A Balanced Math Program

With the endless humdrum of the math wars happening, 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. Even more realistic 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:

Screen Shot 2017-10-16 at 12.07.09 PM.pngWhat 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 some where near the middle. For myself 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’ve had some 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):

  1. Guided Mathematics
  2. Shared Mathematics: Students work together to “Mathematize”
  3. Conferencing/ Monitoring
  4. Congress
  5. Reflection
  6. Math Games and Math Facts

Now my opinion about these things haven’t changed I still think you need to have all of these components but I want to simplify a bit and think more about the practical side.  For this reason I want to steal a little line from the 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 have professional judgement to choose what is best. I also don’t expect to have these ideas prescribed like a five day must follow. I just want you to reflect on these components.

I broke it into five days because I really felt that it was easy to look a five day segment in time. Some times these components may take more time or less 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 predominately a place where students are problem solving and exploring math concepts. During this time, the teachers role is to explore the concepts with the students. It is a fine balance between a guided approach for some to a more let kids explore. As a teacher I am also conferencing, questioning and monitoring students work. I am checking it 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? are all 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 and where that shared, guided and explicit instruction is happening. During this time, I am questioning and explicitly linking the math concepts to their problem solving. Where I may allow students to wander a bit in exploration I am tightly keeping the reigns around the big ideas and misconceptions I observed in the 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 developmentally and finding things that may work for them. This year it has been center work, using board games or math games and digital games like knowledgehook and Mpower.  The important part is understanding that it is purposeful and meaningful.

Overall, I think we need to think less of this war between concept and procedure and meet 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.

Using coding to teach mathematics

I have been a proponent of coding for quite some time. I feel that it will be a skill that students need in the future. I know that this may cause some stir in many of you but here is my reasons:

1) Though I do agree with who knows what the future may hold, I do believe that this is a skill all kids will need. At one point in time no one knew how to read. In fact it was only geared to the clergy because they had to read the bible. Now that skill is in every classroom. We may not be there yet but I think we are very close. Coding is a part of everything that we do and our everyday. I think that it is important to know how things operate. Yes that does mean changing our oil and fixing our cars. We may not have the time but I think as adults these are important skills. Students now should learn about how their electronics work. How do we make them do what we want to do? I am not saying that all of them will become computer programers but we should understand the basics.

2) Coding does more then just teach programming skills. Students learn critical thinking, problem solving, and being creative. As students try and code they learn to research, ask questions and work through till they get a solution

3) Most kids if not all, love to code. Now I say this with a side note. I do find that when the task is meaningless then some kids are not as engaged with coding but if they are creating something and the right differentiation is in place then they are all in. To be honest you can say this with most ideas but it does apply here.

4) Coding teaches logical order and research skills. I don’t have numbers yet but the more that I have done coding the more that I have noticed my students critical thinking and sequencing skills improve. I have noticed my students improve in making connections and seeing how all the big ideas link together. Again I cannot say this is all coding but I believe that this is a major reason. 

For me coding fits naturally with mathematics. I mean the main idea of spatial sense is right in our curriculum. However, that is not the only area you can use it for.

Today I thought of turning a quite boring lesson of order of operations into a coding exercise.  It was really cool to see the students take a foundational lesson and a very procedural lesson and apply some creative and problem solving skills.

The challenge was to create an app that can test students understanding of order of operations. Students had to also have their users think about misconceptions and possible errors.

Here is what the code looked like for most:

The students still need more time but here is the sample that we have been working on, link. 

//scratch.mit.edu/projects/embed/104333621/?autostart=false

Throughout this process the main purpose was not to teach coding but to understand the basic idea around order of operation. Sure I could have just told them the answer but they have now started to work through the procedure and how students can make mistakes. I hope that when we debrief they will forever have an understanding of order of operation.

This is just one example of how coding can fit into our everyday math lessons. The main focus should always be the concept and idea of math and then the tool. By teaching this way I have allowed my students to explore order of operation and to critically think about the concept.

I encourage you to try coding in your math classroom

Are we reluctant to share?

