Collaborative Math Inquiry: Algebra & Geometry Task

During our last Collaborative Math Inquiry workshop, we began to discuss patterns to algebraic reasoning and uncover some struggles and misconceptions.

Why do students struggle with patterning/algebraic reasoning?

• Junior students struggle with the concept of the equals sign and see it as one-dimensional versus seeing the whole picture.
• Kindergarten students struggle with repeating a pattern because they are only focused on some parts of it but again, don’t see it as a whole set.
• Primary students had trouble with growing and shrinking patterns.
• Difficulty with the language/vocabulary to describe the pattern.
• Intermediate students are struggling at solving word problems involving algebra. They had trouble getting started. They were able to see patterns and solve algebraic equations.

Developing Possible Rich Tasks to assess students’ algebraic abilities

Our focus for these tasks are to uncover where the students' misconceptions are and how we will respond to them. We were given a list suggested tasks from Kindergarten to grade 8 from the resource Good Questions: Great Ways to Differentiate Mathematics Instruction by Marian Small. With our team, we chose the following Algebra and Geometry tasks:

Kindergarten Task

• Provide a pan balance and linking (snap cubes.Ask students to use the balance to model the equation 10 = 6 + 4. Then ask them to move cubes to demonstrate a different way to show 10.

Grade 1/2 Task

• Provide a selection of equations with missing values, such as the ones below. Ask students to explain how they know the missing value is less than 5 each time.

7 + ___ = 10

8 = 4 + ___

9 - 6 = ___

10 - __ = 8

Today we decided we use our PLC time to pull our target students to do our task in the hallway. One educator will guide the students through the task and the others will use video and notes to document.

Questions we asked while planning for the tasks:

• What are the best materials to provide to do this task?
• What questions we will use to guide their learning?  And in what order?

Kindergarten Task: 

The educator will provide a balance scale and three containers with 10 loose snap cubes in each (red, green, and orange) and ask: Show me how you can make 10 = 6 + 4? Can you show another way of making 10?  If necessary, educator will prompt by providing the written equation a worksheet with the visual  ◽+◽=◽.

Our Findings:

• Kindergarten students needed to have a visual of the equation in order to get started.
• They eagerly used the manipulatives provided to work on the problem.
• They both demonstrated persistence in the given task and were engaged likely due to the manipulatives.
• Both students needed lots of direction to understand the task

What helped the students move forward in their thinking?

• The educator wrote out the equation or encouraged the student to write out the equation to help remember the numbers.
• When the kindergarten student A only put in numbers 10 and 4, the two end numbers of the equation, the educator pointed to the missing number.
• Stacking the cubes into towers for each side of the equation helped them compare the amounts.
• Educators rephrased the questions and broke them into smaller steps.
• For example: Can you may 10 using a different number of the orange cubes?

What do we need to rethink?

• Balance scale was distracting and not perfectly accurate. Perhaps, it should be introduced later on in a different way.
• Students need more exposure to the vocabulary of the equation and the name and function of the symbols : add/plus, equals.
• Educators must be more conscientious in using math terminology consistently when working with students.

Grade 2 Task: 

Students were given the equations 4 + ___ = 9 and 9= __ + 4 and asked to find the missing numbers.

Our Findings:

• Student read the question backwards when it was displayed as follows: 9=5+4
• The student said it was wrong.
• In the end the student made up a new way to show 9 as: 7+2=9 and then wrote 9=7+2 but physically wrote it from right to left.
• The students used their fingers to help count versus the cubes.

What helped the students move forward in their thinking?

• Rephrasing the questions: Can you write the equation in a different way?

What do we need to rethink?

• Students need more exposure to the equations shown starting with the sum.

Car Caravan: How many minutes will it take to create this?

After watching a video of the students’ discussion from Day 1, we decided to use our target student’s one mathematical wondering below and present it as the challenge of the day.

The challenge: How many minutes will it take to create this?

We presented this challenge at the end of community time and invited those interested to work together to solve it. I brought the students to the drama area since it had the most room. We cleared out the furniture and sat down to get to work. Here is a documentation of our conversation below.

