# Hilldale P.S.

### Mathematical Thinking Made Visible

## School Profile

As a Student Work Study Teacher, I began working at Hilldale P.S. in January of 2015 in a combined 2/3 classroom. For four months the collaborating teacher and I explored the use of manipulatives and developing a problem solving framework to assist students in communicating their thinking in mathematics. We were unable to complete our inquiry due to tension between teacher unions and the ministry of education in settling teacher contracts.

In September, the collaborating teacher and I reviewed our initial theory of action and made adjustments in response to student needs and professional learning goals. The problem-solving framework that we initially utilized supported some students however, it wasn’t meeting the needs of all learners. Collectively, we decided to add the four square model or framework in order for all students to be successful in communicating their thinking and understanding of a problem. This framework was also utilized by teachers in grades 2 to 5.

We officially began our inquiry in November as work to rule sanctions were lifted. Documentation of student thinking expanded from one classroom to three. As a SWST, I spent two days a week at Hilldale, which allowed me to build professional relationships. As a result, I was invited into additional classrooms to document student learning and support student success.

While working in conjunction with the Instructional Coach, the collaborating teachers and I engaged in professional learning about Number Talks as one approach to math instruction. As a balanced mathematics program addresses both conceptual and procedural understanding of math concepts, number talks aligned with our collaborative inquiry as it focuses on conceptual understanding and mathematical reasoning.

## Anchor Charts

## Theory of Action

**Initial Theory of Action for Grade 2/3 **

**If** we co-construct how to unpack a problem with the class to help students engage with problem solving and provide opportunities for students to meaningfully share their thinking **Then **students will be able to communicate what the problem is asking them to do, identify the information needed to solve the problem and confidently communicate their thinking about how they solved the problem.

**Grade 2/3, 3 and 3/4 Theory of Action**

**If** we model and practice how to deconstruct a problem and provide opportunities for students to engage in number talks, math activities and the use of thinking tools,**Then** students will be able to use a problem solving approach/framework to effectively communicate and identify important information needed to solve a problem and explain their thinking.

## Problem Solving Questions

**Diagnostic Assessment**

Question: You have squares and triangles. Altogether, there are 20 sides. How many squares and how many triangles are there? Show all your work.

**Formative Assessment-**A formative assessment was administered whereby the collaborating teachers and the SWST provided students with descriptive feedback outlining their next steps in order to explain their thinking clearly.

For instance:

- I like how you answered both questions, next time show all the steps you took to answer each question.
- What information was important to solve this problem?
- Please use the four square framework to organize your work.
- I like how you labelled the perimeter. How can you figure out the total perimeter?

**Summative Assessment**

Question: Third graders are using number cards from 10 to 100 to play a math game called “Greatest Sum Wins.” Each player takes three cards and adds the numbers together.

Julia has these cards: 21, 80 and 52

Margaret has these cards: 17, 43 and 90.

Who wins the game?

By how much does the winner beat her opponent? (Grade three task)

Grade two students completed the same question only the numbers were Julia had 4, 59 and 33. Margaret had 17, 10 and 62. (This was also a parallel task for grade 3 students)

The formative and summative assessment tasks were taken from the Hands On Problem Solving resource.

## Student A-Diagnostic AssessmentOn the top left side of the page the student wrote 10 plus 10 vertically with the sum of 20. On the bottom left side of the page the student wrote 10 + 10 = 20 horizontally. In the center of the page is a t-chart with the letters S and T. Below the S the student drew 10 squares. Below the T, the student drew 10 triangles. Beside the t-chart the student wrote –Altogether are 20 squares and triangles. | ## Question of the week | ## Student A-Summative AssessmentThe left side of the page is divided into four sections. Information section states: each player take three cards and adds the numbers. The question states: who wins the game by how much does the winner beat her opponent? The Work section has the letter J followed by the numbers 21, 80 and 52 written vertically (standard algorithm) with the sum of 153. The work section also has the letter M followed by the numbers 17, 43 and 90 written vertically with the sum of 150. Finally, the words section states: Who is the winner, the winner is Julia. Julia has 3 numbers, the numbers are 153. Margaret only has 150. On the right hand side of the page the student has written the name Julia followed by the numbers 21, 80 and 52 with a sum of 153. Then Margaret 17, 43 and 90 with a sum of 150. |

## Student A-Diagnostic Assessment

On the top left side of the page the student wrote 10 plus 10 vertically with the sum of 20. On the bottom left side of the page the student wrote 10 + 10 = 20 horizontally. In the center of the page is a t-chart with the letters S and T. Below the S the student drew 10 squares. Below the T, the student drew 10 triangles. Beside the t-chart the student wrote –Altogether are 20 squares and triangles.

