Learning by Interacting with Student Thinking
This wasn't interacting with one of my student's but rather interacting with one of my classmates who is a student but the outcome is similar. Learning to tackle this problem in a different way helped me to see a better way of illustrating some fundamental math principles (substitution, keeping an equation in balance).
It starts with the problem that our professor posed to the class:
Two tug-of-wars have occurred and resulted in ties.
- 4 frogs went up against 5 Fairies.
- 1 Dragon tied 2 fairies and a frog
Who will win when 1 dragon and 3 fairies go up against 4 frogs?
The class sat down to tackle this problem, largely individually. For me, the solution seemed fairly straight-forward. I would simply apply some standard mathematics substitutions. I didn't spend much time thinking about the best way to do it, I just used brute force because I felt fairly confident in my mathematics ability to solve the problem. I solved the problem but I wasn't actually very confident in my solution. So much for my confidence.
Taking second battle result
1 Dragon :: 2 Fairies + 1 Frog
I multiplied both sides by 4.
4 Dragons :: 8 Fairies + 4 Frogs
Then I substituted in the first battle result of 4 Frogs :: 5 Fairies
4 Dragons :: 8 fairies + 5 Fairies
4 Dragons :: 13 Fairies
1 Dragon :: 3.25 Fairies
Putting that information into the final battle:
1 Dragon + 3 Fairies :: 4 Frogs
3.25 Fairies + 3 Fairies :: 4 Frogs
6.25 Fairies :: 4 Frogs
Since we know from the first statement that 4 Frogs tied 5 Fairies, the final battle has the power of 6.25 fairies against the 4 frogs so 1 Dragon and 3 Fairies would win against four Frogs.
So, I got a solution that was consistent with students around me, but I wasn't really 100% sure that I had done everything correctly. I wasn't sure if I'd done the correct substitutions or if I hadn't made some calculation errors.
Visual Arts Leads the Way to Better Math
Another student (Teachable: Visual arts), didn't have a mathematical solution but instead started off with a visual representation of the three tugs of war.
She then reasoned that Frogs are just slightly stronger than Fairies because of the balance of the first tie. Then she reasoned that the Dragon is much stronger than a Fairy or a Frog based on the outcome of the second tug of war and then in the final battle, because the Dragon is so much stronger and the Frogs were just a bit stronger than the the Fairies and there are three Fairies on the side of the Dragon, she reasoned that the Frogs would lose. (I believe that was the flow of the logic). She got the right answer but it's not a very definitive proof.
When I looked at this visual solution, I was struck by the clarity of the visual set-up of the problem as opposed to my solving for Fairies approach, so I wondered if it could lead to a more definitive proof. The natural tendency to look for patterns quickly revealed a solution.
There is an obvious substitution of 5 fairies for 4 frogs which jumped out at me in the visual representation of the problem but I had missed when I did my original solution.
which gives us
thereby eliminating frogs from the tug of war (equation). Now, we can easily see that we can cancel Fairies off both sides in equal number to maintain balance in the equation.
Which leaves us with the equivalent of one Dragon against two Fairies.
But, we know that one Dragon tied 2 Fairies and 1 Frog
It's obvious that two fairies is less than two fairies plus a frog
so we can conclude that
The Frogs will lose.
Take-away
Setting up the problem visually made the correct substitution obvious and made using cancellation to get to the simplified tug of war of one dragon against 2 fairies very easy and more intuitive. It's easy to imagine that if you put one person of equal strength on both ends of the rope, they will cancel each other out.
When the process was completed, I had much more confidence in this solution than I did when I had done the brute force mathematical method. So, collaboration led to clarity of solution and, I think, a better understanding of substitution and what is actually happening when solving equations with multiple variables. Perhaps the next step would have been to then do the mathematical setup of the problem so that the visual representation would then lead to an equation solution.
The argument against this approach is that it obviously doesn't scale beyond very small numbers; however, it is very illustrative of the process of substitution, keeping an equation in balance (subtracting 3 Fairies from both sides) and solving for the remaining variables.
Exploring and learning this way before jumping into the mathematics would give me a better understanding than just jumping to the more abstract x =, y =, etc.