Teacher: What do we need to create this?

Student 1 : Marbles

Student 2: Little pieces of red/orange

Teacher: Of what exactly?
Student 2: Paint

Student 1: Wood

Teacher: We need to use what we have in class so have a look around or if you’d like we can look in our storage room as well.

Students were very excited.

Student 2 ran to get a picture of pattern blocks made into the alphabet.

Teacher: Oh wow that`s interesting.

Student 1: It`s too big

Student 3 went to get more pictures of the pattern blocks and talked about making a rocket.

Teacher: Do we need the pictures of the blocks or the blocks?

I noted they were getting side-tracked

Teacher: How do we know the pattern blocks are good materials?

Student 1: No, we don’t have pink.

Student 2 grabbed the chips of red and yellow.

Student 1: It`s not the same colour.

Student 3: Grabbed a puzzle.

I noted they were very unfocused and needed guidance to choose a material and move forward.

Student 1: It`s too big (shouting).

Student 3: Same size

Student 2 got the pencil crayons.

Student 3: Yes!

Student 1: No, it`s too big.

Teacher: Let`s compare!

Student 2 got more pencils and started to build.

Student 3: Let`s get started.

Teacher: Prompted to clean up what wasn't being used and help gather more pencils.

Student 1: Pencils are not good because we don’t have a lot of yellow.

Teacher: Where else can we get pencils?

Prompted them to make a list of what they needed using the colour words in class to copy the name in French. They only listed yellow and we went to ask two other classroom teachers to borrow their yellow pencil crayons. I encouraged them to count how many they took and write it down so we could make sure to return the right amount. During this time one student lost interest but two others joined to help start to create the image.

We took a snack break and then it was almost clean up time so I decided to take them in the hallway so we weren't distracted and I could take a video. I then took a video of them building it and gave my phone to one student to keep track of the minutes. In the end, I said all students needed to give a thumbs up to show they agreed it was complete. One student was not happy so I prompted him to explain why and for them to fix the issue before stopping the time. He claimed they needed purple.

In the end I encouraged one student to write down the number. I noticed he had trouble writing it down because of the colon and decimals. So I asked all four of them to try to copy down the number.

Next step: Present our video to the class and the next challenge will be to debate the answer to our challenge question after looking at their written results of the time it took to create the image.

Collaborative Math Inquiry: Car Caravan Math Task Results

After completing the Car Caravan task with our students, our primary team gathered to discuss the results. Click on the titles below for the detailed Kindergarten learning stories.

Kindergarten Class A: I see, I think, I wonder...

Kindergarten Class C: How many minutes will it take to create  this?

What similarities did we encounter when presenting this problem?

• The problem was very open-ended and we had trouble getting students to developing mathematical wonderings.
• Students both in Kindergarten and Grade 1/2 needed lots of prompting to develop mathematical wonderings.
• Most primary students noticed the colours first, the floor and some talked about the shape.
• The main observations were about the colours and shapes.
• Regardless of the Grade, students in both Gr. 1/2 and in Kindergarten came up with wonderings that were mathematical at all.

• When the Car Caravan was presented in a small group to target Kindergarten students, educators found that they were distracted easily.
• Some students were overwhelmed by their own questions.
• Students observations demonstrated their fixed mindset Ex. Insisting that their observation was correct and was not open to others’ ideas.
• We discovered students’ comfort level in using counting strategies.

What strategies worked?

• It was best to take the small group out of the classroom to help students focus when presenting the Car Caravan photo and to be able to discuss their wonderings.
• Students needed prompting to get them thinking more mathematically:
• What mathematical question do you think I’m going to ask you?
• You see circles? How many circles do you see?

