In this post I’ll focus on the connections between geometry and the Game of SET. Specifically, I’ll run through the key questions and moments of workshops I run, and roughly follow the slides from my workshop at Twitter Math(s) Camp in 2018. They are in this folder, which is also jam-packed with other resources.

**Warning: This post will contain lots of spoilers. **

When introducing SET, I like to start with spreading out the cards and asking ‘What do you notice? What do you wonder?’ This is a great way to draw out the attributes, along with some other observations and questions that people might initially have.

There are four attributes of a SET card. The three cards below show all possible values of the attributes.

**Question: If a SET deck contains exactly one card of every possible combination of attributes, how many cards are in a deck? **

I like to get to this question by asking people to find a way to organise the cards so that they are convinced they have exactly one card with each combination of attributes. We can count how many cards there are, and also develop a combinatoric/counting argument.

I’ve noticed that when people organise the cards in this way, the idea of a SET tends to naturally arise. This is so much more powerful than me telling them!

**Question: Can you come up with a definition of a SET?**

I show four carefully chosen examples of SETs and ask people to come up with a definition of what constitutes a SET. This is a great way of exploring precision in definitions, as well as examples and not-examples.

A **SET** is a group of three cards where, for each of the four attributes, the features are the same across all three cards or different across all three cards. The three cards below are a SET because shape is all-same, fill is all-same, number is all-different, colour is all-different.

**Interlude: Play the game**

To play SET with others, lay twelve cards out on the table. The first person to find a SET calls ‘SET’, shows it to the rest of the players. The empty spaces are then replenished from the deck. If the players agree that there is no SET on the table, three more cards are dealt. At the end of the game, the person with the most SETs wins. David Butler has a nice variant of the game that involves playing in teams, and levels the competition when some participants are quicker at finding SETs than others.

Playing the game will allow participants to practice finding SETs, which will be useful later.

**Question: Pick two cards at random. How many cards, if any, can be used to complete the SET?**

Spoiler: The answer is 1. (Can you reason why?) This is called the Fundamental Theorem of SET, and is necessary for many geometric arguments that arise later.

**Question: How many SETs are possible (including those where a card is used in more than one SET) among all cards in the deck?**

Spoiler: a quick explanation. There are 81 choices for the first card in a SET, 80 choices for the second card, and 1 choice for the third card (by the Fundamental Theorem of SET). This gives 81*80*1. But! The order in which our cards are drawn doesn’t matter, so we have over-counted. We need to divide by the number of ways to arrange three cards, which is 3*2*1 = 3! = 6 ways. Thus, the total number of possible SETs is (81*80*1)/(3*2*1) = 1080.

**Question: What is the most number of SETs among nine cards?**

It’s perhaps better to pose this question as a challenge: Find nine cards with the most number of SETs among them. This question neatly segues into our geometric representation.

Suppose we represent each SET card with a point in space. A line is a collection of three points that form a SET. We can visualise the 12 SETs among our nine cards as shown in the diagram below. Notice our lines bend; that’s okay!

What we’ve constructed here is a plane in space. There are a few ways to help people explore this fact. One is to start with familiar ideas about a geometric plane, that is, ‘three non-collinear points determine a plane’ and ‘if two points lie in a plane, then any line containing those two points also lies in that plane’.

In SET language, this means we should start with three cards (points) that do not form a SET (are not collinear). Then, for any two cards in our plane, the SET determined by those cards should be in the plane. This means that the third card required to complete the SET should also be a part of the plane. Starting with our three non-collinear points, we continue to add cards to the plane until we have exhausted all options.

We end up with nine cards in our plane. This collection of nine cards is called *closed. *That is, if we pick any pair of points, the point that completes the line lies in the plane.

**Question: What is the most number of cards we can remove from our collection of nine and still be able to reconstruct all twelve SETs?**

Rather than launch from the definition of a plane, I like to start by posing the question above. Eventually people arrive at three cards that don’t belong to the same SET. When we start making connections from here to geometry, it feels a little more concrete.

**Interlude: Finite Affine Geometry**

Did you know that there are lots of different kinds of geometries? To give an extremely non-technical explanation, Euclidean geometry (the one that we are most familiar with) is build on a set of axioms (statements that we take to be given truths). Axioms are the building blocks of mathematics and of geometries. Each branch of geometry has its own set of axioms or, put another way, each set of axioms gives rise to a unique branch of geometry (quoted from page 99 of ‘The Joy of SET’).

Our SET geometry belongs to a non-Euclidean branch of geometry called finite affine geometry: ‘finite’ because we have a finite number of points and lines, and ‘affine’ because parallel lines are allowed. (You might like to think about what it means to have two parallel SET lines!)

So far we have explored an affine plane called AG(2,3). (The 2 is for two dimensions; the 3 is for three points in a line). We can count objects in this geometry: 9 points, 12 lines, 1 plane. And remember, we just need 3 (non-collinear) cards to build our plane.

Let’s now move to three dimensions!

To go to three dimensions we use a SET plane and one point off the plane. We then complete our hyperplane as we did before. That is, we select any two points in the hyperplane and add the third point that completes the line. I’ve started constructing a hyperplane below. Can you see a card that could be placed next?

I’ve used the ‘two empty green ovals’ card to place three more cards to the ‘topmost’ plane.

What cards could you add next? Well, we could finish off the ‘frontmost’ plane (the one formed by the three cards on the top level, the single card in the middle level, and the three front cards on the bottom level).

We could also use the top front left card (two striped purple diamonds) and the bottom back right card (three striped green ovals) to fill in the centre of the middle level (one striped red peanut/squiggly thing).

If we continue in this way, we’ll build AG(3,3). (The first 3 is for three dimensions; the 3 is for three points in a line). We can count objects in this geometry: 27 points, 117 lines, 39 planes, 1 hyperplane. (These numbers aren’t obvious; you’ll need some crafty counting/combinatorial constructions.)

What is the minimum number of points needed to construct the hyperplane? Well, we need three non-collinear points to construct a plane, and then one point off the plane. So we used four cards to build our hyperplane.

We can move into four dimensions in the same way. We just need our hyperplane (from before) and a point off the hyperplane. The photo below was taken by Chrissie Newell at games night at the 2018 Twitter Math(s) Camp.

