Number Talks:

Developing Number Sense and Fluency in Addition in Grades 3 and 4 Through Building a Variety of Mental Math Strategies

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Our Collaborative Inquiry ...

Collaborative Inquiry Partners

McHugh Public School:


Colin Burleton- Grade 3 Teacher

Carolyn Satchwell- Grade 3 Teacher

Lada Sergeeva- Grade 4 Teacher

Sandra Fobert- Student Work Study Teacher

School Overview

McHugh Public School is located in Brampton, Ontario. It was built in 1923 and currently has approximately 280 students in Kindergarten through Grade 5. The school has one Principal and no Vice Principal. It has been designated as OFIP 1 for the 2014-2015 school year and receives the support of a Student Achievement Officer.
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Method

As the SWST, I was generally at the Host School for two full days each week, every Wednesday and Thursday, from mid-September to mid-December, 2014: a period of approximately thirteen weeks. During this time, I collaborated in three teachers' classrooms: two Grade 3 classrooms and a Grade 4 classroom. My visits to these classrooms took place during scheduled Math periods and Math blocks.


I spent the first three weeks of visits to classrooms observing all students, recording pertinent data and building relationships with students. It was during this period of time that my observations of and conversations with students indicated that the predominant strategy used by most students in all three classes for solving an addition equation was to use finger counting and to count on by 1's. Only a couple of students in each class demonstrated an awareness of any other strategies for solving the equation. In discussing this with the Collaborating Teachers, we recognized that, while finger counting and counting on by 1's were valid strategies, they would not be particularly efficient strategies for students as addition equations became more difficult. We recognized that students needed to build a variety of flexible strategies that would allow them to mentally solve more complex addition equations. It was from this discussion that the focus of number sense and fluency in addition was decided upon for our collaborative inquiry. Having completed the Jo Boaler course "How to Teach Math for Students", one of the many things that I learned was that number talks are one of the best ways to teach number sense and Math facts at the same time. I set out to do some research about number talks and shared it with the Collaborating Teachers. Each of the Collaborating Teachers committed to conducting number talks in their classrooms a minimum of three times per week for the duration of our collaborative inquiry.

Our Theory of Action

If students engage in regular number talks that encourage them to collectively reason and communicate about numbers while building connections to key conceptual ideas in mathematics in order to make sense of math, construct strategies and build upon numerical relationships, then they will be able to see, conceptualize, reason and communicate about quantities in mathematics accurately, efficiently and flexibly when engaging in rich tasks.

Embedding the Math Processes of Communication and Reasoning and Proving into Our Collaborative Inquiry

Grade 3:


Reasoning and Proving-

  • apply developing reasoning skills to make and investigate conjectures


Communicating-

  • communicate mathematical thinking orally, visually and in writing, using everyday language, a developing mathematical vocabulary and a variety of representations


Grade 4:


Reasoning and Proving-

  • develop and apply reasoning skills to make and investigate conjectures and construct and defend arguments


Communicating-

  • communicate mathematical thinking orally, visually and in writing, using everyday language, a basic mathematical vocabulary and a variety of representations, and observing basic mathematical conventions
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Linking the Grade 3 Mathematics Curriculum and Our Collaborative Inquiry

Number Sense and Numeration:

Overall Expectations-

  • demonstrate an understanding of magnitude by counting forwards and backwards by various numbers and from various starting points
  • solve problems involving the addition and subtraction of single- and multi-digit whole numbers, using a variety of strategies, and demonstrate an understanding of multiplication and division


Specific Expectations-

By the end of Grade 3, students will:

  • count forward by 1's, 2's, 5's, 10's, and 100's to 1000 from various starting points, and by 25's to 1000 starting from multiples of 25, using a variety of tools and strategies (counting)
  • count backwards by 2's, 5's, and 10's from 100 using multiples of 2, 5, and 10 as starting points, and count backwards by 100's from 1000 and any number less than 1000 using a variety of tools (counting)
  • solve problems involving the addition and subtraction of two-digit numbers, using a variety of mental strategies (operational sense)

