###### Mathematics

#### The Continuum of Teaching Mathematics

With the endless humdrum of the ongoing Math Wars, it is easy to forget that what teachers really need is simple help to understand practical ways to improve or acknowledge their math program. We have heard both sides for decades. One side is about the context and deeper conceptual understanding while the other side worries about the basics. To be fair there really shouldn’t be any sides. Mathematics is a combination of both concepts and procedures. You will never find a teacher that doesn’t do both.

I love this picture from one of the presentations that Matthew Oldridge and I do on this topic:

What it shows is a continuum of teaching. At times, we may be closer to the fully guided while at times we do some unstructured unguided lessons. However, most of the time we are somewhere near the middle. I lean more towards the 3/4 mark of the line.

A couple of years ago I wrote a post about a balanced math class but since then I have made small tweaks that I thought would be useful to highlight.

When I first thought of this subject I thought of six things that should be in the program (you can read about each section in my post):

- Guided Mathematics
- Shared Mathematics: Students work together to “Mathematize”
- Conferencing/ Monitoring
- Congress
- Reflection
- Math Games and Math Facts

My opinions about these things haven’t changed. I still think you need to have all of these components, but I want to simplify and think more about the practical side. For this reason I will steal a line from the Toronto Maple Leaf’s Head Coach, Mike Babcock, “Think of a five day block of time”.

Now, before I go into detail I want to preface that this is just my opinion and in no way is this the only way. I think as teachers we need to use professional judgment to choose what is best. I also don’t expect to have these ideas prescribed like a five day formula that you must follow. I think it is interesting to reflect on these components.

I broke this into five days because I felt that it was simpler to look at a five-day segment in time. Sometimes these components may take more or less time, but on average I try hard to stick to this.

## Day 1: Problem Solving

I am a firm believer that our math program should be predominantly a place where students are problem solving and exploring math concepts. During this time, the teacher’s role is to explore the concepts with the students. It is a fine balance between a guided approach for some to one that allows the kids to explore. As a teacher I am also conferencing, questioning and monitoring student’s work. I am checking the work in relation to landscapes of learning and thinking about how I will debrief the learning. What misconceptions are students having? How are they tackling the problem? What collective conclusions are they making? These are some of the questions that go through my head.

## Day 2: Congress

This to me is one of the most important things we can do in a math class. It is where the shared, guided and explicit instruction is happening. During this time, I am questioning and explicitly linking the math concepts to student’s problem solving. Where I may allow students to wander a bit in exploration, I am keeping the reins tightly wrapped around the big ideas and misconceptions I observed in the original problem.

## Day 3: Number Talks

These have been one of the best decisions that I have made as a teacher. Number talks allow me to discuss strategies, talk through misconceptions and help students visually see the mathematics that is happening around them. Number talks is also a 15 to 20 minute exercise so they happen frequently and often in the classroom. Another great aspect is that it allows students to communicate and talk about math in a meaningful way.

## Day 4: Reflection

The more I read about this topic the more I believe that this needs to be integrated more in the classroom. We need to explicitly show students how to reflect about their learning and how to set goals in order to improve. This year in my class I have purposefully set time aside for students to regularly talk about their math learning.

## Day 5: Purposeful Practise (Math games, Centers and regular practise)

Yes I said it! Purposeful practise. This may be in a worksheet, but if it is, I hope it is geared toward each child’s needs. For me purposeful practise is about seeing where a child is in their development of the skills, and then finding the strategies that may work best for them. This year it has been center work, using board games or math games and digital games that challenge student’s problem solving abilities. The important part is understanding that it is purposeful and meaningful.

Overall, I think we need to think less of this as a war between concept and procedure and meet somewhere in the middle. How can we help our students learn and build bridges mathematically?

I would also love to hear your thoughts. If you have any opinions or questions please feel free to leave a comment.

Here is my slide deck on a balanced math approach.

**Jonathan So** is a Proud Peel Teacher. He is currently teaching Grade 6 at Ray Lawson Public School. He has also taught grades 2 to 5 and is one of the lead instructional technology and math coaches at his school. He is a proud parent of 3 young children Izzy, Micah and Levi. He is always looking to promote creativity and exploration in his family, students and colleagues. Jonathan completed his Masters of Education looking at how his Questions impacted his students’ learning of part-whole relationships in Fractions. His interests lie in math, assessment, mental health and technology (not in any particular order) but he is also passionate about inquiry and the endless possibilities it has for his students. Jonathan is also a keynote presenter and workshopper. Connect with him via twitter (@MrSoClassroom).

## 4 Comments

## Aviva

Thanks for outlining the choices you make and why you make them. I liked your comment under “purposeful practice.” This makes me think of the key question in the K Program Document: “Why this learning for this child at this time?” When deciding on this purposeful practice, how do we ensure that it’s targeted for each child, and links to the learning that we’re doing on our other four days of math? Thanks for giving me more to think about!

Aviva

## Jonathan

Aviva, I am always happy to read your comments. You always have a way to push the thinking forward. I think whenever we think about balance I think we must always think of the individual child. When we think about purposeful practise I think it is important to remember two things. 1) Practise can happen in many things and in many ways and 2) it must be individualized and meet the very different needs of each child. So to answer your questions, practise will always be targeted, and it is always linked to what they are doing through games, number talks, conferencing or anything else you choose.

Thanks again for your comments.

Jonathan

## jessica sachdeva

I found this to be a very insightful post Jonathan. When you write about number talks, after the problem solving and congress, it got me thinking about how/if the number talk is supporting the problem solving. I am now reflecting on how I might better tailor my number talks to draw out key understandings from the problem solving. Thanks again for the insights and ideas to ponder.

Jessica

## Marie-Line Couture

Merci Jonathan! This is exactly how I am feeling about this left-right war in education… In french we say “le juste milieu”… The answer is often close to the middle indeed.