We’ve built AG(4,3), which has 81 points (the number of cards in a SET deck), 1080 lines (all possible SETs in the deck), 1170 planes, and 120 hyperplanes.

What if we added another attribute to our SET deck, like size? Can you imagine what our five-dimensional hyperplane would look like now? I’ll leave it to you to try!

I run SET workshops with large groups, sometimes in different cities. I wanted a quick, cheap and portable set of materials for pairs/trios to use. My method of plastic cups and heavy card (or even clear perspex) works fairly well, and is relatively easy to lug around. (The heavy card and the SET cards themselves are a bit heavy in my suitcase, though.)

I have also made these gorgeous SET cubes.

The cubes are a labour of love. They involve cutting, folding, and taping every side of 81 little cubes. They take **ages** to make (many hours), but they are beautiful. The nets are in the folder I shared earlier. There are no instructions, but I think you can figure it out. You could also try affixing stickers to pre-made cubes, or even 3D printing cubes. If you find something that works, let me know!

The cubes are a lot of fun to play with …

]]>Nat’s original post is about a quadratics functions menu. My attention was captured by Dylan Kane’s tweet in which, inspired by Nat, he wrote a trigonometric functions menu. Later, Sheri Walker was inspired to write an exponential functions menu.

I was similarly inspired, and so included the following linear relationships ‘menu’ in our most recent #math1057 tutorial.

Not only does this style of task elicit strategic thinking and emphasise understanding, it also helps students make connections between certain properties of linear equations.

My favourite tasks have a variety of ways for students to start. With the ‘menu’, I noticed a few strategies. For example:

- Some students found the equation of the line through the two given points (2,-3) and (4,0), and proceeded from there.
- Some started with the equation of a specific line (like from the previous tutorial question) and worked out which requirements it met. Later, they realised they could make adjustments to the line so that it met more of the requirements. Once they met as many requirements as they could, they thought of another specific line with at least one of the remaining requirements, and repeated their process.
- Some started with a rough sketch of a line with the first requirement, and made adjustments as they progressively considered additional requirements. Once they could no longer alter their line, they chose specific values (like two points, or a point and a slope) from which to build a specific linear equation.
- Some students identified requirements that can not be paired, and put them in separate lists. Later, once all requirements were assigned to a list, they built specific equations that satisfied the properties on the list.

Like Nat, I suspect this style of task will become a staple in my classroom. I look forward to expanding it to other topics, like geometry, statistics and probability. If you design your own, please share!

]]>This year I’ve been making a concerted effort in my Mathematics for Primary Educators course^{2} to have students articulate and justify their thinking, including on problems that I used to think were ‘straight forward’ (like this one from our most recent quiz).

The breakdown of responses from 43 students are as follows:

(a) 3 students** (b) 31 students**

(c) 7 student

(d) 1 student

(e) 1 student

There are 12 students who clearly don’t know how to manipulate negative exponents yet^{3}. Am I right to conclude that there are 31 students who do? Have a look at three of the responses (rewritten in my handwriting).

Whoa. Their misconceptions are just as interesting as the 12 students who circled an incorrect answer *but I would never have known if I hadn’t asked them to explain their thinking.*

[1] Remind me again why a PhD is deemed sufficient for teaching at university level?

[2] A content course for pre-service maths teachers, currently across primary and secondary school.

[3] At least, as demonstrated on this problem.

I began the first lecture with a WODB (Which One Doesn’t Belong?), presented as a Stand and Talk. (I’ll link this in another blog post.) In this post I’ll talk about the second WODB I presented in the lecture, developed with input from Twitter.

In case you aren’t familiar, ‘Which One Doesn’t Belong?’ is a great prompt for looking for points of commonality and difference. The idea is to identify a property three items have in common that are not shared by the fourth item. Ideally, we can find reasons why each item doesn’t belong. WODB helps students to see that there is more than one ‘right way’ to think about maths. It also provides a good launch point for terminology and for more closely examining particular mathematical properties. For more, look at #wodb, wodb.ca, and Christopher Danielson’s books.

To begin, I simply asked ‘Which One Doesn’t Belong?’ and collected their responses in Desmos. I wanted to elicit the different representations of numbers, as well as terminology and misconceptions.

[Note: I’ll write the top left as 0.(3) for convenience.]

**The top left doesn’t belong**because the others show 0.3. This was a prevalent response. We talked about the meaning of the vinculum, and recurring decimal numbers.**The top right doesn’t belong**because it’s the only one written as a percentage. G said that it suggests it is out of 100 (and presumably that the others aren’t), which is an interesting idea I didn’t explore in class.**The bottom left doesn’t belong**because it’s the only one written as a fraction. It’s also the only one that involves two separate numbers. (No one picked the bottom left in class.)**The bottom right doesn’t belong**because it’s not displayed as a number. K said that it ‘could mean anything’. S said that it is open to interpretation: it’s either 3/10 or 7/10. I volunteered that it could be the only one with a value more than 1 (that is, 3). We briefly talked about ten frames.

I asked: **Are these all numbers?**

I asked students to put 0.(3) , 30%, 3/10 and 3 on a number line.

I revised the number line I planned to use several times. My first version had tick marks and labels with different colours for positive and negative numbers. Then I removed the labels. Then I removed the zero. Then I removed the tick marks. I settled on a horizontal line with arrows at either end.

We discussed needing to identify the largest and smallest number so that we could make sure all the numbers fit on our number line. This led to the question: ‘**Which is bigger: 3 or 30%?**‘ Someone helpfully asked ‘30% of what?’ This led to a need to put all our numbers in the same representation.

We placed 3 on the right of our number line, and found a need to put 0 at the left of our number line.

We talked about where to put 3/10 = 0.3 on our number line. H suggested we first mark 1 and 2 on the number line. (We talked about placing them at equal intervals.) Someone then placed 0.5 as a reference point. We then partitioned the interval from 0 to 0.5 so it showed tenths, and placed 3/10.

Finally we discussed where 0.(3) belongs. Should it go to the left or the right of 0.3? We started connecting this to place value.

