Answering the right question
Kids often ask ill-posed questions. Are you answering the right question?
Questions happen at the edge of knowledge
It is a given that when your math kid is learning something new, they are working on the edge of their knowledge.
If they get stuck and ask a question, they may use words or concepts that are close to correct but not quite correct.
The question will make sense to someone who is inexperienced, but it will make no sense to someone who is experienced.
For example:
Does a square root only work for squares?
This came up when we explored a geometric example of squaring a number and then working backward to get a square root.
The question doesn’t quite make sense if you've done enough math.
The words are correct, and the concepts are pretty much there, but it doesn’t make sense unless you know the context in which the question was asked.
The right answer to the wrong question
At this point, it would have been very easy for me to say
Me:
What. No. A square root works for any number. It just means a number that multiplied to itself gives the number you are taking the square root of. Does that means sense?Kid:
Umm. Kinda of.Me:
Okay, let’s do another example
Kid:
[Still confused, but hopes to get their answer answered in perhaps another example or two]
Rather than staying in the geometric example, I assumed that the child had already understood the concept of mapping a number A to a different number B that gives the original number A when multiplied by itself.
I gave a somewhat tangential answer to the kid’s question.
Questioning the question
My kids generally hate it when I ask it, but it’s what I have found to be useful:
Me: What do you mean by that
Kid: Well. We are working with a square. So a square root has to do with a square. Are there triangel roots?
Me: So what you’re asking is if we get the area of a triangel by multiplying two numbers is the operation that reverses that the triangel root?
Kid: Um. Kind of. Or other things.
Me: Okay. So it sounds like two questions. First - do other shapes have names for getting their area and then getting back the things we multiplied together. In this example, in a square, we “square” a number and taken the “square root” to get back the number we multiplied by itself. Second - can you use the “square root” on other shapes?
Now that I have a better idea of their questions, it’s more obvious to me that the confusion was in how I used geometry to give an example of a more complicated mathematical procedure.
This is okay as an example, but I didn’t explain enough context to demonstrate that we were looking at an example and not the definitive definition.
Questions lead to more questions
The original question
Does a square root only work for squares?
Gives us several new questions to explore
How do you find the area of other geometric shapes?
Are there names for finding the area of other geometric shapes?
Are there other mathematical concepts that have shapes in their names? (“triangular numbers,” “cubes,” “cube roots,” etc.)
Is using geometry for examples helping or hindering this specific kid?
Is using geometry to explain squares and square roots the best thing for this specific kid?
When I go to teach (x + y)^2 can I show it using a geometric example?
When I show how (x + 1)^2 can I show it using a geometric example?
If I know 575^2 is 330,625 can I show them the trick to get 576 squared without having to plug it into a calculator? (Trick: if I know x^2 for some x, then to get (x+1)^2 I just have to add x and x+1. So if 575 squared is 330,625, then I have to add 575 to that and then 576 to that. Geometrically it’s a simple example of showing a square with unit boxes and adding another row of boxes and then a column with 1 more square. Algebraically, (x+1)^2 = x^2 + 2x +1 = x^2 + x + x + 1 = x^2 + x + (x+1). )
and so forth.
Next time a question comes up, respond with a question
The next time a question comes up, view it as an opportunity to have a discussion and to explore the child's understanding.
Keep going until you are sure you are accurately interpreting their question.
Sometimes, you might answer it correctly the first time.
Other times, it might take up all the time you allotted to studying math that day.
Often, it will lead to a magical mathematical rabbit hole for you and your child to explore.
The more often you do this, the more in-tune you'll become to your kid's way of approaching questions, which will a) make you both closer and b) make it easier for you both to understand each other.
A win for all!
That’s all for today :) For more Kids Who Love Math treats, check out our archives.
Stay Mathy!
All the best,
Sebastian Gutierrez