Are we scared to share? Do we often feel like what we have to say isn’t worth reading? I know that I do. But we have to remember that what may seem as silly to us is often not for others.
It reminds me of this video that I have seen numerous times: 
I was reminded of this a couple of days ago when I reluctantly posted this post on “What does it mean to be a Teacher?”
I say reluctantly because I wasn’t too sure about the topic. I mean I loved the idea, it is why I decided to write but I wasn’t too sure if anyone else would have cared, or even wanted to read it. It was more of a reflection or response to the comments I was hearing about our profession of teachers. However, I decided to post and the next day a good colleague David Petro posted my blog on his daily Math Reviews (on a side note if you are not following David or reading his weekly blog you are missing out). The funny thing was my post wasn’t even about math but on what it means to be a teacher.  It did comment on teaching being more than telling, which I often think this is the preconceived notion of a teacher and why many think they can do our job but I digress.  The fact was that David saw something in my post and then felt the need to share it. Even though I didn’t think that the ideas were decent someone else did. 
To be honest I think that as a profession of teachers we all have amazing ideas and need to share them. We only get better by reflecting and learning from one another. So if you ever feel like your ideas don’t matter remember that an idea that may seem trivial to you, may not be for someone else. If you don’t share that knowledge or thought than you are depriving the profession of some amazing ideas. 
So I encourage you all as teachers, readers, parents get out and blog, Share your amazing ideas and connect with amazing people. And a big thank you to David for reminding me about this.

Blog Hop: Digital Learning in Math

Tonight’s Blog Hop is about digital learning in Math and I want to share two of my favourite tools.

First I think that with any technology it is not the technology but the teacher that makes the difference. I have seen the exact same tool used in various classrooms with various degrees of success; all depended on how the teacher implemented tools.  It’s very important to note that the teacher has the impact to turn any lesson into a great mathematical success or a great failure.  A lot is dependent on how you plan, the thinking and linking to curriculum and big ideas and what questions are being asked in the rich task.

That being said:

The first tool I want to share is an app called explain everything. This app is fantastic to use for explaining students mathematical processes. In my class after students are done their math problem they grab their iPad and then they take a picture of their work and talk over the strategy. They use the laser pointer to describe what they did. You can also have them write on the screen and screencast as you go. This is a great idea for presenting students strategies outside of the classroom. You also have the explanation to go with the written work.

The second tool has been my journey using Minecraft in math.  It has been amazing how this gaming app can be used to explore endless possibilities in mathematics. Take a look at some of the ideas that we have done:

Overall, tech can be used in a variety of ways to enhance the learning of the students. The important part is that as a teacher we bring out the mathematics.  It can be used to engage, promote and discuss math concepts but more importantly it is a way to record and reflect on student thinking.

Don’t forget to check out these other amazing peel teachers:

Graham Whisen

Shivonne Lewis-young

Jay Wigmore

Don Campbell

Jason Richea

Tina Zita

Phil Young

Making kids doers of math instead of Doing math!

I have been thinking about this topic for quite some time now and then when asked to do the OAME 2015 ignite I thought this would be an amazing topic to push thinking.

My biggest fear in education right now is that we are having our kids go through the paces of doing school. We our turning our students through the drudgery of school.  Before I started to really question this thinking I saw it in my own class. My students were coming to school going through the paces and then leaving. Sure they enjoyed it but was I really making them think? What type of work was I making them do? Why was I teaching them these skills?

I then came across this statement by Fosnot:

The purpose of teaching is to learn, but without learning there is no teaching!

I was shocked. Was she saying that if my students didn’t learn then I wasn’t being a good teacher. The answer was yes! And the more that I reflected on this the more I agreed with this statement. Over time, i realized that even though I was teaching different kids the common denominator was still me. So when I asked questions like, why don’t they get this? The answer was because I am not doing a good enough job. I wasn’t making them understand because I was just making them be there instead of embodying the learning.

I see this a lot in math and this is partially because of my lens. In a math class we traditionally stand I front of students, give a lecture, let them work and then test them to see if they understand. B how many of our students are really learning? How many of them become mathmaticians? 

VanDeWalle suggests that The goal is to let all students believe that they are the authors of mathematical ideas and logical arguments.

So then how do we go about doing this?
I would like to propose three key points to this:  Link back to my thesis always link back

1) Role of the teacher
2) Environment of Learning
3) Accountable Kids

Role of the teacher

I want to first preface that teaching to me is about turning my kids into mathematicians through inquiry and exploration but I start with this point because as a teacher we have the most critical role to play.We are not to sit back and allow our students free reign but to ignite (lol) and actually talk about math. I know really insightful!

