Using a resource for a different concept

Demitra's school has a copy of the resource "Puddle Questions:  Assessing Mathematical Thinking."  In it there is a question about how you would measure a puddle.  Demitra decided to modify the question to meet her needs for assessing what the students know about area.  She also wanted them to self-reflect on their work and she wanted to be able to give them some descriptive feedback on what they could do better.  Sounds like a tall order, but when it fit together nicely and the students really got a lot out of the lesson.

Minds-On:  Each group was given a large triangle made out of construction paper.  They were asked how they could find out the area of it.  They then set to work discussing and trying different ways.  What was great was that each group came up with a different way, and they all knew that the units that they used must all be the same.

Action:  After being given the task, Demitra reviewed the success criteria that had been co-created together.  The students gave examples of what the criteria might look like in terms of the task.  For the actual task, the students were asked to draw a puddle and find the area of it.  They could use any way they wanted to.  Some students drew lines, others used cubes, pentominoes, finger spaces, and some even drew shapes that they thought would be "easy" to count.

Consolidation:  Demitra had one-on-one conferences with each of the students to not only assign them a mark (based on the four categories in the achievement chart) but also to give them some feedback on what they had done well on, and what they could improve upon

When this student brought his work up, he came up ready to chat about why he thought he needed to make a smaller puddle.  He said it was because a smaller puddle would have a smaller area and that would be faster to count than the puddle he had drawn.  He also said that he would have used a bigger unit to measure the area with and this would also cut down on the number of units he had to use.  This is why his feedback was to think about how he could have shared his oral knowledge in a written way.

This student used a K-W-C chart to help them make sense of what they had done.  You can see there was a bit of a thinking error in their work as they used pentominoes to answer the question, but when they couldn't successfully draw them onto their puddle, they jumped to just drawing lines on the shape they had traced.  When probed, this student was unable to share why it was important that all tools they used to find the area needed to be the same size.

This lesson left Demitra feeling pretty confident in her student's abilities to find the area of a shape.  The one-on-one conferences also helped her ask some deeper questions to help determine the students knowledge and understanding of area and how well they were able to apply that knowledge and understanding to the problem at hand.

Critically Looking At Questions and Tests

The Grade 7 FOS PLC's used their time together today to look at the resources that we use, and then evaluate them on how effective they are.  We know that in our schools we have textbooks, ministry guides and supplemental activities, but there is not one "just right" resource.  So what do we do?

We started our session by looking at an activity sheet from one of the textbooks that are in our schools.

 Each teacher then looked at the questions (individually) and then we shared how we would mark each question.  When sharing, some people shared why they gave a question a specific mark (e.g., There were 4 things, so I gave 4 marks).

After doing this, we then used the achievement chart to decide what type of question each of the printed questions were.  It became very obvious to us that we had a lot of KU type questions and not a lot of communication type questions.  When we converted this into percent, it turned out that at least 40 - 50% of our questions were just KU.  We had some really good discussions which included things like:  If the question did.....or In my class they would..... - But, we were only taking things for face value - If it only shows Knowledge, then it only shows Knowledge.

After noticing what types of questions were on the sheet, we then turned to our curriculum to then have us see what expectations we were covering.  The questions that were not covered by the expectation were then crossed off. 

What we also discovered was that one of the 4 expectations found under Location and Movement in the GSS strand was not covered at all in this activity. 

The teachers then broke off into groups (1 group in the AM, 2 groups in the PM) and then picked one of the questions that was in the curriculum and then "opened it" up more.  This allowed us to have a really good question that focused on more than just one type of question - it was a question that could be used to help assess all four quadrants in the curriculum. 

The four categories at the bottom are what the teachers would be looking for when they mark each question.

Orange - Minds On,  Blue - Action, Green - Home



After the Diagnostic, Starting The Decimal Unit

Kristen began her teaching of decimals with her Grade 5/6 class by using a problem that is based on one from Making Math Meaningful by Dr. Marian Small.  (MMM, p.160, Activity 8.1).  In MMM the problem is about estimating using whole numbers, and in this case Kristen changed the problem to involve decimal numbers to hundredths and thousandths. 