I asked: **What role do different representations play in how we perceive a number?**

I asked students to find 0.(3) , 30%, 3/10 and 3 on these posters courtesy of David Butler: www.adelaide.edu.au/mathslearning/handouts. I love these posters because they show numbers in different representations, and because numbers appear in the same place when they are on multiple posters. (I slightly modified them to include our numbers.)

I asked what it means: To be on a poster? To be on more than one poster? To not be on a poster?

From here we progressively discussed our numbers:

- The number 3 prompted discussing the difference between natural numbers and integers.
- Our different representations of 0.3 (as 3/10, 30% and 0.3) prompted a closer look at the definition of rational numbers. The ‘quotient of two integers’ connected to representing 3 as a rational number. The ‘can be expressed as terminating or as repeating decimals’ helped us consider the placement of 0.(3).

We drew the Venn diagram representation and probed what it means to be an irrational number. (Every eye was drawn to pi on the poster!)

We wrapped up the discussion with some ‘give an example of’ statements:

- an integer that is a natural number
- an integer that is a rational number
- a rational number that is not an integer
- an integer that is not a rational number

In all, this was a good elaboration of WODB into some of the key concepts of this first lecture.

]]>*The overall brief for the essay was to either: (a) choose a theoretical framework to discuss an Indigenous artist’s work and the standpoint represented in their work, or (b) develop a research paper on a topic of great interest in relation to professional or personal practice.*

*I chose to use the essay as an opportunity to read more deeply into issues of mathematics education in Australia in relation to the experience of Indigenous peoples. The essay had specific requirements in terms of length, inclusion of course readings, and skills to be demonstrated. The style is necessarily impersonal. I humbly offer this for broader consumption as I think that sharing my evolving understandings will provide an opportunity for others to contribute to my education on this topic.*

The participation and achievement of Indigenous students in science, engineering and mathematics is significantly lower than their non-Indigenous peers, leading many to believe that Indigenous students do not have an aptitude for scientific disciplines. This belief is a myth.

This paper dispels the misconception by presenting several examples of the ingenuity of pre-colonial Indigenous science (a term which includes cognate fields such as engineering and mathematics). However, setting history straight is only part of the equation. By examining the origins of the myth, this paper hopes to understand how to rectify inequalities in mathematics education in ways that draw on and elevate Indigenous knowledge systems.

Indigenous people have inhabited Australia for more than 65,000 years (Clarkson et al, 2017). Aboriginal Australians are well established as the first astronomers. Many sources, including oral traditions, art and artefacts, indicate that their detailed understanding of the sky enabled navigation and an ability to predict tides, eclipses and the motion of celestial bodies. (See Norris and Harney (2014) for a lengthy reference list.) For another example from science, Bruce Pascoe’s acclaimed book ‘Dark Emu’ (2018) disrupts conventional thinking by showing that pre-colonial Aboriginal Australians possessed sophisticated knowledge of agriculture. Pascoe (2018) gives powerful evidence of Aboriginal people pursuing the five activities that Europeans deemed as signifying the development of agriculture: selecting seed, preparing the soil, harvesting, storing surpluses, and erecting permanent housing for large populations.

Engineering knowledge is apparent in the range and complexity of Aboriginal-designed houses built from a variety of materials, including stone (Memmott 2007). The fish traps at Brewarrina are one of humankind’s oldest still-standing man-made structures, and are ‘unequivocal evidence of a deep understanding of engineering principles, applied physics, ecology and hydrology’ (Ball 2015, p. 17). While mathematical knowledge is evident in astronomy, agriculture and engineering, it also extended beyond physical applications. For example, intricate kinship systems, developed over thousands of years, are complex arrangements that rely on ‘cyclical, recursive patterns’ which can be found within numbers and other areas of mathematics (Matthews 2005, p. 5).

This handful of examples demonstrates a breadth and longevity of Indigenous knowledge in science, engineering and mathematics. And yet, the response to Pascoe’s book has been widespread astonishment amongst non-Indigenous readers, with the evidence of agricultural societies and of stone engineering techniques sharply inverting their beliefs about pre-colonial Australia. How did this paucity of knowledge regarding Indigenous scientific achievements and the myth that Indigenous people ‘don’t do science’ come about?

Eurocentric perspectives have dominated scientific discourse in Australia since colonisation, determining whose knowledge is legitimate. The innate superiority with which Europeans and the British came to see themselves emerged during the Enlightenment (Smith 1999, p. 58), the period of European history known as the ‘Age of Reason’ for its emphasis on the scientific method. The development of modern science is often used to indicate how progressive, rational and civilised the West is in contrast with the primitive ‘rest’ (Harding 1993, p. 7).

Imperialism was also borne out of economic, industrial and military dominance, and came to be interpreted as the difference between the ‘savage’ and ‘civilised man’ (Hollinsworth 2006, p. 30; Pascoe 2018, p. 3). This was the prism through which the British approached first contact. Ironically, prejudicially framing Indigenous Australians as ‘savages’ and ‘primitives’ overwhelmed the capacity of Anglo-Europeans to apply proper scientific methods; for instance, Nakata (1998) provides detailed examples of questionable scientific practices and inference in the 1898 Cambridge Anthropological Expedition to the Torres Strait.

A view prevalent until the early 1900s was that biological differences placed different groups of people (‘races’) on a single path of evolutionary development, from primitive to advanced (Hollinsworth 2006, p. 37). A similar construct held (and arguably still broadly holds) for scientific and technological achievement, that is, a single line of progress exists from the most primitive societies to the most advanced. It was thought that every culture would eventually reach the same milestones. Conversely, if a culture does not achieve a milestone, the implication is that it is inferior. That there were no wheels in Australia until Europeans arrived was used to conclude that Indigenous cultures were technologically simplistic (Pearcey 1998).

A more unscrupulous demonstration of this way of thinking comes from mathematics. The general perception by Western researchers until the mid-twentieth century was that Aboriginal counting systems were simple and, therefore, that Aboriginal cultures were mathematically naïve. However, Harris (1987) provides proof from several Aboriginal languages to dispute the misperception, declaring that scholars deliberately ignored the evidence so as to promote a view of cultural and intellectual Aboriginal inferiority.

