Using the Five Practices to Focus our Consolidations (Learning to Multiply Focus)

At two of my school’s recent PLC’s they have been focusing on building content knowledge in the area of multiplicative thinking. The one school has also been going pretty in depth with using the Five Practices to plan their math lessons and focus their consolidations. I wanted to share some of the work the PLC group did recently looking at practicing  the last three parts of the Five Practices, selecting, sequencing and connecting. The group had asked  for this to be part of their last learning session. Since we have also been working on building multiplicative content knowledge the task used was a multiplying task. The class that did the task was a grade 1/2 class. They did this task in groups but even though grade ones don’t have to learn multiplication the teacher has many students that are exceeding where they should be and are starting to pick it up. I think this clearly shows when students are challenged they don’t just meet the bar but can raise it!

The learning goal for the task was this: We will understand that “groups of” can be counted as units to tell us how many in total.

Why: This will help us understand how multiplication works.

The “Why” part is something that we are starting to add in. feedback from our district support visit told us that our students could say what they were learning but often could not say why they were learning it.

Here is picture of all the finished solutions from the task that was chosen:

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Close up of the task:

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Selecting and Sequencing

As a group we discussed the learning goal and  how we wanted to select samples that would really help us bring out the mathematics that we wanted the students to learn. Since the focus was on unitizing (seeing groups of) and then counting the groups we thought that we would start with these solutions:

We agreed that these examples all showed groups that had really identified “groups of” and used them to find the total. Two of the groups used groups of 4 and one group used groups of 2 to find the total squares.  The first example on the left is what Alex Lawson describes as a students using the composites units inside of a composite unit to make it more manageable for them to count the equal groups. They used the 2 groups of 2 in each group of 4 to get to their total. We chatted about how this is ok to do especially at this stage in grade 2 but later we may challenge a student that is doing that to move to more efficient way of counting the groups.

We also selected this solution:

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We selected this one because this student listed both ways you could solve this by using the commutative property. This little guy is whole other blog post, he is in grade 1 but is picking up multiplication very fast and has already reached auto-recall for some of his facts. He totally understands the situation and can model it. We chose this one because we were thinking that after we used the first three solutions and asked our prompting questions that we would follow with this one. Our idea was to use his number sentence of 4 groups of 5 and ask this prompting question: How could Loic’s number sentence of  4 groups of 5 help us solve this problem more efficiently? Our thoughts were that most of the solutions used 5 groups of 4 and either skipped counted by 4’s or 2’s. No one but Loic saw the commutativity of the array and thought to count by groups of 5.


We now had done our selecting and we had decided on a sequence. Now came the chat about how we are going to connect the consolidation to our learning goal. The plan was to use this prompting question once the first three solutions went up:

What do you notice about how these groups figured out how many pieces it took to make the quilt?

Then follow with these questions:

How many groups of 4 did it take to solve the problem?

How many groups of 2 did it take to solve the problem?

Which one do you think is most efficient? Why?

Then the last solution would be put up and the question I mentioned above would be asked:

How could Loic’s number sentence of 4 groups of 5 help us solve this problem more efficiently?

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Then follow with these questions:

What do you notice about the number sentences 5 groups of 4 and 4 groups of 5?

Connection to past tasks: How does the array model help you solve this problem?

The class has been working with arrays in past lessons so this question can help link previous lessons to today’s. That is the plan we came up with, will see how it goes when the teacher does the consolidation. One thing the teachers shared with me is how much more focused their consolidations have been since they started working with the Five Practices. Still they feel they need more practice doing the selecting, sequencing and connecting part of the process to gain more confidence. Using your colleagues to bounce questions off of and to ask for help to look at student work also came up as being super important in this process. Hope you enjoyed a little peek into our journey with the Five Practices. It is work in progress but we are getting their and it is all based in student work. Any comments or thoughts would be welcome!


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Using Tools to make Representations (And the Importance of Annotating)

I recently was helping to create the last abode presentation on Tools and Representations for our board, so I have been thinking a lot about this pillar of the pedagogical systems. We have also been working on this at the schools I work at and a recent lunch and learn brought up a great example to help highlight the importance of annotating the representations of the student thinking no matter what tool they have used to do it.