The lesson was set up in Three-Part Lesson format.  Here is the lesson template:

The Minds On focused on having the students order numbers.  She chose to do this to get the students thinking about how decimals can use the same numbers, but mean different things.  We had thought that perhaps some of the students may choose to use ordering as a strategy for solving the problem, but it was not one that we saw.

The action was this problem:  (You can see the minds on at the bottom of the page...It is an Eco-school after all!)

The students also completed a K-W-C as a class to help them flush out the information that was in the problem.

What was great about doing this was that the students got a chance to ask questions that were important to them.  There was some great discussion as to if the animals could only go in 2s or if they could go in groups of 3 or 4 (the answer was yes - but they could NOT go alone).  As well, one student touched upon the fact that the question was asking you to not only solve the problem, but EXPLAIN how you got your answer.  The second step that many students (and teachers) forget to do when solving a problem.

The students quickly got to work in pre-assigned pairs (based on optimal mis-match).  They used markers (no pencil's allowed) and didn't use our "banned" black and yellow (Yellow because you can't see it that well when your work is presented, and no black because that becomes our class editing pen). 

When the students were done their markers were collected.  Kristen really wanted to use the consolidation part of her lesson to help the students make sure that they were giving a really good / Level 3 answer.  So she asked the students the simple question of "What makes a really good answer?"  Using examples from their own work, the students then shared what they thought made a really good answer and why.  These thoughts were recorded on a large piece of chart paper:

The students now have an anchor that they co-created and that makes sense to them.  From it, informal success criteria was created about what makes a good answer.  A follow up lesson to this will be actually putting that success criteria on paper.

The students now were asked to add something that they thought was missing from their work.  They did this in pencil so Kristen could see what the improvement was.  (Hence why the markers were collected earlier on).  This was their independent practice.  They handed in their "updated" work and Kristen went to work providing descriptive feedback to them. 

Since we co-taught this lesson, Kristen was able to record what the students were saying about what made a good lesson as I was putting it down on the chart paper.  The student's focused on 5 different things.  To help her keep track of what the descriptive feedback she was giving to the students she numbered each item in the success criteria from 1-5.  As she gave each pair the descriptive feedback, she then recorded what she had given them feedback on, and what else she noticed that they need to work on.  Here is an example of how she recorded this information:

The first number is what she commented on, and the second or third numbers are what they still need more support in.  Its helpful to keep track of what you are saying to the student because if you notice that you are always writing the same thing, then they are not using the descriptive feedback that you are giving them and putting it into practice.

Here are the student samples showing not only their original answer, but their pencil additions and our black "editing" that matches the descriptive feedback that was given:

Start to finish, this entire lesson (Minds on, Action, Consolidation, Independent Work) took about 65 minutes.  All of the students were engaged, and all were participating in our class discussions. 

Why Minds on Matters

Part of Sarika's Annual Learning Plan this year is to become more familiar with, and use more regularly, the three part lesson in her math class; and to become more comfortable with (and consistent with) teaching through problem solving.  Her Grade 5's are currently learning how to make a table of values and how to recognize patterns.

She has modified problems from her text to use as the "action" part of her lessons for a week.  She was able to break down the problem into multiple parts so that the problem is not only extended, but the students can use information they already know from a previous problem and apply it to the problem they are working on.

In this lesson her minds on activity was having the students identify the pattern and extend it on a table of values.  The first table was a simple growing pattern that increased by 5 each time.  The class talked about what the pattern rule would look like in words and recorded that beside the table.

For the second table of values the students were only given two terms then they were asked to talk to their partner about what thought the next term value would be.

When the students were done they gave us a "thumbs up" sign, and when the majority of the class was showing "thumbs up" the students shared what they thought the next term value would be.  One student said it would be "93" and another justified that by saying that "the pattern rule would be start at 31 and add 31 each time".  This was recorded on the board.  (We used initials and a chart to record their answers and thinking). The students were asked if this could be the only possible number.  They were then shown 124 and asked to discuss if they thought it was possible or not to be used as the next term number.

At first there were a few puzzled looks.  The majority of the class could not see that multiplication was also a way to extend patterns.  A few then began to think of "doubling" and then one student put up his hand and said that it could be the next term because if we "multiplied 31 x 2 we would get 62...62X2 we would get 124..We could then do 124x2 and get 248 for the next term."  The light bulbs went on.

