Learning Through Talk

Building understanding and community in mathematics

Student Work Study Overview

The Student Work Study Initiative (SWS) is a collaborative inquiry between classroom teachers (Host Teachers) and Student Work Study Teachers (SWST) involving the co-planning, co-teaching and co-debriefing of student learning. Our professional goal is to positively impact students by applying a growth mindset when helping them build upon their existing and developing abilities. It is student need, determined through the collection of data (observations, conversations and products), that drive the instructional strategies then used by teachers to deliver the appropriate Ontario Ministry of Education curriculum content.


Methods:


During the 2014-2015 academic years, there were seven SWS teachers in the Peel District School Board. Each teacher was assigned to three or four schools which included both the elementary and secondary panels. SWS teachers spent two days in each school for a half the year, or in some instances, spent 2 days in one school for the whole year. The SWST had access from the system to 13 supply teacher release days to facilitate the co-planning, and co-debriefing components of the inquiry. At the end of the collaborative inquiry, the SWST wrote a digital report for the Board and the Ontario Ministry of Education that encapsulated the applied research experience.

Socio-educational Context

The School

The fieldwork for this study was conducted in a K-5 public school in Mississauga, Ontario from September to December of 2014. The SWST spend two days a week at the school. 342 students attended the school from a wide range of cultural backgrounds. Over half of the students, 55% spoke a Home Language that was other than English or French.


The Students

The observations occurred in one grade 3, and two, grade 2 classes. There were twenty-one students in each of the grade 2 classes, and twenty-three students in the grade 3 class. The grade 2 classes shared one large room, and for most of the day identified as one united learning community that was supported by 2 teachers. The two grade 2 teachers functioned as a co-teaching team. 56% of the group was boys and 44% was girls. The Grade 3 class consisted of 65% boys and 35% girls. From early in the inquiry, the grade 3 class included several students (five boys) with challenging behavioural needs. These had been identified from the previous academic year, and required constant monitoring and attending to by the classroom teacher, and support staff.


The Teachers

4 teachers participated in this collaborative inquiry, including the Student Work Study Teacher (SWST), and three classroom teachers. One of the grade 2 Teachers was participating in this collaborative inquiry process for a second semester. These classroom teachers were responsible for delivering most curricular areas, with the exception of physical education, music and drama.

Where did we start?

After an initial period of observation by the SWST, a co-debriefing session was scheduled for the team. We discussed the data gathered (observations, conversations and products) to identify student strengths and interests. We then considered how to build upon these assets in order to address the areas of need we had decided to target.


Student Strengths/Interests:


  • Increased engagement during structured whole class discussion times
  • Increased student engagement during visual art making activities
  • Attended well to visual information/stimulation (images, videos, picture books, graphic organizers)
  • Responded to open-ended challenges with curiosity and eagerness
  • Enjoyed competitive game situations

Student Needs:


  • To develop purposeful/accountable talk protocols
  • To use talk to make student mathematical thinking visible
  • To build mathematical vocabulary
  • To use talk to build/develop deeper conceptual understanding and to aid clearer communication of said understanding
  • To build student confidence/initiative in order to aid the gradual release of responsibility and student dependence on constant teacher validation and direction
  • To empower students self-monitor their own progress
  • To develop a stronger sense of belonging and community
  • To build a Community of mathematicians

Why focus on Talk?

By explicitly developing positive communication skills we believed that social and academic needs could be met. Creating a vibrant learning community was a priority and integral to having students feel safe to make mistakes and engage in building collective understanding. This focus on the community was a theme that had also emerged during the previous year in SWST research conducted in one Grade 3 and a Grade 3/4 class (see link below).

Glossary of Key Terms Used

Accountable Talk:

In this report accountable and purposeful talk are used interchangeably and refer to the use of focused, on-task student talk as a means of deepening, refining, and making student thinking visible.


Growth Mindset:

The belief that intelligence is plastic, and that with effort and persistence, can be grown. With this mindset, the making of mistakes is understood as an integral, and valuable part of the learning process. The focus is on the learning process and not just on the product.


Math Congress:

The teachers select student problem-solving strategies to share with the whole class. Students present their thinking in a sequence that highlights the learning goals of the lesson, or clarifies common misconceptions that emerged.


Mathematical Landscape of Learning:

This term is used to describe the developmental, and non-linear journey of learning in mathematics. The focus is on understanding the interplay of big ideas, key strategies, and the use of models.


