I was happy watching Shaman King (2021) when right in the first episode, I saw something that I couldn’t ignore…
Maybe most people didn’t care what the teacher was going through on the blackboard in the classroom where the anime characters study. But that made me stop and take a closer look.
Finally, on the left side of the blackboard, the professor used the Pythagorean theorem to calculate the value of the hypotenuse of a right triangle, with sides 4 and 3. The calculation is correct.
Then in the center of the blackboard is a new right triangle. It certainly doesn’t refer to the calculations on the left, as its “a” side appears to have more than 4/3 of the “b” side.
On the right side of the blackboard the teacher starts a new calculation: 8² + 5² = c².
Again it seems that there is nothing wrong, because even if we do an approximation of the size of the triangle on the blackboard, we have that the “a” side seems to be 60% larger than the “b” side. The triangle in question is correctly represented, so what am I complaining about?
Well, for a start, in the calculation on the left, the solution came out conveniently, since 25 is a perfect square, that is, its square root is a Natural number. Because from the 5th to the 6th row, the jump was made from: 25 = c² to c = 5.
On this occasion, the step of placing the square root over the 25 was omitted and a transition between the left side of the equality to the right side was also made. Notice that the “c” appears on the 3 lines above the right side, and suddenly shifts to the left side. Mathematically speaking, there is no harm in this operation, but if we think about it from the didactic point of view, it is an unnecessary procedure and it can confuse those who are following the procedures with difficulty.
On the right side of the blackboard, we have the omission of 2 lines above the calculation. Where would the theorem be expressed from generic components “a² + b² = c²” and would define who is “a” and who is “b”. Note that this was done on the left side of the board, but now omitted on the right side. But you might be thinking, it’s okay to omit a few steps, but note that now the calculations won’t come out as pretty as on the left side.
Because when we do 8² + 5² = c², we will have 64 + 25 = c². This brings us to 89 = c². But now, 89 is not a perfect square, that is, there isn’t a Natural number for which we can simply “square off” c. This means, that on the right side of the board, we will have the square root insertion at 89. What was not included as a step on the left side.
Thus, it is possible that a student based on the procedures on the left side, will not identify the right side as analogous (since the two lines above the equation were omitted). The same way you can crash when you get to 89 = c², since there is no Natural x such that x² = 89.
Yes, these are generally unimportant details when we are already proficient in these calculations. However, when thinking about the period in which these concepts are learned, some omissions, or even adjustments from one side of the equality to the other, can appear as distracting or gaps in the notions and ideas that surround these concepts.
I was happy at least to see that the representation of the triangle on the blackboard was consistent with the measurements of the collared peccaries, as this is something else I often see even in teaching materials and educational games. Geometric shapes are passed with measurements that do not match (or appear to match) those given. So again, when we have mastery in the subject, it ends up being ignored, but if we think of a beginner in the subject, when finding a triangle with a shallow angle, but the material says it is an obtuse angle, what an unnecessary mess that will make your head?
I discuss this in my Master’s dissertation, giving this educational math game as an example of visual clutter. Because if we put a protractor in the triangle on the left, we have that the given angles 57o, 61o and 62o would actually be 49o, 73o and 58o. In the right triangle, the given angles 68o, 56o, 32o and 92o would actually be 65o, 47o, 61o and 50o. Interesting in this case, it is to notice how discrepant the differences are, since the algebraic solution is 92o, that is, superior to a right angle. However, visually we can identify in the geometric representation that the solution should be an acute angle.
Cropped cover image from the first episode of the anime Shaman King (2021).