Inventing Mathematics as a Kid
Using a child's creativity and critical thinking to introduce a new mathematical technique
Doing math the standard way:
technique, then practice problems
The teacher stands at the front of the room and tells the students, "It's now math time. Let's open your book to page 57, and today we're going to be learning about...".
Then, the teacher starts teaching the students a mathematical technique. The teacher explains it. Then, the teacher covers how it is applied. Then, the teacher shows what problems can be solved with it while solving a few practice problems. Then, the teacher passes out a worksheet (or has students copy from the board) to try solving a few problems independently.
Involve the student, but do it backward:
practice problems, then technique
“Tell me, and I forget, teach me and I may remember, involve me, and I learn.”
- Benjamin Franklin
For a kid who loves math, try learning backward if you have the time.
That means you share a problem to solve with the kid of medium difficulty.
But because they don't know the technique you will teach them, they will struggle to solve it.
So let them struggle and encourage different types of exploration, like
various substitutions
rearranging the elements
try thinking outside of the box
While they are doing this, please chat with them regarding the ideas they are trying and why.
You want to see if they can devise their technique to solve it.
As they work through it, even if you're not mathematically inclined, you can also share your ideas.
If they are still struggling, try a second example problem, but with a much easier problem this time.
See what they can come up with.
The struggle is the point and is real
The idea of the problem first and technique second is to have the student struggle.
They might not be able to solve it, or they may find it very long or tedious to do because of the calculations, or they may need help understanding what they are trying to do.
That's okay.
This is very beneficial because it will help them in myriad ways.
Some of those ways are:
helps them understand where their current knowledge boundary is
helps them realize that there is more to know
helps them build tolerance for not knowing how to solve a problem
helps them build a "let's try everything we know to try to solve it" muscle
helps them dig deeper into trying to solve problems
helps them get used to finding a problem and not knowing how to solve it
helps them build up confidence that I can tackle hard things
since they know you'll show them the technique, it helps them try to figure it out
helps them be creative
if they solve it, they can compare/contrast how someone else solved it
professional mathematicians operate this way when reading research from other people
read title, abstract, and conclusion
try to prove it themselves
once they've struggled for a while
read the full paper to see what the other math researchers did versus what they did
The idea of the struggle is to select two problems (medium and more accessible toughness) from the "next section" to have them try to devise the technique you will teach them.
Let's look at an early elementary school example
If your kid is about to learn multiplication as "Fast Addition," you could present them with the following problem.
Medium Problem: "Let's add 100 2s"
Questions:
How would you do that?
What is the result?
Is there a way to write that so that it doesn't take up so much room on the page?
Struggle:
Have them write it out
Have them solve it
Have them think of how to write it in an easier way
Easier Problem: "Let's add ten 2s"
Struggle again:
Have them write it out
Have them solve it
Have them think of how to write it in an easier way
Technique:
Introduce "x" (times symbol) to add this many copies of the number
4 x 5
4 x 5 = 5 + 5 + 5 + 5 = 20
4 is the multiplier
5 is the multiplicand
20 is the product
Observe what the kid did: Did the child write 2 + 2 + 2 + 2 + 2 + 2 + 2? Did they solve it? Did they lose count? Did they get tired? Etc? Did they invent some new notation?
Hopefully, by the end of the example, due to the involvement of trying to write out all the 2's, they'll have a better understanding of the time-saving compression that the times symbol allows.
Let's look at a simple geometry example
If your kid knows about multiplication, is studying the area and perimeter of simple geometric shapes, and is learning about the area of a rectangle/square, you could present them with the following problem.
Medium Problem: "What's the area of a rectangle with a side of 2 and a side of 50?"
Questions:
How would you do that?
If you draw 1 by 1 squares, how many squares do you have to draw?
What is the result?
Is there an easier way to solve the problem?
Struggle:
Have them draw a rectangle with 1 by 1 squares
Have them solve it
Have them think of how to write it in an easier way
Easier problem: "What's the area of a rectangle with a side of 2 and a side of 10?"
Struggle again:
Have them write it out
Have them solve it
Have them think of how to write it in an easier way
Technique:
The area of a rectangle is the length multiplied by the width
2 x 10 = 20
2 x 50 = 100
In both cases, the area of the rectangle is the product
Observe what the kid did: Did the child draw the 1 x 1 squares? Did they lose count of the squares? Did they figure it out before you showed them the easier problem? Did they invent some new notation?
Hopefully, by the end of the example, due to the involvement of trying to draw all the 1x1 squares and then keep track of them, they'll have a better understanding of the time-saving compression that the area of a rectangle formula gives you.
Let's look at a simple high school Algebra 1 example
If your kid knows about factoring polynomials to get the roots and is ready to complete squares, you could present them with the following problem.
Medium Problem: "Given you know how to solve x^2 = 16, how could you use that knowledge to solve the following polynomial: x^2 - 2x - 15 = 0"?
Questions:
Since you know the technique for factoring polynomials to get roots
x^2 + 5x + 6 = 0
(x + 2)(x + 3) = 0
But don't know the quadratic formula yet
How would you do that?
How can you get something that is squared on both sides?
Is there a way to transform the problem into a "Square" = "Square"
What is the result?
Is there an easier way to solve the problem?
Struggle:
Have them try to come up with a square on both sides
Can you rearrange the equation to try to get a square?
Can they change the problem to make it easier?
Have them solve it
Have them think of how to write it in an easier way
Easier problem: "How would you solve (x-1)^2 = 16?"
Struggle:
Try solving it
Try expanding it
How is it related to the earlier medium problem?
Have them solve it
Have them think of how to write it in an easier way
Technique:
Complete the square with the "x"-related terms and leave the constant alone
Show how the original equation can be transformed by adding and subtracting a number
Then completing the square and rearranging
x^2 - 2x - 15 = 0
x^2 - 2x + 1 - 1 - 15 = 0
(x - 1)(x-1) - 1 - 15 = 0
(x-1)^2 - 16 = 0
(x-1)^2 = 16
x-1 = +/- 4
x = 1 +/- 4
Observe what the kid did: Did they rearrange the equation to isolate the x terms on one side of the equal sign and the constant(s) on the other? Were they able to get squares on both sides? Were they able to expand the easier problem and see how it was closely related to the more complicated problem?
Hopefully, by the end of the example, having played around with rearranging and adding/subtracting constants, they'll better understand how you can complete a square to give you an easier way to arrive at a solution.
Challenges to be ready for
This is going to be hard for your kid and you. The whole point of the exercise is that they won't know what they are doing and have to proverbially grasp at the straw in the darkness while fumbling around with the mathematics.
You may find that your kid gets frustrated and cries (I've been there). Or that they refuse to try (also been there). Or that they stare at the paper for 5 minutes, not saying a word or putting anything on the paper (also been there), or they do the bare minimum (also been there).
It's okay if the kid doesn't want to do it. The point is to spend some time not knowing how to do it so that when the technique is shown, it's clear why it is essential to learn.
Something we've found in our household is that setting a 5-minute thinking window for the struggle is helpful and enough time for the kid to stare at the problem.
Next steps
Try it the next time you are learning new math with your kid who loves it. See if you can get them to grapple with the problems before introducing the new technique.
Try it out and let me know how it went :)
Until next time,
Sebastian.