Number Talk:

A whole group discussion of mental computation strategies that is typically used at the start of a math lesson and is dialogic in nature. The purpose is to encourage students to collaboratively build upon each others thinking while developing their ability to accurately, flexibly and efficiently manipulate numbers.


Number String:


A planned sequence of mental math problems that students solve that gradually increase in complexity and scaffold towards the "discovery" of efficient computational strategies.


Talk Moves:

A set of talk strategies used by teachers to promote rich, purposeful talk among all learners.


Talk Prompts:

Explicit sentence frames practiced and used by students to encourage in constructive discussion: agreement, disagreement, seeking of clarification, and elaboration.

Questions that Guided the Inquiry

  1. What does purposeful talk sound like?
  2. What impact will establishing a purposeful talk culture have on student learning?
  3. What impact will it have on classroom climate?
  4. How do we ensure that student talk is purposeful and on task?
  5. What type of data will we collect?

Theory of Action

If we help students develop accountable talk routines and a variety of strategies to make their thinking visible, then students will be able to collaboratively problem-solve and co-construct their understanding of key mathematical concepts/big ideas.

Curriculum Expectations:

Mathematical processes with a focus on communication of problem-solving


• Apply problem solving strategies to deepen mathematical understanding

• Communicate mathematical thinking in a variety of ways using developmentally appropriate mathematical vocabulary, and a variety of representations

Resources and Research Referenced

Ontario Ministry of Eduation Documents

Leaders in Educational Thought Webcasts

Learning Goals

Overall Learning Goal (cross-curricular):


We are learning to talk about our thinking and problem-solving. We are evaluating our solutions, and then choosing to develop the one that best expresses our ideas. We are using talk to build a strong learning community.


Math Learning Goals:


We will talk about our learning and how to solve problems/challenges in math. We will listen carefully, ask questions, and elaborate on ideas to help us understand our group’s thinking. We are using talk to build a community of mathematicians.

What were the Talk Moves We Tried?

We started with Turn and Talk and intentionally used it in our daily teaching to counterbalance the iconic model of classroom discussion, where the teacher asks a question, and one student is selected to answer. We recognized that it was more powerful if all students had the opportunity to respond so, we periodically asked them to briefly discuss their ideas about a topic with an elbow partner. Although this routine originated within a mathematics context, we quickly recognized the increased level of student engagement across the curriculum.


Students worked on paraphrasing one another next. We hoped that by being able to rephrase someone else's thinking they would improve their listening skills while learning to value another point of view.


Adding On was added to the repertoire of talk moves with the intention that it would help students continue to strengthen their listening skills. In order to add on logically, everyone was accountable for attending to the information they were adding on to.


Agreeing or Disagreeing was also explicitly Teacher modeled for students. Early on, we noticed that many students seemed to be under the misconception that it wasn't "nice" to disagree. How to disagree constructively was then often modeled by teachers, and over the course of the inquiry, students became more and more skilled at it. In addition, more students started to see that "mistakes " were important to help them clarify understanding.

Co-Constructing Purposeful Talk Success Criteria

We co-constructed the following criteria to help us monitor our progress. They were posted for students, and a checklist version was used by teachers as "Look-for" tool.


o I listen (eye contact, face the speaker, sit still) carefully to what my partner has to say.
o I can share my group’s thinking using my own words.
o I can give examples of my thinking.
o I have used the “Talk Prompts” to help me understand our learning
o I have used math words to explain my thinking.

Talk Moves Launch

Is it polite, neutral or impolite?

We started with an explicit classroom discussion on how to politely and constructively speak with others. Students were given several statements to read and then classify as either, polite, neutral or impolite. One "aha" that came out of this activity was that students concluded that as well as the words themselves, the tone of voice and body language used by a speaker conveyed as much meaning.

Trying out the Talk Prompts

Talk Prompts were prominently posted in the classrooms so that they could be visually accessed by students at any time. Teachers modeled their appropriate use during class discussions, and frequently reminded students to refer to them to help them with their group work.


Early into the inquiry we noticed:

  • Students were consciously using talk prompts when reminded
  • Students were visibly referring to the wall, but prompts were used indiscriminately, and therefore not always in the correct context
  • Students appreciated sharing their ideas, but often seemed to be waiting for their turn to speak, instead of listening to one another
  • Although students had learned the mechanics of "Turn and Talk", "Adding on", "Agreeing and Disagreeing" they required more authentic opportunities to go beyond the mechanics and internalize the prompts


Later into the inquiry we noticed:


  • Talk prompts were used across the curriculum, and not just in math
  • Students were more naturally using the talk prompts without as many teacher reminders

(Discussion that emerges out of a Number Talk)
HT: How many groups of 10 are there in 89?

