Something I never would have done before the Fraction Pathway Action Research Project!

Today I was working with my principal on the next assignment for the RMS Virtual Learning session we are currently doing as a school. Part of the session is  solving some problems individually and then sharing with our colleagues how we solved the task. It is in preparation for the next session. The tasks have been focusing on how students develop towards multiplicative thinking and helping us to look at it from an LD learners perspective. This was the task for this part of the session:

Bike-A-Thon Problem:

In a bike-a-thon, cyclists will find

-water stations every two-thirds of a kilometre

-medical stations every three-halves of a kilometre

-bike repair stations every three-fourths of a kilometre

John has reached the first medical station, Alexa is at the second water station, and Sima is at the first bike repair station.

Who is further along the course? By How much?


Here is how I, my principal and one other colleague solved this task:


Mine is the top solution. The reason I wanted to share this blog is because recently on Twitter some educators had been sharing some fraction tasks with student work. People were discussing that they didn’t know what they didn’t know and the fact fractions have so many key ideas and levels of understanding that they were beginning to have their eyes opened to the content knowledge you need to have and understand to teach fractions is effectively.

Well I certainly would have fell in that boat, my schooling left me with a severe lack of understanding of fractions, a hate of fractions may better describe it. You can imagine how confident I felt then when it came to teaching it, which I feel many teachers are in that boat. When I had my math awakening and started researching and learning myself I began to feel more confident with fractions, then I moved into a Instructional leader position for my board. In my second year I had the opportunity to be part of the Fraction Pathway Action Research project that was lead by Dr. Cathy Bruce, Tara Flynn and Shelley Yearly and came to realize that I still didn’t truly understand fractions until I met and learned with this crew! Wow to say I made some leaps in my fraction sense is an understatement. If you haven’t seen their research check it out on under the Fraction Pathway.

This leads me back to the point of this post, when I looked at this question the fraction parts just made sense to me. I didn’t need to convert them to decimals as I was taught to do in school. Even the part that said the medical stations were every three-halves made total sense. I also kept everything on my number line mostly in proper and improper fractions to keep the unit fraction visible. I find it easier now see 4/3 and 6/3 then to convert to mixed numbers. I went straight to a number line which was never in my fraction vocab until I was part of the pathway research. I visualized my partitions and even for the last part subtracted fractions to find how much further John was along the course. It just makes me giddy to see how comfortable I am using fractions now when even 6 years ago I would have said it was a weak spot for me. And if you went back to when I was in school I would have been the student so frustrated with fractions that I wanted to cry and run for the hills.

As you can see both my colleagues went straight to converting the fractions to decimals and they both shared that they were confused with the wording of “every three-halves of a kilometre” in the beginning. They both shared they felt uncomfortable working with the fractions and it was because they felt they struggled with fraction sense in school and feel like they shy away from fractions. Many of our students are also in this boat but I feel many that are being taught using ideas from the Fraction Pathway, for example, focus on unit fractions, use visual models and build their fraction sense are going start to be more confident and hopefully not have to wait until they are in their late thirties to have a fraction awakening like I myself did!

Like so many concepts I have re-learned since I have become a math  geek and started down this math journey to show students that math is awesome and that there are better ways to learn math then the rote procedures and memorization tasks I was forced to do and failed miserably at during my schooling, fractions can me learned in a way that makes total sense!

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The Art of not Stealing the Learning!

There is a very fine line when working with a student or group of students between prompting them to move their learning forward or stealing the learning from them by giving to much info away or doing it for them. I tell teachers I work with that it is an art form and takes a lot of practice. I am by no means a master of doing this or even close to where I want to be in my ability to move students learning forward when prompting. Still I do feel I am getting better at it and am in a position to help people learn to do it better. One of my principals I work with is moving along the journey too, and she would be the first to tell you she use to shut the learning down or steal it from students when she sat and worked with them. She is so proud of herself now when she sits with a group and is able to push thinking forward with timely prompts. She often takes notes and has questions for me when I return to her school about interactions she has has with students. We discuss the moves and prompts she gave and how it could be improved or not. More teachers in my buildings are doing this too which has been awesome.

