ComPutational modelling in Grades 1-3 mathematics
George Gadanidis, Janette Hughes, Immaculate Namukasa & Ricardo Scucuglia
Western University, Ontario Tech University, Western University & UNESP
In our work in elementary classrooms, we have been designing, developing and using topic-specific computational environments to engage young children with dynamic modelling of mathematics concepts and relationships. In this paper we share and discuss the design and application of three such environments, focusing on repeating patterns with visual and aural attributes, on symmetry and transformation, where a symmetry is defined as a transformation that leaves an object looking unchanged; and lastly on sets and subsets of natural numbers, in the context of artificial intelligence. While we continue to use more generic computational environments, such as Scratch, we find that they have limitations for certain mathematical modelling purposes (for example, Scratch does not provide coding blocks for performing reflections of shapes). The focus of our work is unique in two ways: (1) unlike the current trend to teach young children coding, our primary focus is on engaging young children with mathematics + coding with minimal prerequisite knowledge of coding; and (2) in contrast to the current focus on computational thinking, our aim is to shift attention to computational modelling.
In today’s society, computational tools are pervasive. They are used by scientists and professionals to model events and processes, to better understand, to communicate and to make progress in their area of work. Most fields have a computational side, such as, computational finance, computational biology, and computational medicine. The authentic computational modelling practices of scientists and professionals involve solving real-world problems and building knowledge – to learn – through computational “conversation” and “interaction” with their field (Barba, 2014) “with and across a variety of representational technologies” (Wilkerson-Jerde, Gravel and Macrander, 2015, p. 396). “It’s a source of power to do something and figure things out, in a dance between the computer and our thoughts” (Barba, 2016).
In our work in mathematics education, in elementary as well as secondary school classrooms, we are shifting our focus away from computational thinking to bring attention instead to computational modelling. diSessa (2018) offers an elaborate critique of recent definitions of computational thinking. For our purposes, computational modelling brings attention to the “visible” actions of students rather than their “hidden” thinking, and it positions computation as a tool in mathematics teaching and learning rather than a goal in itself or an object of study. In addition, computation as a dynamic modelling tool supports student agency in mathematics learning. Children need opportunities to experience the freedom to make choices, to investigate, and to discover (Papert, 1980, 1983). Noss & Hoyles (1992) ask, “How can we build settings that structure pupils’ learning without artificially fragmenting the activities, destroying pupils’ joy and motivation, and threatening teachers’ respect for pupils’ own goals?” (p. 466).
Below we describe and discuss three computational modelling environments we have designed, one focusing on repeating patterns with visual and aural attributes, another focusing on symmetry and transformation, where a symmetry is defined as a transformation that leaves an object looking unchanged; and lastly one on sets and subsets of natural numbers, in the context of artificial intelligence. These environments are each accompanied with an illustrated story (see Figure 1), which sets the context for the classroom activities and also serves as a lesson outline for the teacher, with places to pose the story and investigate math ideas hands-on and with coding on the screen.
Figure 1: Three stories.
repeating patterns in grades 1-3
The repeating patterns focus grew out of a Kindergarten teacher’s request for ideas for teaching “AB patterns”. Her students were able to reproduce repeating patterns she created but had difficulty creating and extending their own. We collaborated to create a series of hands-on activities, which were used in that school in Grades K-3, where students performed their patterns using xylophones, stamped them with colour dabbers, sang them as songs, and “danced” them on colour mats (see Figure 2).
Figure 2: Grades 1/2 students “dance” their repeating patterns on colour mats.
This experience gave us the idea of creating a computational environment where students could model their repeating patterns with code (available at https://learnx.ca/patterns-v2) (see Figure 3). Two lesson studies, in Grades 1/2 and 2/3 respectively were conducted, and documentary videos are available at http://mkn-rcm.ca/repeating-patterns/. We have also used this environment with older students to investigate more complex patterns (Scucuglia, Gadanidis, Hughes & Namukasa, forthcoming).
