# How the Theme of ‘Doing and Undoing’ Applied to the Action of Exchange Reveals Overlooked Core Ideas in School Mathematics

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## Abstract

**:**

## 1. Introduction

## 2. Themes, Actions and Awarenesses

## 3. Method

## 4. Exchange

#### 4.1. Early-Exchange Tasks

Task 1: supplementing Start with a pile of red counters. Now supplement this with some blue counters. What is the relationship between the number of red counters, the number of blue counters, and the final number of both sets of counters altogether? |

Task 2: undoing supplementing Start with a pile of red and blue counters. Ask participants to separate into two piles, one all-red and the other all-blue, and to talk about the relationships between these two numbers and the number of counters altogether. |

#### 4.2. Exchange as First Encounter with Arithmetic beyond Adding and Subtracting

Task 3: elementary exchange Make a pile of red counters. Now exchange each red counter for 3 blue counters. |

Task 4: undoing an exchange Use the layout chosen in task 3 to see how collections of 3 blue counters could each be exchanged for a single red counter. |

Task 5: arithmetic meaning What arithmetical action corresponds to the action in task 3? What arithmetical action corresponds to the action in task 4? |

Task 6: more complex exchange Make a pile of red counters. Exchange 2 red counters for 3 blue counters as often as possible. Rehearse the form of task exchange with the new situation. |

Task 7: compound exchange First, exchange 2 red counters for 3 blue counters as often as possible. Then exchange 2 blue counters for 1 green counter as often as possible. What is the same and what is different about this exchange, and directly exchanging 4 red counters for 3 green counters? |

#### 4.3. Commentary

## 5. Four Operations

Task 8: additive actions What arithmetic action undoes ‘the action of adding 3′? What arithmetic action undoes ‘the action of subtracting 4′? What single arithmetic action has the same effect as the combination of first ‘adding 3′, then ‘subtracting 4′? What is the same and what is different about the compound action ‘adding 3′ then ‘subtracting 4′, and the compound action ‘subtracting 4′ then ‘adding 3′? |

Task 9: multiplicative actions What action undoes ‘multiply by 3′? What action undoes ‘divide by 4′? What is the same and what is different about the compound action ‘multiply by 3′ then ’divide by 4′ and the compound action ‘divide by 4′ then ‘multiply by 3′? |

Task 10: two interpretations What single action undoes the action of ‘multiply by three-quarters’? What compound action undoes the compound action of ‘multiply by 3 then divide by 4′? |

#### 5.1. Exploring ‘Subtracting From’

Task 11: subtracting from What action undoes the action of ‘subtract from 7′? |

#### 5.2. Arithmetic as One Dimension; Geometry as Two or More Dimensions

#### 5.3. Complementarity

Task 12: converting subtraction into addition What action undoes ‘form the nine’s complements of each digit’ of a numeral? Why do the following procedures always work? To subtract one multi-digit number B from another, A, form the number consisting of the 9′s complement of the digits of B (first inserting leading 0s in B to match the number of digits in A). Add this to the number A. Now delete the leading 1-digit of the answer, and add 1 to that answer. To subtract one multi-digit number B from another, A, form the nines’ complement of A; add to B and form the nines’ complement of the result. |

## 6. Equivalence as Permissible Exchange

Task 13: Equivalent Exchanges? Form two piles of red counters with the same number in each. For one, repeatedly exchange 2 reds for 4 blues, while for the second pile, repeated exchange 3 reds for 6 blues. Will the results be exactly the same? What fraction of the blue piles are the original red piles? |

#### Modular Arithmetic

## 7. More Advanced Arithmetic

Task 14: powers and roots What action undoes the action of ‘multiplying 4 copies of some number together’? What action undoes the action of ‘taking the 5th root’? Is there an action related to powers and roots, which is self-inverse, in the way that subtract from and divide-into are self-inverse? |

Task 15: gcd and lcm Finding the lcm (lowest common multiple) of two numbers can be seen as a ‘doing’. The associated ‘undoing’ is: given a number n, for how many different pairs of numbers a and b is n the lcm of a and b? What happens when undoing the gcd (greatest common divisor; highest common factor) of two numbers is considered? |

#### 7.1. Fractional Parts and Percentages

Task 16: Fractional increase and decrease | ||

$\left(1+\frac{1}{2}\right)\left(1-\frac{1}{3}\right)=\left(1-\frac{1}{3}\right)\left(1+\frac{1}{2}\right)=?$ | $\left(1+\frac{1}{3}\right)\left(1-\frac{1}{4}\right)=\left(1-\frac{1}{4}\right)\left(1+\frac{1}{3}\right)=?$ | $\left(1+\frac{1}{5}\right)\left(1-\frac{1}{6}\right)=\left(1-\frac{1}{6}\right)\left(1+\frac{1}{5}\right)=?$ |