## Student A-Summative Assessment

The left side of the page is divided into four sections. Information section states: each player take three cards and adds the numbers. The question states: who wins the game by how much does the winner beat her opponent? The Work section has the letter J followed by the numbers 21, 80 and 52 written vertically (standard algorithm) with the sum of 153. The work section also has the letter M followed by the numbers 17, 43 and 90 written vertically with the sum of 150. Finally, the words section states: Who is the winner, the winner is Julia. Julia has 3 numbers, the numbers are 153. Margaret only has 150.

On the right hand side of the page the student has written the name Julia followed by the numbers 21, 80 and 52 with a sum of 153. Then Margaret 17, 43 and 90 with a sum of 150.

## Student B-Diagnostic Assessment On the left hand side of the page, the student has drawn two triangles followed by two squares followed by two more triangles vertically. Beside each triangle is the number 3 and beside each square in the number 4. | ## Problem of the week | ## Student B-Summative AssessmentThe student has divided the paper into 4 sections. In the information section, the student wrote Julia has these cards 21, 80 and 52. Margaret has these cards 17, 43 and 90. In the question section, the student wrote #1 Who wins the game? #2 By how much does the winner beat her opponent? In the work section, the student wrote Julia 21, 80 and 52, underneath those numbers the student added 21 +80=101 plus 101 + 80=901 plus 901 + 52=1321. Below this work the student wrote Margaret 17, 43 and 90. Followed by 17+43=60 plus 60 +43=103 plus 90+103=1003. In the words section, the student wrote Julia has more and Margaret has less so that makes Julia the winner. I know because I added it up. |

## Student B-Diagnostic Assessment

## Student B-Summative Assessment

The student has divided the paper into 4 sections. In the information section, the student wrote Julia has these cards 21, 80 and 52. Margaret has these cards 17, 43 and 90. In the question section, the student wrote #1 Who wins the game? #2 By how much does the winner beat her opponent? In the work section, the student wrote Julia 21, 80 and 52, underneath those numbers the student added 21 +80=101 plus 101 + 80=901 plus 901 + 52=1321. Below this work the student wrote Margaret 17, 43 and 90. Followed by 17+43=60 plus 60 +43=103 plus 90+103=1003. In the words section, the student wrote Julia has more and Margaret has less so that makes Julia the winner. I know because I added it up.

## Analysis

**Student A- **The diagnostic assessment revealed that the student understood the question to mean the total number of shapes had to equal 20. The student didn’t consider the number of sides for each shape. In the summative assessment task, student A used the four square problem-solving framework and was able to complete each section. However, the final question “How much did the winner beat their opponent by," the student wrote Julia has 3 numbers, the numbers are 153. Margaret only has 150.

Student A has made gains in their ability to communicate their thinking in mathematics. This student initially received a level 1 on the diagnostic assessment followed by a level 2 on the formative assessment and finally a level 3 on the summative task.

**Student B- **The diagnostic assessment revealed that the student understood what the question was asking of them and was able to demonstrate some of their thinking. On the summative task, the student was able to complete the four square problem-solving model making their thinking visible. Student B experienced difficult in the work section of the framework when making their calculations. They still identified the winner but didn’t indicate by how much. As a SWST, I’m left wondering about the reasonableness of their solution. Did the student think about how large the sum was in relation to the numbers being used? They misunderstood how many times they had to add the numbers together and are still developing their understanding of place value.

## Thinking through a problem in small groups

## Student C-Diagnostic Assessment The student has drawn two triangles, two squares and two more triangles horizontally across the page. Underneath the drawing the student sates: You get 2 two triangals and two squares and two 2 more triangles. | ## Question of the week and Problem Solving Framework | ## Student C-Summative Assessment |

## Student C-Diagnostic Assessment

## Student D-Diagnostic Assessment The student has written the numbers 1 to 8 going down the left hand side of the page. At the top of the page the student has drawn one triangle, two squares and two more triangles. Underneath the picture of shapes are two rods (looks like base ten rods). | ## Problem of the Week | ## Student D-Summative Assessment |

## Student D-Diagnostic Assessment

## Analysis

**Student C-**The diagnostic assessment revealed the student understood what the question was asking them to do and was able to show how many shapes were required. On the summative task, the student was able to use the four square problem-solving framework and complete each section. In the work section you can see the student using tally marks to add up the numbers. After awhile the student wrote the numbers vertically using the standard algorithm.