Here are few points I think are super important for providing prompts to students to push thinking forward and not to steal the learning from them. 1) Know your content well, not just well super well! If you don’t know your math content its hard to give timely feedback. 2) Use the Five Practices to plan your lessons. The anticipation part has you do the math first in as many ways as you can. Plus it has you think of any questions they may have or hints that you could provide to move them forward on the task. 3) Ask questions don’t provide answers. 4) Create hint cards before the task that you could provide students with before they access you for feedback 5) Take the time to sit and listen to groups or individual students while they work. Observe and have conversations. If you don’t do this you will never be able to give timely feedback to push their thinking. I know there are more that could be on that list but its a start.

I have uploaded a video below with a quick example of how easy it can be with a simple prompt to push students thinking to other levels.

In the video I all asked these two students were, “Is there another way to arrange your groups of two in way that is easier to see and count?” I didn’t tell them or show them how to put them into lines of ten or eventually into an array like they did when they had finished. I asked a question and then watched as they did the thinking and came up with the idea to arrange them in lines. They didn’t know they were creating an array, they said lines but still they thought about it and came to see that its easier to see and count them when they form them into lines of ten. My prompt was about organizing the counting piles to make it more efficient to count but in doing so it moved them from using equal group piles of 2 to forming an array but also to realize its more efficient to count my tens not two’s! Here is the video of the final count.

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Do Students really Understand the Operation they are Performing?

I recently read this great article from NCTM called “Capturing Children’s Multiplication and Division Stories: Learn the value of having students create their own stories and pictures to represent number sentences as classroom assessments.” Here is link to the article:

It is definitely worth a read and really inspired me to go out and try it in a class. I asked one of my grade 3 teachers I work with if we could try it in her class and give them a diagnostic assessment to see how well they understood multiplication. In the article the study that was done had them do a story and a drawing for a multiplication prompt and a division prompt. We only had the students complete a multiplication prompt. This class has done some work with multiplying but not much this year so far. I have added the curriculum expectations for grade 2 in Ontario below:


This what students in grade 3 have learned previously in grade 2 about multiplication and division. We also know students develop along a continuum for operations so the students in the class have a wide range of where they could be in their understanding of early multiplication.

Here is the prompt we gave them:

Make up a story and a picture about marbles for this number sentence: 4 x 3=

(We said they could change the topic from marbles if they wanted to.)

Here are some of the student work samples:


As you can see all three of these students knew the answer but were unable to make up a story that describes a multiplication situation. The two that were able to make a story up described an addition situation to arrive at 12.

The first pic shows the students understood it can be repeated addition but was unable to draw a picture that models 4 x 3 or make a story up about the situation. The second pic down knew the answer was 12 but drew an equal group picture that represents 6 x 2 not 4 x 3 and also described a addition situation in their story. The bottom pic was interesting in that this student knew the answer immediately. This student said work is done on times tables at home. This is a great example of students memorizing answers to times tables questions without conceptual understanding. The student may know answers but will struggle to be multiplicative if this is not addressed. The pic shows an addition situation as the story and an addition picture not even an equal groups for the picture.

I wanted to also show this piece of student work because I thought it was neat they tried to model it with a number line as their picture of the situation.


There is a few simple errors but they understood its three equal groups of four. The number line doesn’t start at zero and there are two groups that are 3 not four but they tried! I like that they tried to use the number line.

Overall  this assessment showed a clear lack of understanding of multiplication but was very informative for the teacher, she now knows she has a lot of work to do! I am going to show this too my grade 1 and 2 teachers and continue the conversation that multiplicative thinking needs to be developed simultaneously with addition and subtraction. I feel sometimes there is a thought that it needs to start after addition and subtraction and that it is something looked at more in grade 3. The foundations have to start in FDK to grade 2 with learning to unitize (seeing groups of as one unit) and starting explore what multiplication and division mean. Start counting equal groups, looking at equal group situations and solving basic multiplying/division prompts by direct modelling and fair sharing. Students in these grades don’t need to hear the word “times” or multiplication but that doesn’t mean they can’t start learning it by exploring what I listed above. I like the language Kathy Richardson uses in her book “How Children Learn Number Concepts”. She suggests using:  groups of, piles of,  rows of, stacks of, cups of instead of times.

I plan on trying this in some junior classes too, I suspect we will find some students with very similar misconceptions to this grade 3 class. Try it out in your classes, I think you will be very pleased with the information you find out about your students.