$\left(1+\frac{2}{3}\right)\left(1-\frac{2}{5}\right)=\left(1-\frac{2}{5}\right)\left(1+\frac{2}{3}\right)=?$ | $\left(1+\frac{3}{7}\right)\left(1-\frac{3}{10}\right)=\left(1-\frac{3}{10}\right)\left(1+\frac{3}{7}\right)=?$ | $\left(1+\frac{5}{9}\right)\left(1-\frac{5}{14}\right)=\left(1-\frac{5}{14}\right)\left(1+\frac{5}{9}\right)=?$ |

In the table above, possibly using more examples, which you construct for yourself, for each row, express a general relationship of which the three equations are specific instances. What does each expression say about how to undo a fractional increase or a fractional decrease? How could that be connected to percentage increase and percentage decrease? |

#### 7.2. Extending the Palouse Relationship

Task 17: preserving products | ||

$\left(11+\frac{1}{5}\right)\left(7-\frac{1}{8}\right)=\left(7+\frac{1}{3}\right)\left(11-\frac{1}{2}\right)=?$ | $\left(11+\frac{1}{12}\right)\left(7-\frac{1}{19}\right)=\left(7+\frac{1}{14}\right)\left(11-\frac{1}{9}\right)=?$ | $\left(11+\frac{1}{19}\right)\left(7-\frac{1}{18}\right)=\left(7+\frac{1}{25}\right)\left(11-\frac{1}{16}\right)=?$ |

$\left(33+\frac{3}{5}\right)\left(21-\frac{3}{8}\right)=\left(21+\frac{3}{3}\right)\left(33-\frac{3}{2}\right)=?$ | $\left(33+\frac{3}{12}\right)\left(21-\frac{3}{19}\right)=\left(21+\frac{3}{14}\right)\left(33-\frac{3}{9}\right)=?$ | $\left(33+\frac{3}{19}\right)\left(21-\frac{3}{31}\right)=\left(21+\frac{3}{25}\right)\left(33-\frac{3}{16}\right)=?$ |

Extend the rows of the table; generalise! |

## 8. Algebra as an Exchange and as a Notation

#### 8.1. Exchange Underpins Algebraic Notation

#### 8.2. Tracking Arithmetic

Task 18: ThoaN Think of a number; add 3; multiply by 2; subtract 7; subtract the number you first thought of; add 1; the result is the number you first thought of! |

2 × 7 − 1 − 7 = 1 × 7 − 1; 1 × 7 − 1 + 1 = 7.

#### 8.3. Exchange Underpins Substitution

Task 19: area as summation The area of a trapezium with parallel edges a and b and height h is $\frac{h}{2}\left(a+b\right)$. What is the area of a trapezium with parallel edges 3 and 6 and height 4? What is the area of a trapezium whose edge lengths are s and s +2 and height s? What is the area of a triangle as suggested by this formula? What is the sum of n terms of an arithmetic progression whose first and last terms are a and b? |

Task 20: edge and hexagon count How many hexagons will be used to make the nth shape? How many internal and external edges will the shape have? How many external edges? How many hexagons will be used to make the 3n + 2nd shape? If there are 3h hexagons in one of the shapes, how many internal edges will there be? How many hexagons will be used to make the shape which has 20 + 8n external edges |

#### 8.4. Substitution as Part of a Construction Leading to Fractals

Task 21: gaskets At each stage, how many cells in total will there be? How many black? What fraction of the whole is coloured black at each stage? What happens to the density of blacks in the limit? |

Task 22: folded strip Open the strip up and place it on a table so that there is a sequence of peaks and valleys. Predict the number of peaks, the number of valleys, and the peak-valley sequence after a number of folds. Denote by R the folding action (right hand end on top of left hand end) and by L the opposite (left hand end on top of right hand end). Write down a ‘word’ W consisting of a sequence of Rs and Ls. Use this to determine the sequence of folds, reading from right to left (as if the R and L were functions being composed). From the word W, predict the number of peaks, the number of valleys and the peak-valley sequence. Which words will give the same sequence of peaks and valleys (or valleys and peaks)? |

Task 23: pebble arithmetic Each time a rectangle is formed, the pebbles change colour briefly, to indicate a pause. The number of rows slid since the last rectangle is recorded to form a slide-sequence. In the case here, the slide sequence was 1, 1, 1 (the last slide is not depicted in the sequence above). That is the doing. The undoing task asks which sequences of numbers can be the slide-sequence for some rectangle of pebbles, and is it possible then to ask, how many pebbles were involved? |

