When Math Class Matches A Teacher's Dream

It happened today.
And it is for these moments that I keep teaching.
To see the students find excitement for math and to push to see how it works.

It is my goal every day to orchestrate these moments.
But there are so many variables in play.
The students and their demeanor day to day...
I teach middle school.
Game days are tough days to keep math focus.
The lesson content and what the students bring to the table...
The types of questions posed...
The entry point into the problem...

Today's discussion happened by accident.
I didn't plan it.
Nor did I see it coming.
When we got ready to "play" our short Kahoot game,
I noticed that the answer to the last question was showing on the screen.
And that everyone was taking notice.

I first explained how Kahoot works in my math classroom...
...speed is not a factor, I had taken out the point values...
math is not about how fast you can solve a problem but how you process the information,
...it is about gaining feedback on our learning.
What do you understand?
Are you making any mistakes?
If so, can you correct them on your own or is help needed to fully understand?

And as I was explaining my expectations, I was thinking about how I wanted to handle that last question that they all knew the answer to.

When we got to the last question, I reminded them that I knew they already knew the answer.
They could simply put in the answer, but I wanted them to figure out "why" it was the answer.
Everyone got the question right,
Now it was go time.
A student came up to solve the problem.
He did a great job of justifying how the solution was derived from the problem.
He showed his work.
And explained as he went.

It was when he was finished that a student questioned,
"But if we were to work backwards to check our answer, it doesn't seem to work mathematically."
All eyes went to the board.
Mine included.
She was absolutely correct.
I was stumped.
But I grasped onto this moment to model what struggle looks like when we don't know the answer.
Before their eyes I turned back into a student.
I recorded what she had just stated and sure enough, we could not get back to the answer.
We started to talk about what it could mean,
And started playing around to see how we could get it back to the original number.

The math conversation as we dove deeper into this problem is what math teachers dream about.
Why did we have to multiply by 4/4?
Where did the 4 come from?

As some clarification, the analysis to follow is based on the KSDE Math Conference I attended yesterday.  We learned that math education has to change to help raise student understanding.  I constructed a mind map of what all needs to go into a math lesson.  WARNING: It is a little overwhelming but it is doable if you focus on just one area at a time.  I would suggest start with the standards and the teaching practices.

The math practices that were at play...

1.  The students made sense of the problem and persevered in solving it.
In essence, they actually had created the problem by asking "if we work backwards why can't we get back to the original answer?" When we got to the 4/9, the students didn't stop there.  How do we get back to 16/36? They then realized the proportional relationship between 4/9 and 16/36.

2. Reason abstractly and quantitatively. 
Students were demonstrating their reasoning by moving flexibly between the inverse operations of squaring and square rooting, multiplying and dividing.

3. Construct viable arguments and critique the reasoning of others.
Students justified their thinking by coming to the board and working/explaining the problem.  When a questionable line of reasoning was brought up, students in a respectful way countered the math thinking, allowing the conversation to continue with no one shutting down.

4. Look for and make use of structure.
Once we got back to the original number, 16/36, I was thinking it but a student voiced the question in my head, "would it work with different numbers?"  So we tried it.

It didn't take someone long to realize that if the reduced fraction is squared, then the fraction of 1 (16/16) was simply the square of the fraction that was used to divide to get the reduced fraction in the first place!

Teaching Practices Evident in the Lesson

1. Implement Tasks that Promote Reasoning and Problem Solving
Although I had not preplanned that the last problem of the Kahoot game would be the task that spurred on reasoning, I knew as the discussion unfolded this would be a conversation worth pursuing.

2. Facilitate Meaningful Mathematical Discourse
See #1

3. Pose Purposeful Questions
I could have just gone over the problem after they had seen the answers and moved on with our lesson.  But why changing the question, "How is 2/3 the answer to the square root of 16/36?" I was able to advance their thinking and making sense of how inverse operations are used.

4. Build Procedural Fluency from Conceptual Understanding
My purpose with this problem was so students could see that square roots work with not only whole numbers but with fractions as well.

5. Support Productive Struggle in Learning Mathematics
Because the student's initial inquiry, "But if we were to work backwards to check our answer, it doesn't seem to work mathematically", I was put in the role of student and was able to model struggle along with the students.  We were on the same page, offering up ideas to see what would work.

6. Elicit and Use Evidence of Student Thinking
Student thinking was at the heart of this conversation.  As students made the connections, the excitement rose, and more on-task talking was the result.

Standards

8.EE.2 - ...evaluate square roots of small perfect squares...

Growth Mindset

While the Kahoot game was set up as a means to gain feedback on understanding and to address mistakes, the conversation that developed was the result of students in a growth mindset frame of mind.  Risks were being taken, mistakes were clarified, as the focus remained on what we could learn from what Mrs. McCabe thought initially was a mistake on her part.