I wanted to start with some quick pictures of possible tools that students use to create representations of their thinking. (Click on the picture for the names of each tool)


Once these tools have been formed into a representations and annotated they become a representation of the student thinking. Here are some examples:


As you noticed above, the representations are annotated using sticky notes. A student can also annotate their thinking orally by explaining what their model represents. Here are couple pics of two tools that are drawn on paper. The closed array model and the open number line model.

I have included some descriptions of why these tools  become a representation once they are annotated.

The last part of this post is a video that I did just explaining what happened at our lunch and learn last week. I have redrawn the models for this video to put it all on one clear page for when I was recording it. I think it helps make clear why it’s so important to annotate the student thinking in the representations.


I want to thank Shelley Yearley for reminding me of  the importance of annotating the models to clearly show the student thinking in the representations. She made a point of showing us at the last fraction inquiry meeting and it inspired me to make this post. Again thanks for taking the time to read!

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Changing Units- Shouldn’t a bigger unit create a bigger measurement?

On Friday I was helping the 1/2 teacher Mrs. K plan a task to help her students consolidate their learning with changing units while measuring with nonstandard units. They have been focusing on length measurements and have done some tasks out of Marian Small’s resources. They measured their desks with different size rods and have also used their footprints to measure different parts of the room. She felt they were really getting a firm grasp of the fact that the bigger the unit used to measure the less it takes of those units to measure the object.

We brainstormed some ideas and we decided to change the attribute that was being measured to area. It would allow us to see if they were able to transfer the learning and also give us an opportunity to collect some diagnostic evidence of their understanding of area.

The task we chose was variation of a 3 Act Task that started with “What do you notice?” and “What do you wonder?” prompts. We used Graham Fletcher’s big and little sticky note task as inspiration.

Here is pic of what the students saw when they came in from recess:

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Prompt: What do you notice? What do you wonder?

The questions  they decided to explore was this: How many large sticky notes will it take to fill in the tile? How many small sticky notes will it take to fill in the tile? Which one will take the most sticky notes to fill in the tile? How do you know?

Mrs. K had the students do high/ low estimates for both sticky notes. They landed on 100 as their high estimate and 5 as their low for the smaller sticky note. For the larger sticky note the high estimate was 50 and and the low estimate was  also 5. Some just right estimates from the students:

Large Sticky Note: 20, 25, 30, 15, 18, 20

Small Sticky Note: 30, 16, 20, 15, 40, 50, 35

Notice how there were some estimates from students that were similar for both sticky notes. Keeping an eye on those students is a good way to gather formative assessment because it probably means they aren’t consolidating the learning goal of the last few lessons (the larger the unit the less it takes to measure an object).

The students then moved into partners with their own set of sticky notes and went to work solving the problem.

Here are some pics of the students working:

When they were finished their work Mrs. K had them coming up to the smartboard to write some of their findings.

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There was much more discrepancy  between the numbers for the smaller sticky notes. We have a few thought on why that is.

We had these prompting questions ready for the consolidation:

  1. What did you discover? Did it take more of the larger sticky notes or more of the smaller sticky notes to cover the tile?
  2. Can someone share their results for how many it took for each type of sticky?
  3. Why does it take more of the smaller sticky notes to cover the tile?
  4. How close were you to your esitmate for each type of sticky?
  5. What attribute were you measuring today? (diagnostic) (What do they know about area?) This is for the grade 2’s but I believe its ok to extend the grade 1 students!

Here is a video that shows one of the grade 2 students sharing with me what he has learned.

I wanted to show this last video below because measurement is such great context for students to use their number sense. The boy in this video is one of the students in Mrs. K’a room who is beginning to understand multiplying and is using skip counting as a strategy. He recognized his tile as an array and skip counted by 6’s. I love this because it clearly shows even though they are doing a measurement task that we can always be noticing and naming what type of operational strategies they are applying.

 I would love to hear of any other quality tasks that are being used for this idea. Leave a comment or tweet them out!

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Using a Common Task and Concept Maps to Identify School Content Gaps and Build Content Knowledge

What did we do first?

We recently used a common task approach (see pic below) at one of my schools to identify a school wide gap in understanding in the area of multiplicative thinking. We then used concept map building to improve teachers content knowledge and to help close potential gaps while helping to move students from additive thinking to multiplicative thinking. The first common task was given to every student from grade 4 to 8.

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Here is a link to the first common task:

Ants Assessment – English

What was our second step?