Researchers have suggested that children should being engaged in problem versus talk procedures. But our role is to bring out the math not by telling students information and expecting them to regurgitate it but by creating contexts for learning asking critical questions and debriefing the math. In my research I found three types of questions that worked the best for creating these conditions:

1) Interrogation: Just like the title suggests → a lot of why’s and how comes2) Going beyond: Pushing the thinking beyond the schema the student has created. These questions include, have you thought? What about this? Can someone else explain3) Comparing: Often I compare strategies together to see if students can move from one to the next. This includes, what are the differences? Similarities

In order for this process to really work “Teachers must have the [student learning] in mind when they plan activities, when they interact, question and facilitate discussions” ~ Fosnot pg. 24

The key to everyone one of these questions is that it was linked to a big mathematical idea. One that was key to the learning of the student. The same goes to the various talk moves that a teacher can make. These should include: Wait time and revoicing. I cannot stress how important these two items are to the success of building mathematicians. To often we don’t give students enough time.

Creating an environment of learning

In a mathematical environment , students feel comfortable trying out ideas, sharing insights, challenging others, seeking advice from other students and the teacher, explaining their thinking and taking risks. ~ VanDeWalle pg 36. When students do mathematics in an environment that encourages risk and expects participation, it becomes an exciting endeavour. Students talk more, share more ideas, offer suggestions, and challenge or defend the solutions of others. When a context is real and meaningful for children, their conversation relates to the context. They mathematize the situation. ~Fosnot

Making kids accountable:

No one is allowed to be a passive observer ~ VanDeWalle pg 36

I love this quote. I think it is exactly the whole idea around accountable talk. Many teachers may think that just because the student is not talking they are not participating but the key is not to be a passive observer, which doesn’t always involve talking but listening. However, that has not been the case in school. We have been so use to hearing teachers talk that many of our students are use to being told the answer that they are not use to talking. 

In my thesis research I saw that when I asked an Initiate respond Evaluate types of questions (basically questions I already knew the answers) I got no further discussion happening. My kids just sat there. But when I asked going beyond types or comparing questions, basically critical thinking questions, that was linked to big ideas kids talked about math.  They became active users of the information and doers of math not just following the paces.

So I guess I want to ask: How do you make your students into Doers versus just doing? This question doesn’t need to be math as it is a broader problem in education. Love to hear your thoughts and ideas.

Why do we Need to Argue over Math? –> A Call for a Balanced Approach

About a month ago a colleague of mine Kyle Pearce wrote a post “Does memorizing multiplication facts hurt more than help.” It was a very interesting read and I happen to agree with Kyle’s point of view.  As many of my frequent readers of this blog know, I prescribe to the constructivist approach to learning mathematics.  I believe that students through discovery and proper guidance will be able to understand a wide load of big ideas and theories.  Not only do I believe this but I have witnessed this first hand with my students in every grade that I have taught.

However, this is not so for many people.  In fact it was a discussion on Kyle’s blog post (feel free to read the thread) that has me thinking more and more about this topic. And not only thinking about it but trying to fix and insight thoughtful discussion around the ways in which we are teaching math.

Maybe a little background first.  Math has been a hot topic for the past year, if not for the last century.  For many countries, provinces, and states, math curriculum has undergone a significant change from what we grew up with as children.  Some. like myself, believe that these changes are for the better, some have not.  I was recently at the OAME and listening to Brent Davis a professor at University of Calgary.  In his lecture he shared that the reason math curriculum was introduced was so we could have a work force to crunch numbers, nothing more and nothing less.  As we have evolved beyond that (not saying fact crunching is not important) our skills have also changed and I think this is what we need to remember; we have evolved.

For this reason I and many others are proposing a  more balanced approach to mathematics.  Lets stop this war and needless debates and get to teaching good mathematical practises.  One in which our students will push their thinking and really think about the numbers.

If it was up to me this is what I would include:

1) Math should be linked to Big Mathematical Ideas:

I think this is the first step to thinking about our students as mathematicians.  Catherine Fosnot (2002) has some very interesting work around making our students mathematicians.  One of the most interesting facts is there was a study done with so many mathematicians and they were asked to solve a problem.  Not one of those mathematicians solved it the same way.  I found this interesting because that is what I see math.  Math is about the mathematics and there is not one way of doing things.  We have to teach our students the understanding, the flexibility and the patience to be mathematicians.

2) Math is about real numbers:

Students need authentic experiences to learn.  Let’s think about ourselves and how we learn.  Now some do learn through reading and replication but if you honestly think about how you learn a concept the best; it is through trail and error and than guidance from a mentor.  This is the same for our students.  They need real experiences so that they can play and discover the mathematical concepts.  In my personal experience both in tutoring high school students and teaching mathematics in the primary and junior divisions it’s the contexts that allow students to really understand what they are doing.  It’s the context that helps them build models of representation.  In my classroom, these are often done through social justice problems and real life contexts.