Not all evidence of technological and scientific knowledge was wilfully ignored or mischaracterised by the colonisers, but it was also not made more widely known. Many of the journal extracts from explorers, surveyors and pastoralists included in Pascoe’s book describe, sometimes admiringly, the innovation of Indigenous people across the country. It is not difficult to understand why they kept their discoveries private—it is easier to justify colonising and conquering a primitive civilisation. More pointedly, their observations demonstrated land tenure, which ran counter to the portrayal of Aboriginal people as nomads. The notion that Australia was inhabited by nomadic people, rather than occupied by them, underpins the ‘doctrine of terra nullius’ and the right to take possession of the land.

Framing Indigenous people as incapable of scientific thought and refusing to acknowledge evidence of Indigenous science preserved a view that science can only be done and interpreted by dominant groups in the West. The ramifications of denying Indigenous people participation in the intellectual community, except as objects to be studied, are far-reaching and will be elaborated later. Many Indigenous people hold the view that they are the most researched groups in the world (Smith 1999, p. 3; Mack and Gower 2001), and yet have rarely seen any benefit from research. Even in the modern world, research projects involving Indigenous peoples are still sometimes seen as a tool for colonisation; Smith (1999, p. 99) gives ten recent and diverse examples, from genetic to spiritual exploitation.

As has already been mentioned, science is affected by political, economic and social interests. These factors influence the way in which scientists interpret data, judge which knowledge is valid, and decide which problems to pursue. Harding (1993, p. 2) characterises a ‘racial economy’ of Western science, in which institutions, assumptions and practices distribute the benefits of Western sciences along racial lines, widening the gap between the haves and the have nots. To give one example, Freemantle and McAullay (cited in Laycock et al. 2011, p. 21), note that ‘prior to 1976, no Australian jurisdiction separately identified Indigenous persons in vital statistics or hospital-based collections’. The shortage of accurate, consistent and complete Indigenous health data, which occurred in part due to a lack of Government interest in collecting it, has directly contributed to health inequity in Australia.

Although the scientific method that underpins most Western research is meant to enshrine objectivity through systematic observation and testing of hypotheses, research is always filtered through the worldview of the scientist. These biases can give rise to an unexpected form of scientific illiteracy suffered by those who are highly educated—scientists and other dominant groups in the West (Harding 1993, p. 1). One consequence is a partial or distorted view of knowledge. For example, Harris observes that the deliberate mischaracterisation of Indigenous concepts of number, which entered the literature in the 1860s, ‘profoundly influenced the thinking of several generations of anthropologists and linguists’ (1997, p. 30). A failure to be willing to learn from Indigenous science impedes progress in many ways. Pascoe (2018) explains how serious consideration of Indigenous farming practices could revolutionise Australian agriculture and benefit future prosperity (p. 63–67, p. 209–217).

Examining Indigenous knowledge systems from a Western standpoint has inherent limitations. In his study of Yolngu mathematics, Cooke (1990) is conscious that the operational definition of mathematics arises ‘from its characteristics and prominence as a basis for the schema of European culture’ (p. 4), noting that the Westerner looking into the Yolngu world may easily overlook evidence of Yolngu mathematical imagery if they are searching for European-style markers of symbols and diagrams (p. 37). Cooke also considers what might be lost in translation, observing that (p. 2):

… by removing words, concepts, and structures from their Aboriginal context and putting them into a European box called ‘mathematics’, I have inevitably lost much of the full significance of their meanings and have certainly not done justice to the intricacy and complexity of the Yolngu world.

Over time, scholars in different fields began to seek an insider’s point of view—that of the ‘native’. Known broadly as ‘Indigenous knowledge’, the ethno- prefix has been applied to describe the study of traditional knowledge in terms of internal elements of the culture, rather than by reference to any existing external scheme. This approach has given rise to new fields of study, such as ethnobiology, ethnochemistry, ethnoengineering, and ethnomedicine.

The term ‘ethnomathematics’ was coined by d’Ambrosio (1985, p. 45) to describe ‘the mathematics which is practiced among identifiable cultural groups’. There is no consistent view of ethnomathematics in the literature, and its meaning has progressively shifted. In his PhD thesis, Barton (1996, p. 3–8) charts the evolution of the term ethnomathematics from describing particular mathematical practices, to explaining the knowledge behind those practices and, eventually, to a research program investigating the ways different cultures mathematise (that is, translate informal concepts into a mathematical framework).

Ethnomathematics has attracted criticism on two main fronts: epistemological and pedagogical (Pais 2011). The first criticism connects to the struggle many people have in accepting that mathematics and culture can be fundamentally linked. In their minds, the universality of mathematics is meant to transcend specific cultures. Barton (1996a, p. 215) concludes that ethnomathematics is not mathematics, but a ‘tool in which we may make better sense of our world, both as we see, and as others see it’ (p. 229). The pedagogical criticism of ethnomathematics relates to the ways in which ethnomathematical ideas are included in formal education. We now consider this point in more detail.

Australia’s Indigenous students are being left behind in mathematics education. Indigenous 15-year-olds are approximately two-and-a-half years behind their non-Indigenous peers in schooling. Indigenous students are overrepresented at the lower end of mathematical literacy, with half of all Indigenous students deemed ‘low performers’ compared to 18 percent of non-Indigenous students (Dreise and Thomson 2014, p. 1).

Chris Matthews, an Indigenous mathematician and prominent voice in this space, argues that Indigenous education must be understood in the context of the historical positioning of Indigenous cultures as a primitive, simplistic society (2005, p. 2), and with recognition of the lasting effects of marginalisation and exploitation. Education can be perceived to have a role in the colonising process by institutionalising the oppositionality of European and Indigenous ways of being (Pearce 2001, p. 5), for example, by requiring students to speak in a language other than their own and to participate in pedagogical practices that are culturally different.

Being ‘smart’ may also be considered by some Indigenous students as a Western attribute (Sarra 2011, p. 109; Pearce 2001, p. 3). This poses an impossible dilemma: either ‘play dumb’ and have the teacher regard them (and, by extension, all Indigenous people) as academically incapable, or speak up and betray cultural identity.

Matthews et al. (2005, p. 1) identify two fundamental problems. The first is that educators have little faith in Indigenous students’ mathematical abilities, blaming poor academic performance on social and cultural complexities (Sarra 2011, p. 108), without proper consideration of the dilemma just mentioned. This can lead to a ‘dumbing down’ or a lighter version of the curriculum (Matthews et al. 2005, p. 2; Pais 2011, p. 213). Research also suggests teachers respond more favourably to students who are culturally similar to them (Pearce 2001, p. 6) and, with few Indigenous teachers, this may be a difficult problem to quickly overcome.