Teachers Successes:

• Presented the challenge so that students had a voice
• Gave students the liberty to discuss their wonderings without any time constraints
• Clarified their answers by repeating their answers so all could hear and think about it.
• Gave everyone the opportunity to join at any point (kindergarten)
• Praised them for the time and effort put into their work or for any participation or idea suggested whether great or small.
• Allowed students to choose any materials to show their thinking
• Set up the materials as a tabletop activity with the intention of allowing them to build the car caravan in their own way

Next Steps:

• Grade 2s responses seemed to list various possible counting strategies but didn’t show any reasoning for choosing those strategies. We decided that they needed to continue to extend their thinking by doing a third lesson and asking:
• Which counting strategy is the most efficient? Why?
• What other strategies /prompting/questions can we use to help students develop their mathematical wonderings?
• Educators will continue to work on using more language to help pull out the proportional reasoning

Collaborative Math Inquiry: Car Caravan Findings K-8

We gathered with teachers from our school board from Kindergarten to Grade 8 to discuss the results of the Car Caravan Math Task. Here is an account of our findings.

What struck you?

• Students seemed very frustrated they couldn't just apply one formula
• students not used to open ended questions and seemed overwhelmed
• Need to narrow down the wondering in order to be able to tackle it and move forward
• Teachers also at times frustrated that students were not thinking deeply and not producing questions or taking the problem very far.
• Some teachers were surprised that some ESL students were very engaged and using math language to talk about the colours and the proportional language used : ex. more than
• The task was so open it pushed us to think about what we were documenting and pushed us to think about how we question students and focus on presenting more open-ended questions.
• Teachers found students were debating a lot and had trouble working in groups and collaborating.

What are the big ideas we discovered by presenting this problem?

• Big idea: Getting students to justify their thinking. Often the questions used were:
• How do you know?
• Can you prove it to me?
• Are you sure?
• Students ended up first trying to use strategies and concepts that were taught in the previous couple weeks.
• Learning context has a huge impact on the students. Some important questions to ponder prior to planning to help focus on educators’ intentions:
• Are we letting them struggle with a problem or giving them the strategies too quickly?
• What kind of student groupings will be best?
• Are we providing materials that are consistent?
• Ex. on the computer it may look different than when it is printed.
• How does the environment affect their wonderings?
• Ex. The lighting

Kindergarten Small Group Discussion:

• Difficulty getting out the math until educator seemed to find the right questions
• Ex. It`s race track! If it`s a race track , which one is winning?
• Some KG students talked about cars, colours, it`s a rug.
• Lots of talk about what the picture represented.
• Some noticed the background first.
• They were trying to figure out what it was versus wondering
• Focus on colours: ex. 2 shades of blue
• Some discussion about shape
• Teacher asked : Are there a lot of car? - provoked their thinking about what is a big number. One student remarked that they would need to “count fast” - and he started to count by 10s.
• Asking questions was challenging for the teacher because the students went on many tangents in their discussion and it was hard to bring them back and focused.

Kindergarten discussions after viewing Students' work from K-8:

-

• You could see all the strands covered across all the grades
• The problem could be presented at the beginning of the year and brought back again to ask other questions and reach other strategies
• EQAO years – affect their answers and wonderings
• We heard other teachers commenting on kindergarten inquiries saying that in Kindergarten it seems to be more of an exploration
• Noticed how kindergarten students were drawn to colours but that math can be integrated later

What would you do differently?

• Some of us wanted to rethink the questions we were using to figure out how to draw out more math
• Need to MODEL more mathematical wonderings

Number Talks, Take 5!

"Do not give up when things don't go perfectly well; make adjustments and keep moving forward" (p. 31)

And that is just what we plan on doing. We did, however take another break and dropped our Number Talks very readily this past month when our new strategies didn't produce the results we had expected. With report cards due, we all felt pretty justified in focusing our attention elsewhere. But now that they are all done, I finally had the energy to read Chapter 2 of Number Talks to find some fresh ideas. This last tip was at the very end; and yet it is the most important one. Staying positive and persistent when faced with challenges, are necessary qualities if we want to accomplish our goals. And it just so happens that promoting a positive mindset and/or GRIT is a board-wide endeavor this year. Keeping a positive mind set is the only way the students will succeed as well as us teachers.