Student G: 7

Student S: I'm not sure I agree with you because the first number tells you how many groups of 10 there are.


  • Most students accepted personal responsibility to use talk to make their thinking visible
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I did it this way because...

I'm still unclear about...

Please explain why that works.

I realized that...

Help me understand...

I like your answer because...

I disagree because...

I wonder ...

I am not sure that is the correct answer because..,

Your answer is different because...

Are you saying...

Please show me how you did that.

What can you add to your answer?

Rich Tasks: Opportunities for Talk

Connecting Math with Visual Art

Rich Task 1: Geometric Compositions in Clay

Why make clay tiles?


We decided that integrating art and math made a great deal of sense when covering the geometry strand of the mathematics curriculum. We hoped this rich, hands-on experience would:


  • Increase engagement as students experimented with novel materials
  • Allow students to envision multiple solutions through the planning process, and to realize that there are many possible "right answers".
  • Invite experimentation with materials and the sharing of expertise among peers
  • Encourage connection-making between math and art
  • Serve as a source of dialogue as students used mathematical vocabulary to communicate their creative decision making to one another
  • Encourage students to persist through with a task over an extended period of time (planning, making, painting)

Anticipating Student Responses

At our co-planning session, we started by conferring with the geometry strands of the Mathematics curriculum to co-create this rich task. As clay was a novel medium for the Host Teachers, we decided to make our own tiles in order to anticipate where students might be most challenged. This helped us prepare for the following potential obstacles:


  • Creators needed to allow for changes to likely occur from the transfer of a 2D idea (sketch) to a 3D medium (clay)
  • Teachers pre-rolled the slab bases to accomodate for the developing gross motor skills of children this age
  • We established clear clean up routines and stations

Setting Our Learning Goals

Just prior to the planning of their designs, the class compared the use of language in both math and art and generated a Venn diagram. Next Learning Goals and Success Criteria were also made explicit and posted for reference.
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Exploring Geometric Abstraction in Art History

Characteristics of Non-objective Geometric Abstraction

In order to provide students with visual references and historical context for their own designs, we showed them a variety of art historical examples that clearly embodied the mathematical concepts of focus: symmetry, congruency, right angles, and a variety of geometric shapes. Using Turn and Talk, students enthusiastically responded to the following prompts:



  • Describe what you see.
  • Consider what the art might be about?
  • Determine the characteristics of geometric abstract art.
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The Creative Challenge

Create an interesting abstract composition with a clear focal point. It should show your understanding of important ideas in geometry about symmetry, right angles, congruency and composite shapes.

Establishing Success Criteria with Students

Grade 2's

In my geometric design I have included:

Math:

  • Some shapes that are symmetrical
  • 2 or more congruent shapes
  • 1 shape that is composed of 2 or more other shapes

Art:

  • 3 or more plans
  • 2 or more textures
  • 2 or more planes/levels
  • a focal point
  • slab and coil techniques


Grade 3's

Grade 3 Geometric Design Success Criteria

In my geometric design I have included:

Math:

  • A symmetrical composition
  • 2 or more congruent shapes
  • 1 or more right angles
  • 1 or more right-angled triangles
  • 1 shape that is composed of 2 or more other shapes

Art:

  • 3 or more plans
  • 2 or more textures
  • 2 or more planes/levels
  • a focal point
  • Slab and coil techniques

The Planning Process

Students were instructed to come up with 3 to 4 plans experimenting with various possible abstract compositions. Coming up with multiple solutions encouraged students to be more flexible with their thinking. They used the co-constructed success criteria while they were making their plans in order to ensure that the mathematical concepts were included. Finally, they conferenced with the teachers to decide on which composition would be most appropriate, both mathematically and aesthetically speaking.
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Exhibiting our Creations at the Arts Centre

What students had to say about the experience:

The following comments were made by the Grade 3's in a sharing circle immediately following the claymaking session:


Ra: I learned how to use clay and how to glue it on.

Jas: How to place on clay

Ru: How a real artist works

Je: How to use use different tools to make textures

Ja: I learned how to design with symmetry

Fa: How to make angles with clay on the edges

Sa: How to make art using clay

Hu: How to score clay

Rah: How to use clay to be restful

As: Right angles and how to shape clay

Is: About congruent shapes and how all of them are the same size and shape

Sa: Design with clay and how to use math to be inspired in art.


