Reflections on digital technologies in mathematics education across cultures

ion process” (p. 1). Relevant to the questions addressed to this panel, she adds that computational thinking can be defined as “the thought processes involved in formulating a problem and expressing (with a linguistic representation) its solution in such a way that a computer – human or machine – can effectively carry it out” (p. 1). Interestingly, Andy diSessa (2018) – one of the two authors of Turtle Geometry back in 1981 – has taken issue with this point and has argued that noncomputer scientists rarely map out exactly how a problem can be solved before actually doing the solving. But is he right? In opposition to diSessa, and more in line with Wing, Al Cuoco (2018) in a paper on mathematical practice offers three examples. The first of these (see Fig. 1) relates to Wing’s emphasis on the process of abstraction and her point about formulating a problem and expressing its solution in a way that a computing being or machine can carry it out. This example involves what Cuoco refers to as “the dreaded algebra word problem,” where he insists that we think of the answer to the algebra problem as an equation rather than a number – in a method that involves abstracting from numerals. The problem is as follows: “Mary drives from Boston to Chicago, travels at an average rate of 60 MPH on the way down and 50 MPH on the way back. The total driving time takes 36 hours, how far is Boston from Chicago?” Figure 1. Arriving at an equation from abstracting the regularity in numerical guesses (Cuoco, 2018, p. 3) The method that Cuoco suggests builds upon students’ ability to solve similar problems in middle school (note: they have already learned the relationship between speed, time, and distance) and is as follows: Take a guess – but the aim is not intended to get closer to the answer with each succeeding guess; rather it is to arrive at an equation, not a number. The idea is to carry out enough guesses so as to see the regularity of the calculations that allow for checking the guesses – in Cuoco’s words: Develop “a generic ‘guess checker’ that is the desired equation”. The processes of mathematical Reflections on digital technologies in mathematics education across cultures 76 practice that are employed here, and which are ones that Cuoco declares he uses all the time in his own mathematical work, are: 1. Abstract regularity from repeated calculations, and 2. Use precise language (and algebraic symbolism) to give a generic and general description – the equation – for how you check your guesses. (Cuoco, 2018, p. 4) The conclusion to be drawn from this example is that these two processes of mathematical practice fit well with the programming and thinking-like-a-programmer characteristics of computational thinking (Wing, 2006, 2014), and that students who are currently engaged in using digital technologies (e.g., laptops, robots) to code with visual (e.g., Scratch) or text-based languages are participating in mathematical practices. Nevertheless, other research (e.g., Bråting & Kilhamn, 2020) suggests that, while the representations used in programming languages may be similar to mathematical notations, the meanings of several concepts in the two domains differ. But that is a whole other story! In any case, digital technologies afford multiple varieties of mathematical activity that can offer experiences that involve coding but also those that do not. Some Canadian research on the use of digital technologies to foster mathematical thinking I take mathematical thinking to include the various processes that have been drawn upon by Wing and others to characterize aspects of computational thinking – but also more than this, for example, its conceptual aspects. While computational thinking is focused toward coding, mathematical thinking occurs within a host of activities that are not coding oriented, but which can clearly be engaged in within specifically-designed digital environments. However, the tricky thing about terms such as computational thinking and mathematical thinking is their overlap when referring to anything mathematical. Moreover, as Cuoco (2018, p. 2) has pointed out: “In real mathematical practice, it is rare that a piece of work employs only one aspect of mathematical thinking” – and, similarly, only one aspect of computational thinking. Despite the obvious intersection between the two terms, I find it helpful when discussing the use of digital technologies in mathematical activity to distinguish between coding-related activity and non-coding-related activity. In line with this distinction, I offer some examples that give a flavour of Canadian research that has focused on these two types of activity, both of which have successfully combined selected aspects of computational thinking and of mathematical thinking. Digital Technologies in Coding-Related Mathematical Activity Scratch coding on laptops. My first example is drawn from the funded, multi-study research project of