Why this Minds on was important for Sarika is that it gave her a lot of information about her class.  She knows that they are able to identify and express in words what a pattern rule is, they can extend a numeric pattern, and are working on making predictions about growing patterns.  She knows that for her next few lessons, she will have to include problems and activities that involve using multiplication and addition in the patterns, and encourage the students to use these tools to help them solve problems.

Cross Strand Problem Solving

In thinking ahead to Valentines Day, Melissa had her class complete a problem on making valentines.  Her lesson was in a three - part lesson format and was the following:
Minds on - Reading a book to them.  Action - was the following problem: Sierra made valentine cards for her friends.  It took her 5 minutes to make one valentine card.  How long did it take her to make 6 cards? (Adapted from The Math Process Standards Series K-2 by Susan O'Connell) Consolidation - To share answers and develop success criteria from it.

Although telling time to the nearest 5 minutes is not in the Grade 1 or 2 curriculum, Melissa has spent a lot of time getting her students to work on composing and decomposing numbers in different ways.  Her goal for the lesson was to have the students apply a known strategy to a problem in a different strand.  As the students were working, Melissa went around and used the Assessment for Learning tool during her discussions with the students.  This allowed her the opportunity to instantly record their strategy, answer and their errors. 

 Once the students were done, she brought them together to create a list of what made their answer good.  This will now be their success criteria for upcoming problems (photos of that to come later).  Once the students discussed what made their answer good, they then had the opportunity to go back and "edit" their work to put more detail into their answers.  You can see this in the pictures by the different colour that they used.

This pair of  students originally had the incorrect answer.  They then went back and checked their work and noticed the error.

This pair drew too many tally marks originally, they were able to see their error, but simply didn't go back to change the written portion of their answer.

I have made a video of this lesson as it really shows not only what a good three part lesson looks like, but also how you can create success criteria with your class.  If you would like to see the video, or want help in creating success criteria with your class, please let me know.

Looking Critically At A Rich Assessment Task

Sometimes we can come up with a really good question or problem to give our students, but when we go to mark it, we find out that it didn’t turn out like we expected. When we give our students a rich task to work on we need to keep our own success criteria in mind when we create the task. When we start to create the task we have many good ideas, and re-word and re-write the question over and over again. When we are done we need to think about the achievement chart, and if our question A) Does allow the students to demonstrate work at a Level 4 and B) Allows us to have the students perform in all four areas of the achievement chart: Knowledge and Understanding, Application, Thinking/Problem Solving and Communication. Some teachers are Wexford are doing just that.


At Wexford, the school is continuing on its second math pathway of the year. This time they have chosen to work in grade teams instead of by division. What’s nice about this approach is that it lets the team focus on the strand, or issue, that they really want to. This week, two of the grade teams put together a pre-assessment task to administer to their students. The Grade Six team is looking at Data Management and the Grade 7/8 team is looking at Measurement.

Once each team had created their question, they then stepped back and took a more critical look at their question. They went through each area on the achievement chart and then predicted what their students would demonstrate for each area. They then created their own success criteria of what they expected to see in each of the categories.

For the Grade Six team, they also wrote out a list of possible questions that they could ask the students about the problem. They then went back and also looked at what areas on the achievement chart they question covered (e.g., was it really a higher thinking question, or was it a more basic knowledge and understanding type question). They also discussed what responses might look like at all four level s of achievement.

                                                                     Grade Six Sample
For the Grade 7/8 team, they too went through each category on the achievement chart, but then found that when they got to communication, the question didn’t allow their students to communicate at a Level 4. So with some quick tweaking, they were then able to fix the question to allow their students to communicate at all four levels.

More information on Success Criteria can be found in the Growing Success document. It’s a great activity to co-teach with either myself or the literacy coach as not only will you provide the students with information on how to improve, but it will help you to clarify what you are looking for in their work.



Organizing Your Mark Book

One of the great things about the Growing Success document is that is gives us a strong picture of the types of assessment we need to be doing on our students:  Assessing for Learning, Assessing of Learning and Assessing As Learning.  Assessment can be recorded in many different ways:  anecdotal notes, descriptive feedback and as marks in our mark book.