After the the assessment was given all of the math teachers at the school met to moderate mark the assessments. We used the Lawson Continuum to assess what strategies the students used to solve the tasks. We colour coded the strategies based on whether they were counting, additive or multiplicative strategies. See the pic below:


A colleague and I worked together to decide which ones belong under counting, additive or multiplicative. We looked at different research to decide this but some may disagree with us which I would love to hear. We did our best and it probably isn’t perfect so feedback would be welcome.

After the moderated marking was done we discovered that we had a huge gap of students in this school who had not made the transition from additive thinking to multiplicative thinking. We actually had many students still using counting strategies all the way up to grade 8. It confirmed our initial diagnostic work from the teachers at the school who thought this was a significant gap in learning at our school. All of the research we read makes it pretty clear that if students don’t make the transition from additive to multiplicative thinking between grades 3 to 5 then it will impede much of their math learning in junior, intermediate and high school.

Here are some work examples from students in the school:

These students used counting strategies: Composite groups and counting, skip counting

These students used additive strategies: repeated addition

These students used multiplicative strategies, the pic on the right is from the second common task that was done after the PD.

The work in the first two groups of pics was most common throughout the school. We figured out quickly when marking the assessments that an interview was the more effective way to hear/see their thinking. Some students wrote a nice shiny perfect equation but after the teachers talked them they discovered the student had skip counted or used repeated addition. Some though did have auto-recall which is the desired goal for any facts based questions.

What did we do next?

Now that we had all this info about our students, school wide, and knew that their was a significant portion of our students who are still additive thinkers we tackled the third part of the common task cycle. We offered the teachers a PLC day to build content knowledge in the area of multiplicative thinking. We did this by having the teachers build concept maps. We sent this article before the PLC day to have the teachers read it ahead of time.

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What was our first step on the PLC PD day?

On the PLC day their first step was to build an initial version of a concept map for multiplicative thinking based on what current content knowledge they had. After they were finished we had a share out and discussion.

Here are some 1st Version Maps before the PD learning (Notice how it varied in how much content was known but overall the webs are fairly small)

Second part of the day was to do some learning about multiplicative thinking. We gave the teachers a choice to choose from these articles below or this site. (The first one is available online and the other two are from NCTM. I think you need to be a member to access them)


Multiplicative Thinking Site

After everyone had a chance to read then we had discussion about what people learned. The teachers were able to ask questions, clarify meaning, build on each others ideas and overall start to improve their content knowledge. There was excellent math talk around the learning.

Here some highlights:

  • the multiplicative situation has three parts (number of groups, amount in the equal groups and the total)
  • the names for the three parts are: factor x factor= multiple
  • each of those parts has specific names depending on whether it is multiplying or dividing but maybe using the above three simplifies the situation for kids
  • depending on the context you are always looking for one of those parts and the context will determine what operation to use
  • there are specific properties for multiplication and division, some carry over from adding and subtracting (associative, commutative and the distributive property, the zero and 1 principal etc.)
  • along with the properties the other  key ideas of multiplicative thinking are: cardinality, unitizing, part-whole, proportional reasoning, place value
  • knowing the situation, knowing the language, knowing the properties, using the array model are key to teaching multiplication and division
  • we can no longer teach them as separate entities but as one situation
  • there is a sequence for the  models used for developing multiplicative thinking, especially the array, the number line is also effective and may be best for kids learning to unitize

What was our last step?

The third part of the day after all the learning was to add to their content maps and explode them out!

Here are some pics of the content maps after the PD learning (You can see by the end of the day the concept maps had exploded out!

Overall I think the teachers found it to be an effective way to build content knowledge. It helped them to better understand the math and to make plans to move their students forward. The maps can also be used to help identify student needs while on-going assessment is happening. These concept maps reminded me of Cathy Fosnot’s landscapes of learning, accept these are co-created by the teacher.

After the PLC learning day we mapped out a plan for the teachers to use to help move their students from additive to multiplicative thinking. At the end of the cycle we did a second common task to see if the students had made progress and help plan where to go next. I like the common task cycle because it identifies school wide trends, helps create common language among the teachers and is based in the student work. Combining the common task cycle with using a concept map to build content knowledge seemed to go together very well. I need to thank Kelli Gates our math consultant in TLDSB for bringing the idea of the concept map to my attention and also participating in the common task PLC cycle.

Here is the link for the second common task we used:


I got the common task cycle idea from Doug Duff. I also love this quote of his..

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If you have chance try this idea at your school, I think you will love it!

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I forgot what Mean means!

