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: I see, I think, I wonder...

One of my teaching partners kindly allowed me to post the learning story she wrote for our Car Caravan math task. We had some very interesting results!

Car Caravan: I see, I think, I wonder...

A small group including our student was gathered at a tabletop to show the picture. It was first thing in the morning which could explain some reluctance in sharing or talking. The other child selected was also a timid child.

The intention was to present the problem as 'I see...I think....I wonder'

When asked what do you see? Responses included:

'wood'

'It's oval' 'there is a little oval'

'Colours' 'red, orange,yellow,green, blue'

'It looks like a swimming pool' ‘It’s a portal’

'Cars'

'I see a piece of the floor'' it's a table'

'I see pink cars'

'I see circles'

It was a challenge to get the children ( all 4 year olds) to comment more on what they see. A prompt was given to further the observation and to clarify what they were observing.

'How many circles do you see?'

This prompt was followed by the child counting by pointing to the colours in the design.

When asked, 'what are you wondering about when you look at this picture?'. There was no response. After a while, a child asked, 'who made it?'

The photo was presented to the class with the same procedure:

'I see that it is on the floor'

'It's a circle'

'I see colours cars, a little bit of yellow, orange...'

'A dirty colourfully carpet on the floor'

' a really big circle'

' a very big circle with cars on it'

' it is an oval'

One child commented:

'One row red, one row orange, one row yellow, one row green, one row blue, one row shiny, one row red...'

What are you wondering? 'How many cars in a row?'

The next day, the question of “how can we find out how many cars are in a row?” resulted in a blank look and “I don’t know”  from the child.

Manipulatives were set out on a table with the picture made visible to see what the children will do.  A child (just turned 6 years old) duplicated the shape with similar coloured blocks.  Our target child (just turned 5 year old) used pencil crayons and duplicated the colour pattern in a rainbow shape.  Another child (just turned 5 years old) started the colour pattern with popsicle sticks in a square shape (using sticks) but gave up and messed up the sticks, however, she named the colours seen in the picture and collected the coloured sticks and put them into a pile in front of herself.  She then started again to replicate the colours, but this time in a inverted ‘V’ pattern.

The children noticed the colours of the picture foremost.  As an educator, support and extension could have come by asking the question “which colour do you see the most of?” then “how can we figure that out?” Leaving the question open, left the children just observing and guessing what the picture of cars was representing (swimming pool, carpet, portal). These children needed more support in addressing the mathematical thinking this picture was trying to provoke.

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