Linking the Grade 4 Mathematics Curriculum and Our Collaborative Inquiry

Number Sense and Numeration:

Overall Expectations-

  • demonstrate an understanding of magnitude by counting forward and backwards by 0.1 and by fractional amounts
  • solve problems involving the addition, subtraction, multiplication and division of single- and multi-digit whole numbers, and involving the addition and subtraction of decimal numbers to tenths and money amounts, using a variety of strategies



Specific Expectations-

By the end of Grade 4, students will:

  • count forward by halves, thirds, fourths, and tenths to beyond one whole, using concrete materials and number lines (counting)
  • count forward by tenths from any decimal number expressed to one decimal place, using concrete materials and number lines (counting)
  • add and subtract two-digit numbers, using a variety of mental strategies (operational sense)

Definitions of Terms Used

Number Talks:


Number talks are five to fifteen minute regular routines that "provide students with meaningful ongoing practice with computation. A number talk is a powerful tool for helping students develop computational fluency because the expectation is that they will use number relationships and the structures of numbers to add, subtract, multiply and divide. The primary goal of number talks is computational fluency." (Math Perspectives, 2007)


Number talks are a "whole group mental Math activity where students find the answer to a Math problem in their heads. then share aloud the strategies they used to find that answer. They help to develop quality discourse in a whole class setting as students are encouraged to explain their thinking, justify their reasoning and make sense of each other's strategies. Number talks build flexibility, accuracy and efficiency with numbers for all students." (Barge, 2014)


"The heart of number talks is classroom conversations focused on making sense of mathematics." (Parrish, 2011)


Number talks "develop conceptual understanding of numbers and of the arithmetic properties that are critical to success in algebra and beyond." (Boaler, 2014)

Number Sense:


The NCTM Standards defines number sense as involving five components:


1. well-understood number meanings


2. multiple relationships among numbers


3. recognition of the relative magnitude of numbers


4. knowledge of the effect of operations on numbers


5. referents (benchmarks) to measure of things in the real world


"Number sense, like common sense, is difficult to define or express simply. It refers to an intuitive feel for numbers and their various uses and interpretations. Number sense also includes the ability to compute accurately and efficiently, to detect errors and to recognize results as reasonable. Good number sense is also recognizing the relative magnitudes of numbers and establishing referents, or benchmarks, for measures of common objects and situations in their environments." (Reys, Lindquist, Lambdin & Smith, 2012)


Research on number sense indicates that number sense is...

  • "...complex. There are many layers to it and it is rooted within all strands of mathematics." (Shumway, 2011)


  • "The foundation for all higher level mathematics" (Feikes& Schwingendorf, 2008)


  • "...inhibited by over-emphasis on the memorization of Math facts." (Boaler, 2009)

Fluency:


"Knowing how a number can be composed and decomposed and using that information to be flexible and efficient with solving problems." (Parrish, 2014)


Fluency is comprised of four categories:


  • accuracy- assessed as soon as the student answers


  • efficiency- observed on how long it takes the student to solve a fact


  • flexibility and appropriate strategy selection- addressed by prompts such as "How did you figure that out?" or "How could you use this strategy to solve this fact?" (Kling and Bay-Williams, 2014)


"The best way to develop fluency with numbers is to develop number sense and to work with numbers in different ways, not to blindly memorize without number sense." (Boaler, 2014)

Mental Strategies:


The Ontario Curriculum, Grades 1-8: Mathematics defines mental strategies as "ways of computing mentally, with or without the support of paper and pencil."