The second fundamental problem is that Indigenous students find little relevance within mathematics, often struggling with Western concepts, content and pedagogies. A common approach is to use students’ ethnomathematical knowledge to construct a ‘bridge’ for the learning of more formal mathematics. While this method is seen as one way to valorise students’ cultures, it can reinforce the hegemony of Western mathematics, especially when traditional knowledge is used as a ‘curiosity, an illustration, a “starter” to the real mathematics’ (Pais 2011, p. 222). Used in this way, ethnomathematics can jeopardise cultural identity and magnify ‘otherness’.

Here I make a third criticism of ethnomathematics: that use of the prefix ‘ethno-‘ will always position Indigenous knowledge systems as separate and inferior to ‘mainstream’ knowledge. Both this and the epistemological criticisms of ethnomathematics can be resolved by redefining what is meant by mathematics. Cooke (1990, p. 5) proposes that we view mathematics as ‘society’s system for encoding, interpreting and organising the patterns and relationships emerging from the human experience of physical and social phenomena’ and goes on to say that ‘whilst this process is common to all cultures the resulting schemata can be fundamentally different’. With this new definition, Western mathematics is only one type of mathematics. This is a revolutionary view, and a point of contention in the literature.

Rather than use a one-way bridge, ‘two-way/both-way teaching’ is a method in which teaching and learning occurs in a neutral, negotiated space—the ‘third space’—in which neither culture presumes superiority (Purdie et al. 2011, p. xx). One of the most cited examples of this culturally responsive pedagogy is Garma Maths, taught at Yirrkala Community School in Arnhem Land. In the local language, the ‘Garma’ is an open meeting place where everyone comes together. Western and Yolngu mathematics are presented alongside each other, tying the formal logical concepts of Yongu life and thought with Western mathematics (Nicol and Robinson 2010, pp. 502‒503).

Culturally responsive pedagogies also draw on Aboriginal ways of learning. Storytelling is included in the eight ways of Aboriginal learning introduced by Yunkaporta (2010), and has been successfully used to explore algebraic concepts in the Maths as Storytelling (MAST) approach (Matthews 2008, pp. 48‒50). Yunkaporta (2010) elaborates on using the eight ways in the teaching of Aboriginal languages in schools. Conceptualising how to bring all eight ways to the teaching of mathematics would be an exciting enterprise!

The scientist Carl Sagan said: ‘*You have to know the past to understand the present.*’ This paper briefly surveyed evidence showing that Indigenous Australians were the first scientists, engineers and mathematicians. Yet despite this strong history, Indigenous students are being left behind in mathematics education. This paper has examined how racial prejudices led British colonisers to ignore, downplay, and misrepresent Indigenous science, with effects that reverberate into the present for both Indigenous and non-Indigenous Australians.

One consequence of disregarding Indigenous science is a Eurocentric education system that marginalises Indigenous peoples, devalues Indigenous knowledges, and discredits the abilities of Indigenous students. With a focus on mathematics education, this paper described how contextualising learning with Indigenous knowledge and perspectives can help redress educational inequities. However care must be taken so that traditional knowledge is not reduced to a quaint cultural artefact. In the most effective approaches, Indigenous and Western mathematics are situated alongside each other, with neither claiming cultural or intellectual superiority. Although this ‘third space’ may challenge entrenched views of what it means to know and do mathematics, it also provides an important opportunity for both cultures to learn from each other.

Ball, R 2015, ‘STEM the gap: Science belongs to us mob too’, *Australian Quarterly, *vol. 86, no. 1, pp. 13–19.

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Barton, B 1996a, ‘Making sense of ethnomathematics: Ethnomathematics is making sense, *Educational Studies in Mathematics,*vol. 31, no.1–2, pp. 201–233.

Clarkson C, Jacobs Z, Marwick B, Fullagar R, Wallis L, Smith M, Roberts RG, Hayes E, Lowe K, Carah X, Florin SA, McNeil J, Cox D, Arnold LJ, Hua Q, Huntley J, Brand HEA, Manne T, Fairbairn A, Shulmeister J, Lyle L, Salinas M, Page M, Connell K, Park G, Norman K, Murphy T & Pardoe C 2017, ‘Human occupation of northern Australia by 65,000 years ago’, *Nature*, vol. 547, pp. 306–310.

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d’Ambrosio, U 1985, ‘Ethnomathematics and its place in the history and pedagogy of mathematics’, *For the Learning of Mathematics, *vol. 5, no. 1, pp. 44–48.

Dreise T & Thomson S, 2014, *Unfinished business: PISA shows Indigenous youth are being left behind*, Camberwell, VIC: ACER.

Harding, S 1993, ‘Introduction: Eurocentric scientific illiteracy—a challenge for the world community’, in *The ‘Racial’ Economy of Science: Toward a Democratic Future*, Indiana University Press, Bloomington, pp. 1‒22.

Harris, J 1987, ‘Australian Aboriginal and Islander mathematics’, *Australian Aboriginal Studies,*no. 2.

Hollinsworth, D 2006, ‘‘Race’: what it is, and is not’ (Chapter 2) in *Race and racism in Australia*, Thomson/Social Science Press, South Melbourne, pp. 24‒39.

Laycock A. with Walker D, Harrison N & Brands J, 2011,*Researching Indigenous Health: A Practical Guide for Researchers,*The Lowitja Institute, Melbourne.

Mack L & Gower G, 2001, *‘*Keeping the bastards at bay: Indigenous community responses to research*’*, in P Jeffrey (ed.), *Australian Association for Research in Education Conference 2001.*

Matthews C, Watego L, Cooper TJ & Baturo AR, 2005, ‘Does mathematics education in Australia devalue Indigenous culture? Indigenous perspectives and non-Indigenous reflections’, in P Clarkson, A Downton, D Gronn, M Horne, A McDonough, R Pierce et al. (eds.) *Proceedings 28th conference of the Mathematics Education Research Group of Australasia*, vol. 2, pp. 513‒520, Melbourne, Australia.

