I recently had two exhilarating conversations with teachers I work with: one with high school teachers and the other with second grade teachers.

First the high school conversation:

I posted this problem on the board and asked the teachers to solve it on their own before sharing their thinking with the others.

A few moments went by and it became clear that some of the teachers were really struggling to solve this problem, so I asked the teachers to pair up and share their thinking with each other. Some of these conversations went smoothly, while others not so much.

What was the difference? The partners that has smooth conversations tended to use similar solution methods. The conversation went smoothly because the two mathematicians were on the same wave length as each other. Meanwhile the partners who struggled with their conversation often talked past each other because each person was using a wildly different strategy from the other person. This led me to add the question, “Can we find at least six solution methods?”

Clearly this is important, because in order to have a smooth conversation with others (I’m imagining having conversations with my students), we need to be able to anticipate and understand their solution method even if it is a non-standard strategy or a solution method other than the one I preferred.

Teachers appreciated the importance of not only knowing how to carry out common procedures, but also of being able to interpret and understand the non-standard or incomplete thinking of students.

## The surprising sophistication of 2nd grade math

The second conversation occurred with second grade teachers about the addition and subtraction algorithms. I was showing the algorithms side-by-side on the board when suddenly a teacher pointed at the two examples and practically yelled, “Are those ones the same?!?!”

I drew arrows to the ones she was pointing at and asked the rest of the room, “How are these 1’s the same or different?”

The conversation that ensued was exhilarating!

- “They are the same!”
- “No they are not! But I don’t know how to explain the difference?”
- “They are both in the tens place!”

The comments exploding from the teachers were coming fast and furious. I just watched as they grappled with the meanings of the 1 in each example.

Eventually we can to the conclusion that the 1 in the addition problem represented 1 ten, while the 1 in the subtraction problem represented 10 tens.

What is so important about these teacher conversations?

## Mathematical knowledge for teaching (MKT) was at the heart of each conversation.

MKT is the teacher’s ability to

- identify incorrect answers and faulty methods and analyze errors efficiently and fluently, just like mathematicians do in the course of their work.
- make sense of students’ non-standard procedures, even when they have never encountered them before.
- provide justification for the steps in algorithms and procedures, meanings for terms, and explanations for concepts.
- make strategic choices of mathematical representations and examples when illustrating ideas.
- sequence student examples to create a trajectory towards the teaching of algorithms.

As we saw with the high school teachers, teachers with MKT would be able to make sense of a student’s non-standard approach to finding the measure of angle X and then provide guidance to that student as needed. Without MKT a teacher might look at a struggling student’s strategy and dismiss it entirely by showing a completely different solution strategy.

With the second grade teachers, they needed MKT to provide the mathematical reasoning for how the seemingly identical 1’s were actually very different.

### Here are some other questions that require teachers to have a healthy dose of MKT:

Not exactly sure how to answer some of these questions? THAT is why having mathematical knowledge for teaching is so important.

To be the effective teachers we want to be and to improve the mathematical achievement of our students, our first move is probably not to reduce class size or purchase an expensive math curriculum. Rather, we will get the biggest bang from our buck by providing teachers with the mathematics professional learning they deserve to increase their mathematical knowledge for teaching.

I will write more about MKT in the near future.

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