Subitizing and Quantity- Important Pieces to Operational Development

I have done a couple video blogs before and they are challenging for me. I am not very comfortable filming myself! This is a short video blog on subitizing and how it fits in with operational development. I try my best to make them easy to understand as sometimes math edu-babble can overwhelm people when they are first learning. Its another reason why I thought seeing it as a video may help. There may be mistakes, still I think I have caught the essence of why subitizing is so important. There are four parts to the videos. I also try to make the connection to purposeful practice. Hope you enjoy!

Part 1

Part 2

Part 3

Part 4

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Learning Basic Facts in Classrooms Today!

I blogged about this last year with some posts about using guided math groups where the students were learning key math ideas that were helping them to develop basic fact fluency. With so many articles in the news lately about “back to basics” etc. because of the newly released EQAO scores for grade 6 in our province I inspired me to make this my first post of the new year. Most of the articles from the media want to create drama that gets people to click on their article of buy the paper by using terms like “discovery math” and not enough “basics” in the article title to try and highlight that as the problem with our math classrooms. That is another post completely but I want to quickly point out that there has never been a label called “discovery math” in any document that I can find  released for math education in the province of Onatrio. They all promote a balanced math program.

Here are  two pics are from our Effective Guides to Instruction; Volume #1-Foundations of Math Instruction. I wanted to share these two pics because it again shows our Ministry documents promoting a balanced math program with Guided( which includes some direct instruction), Shared and Independent math.


I wanted the focus of this post to highlight that teachers are still teaching basic facts or fact fluency in their classrooms. The difference is many are now doing it in a way that is foreign to most of the classrooms our students parents and myself would have attended. I want to share personal story before I detail some of the differences we now see. I hated math in school, despised it with all my heart, now it is passion and I see it in such a different light. One of many areas I struggled in was memorizing my times tables and simple addition and subtraction facts. I couldn’t learn from flash cards or endless games of “travel around the classroom” that would have seen me shaking and sweating with nervousness as the person who was flying around the class beating everyone moved closer to my desk. First of all my mind doesn’t think fast like that, sometimes I need process time and even if I did know it I would cease up with anxiety. This was majority of my math experiences, memorize this fact, this formula and then watch the teacher do it and then try and repeat it many times even though I knew I didn’t understand it in the first 10 questions! I was never shown other ways to do problems or math concepts and I was certainly never shown that there are strategies, patterns and games that can help you develop auto-recall of your facts. Surprise! Writing math facts out hundreds of times didn’t work for me. Maybe it did work for you and that is great but for many students it didn’t work and even for some that it did work for they maybe didn’t understand what it meant. Here lies the focus of this post, and that is to highlight how we are getting students to develop auto-recall of their basic facts and have a strong number sense at the same time. Let’s make it clear nowhere in our curriculum or in our ministry documents has it said students don’t need to have automaticity of their facts. If there is I have never seen it.

Here are some pics of some of the Ontario Ministry documents that shine a light on the focus of learning basic math facts and computational fluency :

Below is a pic of one of the Effective Guides to Instruction: Volume 5- Teaching Basic Facts and Computations. This whole guide is dedicated to teaching students how to become proficient at basic facts.

This next picture is from our curriculum where it clearly states multiply to 7×7 and divide  up to 49 by 7 by the end of grade 3. Some would argue it doesn’t clearly state memorize but after my post I hope it will be clearer that it means the same. We want students to be able to do this by the end of grade 3. Some teachers think that the second part, up to 9×9 which is for the end of grade 4 probably should be included with the grade 3 expectation and I would agree with that. Why stop at 7×7 when kids are moving to proficiency!


What is the difference between how we learned facts (in most cases, I have heard of some teachers in the past teaching strategies) in the past compared to now?

The simple answer is students are allowed to develop strategies that help them access facts and answer problems until, with enough practise they become automatic. Alex Lawson’s book “What to Look For” has been a valuable asset to help guide teachers to give students the best learning opportunities to move from direct modelling to proficiency. Lawson has an article on edugains where she explains this very clearly and compares it to how most of us were taught. Here is pic of her developmental continuum for moving from direct modelling to proficiency:


Dr. Lawson explains it like this, imagine if all the strategies on the above continuum between counting three times (direct modelling) and proficiency were covered up. That is what most of our past teaching and learning of basic facts would have looked like. We would start in Kindergarten/Grade 1 by answering simple addition and subtraction problems by direct modelling the problem on counters or our fingers and then counting to get the answer. Then we would have quickly moved to trying to memorize the facts through activities like flash cards, worksheets or writing them out multiple times until we hopefully became proficient. We would therefore not learn or be introduced to all these strategies that research now shows kids actual develop through on their way to proficiency. You often talk to many adults who developed many of these strategies on their own because it makes the math easier to do! The same principle applies to multiplication and division. I chose Lawson’s continuum because our board is using it a lot currently. There are other resources that promote this same instructional approach. (Clements, Richardson, Small, Van De Wall are some other good examples). Here is a pic of Lawson’s continuum for multiplication and division:


How do we develop students through these strategies until it becomes automatic for them?

Teachers are using many different instructional strategies for this but two that stand out are number talks and math games. Number talks allow students to share their thinking and strategies with their fellow classmates while seeing different strategies modelled on the board with an appropriate model to support conceptual understanding. Math games which is just as important as the number talks allows the students to practice their strategies and computations in fun way. There  are others but I wanted to highlight these two especially games because in essence it is drill (practice) but in much more  effective setting then say flashcards or drill worksheets.

The last piece of this post is I wanted to share three videos that show one student’s journey along the continuum from counting on to proficiency. The journey goes from beginning of grade 1 to near the end of grade 1. He now knows all his addition/subtraction facts to 20 automatically  and is now applying the strategies to larger number combinations while also beginning to work on the multiplication continuum. Mrs. Kennedy had her students playing games, doing number talks and also working in small guided groups with her on key ideas that help students develop these strategies. Enjoy his journey!

Video #1- October 2016 Beginning of Grade 1

Video #2- January 2017 Middle of Grade 1

Video #3- April 2017 End of Grade 1

Here is pic of a continuum that highlights his journey with each strategy he used along the way.

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Hope this helps show that students are most certainly still learning their facts and becoming proficient to the point of auto-retrieval. Maybe this isn’t happening in all classrooms but it is in the schools I work at and I certainly see a lot of like minded educators on twitter teaching this way. It matches what our Ministry of Education in Ontario  and our board promotes. It also works! The journey is different now which opens it up for all kids to learn their facts!

I am not afraid to admit that I did not know all my facts when I became a teacher, I do now! It comes from being introduced to these strategies and just playing with the math! I still have to think about some, for instance 8×7 and 9×7 confuse me sometimes but I now can quickly do 8×5=40 and 8×2=16, put them together and you get 56. I just used the partial product strategy (distributive property) in under 5 seconds to access my fact I was stalled on. Think of the power this gives a student who use to think if they didn’t know the answer to the flash card/worksheet they just moved on and didn’t get it! That was me! Not anymore and I work now to give all students this opportunity. If you were a memorizer and you got it, congratulations but in my experience that just isn’t the case with most students. Teaching facts may have prevented a lot of math phobia that many parents show today. My 75 year old mom shared a story last week with me that highlighted the anxiety and sick feeling she would get when her teacher use to draw a clock on the board, then point to two numbers and pick someone randomly in their seat to answer. If you didn’t get it then you got centered out in front of the whole class. She still remembers it to this day 60 plus years later! That is just another reason why the process we use now makes me so happy. Thanks for reading.

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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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