Scratch programming and student’s explanations
Excerpt 1. Tor finishes the program for the crab (Figure 1), and Per starts the animation. The crab leaves the screen and then reappears on the left side to proceed
almost to the centre where it stops. Tor says, “No! We have to take eeh … I guess we have to take away …” Tor drags the last two blocks aside and, while holding them, asks, “How do we delete these?” Per moves the blocks back to the menu and they disappear. Tor then runs the remaining program (Figure 2) and concludes that the crab stays on the screen.
Figure 1: Code when crab leaves screen Figure 2: Code when crab stays on screen When the code makes the crab leave the screen and reappear on the other side, the students says “No!” and start immediately to revise the code. The animation mediates an important idea of measurement, namely the indirect comparison of length. The code can mediate the idea of a function for the total path length expressed as the number of steps per block as it consists of four move right blocks grouped according to the multiplication tables for 2 and 4. It does not though mediate the idea of measurement as clear as the animation because there is no limitation to the numbers to be used in the code due to the circular movement. Combined, the code and the animation help them to see how to keep the crab’s movement on the screen. Tor says “I guess we have to take away …” and he disconnects the two last code blocks. This can be regarded as a procedural mathematical explanation because it, although it requires knowledge about the solution, mainly concerns how to solve the problem. Per deletes the two disconnected blocks by moving them back to the menu, and this manipulation of the blocks is also a procedural explanation, though non-mathematical. When they had done the code adjustments, they checked if they had solved the problem by running the code again.
In this excerpt, the differences between the ideas mediated by the code and the animation invite the students to solve a mathematical problem and provide explanations. ScratchJr provides visual modalities both to understand the problem and to communicate the solution. After 19 minutes, a teacher educator turns on the grid function to help the students program a boat approaching the beach, so that it includes perspective effects. They try out the enlarge and normal size functions from the control menu, but no use of the grid is evident.
During the introduction on the second day, the PT asks some of the students to explain the functions in the different menus of ScratchJr. He comments and adds several
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examples when the say command (from the looks functions) and the sound and motion functions are discussed. Then Per and Tor get the task to create an animation using the multiplication tables for 6 and 8. They choose a birthday party as a topic. After two minutes, Tor turns on the grid function.
Excerpt 2. During the first nine minutes, Per and Tor program the pink alien to move on the scene, add sound at several steps, and review the result several times. The PT approaches the students, looks at the screen, checks his notes, and decides to take part in their discussion:
PT: Now you actually have to start including bigger numbers [points at the screen]. Is it big multiplication tables, the multiplication table for 6 and … [checks his notes] 8?
Tor: [Looks at PT] Yes, we have 12 [points at the rotation block] (see Figure 3).
PT: But 12 in itself is not how the multiplication table for 8 is built up.
Tor: But it is very difficult with eight because then we have to go so far, then we have to do something else.
Figure 3: Code for Pink alien Figure 4: Code for Blue alien The students seem to try to avoid the problem of the limited width of the screen by only using the number 2 on their motion blocks (Figure 3). This is not in line with the task of using the multiplication tables for 6 and 8. When the PT points this out, Tor argues that they cannot use larger numbers from the multiplication table for 8 because it will move the alien too far. He answers the PT’s why-question by explaining their decision not to use the multiplication table for 8, and this can therefore be considered as an example of an intentional explanation mode. The PT’s involvement makes the students look for how actions, other than movements, can be used in the animation in order to follow the pattern of the required multiplication tables.
Excerpt 3. After the PT leaves, Per and Tor add a new page with the blue alien as the character. They disagree whether they should use the multiplication table for 6 or 8, but then Per makes a program (see Figure 4) and the blue alien stops having just a few more steps left before it might disappear from the screen:
Tor: We have to think about how many steps it is.
Per: Six.
Tor: One, two, three, four, forward [points at the four squares of the grid ahead of the alien]. So, we have to take four [types 4]. We have to take four. Look, it will be as here. And start!
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While Per makes the code, Tor counts and presents an argument for how many steps the alien has left before disappearing from the screen. The mathematical problem is similar to the one in the Excerpt 1. However, this time the grid function has been turned on, and this adds an iconic representation for the number of steps in the animation.
Together, they mediate the idea of the measurement by counting. This supports Tor’s argument by providing evidence for his point of view. The combination of gesture (counting by pointing at the grid) and language can be considered a mathematical explanation. Although it has a clear procedural goal (how to program the alien to move to the end of the scene), it is a deductive explanation, because Tor first counts the number of squares between the alien and the goal, and then concludes that they have to write four in the program.
On the second day, Per and Tor use the looks (only say), sound and motion functions (except go back). They use reset characters several times, but do not use the control functions as they did the first day. Although the grid function is turned on the first day as well, it is not used to solve the mathematical problem. On both days, the functions the PT emphasises in his presentations are the ones most frequently used by the students. The motion, grid, sound, and looks functions seem to foster explanations and justifications. Procedural and intentional explanation’ modes were identified both days, often related to ScratchJr functionality, but the deductive mode of explanation occurred only on the second day. The students concentrate mostly on forming meaningful storylines, but they adhere more closely to the assigned multiplication tables on the first day. The communication moves from focusing on the exploration of ScratchJr functions during the first day to become more mathematical and result-oriented on the second day as they get more familiar with the program. This is in line with how the PT organized the two introductions, from brief comments to discussion of the functions.
CONCLUSION
What makes grade four students engage or not in mathematical explanations and justifications when programming in pairs with ScratchJr? The focus in this paper has been on which program functions were used and if and how they fostered or hindered such argumentation. Summarized, the study provides results showing that students’
mathematical argumentation can be mediated (cf. Albano et al., 2017) by: conflicting differences between modalities in line with Morgan and Alshwaikh (2012) (cf. excerpt 1); thinking on the solution of the task mediated by the program or one of the modalities (cf. excerpt 3), and, in some cases, negotiating the task when challenged by the PT about how well they addressed it. ScratchJr mediates explanations and justifications by making several representations of mathematical ideas simultaneously available through different modalities.
The study contributes, like Kaufmann and Stenseth (2020), to documenting an educational potential of programming in mathematics education. The affordances and constraints of ScratchJr have some didactical implications for mathematics teachers who plan to use it. For example, ScratchJr allows sending messages and wait a certain
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time for an action, but it lacks conditional blocks. The animation with the grid visualises well that one step forward increases the x-coordinate by one. The circular nature of the grid surprises the students and can trigger explanations and justifications.
Combining the motion and control functions can create a perspective effect for a moving character, while the rotation of a character is limited and only works for some of them. These aspects should be taken into consideration during task design and other didactic actions aiming to facilitate student’s argumentation.
Although using multiplication tables when making animations might not seem challenging enough mathematically for year four students, in a programming context with ScratchJr, it provides opportunities to investigate measurement ideas. A more explicit focus in the task formulation on number sequences in the multiplication tables could invite students to focus more on functional reasoning. Discussing the mathematical ideas students experience while programming and how the program can be made to do what it does, can facilitate students’ mathematical argumentation.
Teachers' preparation, and the presentations and follow-ups in class, influence to what extent students explain and justify with mathematics, but there are also other aspects that deserve further research. Investigating the role of students and teachers’ gestures, how teachers can develop and use knowledge about ScratchJr and task design to facilitate mathematical argumentations, and how teacher educators can help PTs gain sufficient knowledge about how to include programming in their mathematics teaching, are just some examples.
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