The Junior Division at Manhattan Park looked at different forms of assessment during their PLC for math.  What really "clicked" for them was an idea that they saw in the Damian Cooper book "Talk About Assessment" which all of your schools have a copy of.  In thinking of the types of questions they were asking, they also started thinking about how they were assessing the questions.  They saw a sample of a mark book where the students were marked in each of the areas of the achievement chart and decided to try it for the unit they were working on.

What Doris and Diana said they liked about this chart (which they modified to meet their needs) was that they could look and see what types of questions they needed to assess to make sure they were getting the true picture of their student's progress.  As well, the area at the bottom really allowed them the opportunity to record their observations of each student.  This allowed them to have a bank of comments for the report card that was not only student specific, but in parent friendly language.

You can also help parents understand why there is more to a mark in math than just answering questions like 8 x3 = 24, as you can show specific question types their child was successful in, and struggled in.

How To Organize A Unit

In Ontario, we have a specific Math curriculum that we follow.  At times, the expectations seem overwhelming, and knowing where to start can be tricky.  Backwards Mapping is one effective strategy that helps us keep track of not only where we want the students to end up, but also where we want to take our students.  Creating a unit plan is another strategy.

Maryna used the template below to plan out her unit on Patterning. 

She laid out a progression to her lessons, being sure to include tasks that were purposeful, used technology, and allowed her students to demonstrate understanding at all four levels (and categories) of the achievement chart.  The plan is not set in stone.  As she progressed through the unit, she made changes based on what her observations of the student were - Assessing for Learning, Assessing As Learning - and made sure that she helped close the "gaps" that her students may have had in their learning.

Having a plan not only allowed Maryna to focus more on the students, but it also allowed her the opportunity to make sure that she met the curriculum expectations for her grade, thus preparing them for the future.

Congress and Assessment

In Lisa's Grade 4/5 class I was invited in to co-teach a lesson with her. Her school is doing a math pathway this term and her pre-assessment task is "The Checkerboard Problem." The question asks students how many squares are on an 8 x 8 checkerboard. We posed the question to the students, handed them checkerboard templates, and then set to work making notes on our Assessment for Learning (A for L) tool.

All of the students quickly got the answer of 64 - none of them got the correct answer of over 200.  But what amazed us was how many different strategies the students were able to use in order to come up with the correct answer.  Lisa had told them she wanted them to try to solve it in at least two ways.  Many of them were able to solve it in more ways than just that.

Many of the students shared their strategies with us.  We were able to see that the students had a lot of different strategies to call upon when solving a problem.  After they described what they did, we used a different colour marker to give that strategy a name that they could remember it by.  One key strategy was missing, so at the end of our congress I shared with them how to make a table to help them find an answer. 

When our congress was over, we then let them know that although they had really good strategies, none of them got the answer correct.  We told them that there are more than 200 squares on the checkerboard.  At first they looked at us with an "Are you kidding" face - But then one student piped up that she could see some of them that she hadn't before. 

The congress allowed us a chance to see what strategies the students already had in their schema, and then the A for L tool helped us see which ones were most commonly used.  At the end of the pathway, it will then be interesting to see if the students are still using the same strategies, or if new ones have become a part of their problem solving tool kit.

Assessing For Learning and Consolidation

Nancy and I have been doing a lot of co-teaching in her Grade 5/6 class. All of our co-teaching sessions have been focused around teaching through problem solving in a three part lesson. This past week we had the students work on a problem around conversion of fractions, decimals and percents. (Picture #3)

We used the Assessing for Learning tool(A for L) as a way to document not only what strategies the students used to solve the problem, but also what errors they make. What came to our attention was a rather large error which became the topic of our consolidation or the third part of our lesson. (Pictures # 1 & 2) Nancy has also modified the A for L tool to meet her needs by making the squares a bit bigger, and also including a box for her to level the students work as well. (Not every time that we use the A for L sheet do we give the students a level - it is mainly used to simply assess for learning)

The students were quite easily able to figure out 1/4 of an amount and then find out what 50% of the remaining amount was. We saw several strategies and heard several different explanations as to how they got through steps one and two. However, where the misconception occurred was when the students got to the part of the question where they had to solve for 0.3.