In Grade 2, each student conferenced with a Teacher after completing their tile so that we could ascertain whether or not he/she could speak about his/her work both mathematically and artistically. Students referred to a list of prompts to help guide them through their thinking.


The symmetrical shapes in my work are...

The names of these symmetrical shapes are ...

The congruent shapes I made in my design are ...

I know they are congruent because ...

The shape that is composed in my design is ...

The shapes it is made up of are ...

My focal point is ...

The math I learned by making this design is ...

I learned ...


The following video recording is an example of one such Student-Teacher conference.

Liyan's geometry reflection

Our Reflections on Expanding Math Vocabulary

Teacher-Student Conferences (1 week after the making the tiles)


When working in groups during the creative process, students were able to refer to one another, and access the collectives' understanding of the math content language. However, once students were expected to individually discuss the same math concepts in the form of an interview with a teacher, the academic words were no longer easily accessible to them.


The amount of prompting that our students needed baffled us. We had expected them to be able to more fluidly use the language we had targeted in our success criteria. Particularly after, having successfully built the concepts into their personal creative designs; having access to the prompts; and after having observed Teacher modelling of the process.


We realized that it was the terminology that fell specifically within the math domain (congruent, composed) that gave students the most difficulty despite their having participated in a highly engaging task (clay work) where the math was heavily integrated, a week earlier. We decided that students would require more opportunities to actively work with the words, both with peers and with Teachers before they would be able to a fluidly use them. We hoped that setting up math centres might provide the context for students to do this. We wondered whether the group talk that naturally arises while working through centres facilitate more rapid math vocabulary expansion.

Grade 3 Artist Statement

Five months after creating the tiles, students wrote a group artist statement to summarize their creative experiences. The statements were submitted to the Living Arts Centre, along with the creative work as part of the annual student art exhibition, Visual Voices: Through Our Eyes. Writing for this authentic audience gave our students yet another opportunity to share and review both their mathematical and art thinking.



In the process of creating plates, we explored the concept of building a multi-leveled structure with a focal point and a few different textures. We used a slab technique to shape our plates, and a coil technique to add more dimensions. While working on our designs we explored ways of showing symmetry. For that reason we divided plates into four quarters and we used congruent shapes. We also explored the concept of shapes with right angles. Our purpose was to combine artistic and mathematical concepts together.


During our work we learned new techniques, and ways of artistic expression. We found it challenging to stick together different pieces of clay, but learning how to score helped with this problem. We had to be very careful not to squish small parts when creating multi-leveled planes because wet clay can change its shape easily. Overall, it was a lot of fun and we loved it.

Rich Task 2: Math Learning Centres

Our Reflections on Centre Work

A variety of geometry centres were set up for the grade 2's to continue to explore the concepts of composing and decomposing shapes, as well as congruency. Teachers circulated the room and listened in on the talk between peers. We also actively engaged the students in conversation. We found that over the course of the period, the same students who had had trouble using specific terms at their math conference, were able to more comfortably use them with understanding during the centre work. While moving through the centres, our students were making explicit and concrete connections between the vocabulary listed in the success criteria and their activity work. This reinforced for us that students needed multiple and diverse experiences for the math vocabulary to "stick".

Rich Task 3: Measuring for the Art Show Activity

In order to meet the Grade 2 Measurement curricular expectations, we decided to try the two week unit from Catherine Fosnot's, Contexts for Learning kits, Measuring for the Art Show Addition on the Open Number Line. These sequenced activities gave students many opportunities to work with peers to co-construct their understandings of some of the big ideas around measurement. The unit focus was on the development of the open number line model.Through the course of their investigations students developed a sense of efficacy as groups compared their measuring strategies at math congresses.

Measuring for the Art Show, (Addition and Subtraction, Fosnot, p. 5)

Big Ideas addressed in 2 week unit:
  • Distance is measured as a series of iterated units
  • Units used in measuring can vary in size, but the results will be equivalent
  • Numbers can be decomposed and the subunits or smaller amouns can be added in varying orders, yet still be equivalent
  • There are place value patterns that occur when adding on groups of ten
  • Unitizing


Strategies

  • Counting on
  • Counting 3 times
  • Using the 5 and 10 structures
  • Splitting
  • Keeping 1 number whole, landmark numbers, and/or taking leaps of 10


Model

  • Open number line

This Led Us to Number Sense Landscapes and Number Talks

Developmental Overview for Addition and Subtraction

We had started to use the graphic below to begin to map the various developmental stages our students fell into. Although, complex at first, the observational data we gathered was powerful assessment for learning data, as it suggested pathways for student growth that was individualized.
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Developing Efficient Measuring Strategies

One of the early activities required pairs of students to use linking cubes to measure a variety of paper sizes. Some were small and manageable on a desk, but other sizes were quite large, and so needed to be measured on the floor.