## 9. Other Examples of Exchange

#### 9.1. Statistical

#### 9.2. Multiple Masses

## 10. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Gardiner, A. Discovering Mathematics: The Art of Investigation; Oxford Science Publications, Clarendon Press: Oxford, UK, 1987. [Google Scholar]
- Gardiner, A. Recurring Themes in School Mathematics: Part 1 direct and inverse operations. Math. School
**1992**, 21, 5–7. [Google Scholar] - Mason, J. Learning & Doing Mathematics, 2nd ed.; QED Books: York, UK, 1988. [Google Scholar]
- SMP 11-16. Doing and Undoing; Cambridge University Press: Cambridge, UK, 1984. [Google Scholar]
- Mason, J. Bartering Problems in Arithmetic Books 1450–1890. Br. Soc. Hist. Math. Bull.
**1984**, 22, 160–181. [Google Scholar] [CrossRef] - Chevallard, Y. La Transposition Didactique; La Pensée Sauvage: Grenoble, France, 1985. [Google Scholar]
- Bosch, M.; Gascon, J. Twenty-Five Years of the Didactic Transposition. ICMI Bull.
**2006**, 58, 51–65. [Google Scholar] - Simon, M.; Tzur, R. Explicating the Role of Mathematical Tasks in Conceptual Learning: An elaboration of the hypothetical learning trajectory. Math. Think. Learn.
**2004**, 6, 91–104. [Google Scholar] [CrossRef] - Deleuze, G.; Guattari, F. A Thousand Plateaus: Capitalism and Schizophrenia; Bloomsbury: London, UK, 1980. [Google Scholar]
- Vergnaud, G. The nature of mathematical concepts. In Learning and Teaching Mathematics: An International Perspective; Nunes, T., Bryant, P., Eds.; Psychology Press: London, UK, 1997; pp. 5–8. [Google Scholar]
- Mason, J.; Johnston-Wilder, S. Designing and Using Mathematical Tasks; republished 2006; Open University: Milton Keynes, UK; Tarquin: St. Albans, UK, 2004. [Google Scholar]
- Brousseau, G. The Crucial Role of the Didactical Contract in the Analysis and Construction of Situations in Teaching and Learning mathematics. In Theory of Mathematics Education; Paper 54; Steiner, H., Ed.; Institut fur Didaktik der Mathematik der Universitat: Bielefeld, Germany, 1984; pp. 110–119. [Google Scholar]
- Mason, J.; Burton, L.; Stacey, K. Thinking Mathematically (Second Extended Edition); Prentice Hall (Pearson): Harlow UK, 1982. [Google Scholar]
- Cuoco, A.; Goldenberg, P.; Mark, J. Habits of Mind: An organizing principle for mathematics curricula. J. Math. Behav.
**1996**, 15, 375–402. [Google Scholar] [CrossRef] - Schoenfeld, A. Mathematical Problem Solving; Academic Press: New York, NY, USA, 1985. [Google Scholar]
- Mason, J. Doing & Undoing, Project Update; Open University: Milton Keynes, UK, 1988. [Google Scholar]
- Mason, J. Learning and Doing Mathematics; MacMillan: London, UK, 1988. [Google Scholar]
- Mason, J.; Johnston-Wilder, S. Fundamental Constructs in Mathematics Education; RoutledgeFalmer: London, UK, 2004. [Google Scholar]
- Gattegno, C. The Science of Education Part I: Theoretical considerations; Educational Solutions: New York, NY, USA, 1987. [Google Scholar]
- Mason, J. Only Awareness is Educable. In Teaching, Learning and Mathematics; Bloomfield, A., Harries, T., Eds.; Association of Teachers of Mathematics: Derby, UK, 1994; pp. 28–29. [Google Scholar]
- Mason, J. Educating Awareness through Mediated Action. In On Teaching and Learning Mathematics with Awareness; Hewitt, D., Coles, A., Ingram, J., Eds.; Association of Teachers of Mathematics: Derby, UK, 2018; pp. 71–76. [Google Scholar]
- Mason, J. Enabling Teachers to be Real Teachers: Necessary levels of awareness and structure of attention. J. Math. Teach. Educ.
