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.

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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.  http://www.edugains.ca/newsite/DigitalPapers/FractionsLearningPathway/index.html

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 edugains.ca.

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Opening My Eyes to Math Around Me!

Since sites like @estimation180 have began popping up and adding all these amazing resources to our bank of routine math activities I have found myself looking for math opportunities more often in the world around me. Two amazing math coaches (Joe Swhartz and Graham Fletcher) whose math blogs and twitter I follow have wrote about this and also shared some of their ideas. Graham Fletcher calls it mathematizing your world. Now, I must admit, I am already a math geek but since I have started showing the teachers I work with these resources I find myself looking for these situations constantly. What are these situations you ask? Well they can be anything from pictures I take of cool situations, pictures/videos I find on the web or videos I make myself. The potential is limitless once you start seeing them and realizing how cool these opportunities can be for student learning.

Most of these math situations are being used to create math discourse in class, which we all know is absolutely critical in creating a quality math program. Sites like @estimation180 allow students to work on estimating but also create quality math discussions in class while building students ability to reason and prove. They also hit many other math processes but I feel reasoning and proving is front and centre. Looking at @estimation180 as an example, I feel Andrew Stadel only started the conversation. I believe he wants us as teachers to start seeing these opportunities and create our own situations to use in classes. These math situations that we capture in our world also can have a specific math focus such as: a concept, a key understanding or a math process.

I used @estimation180 as an example but there are many more like it for example, wodb.ca, fractiontalks.com and visualpatterns.org! All of them are to get kids talking, agreeing/disagreeing, explaining and building on each others ideas. Ideas to use for these situations appear around us all the time, once you start seeing them and capturing them you may not stop!

Here are two I have captured in the last week or two. I have many more on my phone but thought these two are good examples to share.

I found  the above picture as I was reading an article on yahoo about ice coverage on the great lakes. I used a photo editor to cut out the information on the bottom with the answer creating one picture that just shows the lakes and ice coverage and then the orginal picture for the answer. If I was using this for a number talk I would show the picture on the left first and ask this question. How much more ice is covering the great lakes in February than in January? The photo on the left has the answer to show afterwards. The answer is listed as a percentage but I didn’t ask the question as a percentage question. This would leave it open for students to use fractions, decimals or percents. They might also use add in units, for example square kilometres could be an option. There is just so much math here to get students talking. This is an excellent number talk for fraction reasoning or making the connection between fractions, decimals and percents. It also could be used to look at fractions as an operator. Students could be given the total area of the great lakes and then using  February’s percentage ask: What is 40 hundredths (which is close to 40.4 percent) of the total area?

Here is the second image I want to share. (The treadmill odometer is in miles)

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This came to me as I was doing my run last week. My goal was to reach 5 km (3.15 miles) in 26 min. Showing the students the image you could ask: Using the info provided do you think Mr. Stamp will achieve his goal of reaching 5km (3.15 miles) in 26 min?  Students can use their fraction sense to help solve this, but it also involves number sense (operations, decimals, fraction sense) and measurement (time, distance). Depending on what grade level, this could be a number talk or an actual task.

These opportunities pop up all over the place when you are out in the world. I bought a pound of finishing nails at the hardware store on Saturday, as the lady was scooping them into the scale that hangs from the roof I was trying to get my camera out in time to snap a picture. It would have made a great number talk or estimation talk. I think back to last summer when my father-in-law and myself were squaring up the playhouse we built (he built mostly! LOL) for my kids.  We used the pythagorean theorem to do it, it would have made a great picture prompt to help students discuss its use in real life. Once you start seeing these situations you may never stop! Teachers ask me all the time where do you find tasks, or prompts etc. They are everywhere, just have your camera ready at all times!

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Update for my Responsive Teaching Post

My last post showed the work Mrs. K and myself have been doing to really be responsive to the student need in her class. It focused on our work with early operations using the Alex Lawson’s continuum to really get narrowed in on what key understandings her students need to move along the continuum  from counting three times all the way to proficiency (automaticity). I posted a diagnostic video of one student answering some simple story problems and then described what strategy he was using to solve them. I then added a video of the Mrs. K working in a guided group setting to help teach him the key understanding of the commutative property in addition.

Here is an update video of the same student about 2.5 weeks after the guided group video was taken.

As you can see in the video he is much closer to achieving automaticity. He is using more efficient strategies now and is much closer to proficiency. In my previous blog he was still counting on from the smaller number. Now he is using two strategies from the working with numbers part of the continuum. In the first video he uses a known fact (but also used his understanding of make ten). He knew that 10 and 3 would be 13 so 9 and 3 would be one less, 12. In the second video he used the strategy called using the five and ten anchors. He pulls the five out of 6 to make 5 plus 5 and then adds the 1 more for 11. This is just an amazing example of what some very specific guided instruction can accomplish in a short period of time when you focus on the key understandings that students need to move along the continuum to proficiency.

Here is picture of Lawson’s continuum so you can see where he is now. He was at” counting on” but is now using the strategies, “using a known fact” and “using the five and ten anchor”.

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Using Responsive Teaching in Math to form your Guided Groups

In my travels so far as an IL I have found that teachers I have observed or worked with have a much stronger grasp of how to form guided groups in reading then they do in math. It is an area that we are working on this year in the schools I work at and as board. How well do we know our learners? Are we planning instruction that is responsive to our students needs? In this post I am going show some work I have been doing  with Mrs. K in regards to her journey to be responsive to her students needs in math and the building of her own content knowledge.