As the lengths that needed to be measured got longer and longer, some students decided that instead of counting every individual cube, skip counting was a "faster" way of measuring the lengths and widths of the "art".
Andrew's skip counting by 2 strategy
Nov 28, 2014

Number Talks

In order to reinforce the various measuring strategies that emerged from the daily measuring activities, we decided to start our daily math with a number talk. Students were challenged with a string of addition equations that they were to solve mentally, and then be prepared to share. The number strings were carefully selected to scaffold the strategies we hoped students would discover in their measuring activities.

Consolidation: Math Congress

The Host Teachers decided to use Math Congress to help students consolidate the learning that emerged from the art show measuring activity. Three common strategies were shared with the whole group. The first one, one to one counting was shared by the Host Teacher and used as an opportunity to model for students how to record your mathematical thinking. The second and third, counting by 2's, and counting by 10's was shared with the group using video footage (links above) that recorded their thinking. The students involved were then asked to further clarify and elaborate. In addition, misconceptions around how to line up the cubes to avoid inaccuracy was addressed. A list of class measuring strategies was generated that we continued to expand as the learning progressed.


We also noticed that using video as a sharing medium was also very popular with the students. It proved to be an additional motivator as students were eager to have their own thinking recorded and showcased to the class. This sharing format placed student voice at the centre of the learning, furthering our over-arching learning goal to create a community of mathematicians.

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Grade 2's say, "Measurement, bring it on!"

Celebrating the Journey

We created a book/video as a means of documenting the learning journey for Measuring for the Art Show. This way the students would be able to access it throughout the year during their daily reading. It could also be shared with friends and family.
Measurement, bring it on!

Student Reflections on the Value of Student Talk

Prompt: What do you think about talking about your learning?


Student Ty:

I like to share and talk about learning because it helps others to learn about what you're learning about. And it also helps you hear ideas from others because if you don't know something a friend can tell you, but if they don't know something, you can tell them.



Student Ne:

Turn and Talk is something that you do when the teachers read you a story. We turn to our elbow partner and we talk We talk about things that we do, like if we doing time we talk about time; and when we are doing fractions, we talk about that. We think about our own thing, and then we turn to our partner and share. And then we think about their ideas. Then sometimes they ask us more questions.


Student Shan:

Thinking about my learning is important because, what if somebody else in my class doesn't know what to do? And in my class we have these "thought bubbles" and if someone doesn't know what to say there are nicer words of saying you are wrong or you are right, and you can help them to understand.

Collaborative Inquiry Key Learning

Impact of the Use of Talk Moves

The use of Talk Moves as part of the classroom learning culture provided students and Teachers with:


  • More opportunities to be active participants as it served to "open up" student voice in the classroom
  • Teachers were able to gather more informative diagnostic data about student understandings and misconceptions as peer discussions provided a lens into student thinking
  • Teachers were able to respond to needs it in a more timely manner
  • There was a greater sharing of power between Teachers and students as conversations and consequent learning occurred without being dependent on direct teacher direction.
  • More rich tasks were planned as Teachers noticed that the "best talk" emerged out of situations where students were involved in open-ended, accessible challenges/problems across all content areas
  • Teachers felt that the investment of time in scaffolding group work/collaborative culture early in the inquiry was key to creating a positive climate where all voices were respected and had a far-reaching impact into all areas of the curriculum
  • Maintaining a positive classroom climate corresponded directly to the overall progress in learning
  • Although noticeable progress was made in both Grade 2 and 3 classes, it was not not as linear, or as rapid in the Grade 3 context. Maintaining a positive climate needed to be repeatedly re-addressed throughout the term by Teachers and Students.

New Questions

  1. How can "Talk Moves" be more smoothly implemented in an environment where maintaining a positive classroom dynamic is an on-going issue?
  2. What type of Math Centres will help students to actively use mathematical vocabulary?
  3. How will continued practice of Number Talks impact overall student achievement in mathematics?
  4. Does integrating math with visual art help students deepen their understanding of both content areas?
  5. How will students' emerging identification as mathematicians impact their overall achievement?

Soula Katsogianopoulos, Hon BA BEd, MA

Student Work Study Teacher

Peel District School Board