Figure 3: Grades 2/3 students model repeating patterns with code.
Below we list comments by teachers participating in the lesson studies (Gadanidis & Caswell, 2018):
- The beauty of doing things like this with this age group, is that we removed a language barrier, because it was “Show us what you know.” – Grades 1/2 teacher
- My favourite part was seeing those kids that you’re not sure how they’re going to be in that environment. He was participating and not just superficially. He was very involved and learning. – Grade 1 teacher
- It’s nice to see the flip side, where students who shine in computers and media but maybe struggle in math. I noticed some of the ESL students in my class who seem to struggle when it comes to language and math because they cannot express themselves in a formal context, and they shine here, and become ambassadors and teach other kids who struggle. – Grades 2/3 teacher
- Had we asked him to demonstrate that learning in a completely paper and pencil method, he wouldn’t have felt successful …. that he wouldn’t meet the expectation. But he exceeded the expectation. He was really proud of himself. He wanted his dad to see what he had been doing. That’s exciting. – Grades 1/2 teacher
symmetry in grades 2/3
The symmetry focus grew out of a grade 3 teacher asking for teaching ideas related to symmetry and transformation. Symmetry in school mathematics is most often addressed as an attribute and transformation is not seen as being directly connected. For a mathematician, a symmetry is a transformation that leaves and object looking unchanged. For example, what transformations (symmetries) are possible in Figure 4?
Figure 4: What are the symmetries of the square?
To investigate this phenomenon, we developed a coding app (see Figure 5) where students may investigate rotations and reflections of squares and other shapes (available at https://learnx.ca/sym/rotation-reflection/) and a coding app (see Figure 4) where they may investigate interactions as symmetries “bump” into one another in a sandbox (available at https://learnx.ca/sym/bumper-squares). A Grades 2/3 classroom documentary is available at http://mkn-rcm.ca/symmetry-ct/.
Figure 5: What will happen as rotation symmetries “bump into one another?
We find that it is also helpful to model this “bumping” of symmetries using hands-on materials, to show down the process and draw attention to how the symmetries transform one another, as shown in Figure 6.
Figure 6: Hands-on simulation of “bumping” symmetries.
Below we list comments by teachers involved in the project (Gadanidis, Clements & Yiu, 2018):
- My biggest leaning was about incidental learning. Coding facilitates that.
- It’s really neat because it extends their thinking, but in a natural way.
- I wish you were here to see the kids that never do well on assessments.
- I’ve never seen that part of him. Words coming out were impressive.
- I found that sometimes the tasks we might feel initially [to be] difficult, the kids got just like that.
- Some of them could go beyond what we were showing them. I feel they really surpassed me.
sets and subsets in grades 3-6
The app shown in Figure 7 incorporates rudimentary mathematics of artificial intelligence (the Boolean algebra for decision-making), which students use to create a grid of numbers and use conditional statements to set the background colour of cells and to place characters in the cells. (See https://learnx.ca/sets)
Figure 7: Sets and subsets app.
In a grade 3 classroom, using this app for the first time, students worked on the puzzle of editing the code shown in Figure 7 to produce the result shown in Figure 8.
Figure 8: Coding puzzle.
Figure 9: Grade 3 student solving the puzzle.
Students could easily change the colours to match. The challenge was to also place the characters in the pattern shown in Figure 8. Students who were stuck typically solved the puzzle fairly quickly when prompted to identify the number sequence of the character locations. Noticing the sequence 3, 6, 9, 12 in the grid in Figure 7 and the sequence 7, 14, 21, 28 shown in the grid in Figure 8, they changed “number mod 3 = 0” to “number mod 7 = 0” to solve the puzzle. View a dialogue with one of the students at https://youtu.be/EU4UPvr87lQ.