Matthews, C 2008, ‘Stories and symbols: maths as storytelling’, *Professional Voice, *vol. 6, no. 3, pp. 45‒50.

Memmott, P 2007, *Gunyah, Goondie and Wurley: The Aboriginal Architecture of Australia, *University of Queensland Press, St Lucia.

Nakata, M 1998, ‘Anthropological texts and Indigenous standpoints’, *Australian Aboriginal Studies*, no. 2, pp. 3‒12.

Nicol R & Robinson J, 2010, ‘Pedagogical challenges in making mathematics relevant for Indigenous Australians’, *International Journal of Mathematical Education in Science and Technology, *vol. 31, no. 4, pp. 495‒504.

Norris RP & Harney, BY 2014, ‘Songlines and navigation in Wardaman and other Australian Aboriginal cultures’, *Journal of Astronomical History and Heritage*, vol. 17, no. 2, pp. 141–148.

Pais, A 2011, ‘Criticisms and contradictions of ethnomathematics’, *Educational Studies in Mathematics,*vol. 76, no. 2, pp. 209–230.

Pascoe, B 2018, *Dark Emu, *2^{nd}edn, Griffin Press, South Australia.

Pearce, J 2001, ‘Indigenous students at university: is teaching still a colonising process?’, in P Jeffrey (ed.), *Australian Association for Research in Education Conference 2001.*

Pearcey, G 1998, ‘The Wheel and the Boomerang’, *Ockham’s Razor, *radio program, ABC Radio National, 7 June.

Purdie N, Milgate G & Bell, HR 2011,‘Introduction’, in N Purdie, G Milgate & HR Bell (eds), *Two way teaching and learning: toward culturally reflective and relevant education*, ACER Press, Camberwell, pp. xviii‒xxi.

Sarra, C 2011, ‘Transforming Indigenous education’, in N Purdie, G Milgate & HR Bell (eds), *Two way teaching and learning: toward culturally reflective and relevant education*, ACER Press, Camberwell, pp. 107‒118.

Smith, LT 1999, *Decolonizing Methodologies: research and Indigenous Peoples, *University of Otago Press, Dunedin.

Yunkaporta, TK 2010, ‘Our ways of learning in Aboriginal languages’, in J Hobson, K Lowe, S Poetsch & M Walsh (eds),*Re-Awakening Languages: Theory and practice in the revitalisation of Australia’s Indigenous languages, *Sydney University Press, NSW, pp. 37‒49.

- In 2016, I read 53 books.

Blogposts: Part 1, Part 2, Part 3.

Goodreads: 2016 Reading Challenge. - In 2017, I read 62 books.

Blogpost: #read2017: My year in books.

Goodreads: 2017 Reading Challenge. - In 2018, I read 93 books.

Blogpost: (you are reading it!).

Goodreads: 2018 Reading Challenge.

Out of the 93 books:

- 37 books with Australian authors
- 60 books by women
- 39 fiction books
- 14(?) non-fiction books about political/current events
- 8 ‘Little Books on Big Ideas‘
- 4 books about teaching or maths (or teaching maths)
- 4 authors, multiple books (Terry Pratchett: 2, Veronica Roth: 4, Jessica Townsend: 2, Laura Ingalls Wilder: 3)
- 2 cookbooks
- 6 re-reads
- 3 books aloud (‘Roar’ (Samantha Lane), ‘Minefields’ (Hugh Riminton), ‘Dark Emu’ (Bruce Pascoe), and two substantially underway.)

It’s not a ‘Best Of’ list, but twelve books that I couldn’t put down. (There are probably more in this category on the long list.)

- Nevermoor (and also Wundersmith), Jessica Townsend
- The Book Thief, Markus Zusak
- The Hate U Give, Angie Thomas
- Children of Blood and Bone, Tomi Adeyemi
- Anna and the Swallow Man, Gabriel Savit
- Pachinko, Min Jin Lee
- Eleanor & Park, Rainbow Rowell
- Teacher, Gabbie Stroud
- Any Ordinary Day, Leigh Sales
- Chemistry, Weike Wang
- Becoming, Michelle Obama
- The Life to Come, Michelle de Kretser

I could not stop thinking about these books for the impact they made on me. Most of these also belong on the previous list.

- Home Fire, Kamila Shamsie
- Lampedusa, Pietro Bartolo, Lidia Tilotta, Chenxin Jiang
- The Trauma Cleaner, Sarah Krasnostein
- The 57 Bus, Dashka Slater
- Growing Up Aboriginal in Australia, Anita Heiss (Editor)
- Between Us, Clare Atkins
- One Hundred Years of Dirt, Rick Morton
- Dark Emu, Bruce Pascoe
- On Patriotism, Paul Daley
- No Friend but the Mountain, Behrouz Boochani
- Terra Nullius, Claire Coleman
- Staying, Jessie Cole

- Terra Nullius, Claire Coleman

More and more ‘tells’ were revealed, until ‘BAM!’.

The short reviews are archived at this link: #read2018.

]]>*We hope that others in the Edu-Twitter/blogging community will also write posts that respond to the same six questions. The greater the diversity of responses, the more likely it’ll be that a reader will find an approach that works for them. Write as little or as much as you like. You might also like to read posts by Michaela Epstein (@mic_epstein) and Jeremy Hughes (@JeremyinSTEM).*

*If you write a blog in response, let Ollie, Michaela or me know and we’ll link to it from our posts. If you were at our #MAVCON session, I presented a few slides which can be accessed here.*

**1. What does your average reading/watching/listening day look like? (How much time do you spend reading/watching/listening? Which platforms are you on? Are you reading hard copy or digital, et cetera.)**

I start every morning with a cup of tea in bed, and my Twitter feed. I’ll spend maybe 30 minutes scrolling through my Twitter feed, and responding to notifications. Throughout the day, I check Twitter in snatches — when waiting in line, as a short break between other tasks, while walking to meetings (or even in boring meetings :)). A typical evening has some chill time on the couch. If the TV is on, I’m probably idling through Twitter at the same time. By the time I go to bed, I’ve probably caught up on my tweets, ready to start again the following morning.