**Student**- looked at the SWST, "I'm not sure how to do this." After writing the number 3 under the numbers 1, 0 and 2.

**SWST**- "Whatever way you want to solve it is fine. You know we have counters, the hundreds chart or the number line on the name tag. Whatever you want to use is fine." The student decided to use their fingers. From time to time the student would ask “Am I doing this right?” "I'm not sure where the one goes."

**SWST**-.“You put it where you think it goes.”

The student continued to question and doubt his/her work and looked to the SWST for reassurance. This made me wonder, How much does confidence play a role in the math classroom?

Throughout the inquiry, student C made the most gains when asked probing questions that provoked thinking and reasoning about a problem.

**Student D-**The diagnostic assessment revealed the student was able to draw squares and triangles however, the quantity of each shape was inaccurate in response to the question and their thinking wasn’t clearly communicated as the numbers on the page didn’t correspond to the shapes. On the summative task the student was able to use the four square problem-solving framework and communicate their thinking clearly. Utilizing the problem-solving framework assisted student D in organizing their work. The triangulation of data for student D indicates that he/she has made the most gains throughout the inquiry when asked questions that probe thinking. In addition, participation in small group instruction and working with a math partner has allowed the student to learn with and from his/her peers..

## Student E-Diagnostic Assessment The student has drawn two squares followed by four triangles. On the left hand side of the first square is written sids with arrows above the shapes. Written underneath the picture is altogether 20 sids | ## Question of the Week | ## Student E-Summative Assessment |

## Student F-Diagnostic Assessment The student drew a triangle, square, triangle followed by another square horizontally on the page. Beneath the picture are the words I think sqares a triangles many adition and it will be 41 or 24. | ## Question of the Week | ## Student F-Summative Assessment |

## Analysis

**Student E- **The diagnostic assessment shows the student understood the question as they were able to draw the shapes and the number of sides for each shape. The summative task shows the student was able to use the four square problem-solving model. In the work section, the student is developing their understanding of place value and addition of two and three digit numbers (21+80=90). In the words section the student explained how he/she arrived at an answer of 133 however, the explanation isn’t clear. The student didn’t answer the question to the problem. Reviewing SWST field notes of student participation in class, the student rarely participated in large and small group instruction. During instructional periods, the student would make inappropriate comments like “your dump” to classmates when they shared their mathematical thinking. When the student did participate, the responses were shared with hesitation or as a question, wondering if the response was correct. When working one on one with student E, he/she was reluctant to record his/her own thinking unless reassured that the answer was correct or that his/her information was being placed in the correct place. This lead to a wondering, how does self-esteem affect ones willingness to engage in learning? How do students see themselves as a learner in the mathematics classroom?

**Student F- **The diagnostic assessment shows the student understood they were working with shapes however, it was unclear as to how the student arrived at their answer of 41 and 24. In the summative assessment, the student used three of the four sections of the problem-solving framework. The student was able to answer the first question in the problem however, not the second question. Throughout the inquiry, student F was an active participant in large and small group instruction. While working one on one or with a partner, student F would share his/her mathematical thinking. He/she made the most gains when asked questions that probed thinking and understanding of a problem.

## Thinking Tools and Manipulatives

__How to Make the Most of Math Manipulatives__, she writes how manipulatives provide a visual representation of student thinking.

“*Manipulatives help to make abstract ideas concrete*. *They provide the ability to build and construct physical models of abstract mathematical ideas. They also build students’ confidence by giving them a way to test and confirm their reasoning.” *

The following graphic Benefits of Using Manipulatives is taken from MovingwithMath.com.

## Students using Manipulatives for Graphing

## Pictographs

## Number Talks

__Number Talks Build Numerical Reasoning__, writes:

“*By sharing and defending their solutions and strategies, students are provided with opportunities to collectively reason about numbers while building connections to key conceptual ideas in mathematics.” *

## Student B- Written ResponseI added it up in chunks. I thought the 46 was 40 and the 65 was 60. Then I added the 60+40 and that equals 100 and I just added the 5+6 to it and that equals 111. Student B is comfortable using friendly numbers and is able to use some of the strategies discussed in class. | ## Student E-Written ResponseDemonstrated the standard algorithm. SWST- Walked around the room and noticed student E using the standard algorithm. SWST-“I’d like you to solve the problem using the “number talk” strategies that we’ve used in class.” Student E-I don’t know how, should I erase it?” SWST- “No!” “Explain how you solved it your way.” Student E- I split the columns, I did one + two then I did two + two +five, then six + four =ten. I took away the one from the ten = 0 then 5+0=5. Student E rarely shares his/her thinking during number talk lessons. He/she may benefit from engaging in tasks where the student views him/her self as a life long learner. | ## Student F-Written ResponseFirst, student F solved the problem using the standard algorithm for addition with a sum of 101. Underneath this solution the student wrote 10 + 11+21 followed by a number line of 46, 50, 51 52, 53 etc., to 57. All this was erased when submitted. What appears on the page is 6 + 4=10 and 5 + 6=11. In the right hand corner the student wrote “adding in chunks.”
Student F actively participates in number talks and is developing the ability to apply various strategies. |

## Student B- Written Response

I added it up in chunks. I thought the 46 was 40 and the 65 was 60. Then I added the 60+40 and that equals 100 and I just added the 5+6 to it and that equals 111.

