By Elli Weisdorf,

Student Work Study Teacher,

York Region District School Board

Introduction

During the 2015-2016 school year, Elli Weisdorf, a teacher with York Region District School Board (YRDSB) in Ontario, Canada, participated in the Ontario Ministry of Education's Student Work Study Initiative. As a Student Work Study Teacher (SWST), Elli had the opportunity to embark on collaborative inquiries with homeroom teachers at four different YRDSB schools.

Collaborative Inquiry in Ontario, 2014

http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/CBS_CollaborativeInquiry.pdf

Taking the stance of teacher as researcher and through the use of pedagogical documentation, Elli and the homeroom teachers wondered together about several aspects of student learning in mathematics.

Pedagogical Documentation Revisited - Looking at Assessment and Learning in New Ways, 2015

http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/CBS_PedagogicalDocument.pdf

This presentation describes the journey through one inquiry topic that involved two homeroom teachers at different schools in YRDSB from February to April 2016.

Case Study # 1

Grade 4 - Multiplication & Division

During a test to assess Grade 4 students' understanding of patterning, this was observed

(click video below):

It appeared that this student was using her fingers to count-on to determine the terms that came next in a growing number pattern.

The homeroom teacher and the SWST had many questions about this:

Is this an acceptable strategy in Grade 4?

If this is her main counting strategy, how will this student fare with the upcoming multiplication unit?

How can this student become more comfortable with counting and numbers?

The two teachers researched these topics and found...

Deborah Loewenberg Ball has researched teacher knowledge and how it impacts student achievement. She has determined that teachers need:

knowledge of math content

knowledge of how to break concepts down for learners

knowlege of how to connect concepts to each other

Teachers with these types of knowledge had more successful students.

Maximizing Student Mathematical Learning in the Early Years, 2011

http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/CBS_Maximize_Math_Learning.pdf

The Ontario Ministry of Education suggests teachers use "an instructional trajectory/landscape for planning." A trajectory would break down the stages of development of mathematical concepts. This would help teachers identify student assets and then plan where students can go next in their learning journey.

Differentiating Mathematics Instruction, 2008

http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/different_math.pdf

In April 2016, Stanford University professor, Jo Boaler, along with Stanford research fellow, Lang Chen, publicized their research that found students need to count on their fingers as an essential tool to develop number representation in their brains. She concluded that students should be encouraged to use their fingers in order to develop this model that leads to a helpful visual representation of concepts.

http://www.theatlantic.com/education/archive/2016/04/why-kids-should-use-their-fingers-in-math-class/478053/

The Grade 4 teacher and the SWST wondered about a trajectory for this student's learning in multiplication. We consulted "What to Look For" by Alex Lawson.

http://www.pearsoncanadaschool.com/index.cfm?locator=PS28F9&PMDBSOLUTIONID=25862&PMDBSITEID=2621&PMDBCATEGORYID=25876&PMDBSUBSOLUTIONID=&PMDBSUBJECTAREAID=&PMDBSUBCATEGORYID=&PMDbProgramID=118541&elementType=asset&elementID=Custom%20Bucket%202

So thinking about the Grade 4 student who was counting on with her fingers...

How would she approach mulitiplication?

We asked her to do some mental math.

(Adapted from Parrish p. 269)

What is 5x8? (click video below)

What is 10x8? (click video below)

What is 9x8? (click video below)

(5x8, 10x8)

She uses skip-counting when multiplying by 5s and 10s.

(9x8)

She tries to use 10x (which is a familiar fact) to assist in finding 9x8, but explains that she learned last year that addition and subtraction help you multiply but does not explain why. She is attempting a memorized procedure in order to solve this question. Sherry Parrish's "Number Talks" calls the strategy of using a group of 10 and adding/subtracting groups "making landmark or friendly numbers."

The Grade 4 teacher and the SWST thought that this student (and many others in the class) would benefit from experiences involving connecting familiar facts and partial products to deepen her number sense and move away from her reliance on memorized procedures.

It was decided that in order to gain these experiences, the students would be exposed to Number Talks from Sherry Parrish's book that focused on "making landmark or friendly numbers" as well as "partial products."

http://www.pearsoncanadaschool.com/index.cfm?locator=PS1zOt&PMDbSiteId=2621&PMDbSolutionId=25862&PMDbSubSolutionId=&PMDbCategoryId=25876&PMDbSubCategoryId=&PMDbSubjectAreaId=&PMDbProgramId=82541

Here's an example of part of a Number Talk where students started by thinking about 2x7. (click video below)

Notice that students raise a thumb to show that they have come to a solution. Students raise more than one finger if they are able to solve the probem in more than one way. They share their answers then discuss the strategies they used to get them.

