Conversations with My Son: More Midpoints in Geometry

The other day I published a conversation, a tutoring session, between myself and my son. It was a painful reminder of a problem that is so systemic in math classrooms that students, like my son, are suffering on a day to day basis. It’s called 100% lecture and zero inquiry.

The problem with this method is that lecture does not reach every student. It reaches some…the ones that can see the problem happen visually as they hear it. My son is not that kid. He needs to manipulate, ask questions, manipulate more and ask more questions. Sometimes he needs questions asked of him because there are moments that he still does not see.

When we ignore the necessities of “the hands on learner”, we are sending this message…

You do not matter and you need to learn it as I taught it. If not, that’s on you.

Dear “lecture only up-front-information giving math teacher”, please reconsider your methods for the good of the mathematically thinking universe! The fact of the matter is that if kids are not actively pursuing the what, why, when, where and HOW of math, there is a greater chance that they are not learning.

Regurgitating a given formula isn’t learning. Knowing how to do a problem is one thing. Understanding why you do it and why it works is another. There is a difference.

Tutoring My Son…Continued

When I left off my last post about my son’s pursuit of finding the midpoint between two points, he was at a place where he understood not only the meaning but also how to manipulate the equations. However, when he was presented with a problem where he was given the midpoint and told to find the other point, he was a bit stumped again.

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Son: Ok, so this is different.
Me: What makes this different?
Son: I am given the midpoint and I need to find the other point, like it says.(pure sarcasm)
Me: Alright smarty, what does this mean to you?
Son: Well, it’s like if Ferris (TX) were the midpoint and Ennis (TX) was the other point, I need to find what’s on the other side of that which might be Dallas (TX)
Me: Oh??
Son: Well it makes sense.
Me: So, how would you go about doing that?
Son: Can I use the points that we used before because the pattern should be the same here too, right? (Goes to Geogebra to test)
Son: Wait, I’ve already worked out this problem. (Somehow he chose this series of points last time)
Me: How can you use graphing as you pursue these answers? Do you feel like you understand? If so, are you ready to look at how the formula works?
Son: Mom, when we are at home, can you use “home language” and not “School language”? You sound like a teacher. (insert 5 minutes of off task comedic banter)

My son continued with physically placing midpoints and one endpoint on his geogebra board in order to play around and make sure that he understood. Algebraically, he could easily plug points into the formula and find the other point but on occasion, he made mistakes.

mpWhat he found was that if he “free handed” a graph and points, he could estimate where his answer should be and this gave him a way to check himself.

I must also point out that our formulas can’t always be their formulas if students find a better way.

My son’s way of doing it…

“I feel like this is working backwards. I already have my midpoint so since I added and then divided by 2 to get it, I can work backwards to find either point.

If I take my midpoint and double it, you know….multiply by 2, I can then subtract the given point and I will get the other point every time. See….try it, I bet it works. Or better yet, in your “teacher language”, find a time when it doesn’t work.”

“I bet you can’t”

The student becomes the teacher. That should be our goal…always.


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