I was recently helping a grade 6 student practice some EQAO prompts at one of the schools I work at.  We were looking at some of the multiple choice prompts and going over the ones that the student had misconceptions with. We eventually landed on this prompt:

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I think if you have been teaching math for awhile you have probably run into students who consistently confuse mean, median and mode. You may have also heard this question, “What is mean?” or this statement, “I forgot what mean means?”

I have few thoughts on this, could it be because it is often taught procedurally? Or could it be because it is often taught out of context? I know that just because we put problems in a pseudo-context doesn’t mean they will always be effective but I think in the case of dealing with central tendencies like mean, median and mode it is pretty effective. As for the first thought about it being taught procedurally, I think it needs to be addressed too. Both these two thoughts have to come together in my opinion. We need to have students model mean conceptually and put in context so they know why we use mean (or any central tendency).

I am going to share some pictures and ideas how I helped this student get a clearer idea of what mean is and how to model it. When we chatted about it he remembered what mode and median were but really didn’t understand why we use any of the central tendencies. I started by creating a new context for the student by asking him if there were any sports he enjoyed playing. He stated that he played in a soccer league in the summer.File_000 (25)

(This picture was taken after we had finished everything in our guided mini lesson, the answer for mean was not filled in until the end)

I started by creating some mock stats for possible shots on goal over a 6 game period. I then asked him what he thought it meant to calculate the mean?

He wasn’t sure, but he said, “The mode of the new data would be 5 because it occurs the most often.”

I asked what that meant and he said, “It means that I got 5 shots on goal, most times out of the 6 games.”

I said, “That is exactly what the mode tells us.” I then said, “We know the mean and mode are related and you told me that mode is the number of shots on goal you got most often in 6 games. Now,  how could you use what you know about mode to help you figure out what mean is?”

He said, ” I think it may have to do with my average shots, I think I remember that from last year.”

I think he was starting to access some prior knowledge at this point, he has some idea of what it mean means but hasn’t really understood it or conceptualized its meaning. I let him know that he was correct that it has to do with finding an average but we need to really dig in and model what it means to find the mean of set of data.

We started by modelling the data with snap cubes:

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Then I asked him, “If we want to find out what your average shots on goal per game are what do we need to do with the cubes?”

He thought about that question and finally said, “I think it will be one number.”

He couldn’t offer a reason why but I think he was on the right track but couldn’t explain it. I then said, “You are on the right track, it will be one number that describes your mean (average) shots on goal for the six games just like the mode was one number (5).  I then asked him to make an estimate based on looking at the data displayed with the snap cubes and using other info he knows.

He said, ” I think it will be five which is also the mode.”

I asked him, “Why do you think that?”

He then said, “If I rearranged the cubes I think they will make equal groups of 5. That makes sense because I want to see how many shots a take most games”


He had started to move towards modeling the mean on his own without me specifically telling him to rearrange the cubes into equal groups or me just teaching him to add all the numbers together and then divide by 6 which is often the procedural way mean is taught. The research says, “When students are first introduced to the concept of mean, they should have opportunities to act it out and explore it concretely.” (Small 2013)

Hereis pic of what happened next after he rearranged the into equal groups:

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He moved the cubes into equal groups and then said, “I can make six equal groups of four but have three left over. I know my estimate of 5 is close but it is not going to be 5.” He pondered this for a moment and wasn’t sure what to do next.

I asked him this question, “What can you do with those leftover cubes? You moved all the other cubes into equal groups but don’t have enough to make each group 5.”

He thought about this for a moment and then a light popped in his eyes and he moved the cubes like this:


I asked him, “What does your model represent now?”

He replied, “I think if I was able to cut each cube in half I would have four and half cubes in each group.” ” Does this mean that I would average 4.5 shots per game over these six games?”

I said, “Yes it does mean that. How did you decide it would be 4.5 shots per game?”

He said, “With the cubes it is easy to see, each group is a game, so for each game I have 4.5 shots.”

This was just a quick one on one lesson with this student but I felt an effective way to teach mean. I felt like this student had a much stronger understanding after leaving our little guided lesson. Do I think every student would be able to make some of the jumps he did, no I don’t. Some students may need more explicit guidance.