A little background to what you will see in the series of videos and pictures. At her school we have been focusing on operational sense, more precisely  her students journey from direct modelling addition and subtraction problems to proficiency at solving addition and subtraction problems. We are using Alex Lawson’s book “What To Look For: Understanding and Developing Student Thinking in early Numeracy” as our mentor text to help build content knowledge and for the continuum on numeracy development for addition and subtraction that is provided in the book. Below are two pictures, the one on the right is the cover of the book and the left is the continuum for addition and subtraction.

After you have completed diagnostic interviews with your students by asking them a simple story problem you can then analyze what strategy they are using and then place them on the continuum. Once you have the strategy they are using you can use the resource to help you understand what key ideas the student has developed and what key ideas they still need to develop to move towards proficiency (automaticity for fact combinations/efficient methods for 2 digit by 2 digit problems and beyond). Once you have a class profile of where your students are based on your data, the continuum  helps you to group your students into guided groups based on the key idea they need to move towards proficiency. The continuum is divided into four main sections. Direct Modelling and Counting, Counting more Efficiently and Tracking, Working with Numbers and Proficiency. The series of videos I have below show the process from the initial interview to working in a guided group  for one student in Mrs. K’s class. I have provided her data chart that shows where everyone is at on the continuum. The reason this guided group only has one student is because he was the only student using this strategy at this time.

Here is the initial interview problem:

With this interview Mrs. K determined this boy is using the strategy of counting on. All of her students were assessed and then placed on the continuum. Here is a pic of her data chart that she uses to track the student’s progress. She uses this chart to group her students into guided groups. As you can see there was only one student using counting on at this point.

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Using the interview observations Mrs. K and I chatted about what key idea this student needed to develop in order to move along the continuum to counting on from the larger number. We decided that he had not yet developed the key idea of the commutative property in addition (the idea that the numbers can be flipped: 5+7=7+5). Mrs. K then brought this student into a guided group. It happened to be only one student this time but as you can see by her data she would have a few groups at counting three times, near doubles and up and over ten .

I had to break up the guided group video into parts because of the length of the video.The next series of video’s is of Mrs. K working with this student. This was his second time in a guided group for this key idea. I wasn’t there for the first guided group session so he has moved in development from the first session

As you can see in the video’s this student has now started to develop the key idea of the commutative property and is recognizing that he can switch the numbers in order to count on from the larger number. He shows this understanding by stating, “Makes it faster to count on” and by reversing the rods to show his understanding. He doesn’t say it’s faster because there are less numbers to count up from 7 than there would be from counting up from 5 but he is clearly understanding that it is more efficient. Mrs. K is now having him practice using counting on from the larger number independently with similar story problems and with math games. She will  then add him to the group that is working on hierarchical inclusion (which means there are smaller number inside of numbers that increase by 1 or  the easier way to say it “decomposing numbers”) in order to help him start using strategies that are in the working with numbers part of the continuum, for example: Using 5 and 10 Anchor, Up and Over Ten, Doubles Near Doubles. 

While Mrs. K works with her guided groups the other students are playing math games where they are grouped based on what strategy they are using. This helps them get more efficient using that strategy and helps move them further along the continuum to proficiency. There are many math games out there to use for these strategies but some excellent ones are provided in the teacher kit chapter of the What to Look For resource. Mrs. K also has them work independently answering story problems that use the different problem types (joining, separating, part-part and compare) as research shows this is one of the most effective ways to help them develop more efficient strategies.

This blog is an example of how you can get to know your students well in the area of operations and then be responsive to their needs. The example holds true for all concepts or cluster of concepts. Diagnose, look at your data and then plan instruction with quality math tasks based on your students needs using guided math, shared math, independent math.

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Celebrating Students Shifting Mindsets

As this school year kicks of I wanted to share a success that one of my teachers has already had early in this year. We always wonder what will stick with students over the summer when their minds are often on anything but school! When Damien Cooper presented to our board last year he shared with us that over the summer students lose up to 80% of the content they had learned in that school year. They only retain around 20%, obviously this varies with each student but on average this is true. I was blown away by that statistic and it made me think even deeper about what we teach and how we teach it. What are the enduring understandings that we want the students to take away.

This leads back to my success story that I want to share which isn’t directly related to content but has huge effect on math learning. I am a huge fan of Jo Boaler’s mindset messages and the way she advocates to teach math. I have spent a lot of time in different classrooms sharing her message and having teachers work on mindsets in math. She believes everyone can learn math, that we have to challenge our students while creating environments in our classrooms where students are safe to make mistakes. A place where math discourse is evident, where students build on each others ideas and argue and defend their math solutions. Mr Poropat is one teacher who has really worked hard on creating this type of environment in his classroom.

This year he has started off the year doing Jo Boaler’s “Week of Inspirational Math” from her youcubed.org site. It is a whole week of math tasks that are set up to be high engagement, low floor high ceiling tasks that help build these wonderful math learning environments. Each day starts of with a mindset video to help improve students mindsets and ends with high engagement task. The first day also has part where students create two anchor charts, one for how they like their math environment to be, the other for what they don’t want their math environment to be. Here is the connection to what sticks with students over the summer or becomes a permanent part of the way they learn. Mr P has the pleasure of having his grade 6 students from last year again in his grade seven class this year. Check out the anchor charts they made, these all came from his students!

Mr. P was super stoked when they completed this activity, all the work we have been doing to help build a math learning environment is clearly paying off! The message stuck, they had not done any work with this all summer and his students returned feeling this way about math class. We knew that this group last year had really bought in and understood how having better mindset can improve their math experience. It is still just amazing to see that it has stuck! Let the math learning continue to explode. Mr. Poropat is not the only teacher I work with where this is beginning to happen. It is the first class where we have got to see the same kids the next year knowing they had been exposed to this type of learning environment. If you haven’t seen any of Jo Boaler’s work I recomend you read her “Mathematical Mindsets” book and visit her youcubed.org website. 

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