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:

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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:

MultiplicativeThinkingCommonTask2

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”

Bingo!

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:

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

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

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

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

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

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

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Purposeful Number Talks!

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

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

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

 

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

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

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

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

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

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

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

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

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

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

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

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

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

Here are two videos of the students answering these prompts:

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

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

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

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

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

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Distinguishing the Whole when working with Fractions

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

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

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

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

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

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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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