The class also played several games that would give them more opportunities to become familiar with the relationships between factors and their products, as well as different ways to obtain the same product.

-Boxed Out (YRDSB)

-Four in a Row (YRDSB)

-Salute (YRDSB, variation of Lawson's Salute game p. 167)

-Rollin Rollin Rollin (YRDSB)

YRDSBs "Games, Puzzles and Purposeful Practice" website:

https://bww.yrdsb.ca/services/cis/mathliteracy/Pages/Games,%20Puzzles%20and%20%20Purposeful%20Practice.aspx

After over two months of almost daily Number Talks, the student started to demonstrate the use of additional strategies when faced with mental math calculations. (click video below)

The student is able to apply the "partial products" strategy in order to solve 8x45 as 8 groups of 40 added to 8 groups of 5.

The student is now able to visualize 8 x 45 through "partial products." As can be seen by the prompting the student receives in the video, she continues to work on using her "familiar facts" to help her with this.

Your friend has 9 packages of gum with 13 pieces inside each package. How many pieces of gum does he have altogether? (Adapted from Lawson p. 129)

(Click video below)

The teacher began to list students' strategies as they were shared during Number Talks. A strategy wall was created for the students to refer to.

(9 packs of gum, each with 13 pieces)

She tracks each group of 13 using a table and counts on by ones using her fingers

"Guide to Effective Instruction in Mathematics K-6, Volume 3 Classroom Resources and Management" - Strategy Walls p. 15-16.

http://eworkshop.on.ca/edu/resources/guides/Guide_Math_K_6_Volume_3.pdf

**Conclusions:**

1) Moving students across Alex Lawson's Student Continuum of Numeracy Development takes time.

2) Student progress is not always in one direction along the Student Continuum of Numeracy Development. One of the Grade 1/2 students was able to incorporate new mental math strategies in some contexts (counting on and using known facts) but after 2 months, she relied on her original strategy of counting three times during her interview.

3) Students would benefit from daily practice with Number Talks and regular exposure to games and problem solving in order to make deep connections between numbers and number concepts. Number Talks and games should target specific strategies for students to practice in order to move along the Student Continuum of Numeracy Development.

4) Using a trajectory in conjunction with pedagogical documentation and collaborative inquiry is an effective way to become responsive to student needs, to collect more observations and conversations (instead of mostly products), and help students develop a facility with and deeper understanding of numbers.

1) Moving students across Alex Lawson's Student Continuum of Numeracy Development takes time.

2) Student progress is not always in one direction along the Student Continuum of Numeracy Development. One of the Grade 1/2 students was able to incorporate new mental math strategies in some contexts (counting on and using known facts) but after 2 months, she relied on her original strategy of counting three times during her interview.

3) Students would benefit from daily practice with Number Talks and regular exposure to games and problem solving in order to make deep connections between numbers and number concepts. Number Talks and games should target specific strategies for students to practice in order to move along the Student Continuum of Numeracy Development.

4) Using a trajectory in conjunction with pedagogical documentation and collaborative inquiry is an effective way to become responsive to student needs, to collect more observations and conversations (instead of mostly products), and help students develop a facility with and deeper understanding of numbers.

Upon reflection, the Grade 4 teacher saw many benefits of using games and Number Talks:

student engagement increased during the games

students were able to practice familiar facts in fun ways

students enjoyed thinking about numbers during Number Talks and they would ask for them

The Grade 4 teacher deduced that students would rather talk about their strategies through Number Talks than solving problems independently and silently.

Case Study # 2

Grade 1/2 - Addition & Subtraction

While working with Grade 1/2 students at another YRDSB school, the teacher said to the SWST, "I don't know what to do! My students just don't understand subtraction!"

Together, the SWST and the teacher examined students' understanding of addition and subtraction and using Alex Lawson's Student Continuum of Numeracy Development: Addition and Subtraction.

12 students in the class were interviewed individually and asked to solve 2 word problems (one involving finding a sum, and one involving finding a difference).

You have 15 pencils. Your friend gave you 12 more pencils for your birthday. How many pencils do you have now/altogether? (Adapted from Lawson p. 94)

Click on the videos below to see how some of the Grade 1/2 students solved this.

You have 23 pieces of Lego. You gave 7 pieces to your friend to use. How many do you have left?