Resources Used to Inform Our Collaborative Inquiry

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Focus Students

Focus Students for concentrated data collection by the SWST were identified by the Collaborating Teachers during the second week of October. Each Collaborating Teacher identified three Focus Students who were experiencing difficulty with number sense and fluency:


Class 3B- Student D, Student K, Student V


Class 3S- Student A, Student L, Student N


Class 4S- Student K, Student L, Student S


Throughout the collaborative inquiry a large number of student observations, conversations and products were documented through video recordings, audio recordings, photographs, performance tasks and field notes for all students in each class. Very detailed data for all Focus Students was documented through video recordings, audio recordings, photographs, performance tasks and field notes and analyzed during release sessions with Collaborating Teachers. An analysis of data for the three Focus Students whose names are bolded above is included within the content of this Smore. The analysis of data for all Focus Students is available for sharing by request, but not included within the contents of this Smore due to length.

Focus Student Fluency Interviews

Once Focus Students had been identified in all three classrooms, the SWST individually interviewed each of the Focus Students in order to gather baseline information about their fluency relating to addition. The interview was designed based upon a research article from Teaching Children Mathematics (April 2014) titled "Assessing Basic Fact Fluency". The student data collected during these interviews provided information pertaining to the students' basic fact fluency in addition based on the previous grade's curriculum expectations, specifically:


* accuracy- the student's ability to produce an accurate answer

* efficiency - the student's ability to choose an expedient strategy

* appropriateness- the student's ability to choose an appropriate strategy

* flexibility- the ability to use number relationships with ease

Grade 3 Fluency Interview

Based upon the previous year's curriculum expectations ...
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Grade 4 Fluency Interview

Based upon the previous year's curriculum expectations ...
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Establishing a Learning Goal

In order to establish a learning goal, a variety of Ministry documents, LNS Monographs, LNS Webcasts, professional books and current research articles were consulted.


An anchor chart of the Learning Goal was created, introduced to and discussed with the students and posted in all three classrooms for reference.

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Co-creating a Checklist of Math Conversation Norms

A list of Math conversation norms was co-created with the students in each of the classes. While the list looked slightly different in each of the classes, the common elements of valuing think time and valuing mistakes were included on all class anchor charts. Students were often asked to self-reflect on their participation in a number talk by referring to this anchor chart at the end of a number talk.
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Implementing Number Talks

In order to begin the implementation of number talks, the SWST modelled for the Collaborating Teachers in each of the classrooms. Modeling progressed to a co-teach format and, as Collaborating Teachers felt comfortable, they conducted the number talks while the SWST observed Focus Students during the process.


Initial number talks focused on Choral Counting and Count Around the Circle as identified in Shumway's book "Number Sense Routines". The counting expectations from the previous grade level served as our planning tool. Over time, the number talks expanded to include Choral Counting and Count Around the Circle focusing on the current grade level's counting expectations. In Grade 3, number talks further expanded to include mental computation of two digit plus two digit addition. These number talks were based upon the number strings in the "Minilessons for Extending Addition and Subtraction" book contained within Fosnot and Dolk's "Contexts for Learning: Number Sense, Addition and Subtraction" kit.

Students Were Provided With Multiple Opportunities to Engage in Number Talks

The Emergence of a Math Strategy Wall

During number talks that focused on Mental Computation of Two Digit Plus Two Digit Addition, students were asked to orally share the strategies that they used to solve the given equation. During this sharing, the SWST and the Collaborating Teachers named the strategies used by the students using the language referenced in Fosnot and Dolk's Landscape of Learning for Number Sense, Addition and Subtraction. As strategies emerged through number talks, students created a Math Strategy Wall in the classroom for reference.
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An Analysis of Student Data for Three of the Focus Students

Towards the culmination of our collaborative inquiry, the SWST individually interviewed each of the Focus Students twice:


  • firstly to observe specific mental strategies that students were using to solve a given equation and document progress for each student from pre- to post-collaborative Inquiry

  • secondly to re-administer the collaborative inquiry fluency interview questions and document progress for each student from pre- to post-collaborative inquiry

Student A- Class 3S

Fluency Interview

Results of the PRE- and POST-collaborative inquiry fluency interviews for this student ...
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Collaborating Teacher and SWST Analysis of Student Data:

Student A-


  • provided an accurate answer on both the PRE- and the POST-fluency interviews


  • demonstrated a new awareness of the strategies splitting, taking jumps and algorithm in the POST-fluency interview


  • accurately and independently used the strategies splitting, taking jumps and algorithm in the POST-fluency interview


  • demonstrated an understanding of place value when using an algorithm in the POST-fluency interview


  • demonstrated an understanding that some strategies are more efficient than others in the POST-fluency interview (i.e., "It's the fastest.")