So last week we restarted our Number Talks with some adjustments. Firstly, we decided it was best to continue to do Number Talks in small groups. This time, we did them first thing in the morning during Tabletop Activity Time when students are most alert. To avoid distractions of other activities at the tables, I brought them into a quiet corner of the room on the carpet and I sat facing outward so I had a view of the rest of the room, while they sat facing me. As Sherry Parrish suggests: "I have found it that it is easier to build a cohesive community and a focused discussion when students shift away from their regular routines and are removed from typical desk distractions" (p. 17). This shift in location, made a huge difference in terms of their attention and engagement. And within one week, students began to await their turn to join in a Number Talk. This past Monday morning I saw one student walk in and wander, so I asked if she wanted to join me for a special activity in the corner. She replied: "I've seen you doing the Number Talks in the corner." I asked if she was excited that it was her turn and she just smiled from ear to ear. Parrish notes: "One way to elevate the status of number talk time and signal its importance is to have a specific location in the classroom where the students gather for this purpose" (p.17). 

Click Here for a blog post on Small Group Number Talks in the Junior Grades

First, I reminded each group of the routine for our Number Talks. First, I would show a card of dots very briefly and then ask them "How many dots do you see?" and "How do you see them?". Then they would place a thumbs up on their chests if they were ready with an answer. I reminded them that it is not important to figure out an answer quickly, but rather that we give everyone a chance to take their time to think and come up with an answer. Since I began teaching, the importance of " wait time" has always been stressed; however this strategy of including a gesture is neat because it makes the wait time obvious to the students themselves. By making "wait time" known to students, they may better understand that speed is not the most important part of success in math. As Sherry Parrish explains: "Many students have the misconception that they are not good at math because they are not fast. We can easily reinforce this misconception if we begin gathering answers as soon as the first students are ready. " (p. 18). Using the procedure of giving a thumbs up is important to "send the message that all students are expected to think and contribute during this time" (Parrish p. 18). 

Set students up for success by grouping them
 by ability and providing tasks at their level.

Next, I explained I would invite those interested in sharing their answer to raise their hands. Keeping the groups small, to a maximum of three students, meant that students were not only more engaged but more comfortable sharing their thoughts. I also made sure to group the students based on similar abilities. Because I kept the groups small (max. 3 children) and carefully selected students with similar needs, all students were eager to share and even the shy ones participated, though they still spoke softly. Parrish also notes that by grouping the students based on their needs, it allows for those that are ready to be challenged (p. 25). I am not yet sure what kind of challenge I will present to those that are stronger but for now I will record their answers and then take the time to analyze them and plan for our second set of small group Number Talks. 

One of Parrish's 5 steps towards teaching for understanding (p. 28) suggests limiting number talk time to a maximum of 5-15 minutes in order to maximize engagement. Back in January, we had decided to review all the cards for number 4, so the Number Talk definitely went on for longer than 5 minutes. Even though it was in a small group,I found even I, myself, was quite bored since it was always the same answer. Although I was recording the answers and analyzing them throughout the Number Talk, this documentation took up way too much time and thus, the students were distracted. So finally I told them, this time our talk would be short so no one would get bored and they could go back to a table activity of their choice after only 5 minutes. Before we began, I set a timer so that we would not get carried away. We were able to get through just 3 cards (one set) with everyone sharing each time. The timer has been ringing just as we are discussing the final card. Limiting the talk to 5 minutes in Kindergarten is a rule I would definitely recommend. 

A few other tips I liked from Chapter 2 (p. 30):

• Offer a strategy from a previous student 
• If no one offers a strategy, give an example and ask if it will always work (checking for reliability).
• It is all right to put a student's strategy on the back burner.
• If you don't understand a student's strategy and they've already restated it, you can have students turn and talk about what they understand and brainstorm questions to pose for clarification.
• During this time, listen to students and get a sense if everyone is confused or you can build upon this idea.
• If not, tell the student you need time to think about it before continuing.
• This helps keep the conversation pace versus slowing it down and losing students' focus.

I especially like this last phrase because it reassures the student his/her ideas are still valuable and as Parrish says, you can meet later with the student to work through their strategy.