Grade 3 students did not at this stage understand the meaning of “number mod 7 = 0” but they did understand its effect on the patterns they created on the screen. For the teacher, this creates an opportunity to engage students with further activities to help them develop a more robust understanding. The app offers two other, equivalent forms of “mod” that may help, as shown in Figure 10.
Figure 10: Code blocks with the same meaning.
The sentence “number mod 7 = 0” means that when “number” is divided by 7 then its remainder is 0. The term “mod” or “modulo” can be seen in action on a 12-hour clock where, for example, 7 hours past 8 o’clock is 3 o’clock. On a 12-hour clock, the answer to 7 + 8 becomes 3, at it is the remainder when we divide 15 by 12. Likewise, 24 hours on the 12-hour clock would be 1 o’clock, the remainder when when we divide 25 by 12. This relationship may also be expressed as “15 mod 12 = 3” and it may be modelled using the app as shown in Figure 11.
Figure 11: Modelling 3 o’clock.
Sullivan (2000, p. 211) asks: “What exactly are teachers asking for when they say, ‘Pay attention’?” “Typically, few of our students are actively attending to the mathematics ideas at play in their classrooms. Most students have learned to be passive observers, waiting to be “explained-to”” (Gadanidis, Borba, Hughes & Lacerda, 2016, p. 239). However, this is not their natural state, as “Children begin their lives as eager and competent learners. They have to learn to have trouble with learning in general and mathematics in particular” (Papert, 1980, p. 40). We also cannot directly equate “experience” and “education”, as some experiences can be “mis-educative” by limiting growth from further experience (Dewey, 1938, p. 25). Viewing learning as experience in our work with mathematics and computational modelling raises the status of student engagement, as only when students consider the experience worthy of their attention “will the transaction of experience have its full impact” (Parish, 2009, p. 512).
We have noticed that carefully designed computational modelling in mathematics activities can draw student attention and help both students and teachers focus deeply on mathematics patterns, concepts and relationships. As we have noted in our previous work (Gadanidis, Clements & Yiu, 2018), computational modelling can help to afford student agency, allow for differentiated teaching and learning, and help students experience the pleasure of mathematical surprise and insight. As we discussed in Gadanidis, Borba, Hughes & Lacerda (2016, p. 239) we believe that occasional, well-designed aesthetic mathematics experiences “that are immersive, infused with meaning, and felt as coherent and complete” (Parrish, 2009, p.511), and the associated experience of complex, surprising, emotionally engaging, and viscerally pleasing mathematics, can serve as “a process of enculturation” (Brown, Collins and Duguid, 1989, p. 33) with lasting impact on students’ (and teachers’) dispositions, living fruitfully in future experiences by raising expectation and anticipation of what mathematics can offer, and what the intellectual, emotional and visceral rewards might be when quenching a thirst for mathematics.
In project classrooms, we have also learned to attend to the crucial role played in by teachers in determining the quality of the learning experience. Noss and Hoyles (1992) reflect that part of the reason Logo (which also combined mathematics and coding) did not deliver all of what it promised was due to a tendency to see technology as an end in itself – as “the” determining factor of success – and a failure to address non-technical aspects, such as “the social, cultural, and pedagogical context into which Logo is inserted—an influence that [. . .] is crucial” (p. 431). In our work with educators, we use an engagement and reform model (Gadanidis, Borba, Hughes & Lacerda, 2016) that first asks teachers to identify their personal teaching goals and their students’ needs.
In terms of today’s predominant use of computational tools in education, the focus of our work, as illustrated in the three cases briefly described and discussed above, is unique in two ways: (1) unlike the current trend to teach young children coding, the primary focus of our work is on engaging young children with computational modelling of mathematics patterns, concepts and relationships, with minimal prerequisite knowledge of coding; and (2) in contrast to the current focus on computational thinking, which has been problematic in its definition (diSessa, 2018), our aim is to shift attention to computational modelling, which can been seen in action.
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