  • used anchors of ten in the POST-fluency interview

Strategy Interview

A strategy interview was conducted individually with each of the Grade 3 Focus Students on December 4, 2014. The purpose of this interview was to observe what strategy/strategies the Focus Students would use to solve the addition equation (observing for accuracy, efficiency, appropriateness and flexibility). All Focus Student interviews were captured using the app Explain Everything for later reference and analysis.


During the strategy interview, each of the Grade 3 Focus Students was asked to mentally solve the addition equation 37 + 26 =. After mentally solving the equation, each of the Focus Students was asked to show his/her thinking on paper. ***Note that the equation asked now addresses end of year Grade 3 curriculum expectations and was the equation asked in the Grade 4 Focus Student PRE- and POST-fluency interviews.

Photograph of Student A's work completed during the strategy interview ...
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Notes recorded by the SWST during the strategy interview to mentally solve 37+26= and then show his/her thinking on paper ...


  • accurately and independently solved the equation using an algorithm/used fingers to solve 6+7/counted on by 1’s from 7


  • accurately and independently solved the equation using splitting and keeping one number whole and taking jumps (took 6 jumps of 1/recognized that she could take 1 jump of 6 or jumps of 2’s or jumps of 3’s)


  • recognized place value difference between 1’s and 10’s


  • accurately and independently drew a base 10 diagram


  • favourite strategy is algorithm or taking leaps (“when algorithm doesn’t work”)


  • student confidence has increased greatly since the beginning of the collaborative inquiry and she now indicates that he/she is 'good' at Math (at the beginning of the collaborative inquiry the student indicated they he/she was 'sometimes good' at Math)


  • strategies observed during this interview: algorithm, splitting, taking jumps, using base 10

Student K- Class 3B

Fluency Interview

Results of the PRE- and POST-collaborative inquiry fluency interviews for this student ...
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Collaborating Teacher and SWST Analysis:

Student K-


  • provided an accurate answer on both the PRE- and the POST-fluency interviews


  • demonstrated a new awareness of the strategies splitting and taking jumps in the POST-fluency interview


  • accurately and independently used the strategy of taking jumps in the POST-fluency interview


  • required some teacher support to accurately use the strategy of splitting in the POST-fluency interview


  • used anchors of ten in the POST-fluency interview

Strategy Interview

A strategy interview was conducted individually with each of the Grade 3 Focus Students on December 4, 2014. The purpose of this interview was to observe what strategy/strategies the Focus Students would use to solve the addition equation (observing for accuracy, efficiency, appropriateness and flexibility). All Focus Student interviews were captured using the app Explain Everything for later reference and analysis.


During the strategy interview, each of the Grade 3 Focus Students was asked to mentally solve the addition equation 37 + 26 =. After mentally solving the equation, each of the Focus Students was asked to show his/her thinking on paper. ***Note that the equation asked now addresses end of year Grade 3 curriculum expectations and was the equation asked in the Grade 4 Focus Student PRE- and POST-fluency interviews.

Photograph of Student K's work completed during the strategy interview ...
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Notes recorded by the SWST during the strategy interview to mentally solve 37+26= and then show his/her thinking on paper ...