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The Importance of Making 10 and Connecting across Strategies

As I dig deeper in this journey of operational development and students movement towards number sense fluency (proficiency) I wanted to write a post about how important it is for students to develop the Make Ten Strategy. Some people also call it Break Apart and Make Ten or Using Up/Down over Ten, whatever name you choose it is all the same strategy and the power of this strategy for students developing number sense fluency can not be overstated. It is often said Make Ten is the most powerful strategy in addition/subtraction development because of its longevity or how far it carries over into more complex strategies that are used when the problems become multi-digit. This leads me to the second part of the title of this post and that is highlighting that connection across strategy development.

I was asked to come into a grade 6 class recently and model a number talks for the teacher that would help make these connections for her students. The teacher recorded the number talk so now I am going to share it with you! Below is the number talk broke apart into four videos. Videos 1-3 cover the the 3 problems in the string. The fourth video is the quick consolidation we did to connect the Make Ten strategy across the more complex strategies. The string of problems I chose for this number talk  were these:

8 + 7=

28 + 7=

28 + 17=

Before you see the videos here are some pics from Doug Clements book “Learning and Teaching Early Math: The Learning Trajectories Approach” where he speaks to the importance of Make Ten and highlights how it is used in Japan which I found very interesting. The pages are 96 to 99 in his book if you are wanting to read the whole section.

I also like that he emphasises that students need to also see and learn other strategies too, to build fluency but by focusing on Make Ten it will help with later multidigit computations.

Here are the four videos that help show how Make Ten is connected across multiple strategies. I hope it is helpful and like I say about all my posts its not perfect and there may be some mistakes and I am always open for comments and feedback. Thanks!

Video 1- (8+7=)


Video 2- (28+7=)


Video 3- (28+17=)


Video 4- (Consolidation)

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Breaking Down an EQAO Prompt through the Lense of Multiplicative Thinking

Today as my principal and I were having a meeting looking at our EQAO results we came across a great example of a prompt where the importance of being a multiplicative thinker really becomes evident. As we chatted we discussed how it tied in with our focus at the two schools which is really digging in deep and knowing our learners in their operational development from counting to multiplicative thinking. (Note: There is a stage higher than multiplicative thinking and that is exponential thinking) We choose this prompt to look at because it was the question this group of grade 6’s scored the lowest on. Only 18 percent of the students answered it correctly. Here is pic of the prompt:

1557BD42-109A-49B5-A63D-AE1ECD2227D7The prompt is actually coded as a thinking prompt under the strand measurement. Take a minute and break it down into what the students need to know and do in order to solve this task. Then look at the list I made in the meeting with my principal below. Maybe I have missed some and if I have leave me comment. I would love to see other people’s thoughts.







(Note: I only looked at the concepts needed not the problem solving strategies etc.)

The fact that this measurement prompt sits on top of multiple number sense concepts is worthy of another blog post entirely!

Even if I missed some other pieces in my break down of the prompt I wanted to focus on what I have highlighted in yellow. The idea that students need to move to multiplicative thinking in order understand scaling, unit rates and proportions becomes fairly clear in this prompt. If they can’t compare the quantities in a multiplicative way they are going to struggle with this task. I like Kathy Richardson’s explanations, here are some pics from one of her books:


She clearly states that “Children need more than how to get answers to multiplication and division problems. They also have to develop multiplicative thinking, because multiplicative relationships underpin many number-related concepts, such as fractions, percentages, ratio and proportion, similarity, functions and graphs, rates of change and algebra.’  As you can see by the list of concepts in the excerpt from Kathy’s book, multiplicative thinking becomes one of the most important milestones in students math learning. It can stall much of a student’s learning from junior level on if they don’t successfully make the transition.

This task is great example to break down for teachers during PD to help highlight how students will struggle if they can’t think multiplicatively. It also can be used for an example of how just knowing your facts by auto-recall or using an algorithm to multiple is not enough. The students were allowed to use calculator on this prompt yet still struggled. It could be that the students didn’t  retain the measurement concept of volume or the application piece of using the volume formula, but our suspicion based on diagnostic interviews of the grade 7’s at that school this year is that many are not yet using multiplicative thinking. There are still many counters and early additive thinkers in the class. In the diagnostic interview most of the class were not able to view relationships as “times as many” or proportionally. Its work that we are setting out to address  but I wanted to share this prompt and the break down to highlight why it is so important to know our students developmentally in operations and to start closing gaps with students who are not multiplicative thinkers. I would love to read your thoughts on this so please leave a comment if you like! We are going to give this prompt again to the grade 7’s so we can then look at the student work that we can’t see on the EQAO results. I will let you know how it goes!

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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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