My iPhone tells me that my daily average on Twitter over the last seven days was 2h 48 min. (That’s kind of horrific but, in my defence, I’ve been at a conference all week, and so my Twitter use often skyrockets.) I estimate I’d spend 60-90 minutes on Twitter each day.

I follow many blogs. I use an iPhone app called Reeder to aggregate the RSS fields from blogs (more below). Occasionally I’ll check through Reeder and send any long posts I want to read more slowly to Instapaper. If I want to act on it, I’ll email it to myself. If I want to store it for reference, I’ll send it to Evernote.

I used to check blogs in a similar manner to how I check Twitter. Now I might spend an hour (at most) every week looking at either Reeder or Instapaper.

**2. If you use Twitter, how do you use it? (Do you use lists? Just scroll through your feed? etc)**

I joined Twitter in 2009 but didn’t post my first tweet until February 2013. When I began, I followed mainly followed politicians, comedians, productivity and technology experts, and a relatively few mathematicians and maths educators. I read tweets but never responded.

My first maths-related tweet was in 2014 but it wasn’t until I gave one of the plenaries at the 2015 AAMT conference (#AAMT2015) that I started using Twitter to properly engage with other maths educators. I found that the conference hashtags was a good way to follow what was going on, to talk to other people about their reactions to sessions and, after the conference ended, to have a way of keeping loosely in contact with the people I’d met.

Eventually I was following more than 1200 accounts. As someone who tries to read every tweet in my timeline (more on that later), I was feeling overwhelmed. I tried lists but they didn’t suit my daily reading style. I pared down the list of accounts I’m following to 600, which is manageable. I’m not sure what proportion is currently maths related.

As someone who has a iPhone surgically attached to their hand, that is my primary way of accessing Twitter. The official Twitter website and apps make me want to scream; I use Tweetbot 3 on both my iPhone and MacBook Pro. These third-party apps are missing some Twitter ‘bells and whistles’, but they also suppress ‘features’ I don’t want, like ads and what other people have ‘liked’. Importantly for me, Tweetbot presents my timeline in chronological order, which I find much easier to follow. Occasionally I will use the official Twitter app or website to access group conversations and other features.

When I log onto twitter, I start by working through my notifications, then go to my timeline. If there are a lot of tweets, and I don’t have much time, I start to remove tweets from my feed. I do this by temporarily muting hashtags and people that corresponding to tweets I’m less interested in. For example, lots of the people I follow might be at a conference that doesn’t interest me, so I mute the hashtag for the conference.

You might be wondering why I ‘waste’ so much time on Twitter. I can honestly say that it has transformed my teaching, and made me feel like I belong in a supportive community of educators.

If you are wondering how to get started tweeting, there are three excellent resources I can recommend:

- ‘ExploreMTBoS: Getting started with Twitter’: https://exploremtbos.wordpress.com/2015/04/25/twitter/
- Michael Fenton’s ‘Five things every teacher should know about Twitter’:

http://reasonandwonder.com/5-things-every-teacher-should-know-about-twitter/ - The Math Twitter BlogoSphere directory: find people to follow on twitter or blogs.

http://www.fishing4tech.com/mtbos.html

**3. How do you manage your reading list, and how do you decide what makes the cut? (google keep, tabs in a web browser, evernote, your brain? Do you use an RSS list like feedly?)**

In July 2016, prompted by Tracy Zager’s keynote at Twitter Math Camp, I did an audit of the 311 maths accounts I followed (out of approximately 700).

I like having a breadth of levels, from primary to research level, because those people provoke me to think about different kinds of maths, and ways to teach maths, in different ways. Nowadays, I suspect that I follow more maths accounts, and I hope the range is more even too — I learn a great deal from primary and middle school educators.

At the moment I roughly have a ‘one in, one out’ follower policy, so that Twitter doesn’t get too unmanageable. I’ll follow people who have ideas that I find interesting, or who I’ve met in person and with whom I want to maintain some kind of contact. I’ll occasionally go through and ‘evaluate’ the people I follow. Are they still adding to what I’m looking to gain from twitter? How do I react when I read their tweets? If I find myself consistently exasperated (politicians and news sites excluded!), I’ll unfollow. Life is too short to spend it annoyed by things you can control.

If I see something on Twitter that I want to act on, I email it to myself. If I find a good teaching resource/question/idea, I send it straight to Evernote. Later, when I’m scouting for ideas on teaching a particular topic, I’ll have a quick search in Evernote.

For the blogs I follow, I use Feed Wrangler to aggregate posts. Feed Wrangler has an online twitter-like feed. However, my method for reading those posts is via the iPhone Reeder app. I will typically sort by author, rather than by chronological order. Sometimes I’ll send very long posts off to Instapaper, and read them typically on flights when I don’t have wifi access to catch up on Twitter …

**4. How do you take notes and collect the gems from what you read? (Where do you store? How do you store? Do you periodically revisit? Do you take notes as you go or all at the end?)**

I’ve already mentioned Evernote and Instapaper. If I know an idea is relevant to a particular project (like a talk I’m giving, or a class I’m teaching), I might save a PDF of the blog post or the tweet in the project folder on my computer.

**5. Is there anything you’re still trying to work out in terms of managing overwhelm and the massive amount of edu-info that’s out there?**

I haven’t mentioned books. I read a handful of edu-related books every year (and many more non-edu books). I haven’t worked out how to take notes that are useful, or to work out a way to reference the good ideas to come back to later. I think I have a lot to learn from Ollie’s post.

**6. What are some of your fave tweeters/blogs/podcasts/youtubers etc that you’d recommend to others (please write a sentence or two after each recommendation to say what you like about that source).**

Without knowing a person’s tastes and interests, I’m reluctant to make recommendations. Instead, I suggest following a few people, paying attention to who they retweet, and then slowly finding more people to follow that way.

]]>It is no secret that Quarter the Cross is one of my favourite tasks. I’ve written about it twice before: as a Day 1 activity and in connection with Fraction Talks. The original source is apparently T. Dekker & N. Querelle, 2002,

Quarter the Cross promotes creative thinking, encourages students to find multiple ways of achieving the same goal, and compels students to justify their reasoning.

In the task, students verbally justify their visual representation of one quarter of the cross. This connects two representations together: verbal and visual. So far, students have been considering the cross as one whole.