This idea is not mine it is from Marian Small’s book Making Math Meaningful on page 576. This student still needs more purposeful practice with this concept and we haven’t touched on when to use a certain type of central tendency or when they are meaningful or biased but it is a good start. Like all math concepts we need to start with concrete examples, using models to help students understand the math. I was taught to just add up the numbers and divide by how many numbers are in the set of data with no explanation of what it means or why it works. I hope this post also shines a light on why adding context can make a lot of difference in our students understanding. That original EQAO prompt was just random numbers with a question to calculate the mean. Many students always ask why are we doing this so let’s help them see why we are doing these sometimes random prompts and then maybe we won’t get the question. “What is mean again?”

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Purposeful Number Talks!

Number talks has taken off in my schools like it has in many schools in other areas. It is quality math routine that helps build students number sense, their ability to justify their thinking, helps build math discourse in your class plus many other benefits. I just want to take the time to write quick blog about really making sure these are purposeful. I know they are planned based on strategy if you are using the “Number Talks ” resource by Sherry Parrish which is important but I think it is also important to make sure their are a couple other key points when using number talks. All of these are mentioned in the number talks book or other resources but I am not always seeing them used in classes.

I think when planning to use number talks you need to not only be looking at strategies but also making sure you have thought about these three parts too:

1) Models (array, number line etc.) are being used to support the student thinking you are recording on the board. Cathy Fosnot talks about this in a video on the LeadTeachLearn site. She says it is the most important piece when doing number talks or number strings.  Here are two examples of what that could look like:


2) A key idea is included in your focus (associative property, unitizing, distributive property etc) and that is based on knowing your students and where they are at. For example, if you are using the Lawson’s Multiplying/Dividing continuumfile_000-19to track your students progress through operations development and you have a class that is mostly drawing composite units and counting (see picture below)

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even though they may be in Grade 4/5 we have to craft our number talks to build on key ideas that they need. These students need to learn unitizing (seeing a group of ones as one unit, in the case of the example above it would be seeing them as groups of 3). If you are using the number talks resource to work on doubling and halving or partial product strategies but the majority of your students are still counting composite units, the number talk is not going to be as effective. Those strategies may be to advanced for the majority of your students. I think it is also another reason why we should put the prompt up and see what strategies come out and not specifically instruct a strategy for the day or week. Through their work guide them to highlight strategies that come up and how they make use of certain key ideas. Eventually modeling how to use certain strategies that don’t surface but always connecting them to each other and the key ideas needed to use them. I would love to hear other peoples thoughts on this aspect of number talks.

3) Alway think about how you can improve the math discourse during your number talks. Maybe this is co-creating a math talk anchor chart with words they can use to help build their vocabulary when talking to each other, words like: build on, agree or disagree, piggy back etc.

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I also recently was shown an article that was provided at one of our math lead adobe connects sessions, it’s called “Engaging All Students in Mathematical Discussions” by Bahr and Bahr.File_000 (24) I believe it is from NCTM. It is an awesome article that talks about assigning listening roles during  math discussions. For example, before your number talk you would assign everyone the listening role where they need to listen to the speaker’s strategy and then compare how it is different/same as their own. This will help engage more students in the math discourse during number talks.

I was recently in a teacher’s classroom that I work with often. He does a great job of using purposeful number talks. In this video he was using dot arrays to help students learn and improve on using the distributive property of multiplication. Through diagnostic assessment he realized he had many additive thinkers and is trying to move them to use more multiplicative strategies by making use of the distributive property and dot arrays. I have heard the distributive property called a student’s friend for life in few different articles. Most recently in Chris Hirst’s “The Multiplicative Situation”. I love that statement and agree totally! Here is the clip:

I really like how Ed stated that he was going to model the students thinking. The students in his class are use to having their thinking not only notated with numbers but also modelled using a models like the array or number line. You can also see they are very comfortable sharing their thinking because of the environment he has created in his class. Number talks are great! I just think  we need to be more purposeful in how we do them and be careful not to fall into just following the resource and moving strategy by strategy.

I am always interested to hear peoples thoughts on my posts. Please share any comments or ideas you have to build on this post. Thanks!

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Does getting the correct answer actually let us know our learners?

The past few weeks I have been learning and researching a lot on multiplicative thinking. Chris Hirst’s article “The Multiplicative Situation” has been my go to, it’s an incredible article that makes it very clear on what we need to do to help develop multiplicative thinkers. This post was going to be about multiplicative thinking but something more interesting came up. I was working with a teacher while she was diagnosing her students before starting a learning sequence on multiplying/division. What came up was a truly “aha” moment for this teacher. This “aha” moment was seeing that two students who she thought were way ahead in their multiplicative thinking based on some problems they had solved were actually not.