And finally, Parrish's last tip (p.31): 

Possible Next Steps:

• Use one half PLC day to meet with my colleague from my second class to explain Number Talks and plan to implement them.
• Read Ch. 3 of Number Talks for more specific strategies in the K-2 classroom to be shared with colleagues.
• Use my final half PLC day to meet with my first Kindergarten team and discuss possible next steps below.
• Devise new plans for recording students' thinking so that it can be easily shared with classmates. (Right now we are simply making our own handwritten anecdotal notes to share among the teaching team.)
• Discuss possible activities/questions Parrish suggested to add to develop more accountability in our Number Talks. (p. 25)
• Ask students to use finger signals to indicate the most efficient strategy.
• Require students to solve an exit problem using the discussed strategies. 

Collaborative Math Inquiry Lesson Plans: Car Caravan Math Task

As part of the Collaborative Inquiry in Learning Mathematics (Funded by EOSDN), we have been given some PLC days in order to plan for our math tasks. Today, our primary team met to discuss and plan for the Car Caravan task. This task has been given to all teachers in our board from Kindergarten to Grade 8. We will be gathering documentation on target students and coming together to analyze our findings with the goal of promoting a growth mindset, more positive attitude towards Math in general and encouraging the development of proportional reasoning.

Here are our plans for the Car Caravan Task:

Day 1 

1) Introduce the Car Caravan problem by showing a small group (including the target students) the following photo and asking:

• What do you see?
• What are you wondering?

Document their wonderings with video and/or anecdotal notes. Videos can be shown to the whole group for discussion.

(Gr. 2 students willl all be put into small groups to work on their own.)

Our intention: to allow target students to develop a wondering. By presenting in a small group, the target students may have more confidence to speak and  share. This will allow them to have a voice and develop a more positive mindset versus potentially withdrawing  if the problem was presented in a whole group and feeling intimidated.

2) Present the problem to the whole class and present the video of the small group’s wonderings and discussion. Ask the whole group if they have any other wonderings.

3) Analyze documentation and list of wonderings. Choose a wondering to focus on for the following day, based on target students’ interests. Gather potential materials to assist in solving the problem tomorrow.

Day 2

4) Challenge of the Day: Present the chosen wondering of the day at the end of community time as an optional centre time activity, guided by the educator.

**If the target students do not willingly join the challenge, (perhaps they are hungry and want to go to the snack table right away), the educators will invite them to join later. If they are resistant, then give them time to do another activity and invite them back later. Document their response and engagement.

5) Provide the small group with the photograph and the specific wondering (written on a paper). Ask them what materials they need to solve the problem. And guide them to discuss and attempt strategies and work on a solution.

6) Consolidation: During sharing time, we will have small group of students present their strategies either orally or through a video/pictures.

Pedagogical Documentation Look-Fors:

• Describe what you see and hear.
• Do they have a curiosity about the task?
• Look at their understanding and ability to explain their thinking.
• Look at their persistence.
• What other factors contributed to the students’ engagement : Did anything else happen that morning? or the night before?

• What materials did the students suggest.
• What language are you (the educator) using to pull out the information?

Next week we will meet for another PLC day as a primary team and use the following success criteria to analyze and evaluate both educators and students learning. We are very excited to put our plans into action and discover all the learning that will take place!

Kindergarten Success Criteria

Success Criteria Teachers	Success Criteria for Students
Using descriptive feedback more than just saying wow and passing it along	Having them explain their thoughts and they can have a voice – they can be the teachers too
Planning open ended activities for all entry levels	Demonstrates interests and curiosity by taking risks and attempting to solve problems in play areas
Praise the students based on their abilities rather than their output	Shows willingness to collaborate with others when solving problems.
Allow the students to use their own strengths to communicate their thinking	Displays persistence when facing challenging problems and when experiencing failure
Using intentionality in placing the manipulatives and allowing the students to be creative in their explanations.	Seeks opportunities to extend their mathematical learning and make connections to real life situations
Using your language to help pull out the proportionally reasoning language (You can go and wash your hands if you are wearing more red on your shirt than any other colour)	Allow the students to use their own strengths