We can incorporate a richer symbolic representation by considering the cross as composed of five smaller squares. (I was originally inspired by David Butler’s mention of this in a blog post.) In this second method, if one square is a unit, then what does it mean to ‘quarter the cross’? We can colour 1/4 of one unit and do this five times. This is represented symbolically as 5/4 = 5×(1/4). One way to represent this visually is shown below.

We can now explore different ways of expressing 5/4, and connect them to corresponding visual representations. For example, 5/4 = 1 + 1/4. Colour in one whole unit and a quarter of another. One way to represent this visually is shown below.

In the past, I’ve shown students a few different ways to symbolically express 5/4 and their corresponding visual representations. Occasionally I’ll pose one as a challenge. (‘Show me a visual representation of 5/4 = 2×(5/8)’.)

This year I wanted to try having students actively connect the representations. Inspired by Maureen Hegarty, I decided to have students try a card sort, with the following instructions:

- Cut out the cards.
- Match each visual representation to the symbolic representation. (You can write your own card, if needed.)
- When you are finished, try writing alternative expressions using different operations.

The pages are shown below. You can also download the entire PDF here.

What I found interesting is that because *all* representations show 5/4, students are focused on *which* symbolic representation best suits a particular cross. They also naturally debate whether more than one symbolic expression could be used. Next time I might include a few easier representations (students found some of these to be very challenging). In summary, I loved this way the discussion was mathematically rich, focused on connecting representations, and encouraged students to justify their reasoning to one another.

Responses from my #math1070 students:

- A koala because maths is interesting and unique.
- A spider because of the pattern organisation of their web.
- A cat because it has lots of layers.
- An ant because lots of small parts make up a big part.
- A zebra because it has many patterns and it is black and white with one solution. The patterns are often confusing but has many approaches: black with white stripes or white with black stripes.
- A chameleon cos once you think you’ve got it, it can disappear.
- A snail because it’s slow.
- An octopus because they’re calculated yet inspire intrigue.
- A starfish because there are many different aspects and ways to solve problems.
- A giraffe because it’s leagues above my level.
- A dog because while it has complex intelligence, its commands and responses are consistent.
- A leopard because it has a sophisticated pattern but also because it has an elegant mannerism.
- An eagle because it’s precise and adaptable.
- A turtle because you need to work through slowly and thoroughly.
- A tiger because it has a soft fur but could bite you.
- A monkey because its unpredictable.
- A zebra because there is an unknown amount
^{1}of stripes. - A bear because it looks scary but it’s actually soft and cuddly.
- A sloth because that’s how it makes you feel.
- A spider because we create webs of problems, solutions and challenges.
- A chameleon because just because you can’t see it doesn’t mean it’s not there.
- An ant. Maths underpins almost all concepts, it may be invisible at times but you can always find it if you look hard.
- A lion because a lot of people are scared of it until they know more about it and find the friendly side.
- A puppy because they are fun but then sometimes annoying.
- A penguin because they are fascinating.

[1] I’m transcribing student work. I know it’s ‘number’. Don’t @ me.

]]>As is customary, in the last few days there has been a flurry of reflective blog posts. (Eventually they’ll be linked here on the TMC website.) I’ve been reluctant to write one, and not just because it feels like I’ve been too busy since I got home to even breathe.^{2} Without wishing to sound overly sentimental, #TMC18 feels like a perfect memory that I’ll spoil if I try to capture it in words. I also don’t want to single out any moments for fear that I’ll forget to mention someone.

And, to be honest, I don’t go to conferences for the talks (as fabulous as they were). I go to conferences for the *people *and for the conversations that happen outside of the conference program. And my goodness, TMC has *so* many opportunities for those conversations, thanks to the careful design by the organisers. There is something special about meeting someone in real life after chatting online, sometime for years—the conversation naturally picks up from where it left off, irrespective of the medium. (It’s also fun to find out if someone sounds just like or nothing like you imagined!)

Perhaps more surprisingly, there is also something special about meeting someone at TMC who you *haven’t* yet interacted with online. Members of the #MTBoS community have a shared enthusiasm for maths in many forms, and so it is easy to strike up a conversation about teaching maths, playing with maths, making beautiful mathematical art, or any other variant. I knew the #TMC18 crowd would be my kind of people, I was just delighted at how quickly they felt like my kind of people.

During TMC18, there was some talk about what makes the MTBoS feel different to NCTM (the North American mathematics education organisation with more than 60,000 members), and TMC (limited to ~200 attendees) different to the NCTM Annual Conference (with thousands in attendance). There are, of course, some obvious differences (size, financial imperatives, charter, …) but I think it comes down to this. With a large organisation, you start by *paying*^{3 }to be a member and so it’s easy, subconsciously or otherwise, to automatically seek to get something in return. With the MTBoS, you start by *receiving—*good ideas, encouragement, friendship—and so you automatically seek to give back by contributing whatever you can. The generosity and warmth of the MTBoS community is remarkable.

In this kind of community it is easy to feel like an imposter. (This was a theme of Friday’s keynote.) To this I offer two thoughts. First, we tend to compare the inside of ourselves with the outside of others. Second, we all suck sometimes. With courage, sharing the failures also contributes to the community. Seeing that others are also imperfect helps to alleviate imposter syndrome. I wrote a little more about it here, and am reminded to use the hashtag #lessonfail more. (Because I know I’ve missed plenty of opportunities in the last year to do so!)

Social media is often derided as either frivolous or a hellscape. Mostly, it is what you make it^{4}. It’s fair to say that the #MTBoS has invigorated and enhanced my teaching in ways I’d not thought possible. So, if you are waiting for an invitation to join in, here it is: in the words of the TMC18 song, ‘Tweet at me now!’. I’d love to talk with you.

[1] I’m not going to get into the whole ‘teacher’ versus ‘educator’ debate except to say that not everyone in the #MTBoS is a teacher, and educator feels a little more inclusive. Or maybe we could just dispense with labels and say ‘people’.

[2] Our semester started while I was winging my way back to Adelaide. I didn’t even teach my first two classes. Shhh…

[3] Sometimes a lot. The NCTM conference? Well out of the price range of most everyday classroom teachers.

[4] Admittedly, it might require steady use of mute and block.