The teacher had done some multiplying problems before and both these students were solving the problems and coming up with the correct answers while showing their thinking. We then decided to interview her students as the diagnostic using Alex Lawson’s continuum



and two of the prompts from her book to see/hear what the students shared with us. The two prompts were:

You have 5 boxes of chocolates with 4 chocolates in each box. How many chocolates do you have altogether?

 You have 15 guppies and want to put 3 in each jar. How many jars do you need?

Here are two videos of the students answering these prompts:

As you can see in the videos both students are using a combination of strategies from the direct modelling and counting more efficiently stages of the continuum. Both modelled the groups and then skip counted. These students are still counters and not yet using multiplicative thinking when looking at multiplying and dividing problems. Making the move from being an additive thinker to a multiplicative thinker is one of the most important steps as we move from primary to junior. If they don’t make this leap it can stall much of their math learning from junior through to high school.

During these interviews  she realized that two of her best math thinkers, who get the answer correct most times are actually still using counting strategies. We then discussed what specific instruction they need to help move them into using more advanced additive strategies and then on to multiplicative strategies for the transition to being a multiplicative thinker in the junior years. Most research I have read says they should make this transition between grade 3 and 4 and then extend their understanding into more complicated multiplicative thinking (rate, combinations) as they move into later junior/intermediate years. We realized there is no reason for this teacher to panic, these girls are in grade 3 but it was still eye opening for her to see where their thinking is located on the continuum even though they were consistently answering problems correctly.

Here is some pics of written product assessment that gives further evidence these two students are still counters:

Looking closely at the two work samples you can see they both counted all the squares. If they were more advanced additive thinkers we would have seen something more like this:. 5+5+5+5=20 or 10+10=20. You can see they even wrote the number in each box as they counted.

The math consultant for my school board and I chatted at a PLC last week and she tells me this conversation is starting to take place more around our board as teachers start to know their learners and the math more deeply. Correct answers on assessments can be deceiving especially when looking at bigger ideas like moving from counting to additive  and then to multiplicative thinking. Precision is important too, but seeing and hearing their thinking, going deeper and learning what strategies they are using and where their conceptual understanding lies, to me is so much more important! Please leave some comments, I would love to hear other peoples thoughts on this topic.

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Distinguishing the Whole when working with Fractions

I have been doing some work on fractions with a 7/8 class at one of my schools. The class as with many classes has some significant learning gaps when it comes to fractions. The teacher and I determined that we had to go all the way back to showing that a fraction is a quantity, representing fractions using models/equipartitioning  and determining the whole.

After a well planned sequence of tasks we really feel we have closed some major gaps in terms of representing fractions and they now seem to have solid grasp of equipartitioning. I know you may be thinking that a 7/8 class is in dire straits if we are working on these concepts when they should be ready to add and subtract fractions in grade 7 and multiply and divide fractions in grade 8 (In Ontario). Still we did what we needed to do because you can’t do operations with fractions if you can’t even represent a fraction! The class has made some huge jumps with only a few really precise tasks to help close those  gaps.

With this blog I just wanted to share some student work and show one spot where we are still seeing some misconceptions. I see this actually in a lot of classes I am in where students sometimes get confused in determining what is the whole or wholes they need to use to answer the problem.

Here are some examples of the group task that was done on Monday. There is picture of the task below too. I also love that the teacher decided to use the VNPS whiteboards. Makes the math more visible and the students love to work on them.

As you can see with the work samples the groups were able to see that the juice containers each represent a whole. This was also evident in the other 5 groups except for 1. After the group task was complete the students then did some purposeful independent practice. They completed a task similar to the one they did as a group. Here are some work samples.


If you look at the top two on the right and the middle one on the left you will see that all three of these students treated both gatorade bottles as the whole together and came up with an answer of 2/6. There were three other students who also had this misconception. When the teacher and I reviewed the student work we saw that we still have some work to do with determining the whole. Moving forward some guided groups will be created to look at the key understanding of “a fraction should always be interpreted in relation to the specified or understood whole.” Two things I want to highlight. 1) This is one reason why the exit card is so important, judging by the group work it could have been easy to say they all understood that you have to determine the whole first before representing the fraction. 2) Determining the whole is a key understanding of fractions that I think is sometimes overlooked or not as much time spent on it as other key understandings related to fractions. Remember this is my opinion with what I see in many students work with fractions, doesn’t mean it’s right! LOL.

