**I believe one of the most important things we can do as math teachers is to constantly deepen our understanding of the concepts our students are learning.**

In my 2nd year of teaching I was assigned to teach a grade 6/7 split class with 35 students. Being a new teacher, I taught my students the way I was taught… through direct instruction… and in a very procedural way… with the belief that learning happens by me telling, and students following the rules/steps that I shared.

Now, I *could* look back at my teaching and cringe with disbelief at my actions, however, I was doing the best I knew how! Instead of looking back and thinking about how poor of a job I was doing, or even how far I have come, I think it is far more valuable to recognize the moments that helped me grow as an educator. Here is one such incident:

So, my second year of teaching was well underway when the topic of adding fractions came along. I knew how to add fractions, and so I taught my students the only way I knew how. At the end of the unit I gave a test and was surprised at how poorly many of my students did.

I decided to offer those that did poorly a second chance, but assumed that first I needed to offer some help. I called a student over to my desk and showed him a few of the errors he had made on his test.

His first incorrect answer looked something like this:

Pretty common error right! In fact, for each of the questions I had given where the denominators were different, this was his strategy.

So, slowly I “re-taught” him the steps of how to add fractions… I asked little mini-questions to walk him through each small step in finding the solution. It looked like he understood until he said:

##### **“I get that you are telling me that 3/5 + 2/7 is 31/35, but look, on my test I got 3 of the 5 right on the front and 2 of the 7 right on the back. Why did you put 5/12 on the top of my test?”**

I, of course, wasn’t expecting this, and completely didn’t know what to do… so, I again showed him **my** method of adding fractions, completely dismissing his question.

While I think the example above illustrates an interesting moment where I recognized that I didn’t know enough about the math to understand how to react, it is equally interesting to point out what I did when confronted with something I didn’t understand. Take a look at Phil Daro’s quote below:

##### When **we** don’t know the math deeply, we jump to “answer getting” strategies… we tell the students procedures to remember… provide them with tricks… we focus on notations… we provide closed questions that are easily marked as right or wrong….. and our attempts to help students that struggle includes doing the same things over and over again!

This brings us to where we started:

**I believe one of the most important things we can do as math teachers is to constantly deepen our understanding of the concepts our students are learning.**

If we want to get better at being math teachers, we need to learn more about the concepts our students explore! We need to learn from knowledgeable others about how the concepts develop over time, and the experiences our students need to make sense of the mathematics!

For me, this has included learning from wonderful influential math leaders like Cathy Fosnot, Cathy Bruce, Van de Walle, and Marian Small (check out the links).

When the Ontario Ministry published Paying Attention to Fractions K-12 I finally saw the connection and differences between my understanding of 3/5 + 2/7 = 31/35 and my student’s comment showing 3/5 + 2/7 = 5/12. Take a look. Can you see how my student was seeing fractions?

While a blog might not be the easiest place to deepen our content knowledge, it can be a platform to encourage you to consider how we are challenging our understanding of the math your students are learning.

Do you have a knowledgeable other learning with you? Do you read resources that challenge your current understanding of the concepts you teach? How are you making connections between different representations, or between concepts? How are you learning more about what is developmentally appropriate?

I think we owe it to our students to be continually learning! This learning is often referred to as “Math Knowledge for Teaching” or MK4T. Take a look:

In the beginning of this blog I explained a time when I recognized that while I understood the content myself, I did not know what to do when my students struggled with the math. This is where we need to spend our time learning! This is what our focus needs to be as professionals!

Great post! So true that we need to think about the how and why – and not just the answers! Students often teach me more than I teach them (maybe not in the same content! LOL).

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For me, in the K-5 arena, fractions (especially operating w/ them) is a place where I need to further my understanding. Also, you are one of the people who routinely helps me to deepen my understanding of maths! Thanks for sharing your thoughts, expertise, learnings, and wonderings.

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Mark! I love this post. I appreciate that you focus on the fact that you were doing the best you knew how. I totally cringe at my own previous teaching, but I was trying my best. I agree that we need to continue to develop our own content knowledge. When I guest taught in fifth grade I had to spend a great deal of time anticipating student responses because I wasn’t sure where things would go so I needed to make sure I understood the math well enough to plan questions to help advance the thinking of the class. I think this portion of planning needs greater attention. Thanks for sharing.

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Hopefully we can all find moments where we recognize that our previous best isn’t good enough anymore. Not because it was awful before, but because we know better now!

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If you could go back and respond to the student about the numbers that you put on the front and back of the student’s test, I’m curious about how you think you might respond now. Really interesting that the student picked up that pattern and made sense of it….speak volumes. .

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On my blog matholdiesbutgoodies.wordpress.com you might enjoy “are we adding ratios (rates?) or fractions?” It captures the issue you present.

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No … I don’t see how your student was seeing fractions. Thanks for leaving me in the dark.

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Hopefully you can tell that one of my goals is to get others thinking……

Typically when we add fractions the assumption is that the fraction is a part-whole relationship, and that the whole is the same whole.

This wasn’t true for how the student saw the test scores adding up.

So, how did they see fractions then? Why did their addition make sense in that context?

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