(Adapted from Lawson p. 92 and 94)

Click on the videos below to see how some of the Grade 1/2 students solved this.

http://www.pearsoncanadaschool.com/index.cfm?locator=PS28F9&PMDBSOLUTIONID=25862&PMDBSITEID=2621&PMDBCATEGORYID=25876&PMDBSUBSOLUTIONID=&PMDBSUBJECTAREAID=&PMDBSUBCATEGORYID=&PMDbProgramID=118541&elementType=asset&elementID=Custom%20Bucket%202

addition

subtraction

(attempted)

(used a standard written algorithm that they called "stack and add")

**After interviewing 12 of the students in the class, two patterns emerged:**

**When adding, students were split into two categories.**

those who counted three times or counted on

those who used splitting or counted up from ten

those who counted three times or counted on

those who used splitting or counted up from ten

**Since all the interviewed students in the class were using strategies from the left-most side of Lawson's Student Continuum of Numeracy Development: Addition and Subtraction, it was decided to focus students on making tens then using them as anchors when counting.**

Also, many students were counting on from the smaller number, and the teacher introduced specific mini-lessons to support "counting on from the larger number."

http://topnotchteaching.com/lesson-ideas/mental-maths/

Also, many students were counting on from the smaller number, and the teacher introduced specific mini-lessons to support "counting on from the larger number."

http://topnotchteaching.com/lesson-ideas/mental-maths/

**These are the games from Lawson's "What to Look For" that were played:**

Hide the Counters p.164

I Spy p.169

Handful of Dice p.169

Addition War p.176

Tens and Ones War p.179

An additional game was played:

Counting On Game

http://topnotchteaching.com/lesson-ideas/mental-maths/

Hide the Counters p.164

I Spy p.169

Handful of Dice p.169

Addition War p.176

Tens and Ones War p.179

An additional game was played:

Counting On Game

http://topnotchteaching.com/lesson-ideas/mental-maths/

**The teacher led several Number Talks from Sherry Parrish:**

Grade 1, Counting All/Counting On: Double Ten-Frames p.104-105

Grade 1, Addition: Making Tens: Double Ten-Frames p.115-116

Grade 1, Addition: Making Tens: Number Sentences p.117

Grade 2, Category 1: Making Tens p.126

Grade 1, Counting All/Counting On: Double Ten-Frames p.104-105

Grade 1, Addition: Making Tens: Double Ten-Frames p.115-116

Grade 1, Addition: Making Tens: Number Sentences p.117

Grade 2, Category 1: Making Tens p.126

**Here are some of the strategies students were using after about 2 weeks:**

(click on the videos below)

(click on the videos below)

**The student in the purple shirt counts on from the larger number using her fingers to track as she counts. Her previous strategy was to count three times.**

The student in the black shirt relates 8+6 to a problem she saw earlier in the game (8+5) which she remembered was 13 (using a known fact and adjusting). Her previous strategy was to count on using her fingers.

The student in the black shirt relates 8+6 to a problem she saw earlier in the game (8+5) which she remembered was 13 (using a known fact and adjusting). Her previous strategy was to count on using her fingers.

**The student in the purple shirt sees 4+2 and adjusts it by taking 1 from the 2 and adding it to the 4 to make 5+1, a known fact that she is familiar with.**

http://www.pearsoncanadaschool.com/index.cfm?locator=PS28F9&PMDBSOLUTIONID=25862&PMDBSITEID=2621&PMDBCATEGORYID=25876&PMDBSUBSOLUTIONID=&PMDBSUBJECTAREAID=&PMDBSUBCATEGORYID=&PMDbProgramID=118541&elementType=asset&elementID=Custom%20Bucket%202

**After almost two months of specifically targeted games and Number Talks along with ongoing assessment as learning, students were once again asked to solve two problems: one involving finding a sum and one involving finding a difference.**

**You have 13 baseball cards. Your friend gives you 8 more. How many cards do you have now? (Adapted from Lawson p. 100)**

Click the videos below to watch how some of the students solved this question.

Click the videos below to watch how some of the students solved this question.

**After two months, there were only 3 out of the 12 students interviewed in the class who were using the counting three times strategy to add or subtract.**

Most students were either counting on/back or had begun counting up/down to ten.

Students were more centrally located on the continuum instead of on the left side.

Most students were either counting on/back or had begun counting up/down to ten.

Students were more centrally located on the continuum instead of on the left side.

**You have 17 Angry Bird movie tickets. You give 9 away. How many do you have left? (Adapted from Lawson p. 92)**

Click on the videos below to watch how some of the students solved this question.

