. society |

Modality | Both written, same format | Both written, multiple choice vs. essay | Book learning vs. oral exam | Lecture vs. wine testing | Lecture vs. wood carving |

No | Formulae to Measure Transfer |
---|---|

1 | Transfer rating = z-score for first attempted component − z-score for mathematics |

2 | Transfer index = the sum of transfer scores ÷ the number of paired questions × 50 |

Semester 1 Course Codes & Names | MATH1901 | MATH1001 | |
---|---|---|---|

Semester 2 Course Codes & Names | Differential Calculus (Advanced) | Differential Calculus | |

PHYS1902 | Physics 1B (Advanced) | 67 | 27 |

PHYS1003 | Physics 1 (Regular) | 28 | 136 |

ENGG1802 | Engineering Mechanics | 44 | 382 |

MBLG1901 | Molecular Biology and Genetics (Advanced) | 33 | 72 |

MBLG1001 | Molecular Biology and Genetics | 53 | 190 |

BIOL1902 | Living Systems (Advanced) | 12 | 20 |

BIOL1002 | Living Systems | 6 | 55 |

Total Enrolment for Combination of Two Courses | 243 | 882 |

Math score | 1 | 0 | 1 | 0 |

Non-math score | 1 | 1 | 0 | 0 |

Transfer score | 2 | 1 | 0 | 0 |

Transfer Index | Min | Max | n | Mean | Mode | SD | SE of Mean | |
---|---|---|---|---|---|---|---|---|

MATH1901 Differential Calculus (Advanced) | PHYS1902 Physics 1B (Advanced) | 2.5 | 95.0 | 67 | 48.69 | 26.25/92.50 * | 28.29 | 3.46 |

PHYS1003 Physics 1 (Regular) | 0.0 | 95.0 | 28 | 47.02 | 43.50/68.50 * | 26.30 | 4.97 | |

ENGG1802 Engineering Mechanics | 22.5 | 100.0 | 44 | 67.28 | 70.00 | 20.93 | 3.16 | |

MATH1001 Differential Calculus | PHYS1902 Physics 1B (Advanced) | 7.5 | 85.0 | 27 | 50.49 | 69.17 | 19.41 | 3.74 |

PHYS1003 Physics 1 (Regular) | 0.0 | 100.0 | 136 | 30.15 | 0.00 | 28.76 | 2.47 | |

ENGG1802 Engineering Mechanics | 0.0 | 100.0 | 382 | 74.79 | 77.50 | 18.18 | 0.93 |

**Table 6.**Spearman’s rank correlation coefficient between the transfer indices and final marks in mathematics, science, and ATAR.

Transfer Indices | MATH Final Marks | n | PHYS/ENGG Final Marks | n | ATAR | n | |
---|---|---|---|---|---|---|---|

MATH1001 (Norm) & PHYS1003(Reg) | TI | 0.477 * | 136 | 0.447 * | 136 | 0.423 * | 100 |

ATAR Adj TI | 0.186 | 100 | 0.147 | 100 | |||

MATH1001 (Norm) & PHYS1902 (Adv) | TI | 0.759 * | 27 | 0.753 * | 27 | 0.355 | 22 |

ATAR Adj TI | 0.483 | 22 | 0.619 * | 22 | |||

MATH1001 (Norm) & ENGG1802 | TI | 0.505 * | 382 | 0.711 * | 382 | 0.361 * | 255 |

ATAR Adj TI | 0.239 * | 255 | 0.479 * | 255 | |||

MATH1901 (Adv) & PHYS1003 (Reg) | TI | 0.495 | 28 | 0.706 * | 28 | 0.494 | 24 |

ATAR Adj TI | 0.032 | 24 | 0.489 | 24 | |||

MATH1901 (Adv) & PHYS1902 (Adv) | TI | 0.497 * | 67 | 0.537 * | 67 | 0.438 * | 57 |

ATAR Adj TI | 0.368 | 57 | 0.474 * | 57 | |||

MATH1901(Adv) & ENGG1802 | TI | 0.488 * | 44 | 0.541 * | 44 | 0.315 | 39 |

ATAR Adj TI | 0.400 | 39 | 0.500 * | 39 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nakakoji, Y.; Wilson, R. First-Year Mathematics and Its Application to Science: Evidence of Transfer of Learning to Physics and Engineering. *Educ. Sci.* **2018**, *8*, 8.
https://doi.org/10.3390/educsci8010008

**AMA Style**

Nakakoji Y, Wilson R. First-Year Mathematics and Its Application to Science: Evidence of Transfer of Learning to Physics and Engineering. *Education Sciences*. 2018; 8(1):8.
https://doi.org/10.3390/educsci8010008

**Chicago/Turabian Style**

Nakakoji, Yoshitaka, and Rachel Wilson. 2018. "First-Year Mathematics and Its Application to Science: Evidence of Transfer of Learning to Physics and Engineering" *Education Sciences* 8, no. 1: 8.
https://doi.org/10.3390/educsci8010008