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Unpacking Expectations and Identifying the Specialized Content Knowledge (Key Ideas)

Currently at two of my schools we are working on a PLC that is looking at the planning that has to happen before a lesson or sequence of lessons is delivered. One of the main focuses has been looking at unpacking the curriculum expectation (concept) that the sequence of lessons or lesson is based on. Two statements that are being used a lot lately are “know the learner” and “know the learning”. This post is looking at the later “know the learning”. It is also fairly long so I apologize ahead of time!

We know that we should always start with the curriculum when planning our math lessons but being able to identify the curriculum expectation or expectations that you want a sequence of lessons to be based on is not enough. After you have decided what expectation or expectations you want the lessons to cover you then have to unpack those expectations and really know the learning that is behind them. There is often many mathematical ideas that you need to know and understand to effectively set up learning situations or tasks to help students fully understand a math concept. Many different math people call these by different names: key understandings, key ideas, and sometimes big ideas. They are all basically talking about the same thing although I think big ideas have a different meaning. I use a lot of different resources and you can find these key ideas in all of them. Marian Small’s “Making Math Meaningful” is one such resource that has them laid out in an easy to understand  way. When I reference key ideas later blog these same key ideas are called fraction principles in Marian’s book. Cathy Bruce’s fraction pathway research calls them key understandings so you see what I mean by different people calling them by different names.

There is another resource that I think explains what is meant by key ideas even more clearly.“Teaching with Tasks for Effective Mathematics Learning” by Sullivan, Clarke and Clarke has small section in their book where it discusses the role of teacher knowledge in effective task use. It’s on page 15 and 16 in their book. Here is picture of the book and the page which I am referring to.

In this section of their book they refer to two major categories of  teacher knowledge that is needed for effective math lessons. The first is subject matter knowledge and the second is pedagogical content knowledge. Subject matter content knowledge is then broken down into two sub-categories which are common content knowledge and specialized content knowledge. Common content knowledge is often what many people have, they can solve the problems based on a concept because of the way they were taught, they know how to do the math! Often this is where many teachers reside, possibly because of many reasons, maybe they are a little math phobic themselves.The second part, specialized content knowledge is this: the knowledge that allows teachers to engage in particularly teaching tasks, including how to accurately represent mathematical ideas, provide mathematical explanations for common rules and procedures, and examine and understand unusual solution methods to problems (Hill et all., 2008, p.378). This is where the understanding of the key ideas for a concept lay, along with the other pieces laid out in the definition of specialized content knowledge.

Here is an example of where this understanding of key ideas comes into play in the classroom. Recently in a grade 4 classroom we were doing this guided math task. The students had red and blue coloured tiles and the teacher was asking them to build shapes that represented the fractions being called out. These work examples are from when 1/4 fourth was called out. Here are three of the student solutions that came after 1/4 was called.

At first glance two clearly stick out as representing 1/4 but one is not so clear. I want to focus on two key ideas happening here ( there are others). If the teacher doesn’t understand clearly or has not planned for, then one student possibly could be told theirs is incorrect or a quality teaching moment could get looked over. The teacher needs to see what is evident in these solutions and name the math for all students to see. All of these solutions represent 1/4 because of the key ideas that fraction pieces have to be equal in area and not necessarily the same shape and that the pieces don’t have to be adjacent to one another, which allows for the picture on the right to also represent 1/4. Many teachers do not understand these key ideas or specialized content knowledge and it is so important that we do!

The teacher I was working with here was the first to admit they didn’t understand either of those key ideas before we started planning for this sequence of tasks. We started with this curriculum expectation: represent fractions using concrete materials, words, and standard notation, and explain the meaning of the denominator as the number of fractional parts of a whole or a set, and the numerator as the number of fractional parts being considered. As you can see there is a lot of math understanding in that expectation. If you just pick a task or sequence of tasks you think will help students learn this expectation without planning and understanding the math behind it (unpacking) then you are going to miss opportunities like what came up in this teachers class or possibly teach misconceptions.