Click on the videos below to watch how some of the students solved this question.

counting on

counting three times

using a standard algorithm

splitting

using a standard algorithm

splitting

counting three times

counting back

counting three times

attempted counting back

counting back

counting three times

counting back

counting three times

**Before: counting three times**

After: counting on

After: counting on

**Before: using a standard algorithm**

After: using up/down over 10 with counting on

After: using up/down over 10 with counting on

**Before: using a standard algorithm**

After: using up/down over 10, possibly with splitting

After: using up/down over 10, possibly with splitting

**Before: counting three times**

After: counting back

After: counting back

**Before: attempted counting back**

After: using up/down over 10

After: using up/down over 10

**Before: counting back**

After: using up/down over 10

After: using up/down over 10

http://www.pearsoncanadaschool.com/index.cfm?locator=PS28F9&PMDBSOLUTIONID=25862&PMDBSITEID=2621&PMDBCATEGORYID=25876&PMDBSUBSOLUTIONID=&PMDBSUBJECTAREAID=&PMDBSUBCATEGORYID=&PMDbProgramID=118541&elementType=asset&elementID=Custom%20Bucket%202

**addition**

**subtraction**

**After Almost 2 Months of Number Talks and Games**

**When subtracting, ALL students were either counting back or counting three times.**

One of the 2015-2016 York Region host teachers who participated in the Student Work Study initiative said,

"After speaking with many of my colleagues, I’ve come to conclusion that many of us aren’t adequately prepared to teach math today. Many of us don’t have the adequate knowledge in pedagogy to teach math the way we should. I do try to educate myself by reading books or trying different resources, but I feel like I lack the pedagogy in order to help my students take the next step forward."

Teachers recognize that they sometimes lack the necessary knowledge that Deborah Loewenberg Ball describes in order to help students succeed.

Collaborative inquiry is a powerful method for teachers to work together to explore these concepts.

**References**

**For more information about this research, please contact Elli Weisdorf at elli.weisdorf@yrdsb.ca.**

Boaler, J., & Chen, L. (2016, April 13). Why Kids Should Use Their Fingers in Math Class.

The Atlantic.

Retrieved from http://www.theatlantic.com/education/archive/2016/04/why-kids-should-use-their-fingers-in-math-class/478053/

Games, Puzzles and Purposeful Practice. (n.d.) In

York Region District School Board's Board Wide Web.

Retrieved June 22, 2016, from https://bww.yrdsb.ca/services/cis/mathliteracy/Pages/Games,%20Puzzles%20and%20%20Purposeful%20Practice.aspx

Lawson, A. (2015).

What to Look For: Understanding and Developing Student Thinking in Early Numeracy.

Toronto, ON: Pearson Canada Inc.

Melinda. (2014, October 8). Mental Maths - Counting On Strategy.

Top Notch Teaching.

Retrieved from http://topnotchteaching.com/lesson-ideas/mental-maths/

Ontario Ministry of Education. (2014). Collaborative Inquiry in Ontario: What We Have Learned and Where We Are Now.

Capacity Building Series, No. 39

. Retrieved from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/CBS_CollaborativeInquiry.pdf

Ontario Ministry of Education. (2008). Differentiating Mathematics Instruction.

Capacity Building Series, No. 7

. Retrieved from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/different_math.pdf

Ontario Ministry of Education. (2006). Classroom Resources and Management.

A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6, Volume Three.

Retrieved from http://eworkshop.on.ca/edu/resources/guides/Guide_Math_K_6_Volume_3.pdf

Ontario Ministry of Education. (2011). Maximizing Student Mathematical Learning in the Early Years.

Capacity Building Series, No. 22.

Retrieved from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/CBS_Maximize_Math_Learning.pdf

Ontario Ministry of Education. (2015). Pedagogical Documentation Revisited: Looking at Assessment and Learning in New Ways.

Capacity Building Series, No. 40.

Retrieved from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/CBS_PedagogicalDocument.pdf

Parrish, S. (2010).

Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5.

Sausalito, CA: Math Solutions Publications.

5x8 Video

Transcript

(click to zoom)

10x8 Video

Transcript

(click to zoom)

9x8 Video

Transcript

(click to zoom)

9 Packs of Gum Video Transcript

(click to zoom)

Number Talk

Video Transcript

(click to zoom)

Video Transcript

(click to zoom)

Click on the PDF below each image to view video transcript

Click on the PDF below each image to view video transcript

(Found a sum instead of a difference)

Click on the PDF below each image to view video transcript

**Click on the PDF below each image to view video transcript**

**Click on the PDF below each image to view video transcript**