The students were well aware of how to convert the decimal into a fraction - Except that they were trying to convert this particular decimal into 1/3 which then would give them the wrong answer. Many of the students said that they didn't know what to do when it came time to "deal" with the 0.30. For some that were stuck, we prompted them with open ended questions to them to help make sense of what they should do next.

During our consolidation piece we had one pair come up and share their work with the group. We had pre-selected this pair because we knew they had used the same strategy as most of their peers, and had made the same mistake. As they explained their answer I annotated what was missing from their words and their sheet by using a different coloured marker. When we came to the part about the 0.3 we had them share their answer and asked the class if this looked right. Almost everyone agreed. We then led the students into looking back on the first two questions they solved using the fraction and the percent. We put each one of those into a decimal, percent and fraction form on the board beside of us. We then did the same with 0.3, 30% and then 1/3. It was at this point that a few of the students got the "light bulb" look and put up their hands. They said that 1/3 was not equal to 0.3 but it was 0.33 instead. We then used number lines to help us show the difference between 0.3 and 0.33.

One other group - who were the only ones to correctly get the answer in the first place - put up their hands to share (and show us) how they solved for 0.3 and what it would look like in the context of the problem.

Had we not been doing this problem, this issue might never have come up. The three part lesson is a really powerful tool to help us teach our students better - It allows us the chance to take a simple, and easily corrected, mistake and fix it on the spot. Because of the consolidation piece (or congress) we were able to not only show our students different ways to show a number as a decimal, a fraction and a percent, but also stop them from making the same mistake later on.

Student Centered Assessment

Adrienne's Grade 2/3 class is starting to work towards self assessment. We were co-teaching a lesson together in a three part lesson format (M.A.T.C.H.) and things took a different turn when we were about to consolidate:

Here is how the lesson went -

Minds on - We had the students discuss things they don't do everyday, but perhaps every other day.

Action - The problem we decided to do was about Jake and his sisters and how many times in a month Jake walked the family dog. Before we sent them on their way to solve the problem, we had them complete a KWC chart to help clarify their knowledge. They wrote the items in the "K" column, but for the sake of Time, we scribed their answers for the other two columns.

Consolidation - As we were about to start the consolidation piece. Adrienne brought forth the idea of turning their work into an anchor chart to help the students assess themselves on their own work. This is what her school is focusing on for their pathway for this term (Note: the pathway is rooted in Literacy, but Adrienne wanted to try it out in Math as well).

We brought the students back to the carpet and asked someone to share the answer with us. The answer was written down verbatim. It was "He walked 8 times." We asked the students what kind of answer this was and why. At first, the students thought that it was a Level 1 answer. But then quickly decided that it was not a Level 1, but a Level 2 answer. (The bright pink paper is because of this change). They then provided justification for their levelling: It doesn't explain how or why we got the answer; the answer is a little confusing; not sure how they got the answer. We then began to work ahead and had someone share why they thought that their work was a Level 3. Once again, they went through the same criteria and came up with justification: there is an explanation that is clear; only 1 small mistake; fully answered the question; showed the work - the pattern rule.

At this point we had to stop for recess and then prep. However, later in the day Adrienne went back to the carpet with the students and had them do the same for Level 1 and Level 4 work. Once all of the Levels and justifications were up, students then got the chance to Level themselves according to the criteria that they had come up with. But - They didn't stop here.

The students then went back and, (using a coloured pencil crayon) made additions to their work to help the quality of their work improve. This was called the "editing process."

For Home, the students were then asked to complete a journal question relating to the work that was done in class today.


The picture shows the work displayed on her wall. How it is set up shows both the problem and the graphic organizer (In this case the KWC chart) and it also has the samples of the student work (with the names cut off) of where it would have fallen BEFORE "editing." If you notice, there are a lot of samples in between Level 2 and Level 3 - This is because those students are at a Level 2+ (based on their original work).

This activity was very powerful for both Adrienne and her students. She now will plan further lessons with those Level 2 students in mind and have a visual reference to help guide her in her planning.

Including the inital part of the lesson, she spent about 50 minutes in total on this lesson - Not a great amount of time, but a powerful use of time. Which goes to show, that a three part lesson can be accomplished in one day.