When we unpacked this expectation in planning we identified the key ideas, two I mentioned above. The teacher was ready then when the task was done in class. She was able to see that all of these representations of 1/4 were accurate and then was able to connect the representations (on the fly) together and point it out to the class that they all represent 1/4 by allowing the student to explain their thinking about why the picture on the right also is accurate. There was quite a debate in class between students about the third representation above on the right. Only  one student made one like that and they justified their work helping to teach a key idea to the whole class. This opportunity would have been totally missed without the planning ahead of time. There are other key ideas involved in this expectation but I focused on these two for this blog. I think as instructional coaches it is so important for us to help work with teachers to help develop this specialized content knowledge.

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Being Wary of the use of the Pizza (Circle) Model for Fractions in Primary!

The last few years has seen a huge jump in my content understanding when it applies to fractions. During my elementary years and high school I couldn’t think of an area in math that I hated more or that I had a more difficult time understanding. Come to think of it this was all areas in math. Hated is not understanded here either, I despised math. So I find it funny now that teaching and learning math has become my passion. I love all aspects of it. Now I am not going to get into my feelings about the way I was taught math (a whole other blog post, maybe even a book!) or the fact I suffered because of the way I was taught math. I am just going to take a snapshot of an area (representing fractions/partitioning fraction models) that we have to be very careful about how we introduce to students.

A few years back I had the pleasure to take part in some learning involving the the fraction pathways that Cathy Bruce, Tara Flynn and Shelley Yearley are  developing for the Ontario Ministry of Education. I supported some of the teachers I am working with as they took part in the collaborative inquiry around the fraction pathways. For more info check out this link on edugains.

To say this moved my understanding forward as far as fractions is concerned is the understatement of the year. I now fully understand fractions and the key understandings behind the concepts. I have done a lot of other reading to, the book a Focus on Fractions Bringing Research to the Classrooms by Petit, Laird, Marsden and Ebby is also a go to resource.

During the learning for the fraction inquiry they spoke about how in other countries such as Japan they use fraction models that have longevity across grades. They reference the number line and rectangular area model as two such models. One area model that research has shown can be difficult for students to partition or use accurately is the circle model. This is a quote from the fraction pathway resource “In particular, partitioning circles equally is much more difficult with odd or large numbers whereas rectangular area models and number lines are more readily and accurately partitioned evenly for odd and large numbers (Watanabe, 2012).” They also told us that once this model is shown to students at young age it is very hard for them to access other more efficient models once they are shown them in later grades (more so even for girls). Students will tend to always revert to using a circle as the model of choice which then potentially creates more errors. Please read their research because there is a lot more information and I am only using bits for this blog. All this background leads to the whole point of my blog.

We have been working with representing and partitioning in the grade 2/3 classroom at one of my schools. The pizza model or circle model has already come up. Students are using it from prior knowledge, because it has not even been addressed in the current classroom this year. We have had some discussions about using consistent models like the number line or the rectangular area model.

I brought one student who has pretty strong understanding in regards to equipartitioning. The student knows the fractional parts have to be equal but in class has shown they really like to use a circle model and is struggling with representing the fractions as equal parts when she partitions using the that model.

Here are some videos of me working with her to help her potentially discover why these problems are arising and address the issue I spoke of above.

I prompted the student to see if they could think of a different way to partition the circle. Here is what the student tried next.

As you can see the student was struggling to find a way to partition the circle accurately.

This was happening consistently in class when this student was asked to represent fractions using her own model. Using 1/2 or 1/4 as the fraction to be represented was fine but as soon as the fraction was an odd number like thirds or fifths it became very difficult using this model. I then demonstrated how a circle can be partitioned into thirds and then allowed the student to try again after I modelled it. I wanted it to really hit home that this model is difficult to use  and that there is potentially a better one to use.

The last video shows the student trying to partition the circle after having it modelled for them.

You can see my modelled example on the page the student is using and they still struggled to partition the circle into thirds.

I had to edit the length of the videos so some parts are missing. The student did go back and colour in one third on the rectangle. I also edited out the last question I asked (not on purpose a tech failure on my part trying to edit! Opps), “What model made it easier for you to represent one third?” The students answer was immediate, “The  rectangle!” This also another good example of how a quick guided session can help move a students thinking forward rather quickly. I know as teachers in primary its easy to go to the pizza as a model for context, but maybe it is just as easy to use a rectangular brownie or cake model! There are also fine motor issues to worry about in primary when they are drawing models, I used pre-drawn circles ans rectangles so imagine how much harder it would be to partition the circle if they had to draw the circle too! I look forward to hearing some thoughts on this and please if want to read more about this check out the Fraction Pathway research on

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