George Gadanidis and colleagues from across Canada, titled Computational Thinking in Mathematics Education – a project aimed at researching the use of computational thinking (via, e.g., digital tangibles such as circuits, programmable robots, and coding with Scratch on laptops) in mathematics education, from pre-school to undergraduate mathematics, and in mathematics teacher education (see ctmath.ca/about). In one of the publications from this project (Gadanidis et al., 2017), the initial activity engaged in by the Grade 1 students of a school in Ontario was the use of the blockbased, visual programming language, Scratch (available at http://scratch.mit.edu), for exploring squares by drawing a set of squares rotated around a point (see Fig. 2; see also Gadanidis, 2015). One of the fundamental principles underpinning these study projects is connecting the digital technology work in classrooms to the math curriculum that teachers need to teach. Reflections on digital technologies in mathematics education across cultures 77 Figure 2: Scratch coding in Grade 1 (from Gadanidis et al., 2017, p. 81) Figure 3. Programming a robot using loops (from Francis & Davis, 2018, p. 82) Coding robots. Francis and Davis (2018) studied 9and 10-year-olds’ understanding of number, and the transition from additive to multiplicative thinking, in the context of learning to build and program Lego Mindstorms EV3 robots. The sequence of tasks focused on students’ becoming aware of the architecture of robots, programming the robots to trace a triangle, square, pentagon, or hexagon; and building a robot that could find and douse a ‘fire’ in any of four rooms of a miniature model building. In one of the scenarios that Francis and Davis report on, a student learns how the number of sides and angles of a polygon connects to the number of repeats in a loop, which illustrates a developing shift from thinking additively in terms of a sequence of like actions to thinking multiplicatively in terms of a repetition of a single action (see Fig. 3). The authors argue that coding-related activity with digital technologies can co-amplify mathematics learning, as long as computer programming is seen as “something for” and is integrated into the existing curriculum with well-designed tasks, not as “something more” in a separate curriculum. Digital Technologies in Non-Coding-Related Mathematical Activity Figure 4. TouchCounts App: upper -10 tap; lower -result of 10 single taps (Rodney, 2019, p. 169) Figure 5. “Five Steps to Zero,” with a starting number of 151 (adapted from Williams & Stephens, 1992) TouchCounts – an iPad touchscreen App. The TouchCounts application software, developed by Sinclair and Jackiw (2014), served as a window for the researcher Rodney (2019) to study how a 5and-a-half-year-old, Auden, thought about number. Although Auden was able to say the number names initially, he seemed unaware that the written numeral ‘10’ would appear right after ‘9’ and Reflections on digital technologies in mathematics education across cultures 78 that ‘10’ also represented the number of taps made on the iPad screen (see Fig. 4). Auden’s unsuccessful initial activity with the App revealed that his memorized number chanting needed the further support that TouchCounts could afford in order to reach a fuller understanding of counting and to begin to identify the relational aspect of numbers. Calculators with multi-line screens. Calculators remain a staple in many mathematics classes. This resource, one with a multi-line screen, served as the digital tool underpinning a study that focused on the mathematical practice of seeking, using, and expressing structure in numbers and numerical operations (Kieran, 2018). The study (co-conducted with José Guzman†) involved classes of 12-year-old Mexican students on tasks adapted from the “Five Steps to Zero” problem (Williams & Stephens, 1992; see Fig. 5). Successfully tackling the designed tasks, and subject to the rules of the game, involved developing techniques for reformulating numbers (prime or composite) into other numbers in the same neighbourhood (not more than 9 away from the given number) that have divisors not larger than 9 so as to reach zero in five or fewer steps. Some of the most powerful structural explorations that occurred during the week of activity on the tasks involved the search for multiples of 9. For example, students became aware that “738 and 729 are two adjacent multiples of 9 and, when they are both divided by 9, the quotients are consecutive,” and “in the 9-number interval from 735 to 743 inclusive, there is exactly one number divisible by 9.” In trying to explain the oftensurprising results produced by their digital tools, the students developed several mathematical insights that were new to them. Carolyn Kieran’s concluding remarks My concluding remarks pick up on the interest shown by students in the use of digital technologies – be they coding-related or not. For example, Gadanidis et al. (2017) emphasize “learning experie