3) Students need time to explore:

This goes hand in hand with the above comment. As much as we need instruction, we also need exploration. Students need time to make mistakes, reflect, debate and discuss. These experiences allow students to make connections between concrete and abstract thinking. I was reminded at the OAME that every mistake makes a new synapse in the brain. We need these mistakes I order to solidify  our learning.

4) Students need Mentors:

My most recent research in understanding teachers questions has shown me the importance of teachers, not that I didn’t believe that before.  With a shift towards discovery learning, we as educators have forgotten the importance of our role, or what even our role is.  I truly believe that we should not be the dispenser of knowledge but the mentor of that knowledge.  Though through exploration students will learn (many studies to show this) they may not have tools to reflect or pull together the big ideas.  I think this is where much of the back lash has come from discovery math, reform math or whatever you want to call it.  As students explore and discover there needs to be some sort of guidance.  This is where a teacher can shine and help students with the mathematics.  However, my research shows that for this to happen, a teacher needs to 1) have a good understanding of mathematics, 2) have a good understanding of how children learn mathematics and 3) plan. I know that all teachers plan but this planning involves thinking of big ideas, landscapes and possible questions.  It is these questions carefully placed that can allow students to figure out and make mathematical connections.
Take a look at this video of three of my students thinking about fractions.

They have never been taught fractions from me before this and in fact as a class we haven’t even started the unit. However, that being said think about their learning and the role they play and the role that I play as a teacher.  Where do my questions come from? Why did I ask them at the time I did?

5) Time for debating, conjecturing, discussing and proving:
For me this is the time for a teacher to shine; however not in the traditional sense of standing in front of the class and lecture or tell students how it should be done. Just like students need time to explore they also need time to debate as a community. It is through this debate that students defend their thinking, conjecture, question and solidify their learning. Moreover, it goes beyond just showing and telling.  As a teacher the types of strategies that you show matter. How are you building the learning up, what questions are you modeling? How are you focusing the talk? How are you fostering talk? These are all questions that a teacher needs to be asking.
Also take a look at my most recent grade two conversation of multiplication.

6) Repeated Practise:

Yes I said repeated practice! Students need it, but it’s not just doing procedural learning over and over again.  When I say repeated practise I mean a similar problem for students to continue their exploration.  Students, well most students, cannot solidify their learning through one experience.  In a typical unit my students will solve about 7-8 problems that could take three to four weeks to learn.  These problems build their learning and knowledge from day to day.

7) Skills:

Yes skills are important. They are needed but they are not needed like we use to think about it. For me it’s how are we introducing facts. Do we make students think through facts? Are they taught in isolation or allong with concepts?  In my classroom, students practise facts at home, they play math games in the classroom (here is a file of my math games:  https://drive.google.com/folderview?id=0B4245QONE7HaaHl3M3ZKNWd3SUk&usp=drive_web). In addition before my problems start I often use string lessons which builds on mental math strategies and learning how to be flexible thinkers and playing with numbers. This to me is more important then struck memorization. It teaches students that numbers are not confined facts but that you can pull apart numbers and use known facts to solve other facts. Through this process students often learn all of their math facts, can recall them and use them in problems, which to me is way more important.
These are just a few of my thoughts on what I am calling a balanced math approach. I have a few more  to hash out around integrating and  implementing a center approach within my problem solving approach.

What are your thoughts?  Don’t you think it is better to discuss and fix our problems rather than lay blame about which is better?  Shouldn’t we think about our students first and their needs in their 21st century world?  Love to here your thoughts.

Math Games

In a previous post I mentioned that I teach many of my fact recall through the use of math games.  Math games have always been a passion of mine.  They have so many possibilities that help with mathematics.  Once a week we have a math game focus.  This is two periods in the week devoted to playing math games.  In addition, students who finish problems faster than others can also play math games.

Why use math games?

Math games not only are fun but they teach the basics of fact recall, WHILE, teaching understanding.  They allow students to practice various strategies in addition, to learning their facts.  This is the main reason why I love them so much.  They teach something that is procedural in a conceptual way.

What do you do while students are playing?

Math game time is not just a time to sit back and relax (not to say that you would).  During this time I am looking at how my students are developing, what strategies are they using, and what games they are playing.  I am also asking questions about why they chose a particular strategy and what they were thinking while playing.

Math games at home:

Math games are also important for home.  They bring the family together and help parents practise facts in a fun environment.  At times my math homework is to play a math game.

Here is my google drive link with the games I have on hand:  Math Games.  You can also watch my youtube channel for explanation of the primary games: Math Game Play List

If you have any games you would like to share please do so, would love to find some new ones.