**1998**, 1, 243–267. [Google Scholar] [CrossRef] - Mason, J. On The Structure of Attention in the Learning of Mathematics. Aust. Assoc. Math. Teach.
**2003**, 59, 17–25. [Google Scholar] - Watson, A.; Mason, J. Mathematics As a Constructive Activity: Learners Generating Examples; Erlbaum: Mahwah, NJ, USA, 2005. [Google Scholar]
- Bruner, J. The Narrative Construction of Reality. In Rappresentazioni e Narrazioni; Ammanti, M., Stern, D., Eds.; Laterze: Roma-Bari, Italy, 1991; pp. 17–42. [Google Scholar]
- Chi, M.; Bassok, M. Learning from examples via self-explanation. In Knowing, Learning and Instruction: Essays in Honour of Robert Glaser; Resnick, L., Ed.; Erlbaum: Hillsdale, MI, USA, 1989. [Google Scholar]
- Hamilton, E.; Cairns, H. (Eds.) Plato: The Collected Dialogues Including the Letters; Phaedrus. Bollingen Series LXXI; Princeton University Press: Princeton, NJ, USA, 1961. [Google Scholar]
- Mill, J.S. A System of Logic, Ratiocinative and Inductive, Being a Connected View of the Principles of Evidence, and the Methods of Scientific Investigation, 8th ed.; Harper & Bros: New York, NY, USA, 1882. [Google Scholar]
- Montessori, M. The Montessori Method; Transaction: London, UK, 1964. [Google Scholar]
- NRich. Available online: https://nrich.maths.org/10059 (accessed on 14 February 2021).
- BBC. Available online: https://www.bbc.co.uk/cbeebies/topics/numeracy (accessed on 14 February 2021).
- Gripton, C.; Pawluch, D. Counting Collections in the Early Years. Math. Teach.
**2021**, 275, 6–10. [Google Scholar] - Marton, F.; Booth, S. Learning and Awareness; Lawrence Erlbaum: Hillsdale, MI, USA, 1997. [Google Scholar]
- Marton, F. Necessary Conditions for Learning; Routledge: Abingdon, UK, 2015. [Google Scholar]
- Barton, B.; Fairhall, U.; Trinick, T. Tikanga Reo Tatai: Issues in the development of a Maori Mathematics register. Learn. Math.
**1998**, 18, 3–9. [Google Scholar] - Borden, L. The ‘Verbification’ of Mathematics: Using the grammatical structures of mi’kmaq to support student learning. Learn. Math.
**2011**, 31, 8–13. [Google Scholar] - James, N.; Mason, J. Towards Recording. Visible Lang.
**1982**, 16, 249–258. [Google Scholar] - Thomsen, G. The Treatment of Elementary Geometry by a Group-Calculus. Math Gazette.
**1933**, 17, 230–242. [Google Scholar] [CrossRef] - Hewitt, D. Approaching Arithmetic Algebraically. Math. Teach.
**1998**, 163, 19–29. [Google Scholar] - Gillings, R. Mathematics in the Time of the Pharaohs; Dover: New York, NY, USA, 1972; Reprinted in 1982. [Google Scholar]
- Gattegno, C. What We Owe Children: The Subordination of Teaching to Learning; Routledge & Kegan Paul: London, UK, 1970. [Google Scholar]
- Tahta, D. A Boolean Anthology: Selected Writings of Mary Boole on Mathematics Education; Association of Teachers of Mathematics: Derby, UK, 1972. [Google Scholar]
- Mason, J.; Johnston-Wilder, S.; Graham, A. Developing Thinking in Algebra; Sage (Paul Chapman): London, UK, 2005. [Google Scholar]
- Bloor, D. Knowledge & Social Imagery; University of Chicago Press: Chicago, IL, USA, 1976. [Google Scholar]
- Davis, B. Teaching Mathematics: Towards a Sound Alternative; Ablex: New York, NY, USA, 1996. [Google Scholar]

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**MDPI and ACS Style**

Mason, J.
How the Theme of ‘Doing and Undoing’ Applied to the Action of Exchange Reveals Overlooked Core Ideas in School Mathematics. *Mathematics* **2021**, *9*, 1530.
https://doi.org/10.3390/math9131530

**AMA Style**

Mason J.
How the Theme of ‘Doing and Undoing’ Applied to the Action of Exchange Reveals Overlooked Core Ideas in School Mathematics. *Mathematics*. 2021; 9(13):1530.
https://doi.org/10.3390/math9131530

**Chicago/Turabian Style**

Mason, John.
2021. "How the Theme of ‘Doing and Undoing’ Applied to the Action of Exchange Reveals Overlooked Core Ideas in School Mathematics" *Mathematics* 9, no. 13: 1530.
https://doi.org/10.3390/math9131530