  • accurately and independently solved the equation using an algorithm/used tally marks to solve 6+7/drew 6 tally marks and then added 7 additional tally marks/ counted on by 1's from 6 (though the number 63 was recorded by the student using a reversal for the number 6)


  • accurately and independently solved the equation using splitting (though the number 63 was recorded by the student using a reversal for the number 6)


  • attempted to use the strategy taking jumps/required teacher support to use the strategy accurately/kept 26 whole and jumped 10/miscalculated 35 instead of 36/took a jump of 2 and got to 37/abandoned the strategy


  • recognized that the 6 in 63 is 6 tens or 60


  • favourite strategy is algorithm or taking jumps


  • student confidence has increased greatly since the beginning of the Collaborative Inquiry/the student now approaches addition equations with an "I can do this" attitude


  • strategies observed during this interview: algorithm, splitting, taking jumps

Possible Next Steps for Grade 3 Number Talks

  • continue to provide opportunities for the students to engage in number talks that build toward a generalized use of a repertoire of strategies for addition using the number strings identified in book "Mini Lessons for Extending Addition and Subtraction"


  • as new strategies emerge during number talks, have students add these strategies (word and picture) to the classroom Math Strategy Wall


  • provide opportunities for the student to apply newly acquired strategies while engaging in rich tasks, such as open Math questions
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A Change of Course Occurred in Our Grade 4 Collaborative Inquiry

The focus of number talks in Grade 4 incorporated only Choral Counting and Count Around the Circle. While it was our intention at the outset of our collaborative inquiry to venture into the emergence of a variety of strategies for the computation of addition, as was done in Grade 3, we were unable to do so. There was an unanticipated change in the dynamics of the classroom that necessitated that the Collaborating Teacher to scale back from our original intentions.

Student L- Class 4S

Fluency Interview

Results of the PRE- and POST-collaborative inquiry fluency interviews for this student ...
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Collaborating Teacher and SWST Analysis:


Student L-


  • provided an inaccurate answer on the PRE-fluency interview


  • provided an accurate answer on the POST-fluency interview


  • demonstrated a new awareness of the strategies splitting and taking jumps in the POST-fluency interview


  • required teacher support to accurately use the strategies of splitting and taking jumps in the POST-fluency interview


  • recognized that the 3 in 37 is 30


  • recognized that the 2 in 26 is 20


  • accurately added 6+7 to get 13/used fingers and counting on by 1's


  • accurately decomposed 13 into 10 and 3/added this 10 to 30+20 to accurately calculate 60

Possible Next Steps for Grade 4 Number Talks


  • begin to provide opportunities for the students to engage in number talks that build toward a generalized use of a repertoire of strategies for addition and subtraction (based on Fosnot's landscape)


  • as new strategies emerge during number talks, have students create a classroom Math Strategy Wall and add these strategies (word and picture) to it


  • once new addition strategies are acquired through number talks, provide opportunities for the student to apply these newly acquired strategies while engaging in rich tasks, such as open Math questions
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General Observations About Grade 3 and Grade 4 Student Learning During Our Collaborative Inquiry

From the beginning of our collaborative inquiry the Collaborating Teachers and the SWST observed that students also demonstrated progress in:


  • recognizing and describing themselves as Mathematicians


  • recognizing and describing their peers as Mathematicians


  • recognizing the value of making mistakes because mistakes help their brain to grow


  • supporting their peers when mistakes were made


  • their willingness to take risks and make mistakes


  • recognizing that deep thinking in Math takes time because all brains work differently (a faster answer is not necessarily a better answer)


  • taking the necessary think time they needed to provide an answer during number talks


  • supporting their peers in taking the necessary time needed to provide an answer during number talks by waiting patiently and silently during the peer's think time


  • actively listening to one another's thinking during number talks in order to learn from their peers


  • the use of the addition strategies of splitting, keeping one number whole and taking jumps, algorithm and doubles (Grade 3)


  • using the names of the strategies that emerged through number talks when explaining how they had solved an addition equation (Grade 3)

General Observations About Teacher Learning During the Collaborative Inquiry

From the beginning of our collaborative inquiry the Collaborating Teachers and the SWST noted the following learning:


  • the value of another educator's voice and eyes in the classroom


  • the value of providing think time for students


  • the value of encouraging students to make mistakes


  • the value of actively listening to student thinking and peer sharing as a form of assessment for, as and of learning


  • multiple new strategies to teach double digit addition beyond an algorithm, with and without regrouping, including splitting, taking jumps and doubles


  • an ability to name specific addition strategies and observe for them
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What Do Students Have to Say About Number Talks?