Student B is comfortable using friendly numbers and is able to use some of the strategies discussed in class.

## Student E-Written Response

Demonstrated the standard algorithm.

SWST- Walked around the room and noticed student E using the standard algorithm.

SWST-“I’d like you to solve the problem using the “number talk” strategies that we’ve used in class.”

Student E-I don’t know how, should I erase it?”

SWST- “No!” “Explain how you solved it your way.”

Student E- I split the columns, I did one + two then I did two + two +five, then six + four =ten. I took away the one from the ten = 0 then 5+0=5.

Student E rarely shares his/her thinking during number talk lessons. He/she may benefit from engaging in tasks where the student views him/her self as a life long learner.

## Student F-Written Response

First, student F solved the problem using the standard algorithm for addition with a sum of 101. Underneath this solution the student wrote 10 + 11+21 followed by a number line of 46, 50, 51 52, 53 etc., to 57. All this was erased when submitted. What appears on the page is 6 + 4=10 and 5 + 6=11. In the right hand corner the student wrote “adding in chunks.”

Student F actively participates in number talks and is developing the ability to apply various strategies.

## Math Centers

The collaborating teacher in the combined 2/3 classroom implemented math centers in an effort to provide guided instruction in small groups geared to student learning needs and to allow for independent practice. Students were on task, engaged and learning from and with their peers.

Through the support of the instructional coach and the administration, all classroom teachers participated in a full day professional learning session on **Boxed Cars and One Eyed Jacks**. This company specializes in games and teaching strategies that support mathematics instruction in an interactive hands-on manner.

The staff shared this professional learning by inviting students and their family to a numeracy night. The Hilldale community participated in a variety of math games and how this interactive way of learning could continue at home.

## Math Games

## 3 to 6 Network

The 3 to 6 network was well received by the administration and staff. The professional learning session provoked discussion as we identified the learning goals, engaged in defining mathematical thinking and explored school and grade specific theories of action.

**School Theory of Action**

If we engage in developing mental math and computational strategies (Number Talks) by incorporating problem solving and a 3-part lesson framework, in conjunction with the use of thinking tools and manipulatives,

Then, students will be able to identify and use a variety of strategies to explain their thinking and develop flexibility and understanding in number sense.

Student success will be measured by EQAO results, report card data and grade level testing.

## Teacher Reflection

**Number Talks**

-great opportunity to see students' thinking and practice verbalizing their strategies

-exposed students to many different strategies

-increased engagement from reluctant participants

-increased confidence and participation

-strategies practiced carried over to other math tasks

-multiple entry points for students (complexity and number of strategies students could use)

-good retention of concepts, even after EQAO pause students were fully participating and remembered the routine and strategies

-looking forward to starting in September with next year's class and incorporating the new book, too

**Math Games**

-great motivator and sneaky way to practice and review skills

-students are LOVING the One-Eyed Jacks games

-will definitely incorporate more next year

**Centres (2/3 teacher’s thoughts)**

-good way to practice many concepts

-allows for peer to peer support/tutoring

-provides structure once students learned routines and expectations

-provides hands-on approach

**Problem Solving Framework**

-increased understanding of the problems since students were required to look for important info and restate all parts of the question.

-better justification and proof of answers

-answers better organized (all parts complete)

-communication was improved as students practiced explain their thinking

-good practice for EQAO problems

-gave us (teachers) a good understanding of how the students were doing at the beginning of the year and then showed the progress and improvements made as we completed the 2 other tasks

-good chance to moderate to come to a better consensus and consistency between classes

## Literature

## Key Learning

- A balanced mathematics program contributes to all learners achieving success
- Growth Mindset- How can we foster a growth mindset in students and allow them to feel supported in the math classroom enough to take risks and persevere with a problem?
- Moderating student work of a common task facilitated rich discussion
- Utilizing the triangulation of data to capture the whole student as a learner is key to determining strengths and next steps
- Release time for job embedded professional learning contributed to the success of the collaborative inquiry

## Collaborating Members

Kristen Ratzki

Andrea Colbourne

Shannon Beach- SWST PDSB