All Grade 3 and Grade 4 Focus Students were asked two questions at the culmination of our collaborative inquiry:


  • Do you like number talks?

  • Are number talks helping you to get better at Math?

Focus Student Voices

  • 100% of Focus Students indicated that they like Number Talks

  • 100% of Focus Students indicated that Number Talks are helping them to get better at Math

Class 3B


Student D-

"Yes. You get to talk about numbers and skip count."


"Yes. I'm better at skip counting now."


Student K-

"Yes. I like having a turn and think time."


"Yes, because we're talking about Math and that helps me."


Student V-

"Yes. number talks give us time to think and we can use different ways to get an answer. They're good for your brain. It gets bigger."


"Yes. Number talks help me with (my work) in class."

Class 3S

Student A-

"Yes, especially Counting Around the Circle. I don't like Choral Counting because I like having think time."


"Yes, because we're practising skip counting and I'm getting faster."


Student L-

" Yes, because you learn new math strategies."


"Yes. I've learned splitting, keeping one number whole and taking jumps, making friendly numbers and doubling."


Student N-

"Yes. It helps me to learn."


"Yes. I can skip count in my head better."

Class 4S:

Student K-

"Yes. They're fun."


"Yes. I am better at skip counting."


Student L-

"Yes. I like counting with bigger numbers."


" Yes. At first I didn't know what 25+25 and other stuff was, but now I do."


Student S-

"Yes. I like think time and it helps me get better at Math.


"Yes, because we're practising a lot."

Conclusion

The resulting student learning from our collaborative inquiry supports the power of purposefully constructed number talks when students are actively engaged in them a minimum of three times per week.


At the outset of the inquiry, we anticipated that there would be an increase in students' fluency regarding addition (accuracy, efficiency, appropriate strategy selection and flexibility) and, in Grade 3, there was. In fact, at the culmination of our collaborative inquiry, most of the Grade 3 Focus Students were able to accurately solve the Grade 4 PRE Fluency Interview question using more than one appropriate strategy in an efficient manner; a question which represents Grade 3 end of year curriculum expectations. Additionally, most Grade 3 students were able to accurately use the names of the strategies that emerged through number talks when explaining how they had solved an addition equation. Of importance to be noted is that this progress was documented over a period of approximately only seven weeks.


In Grade 4, as previously indicated within the context of this Smore, we were not able to venture into the emergence of a variety of strategies for the computation of addition. While definite progress for all Grade 4 Focus Students in accurate skip counting were evidenced, there was little progress noted in students' fluency regarding addition and this may represent a focus for future inquiry.


In both Grade 3 and Grade 4 there were very positive unanticipated student outcomes that were realized during our Collaborative Inquiry relating to how the students increasingly viewed themselves and their peers as Mathematicians, valued mistakes and think time and actively listened to and learned from one another's thinking. Essentially, the norms co-created with students for engaging in number talks resulted in a transformed, highly positive climate for learning during Math.

Possible Questions for Further Inquiry

The student data for our collaborative inquiry was documented over a period of approximately seven weeks. What further student growth in relation to student development of number sense and fluency might be evidenced given a similar collaborative inquiry of greater length?


What student growth in relation to student development of number sense and fluency might be evidenced using the Fosnot number strings for multiplication in Grades 3 and 4?


Will the positive class climate during Math that resulted from the implementation of number talks carry over into another Math collaborative inquiry of a different nature within each of the classes?

Contact Information

Sandra Fobert M. Ed., OCT


Student Work Study Teacher

Peel District School Board