In A Collaborative Classroom
Student Work Study Overview
The Student Work Study Inititaive (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. The professional goal is to positively impact students by applying a growth ethos 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 curriculum content.
During the 2015-2016 academic years, there were seven SWS teachers in the Peel District School Board. Each teacher was assigned to two 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 10 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 reflected the applied research experience.
NB: The 2015 school year began with the ETFO union in a position of continued bargaining with the province. As a result of this, SWST did not begin their role in full until the end of November. Prior to that time SWST were in schools developing relationships in a supporting role.
The fieldwork for this study was conducted in a K-5 public school in Malton, Ontario from October of 2015 to January of 2016 (wrap up sessions). The SWST spent two days per week at the school. 468 students attended the school from a range of cultural backgrounds.
The collaboration occurred in one grade 2, three grade 3, and one 3/4 split classes. There were 20 students in the grade 2 class and 20 students in the grade 3 classes. The 3/4 class had 26 students (8 grade 3 and 18 grade 4). The grade 2 class shared a ‘pod’ area (four rooms with some minor dividing walls) with two other classes, one grade two and, one grade one class. The grade 3 and 3/4 classes were ‘stand alone’ classes and shared a hallway in the upper portion of the school.
The grade 2 class was 65% boys, 35% girls, the grade 3 and 3/4 class were approximately 50% each.
One grade 3 class had a student with exceptionally challenging behavioural needs who required constant monitoring by classroom teacher and support staff. The remaining grade 3 classes had varying degrees of challenging behaviour students. The grade 2 class had no students (participating) with challenging behaviours.
5 teachers participated in this collaborative inquiry, including the Student Work Study Teacher (SWST), and four classroom teachers. These classroom teachers were responsible for delivering most curricular areas, with the exception of physical education, music and drama. Three of the teachers were new to the school, as well as their grade level, one teacher was new to the grade level.
How Did it All Start?
At the beginning of the year the SWST was working with the grade 2 classroom as a resource to suggest and assist in the implementing of strategies for two students integrated into the space with an ASD diagnosis. A positive relationship was formed with the future host teacher.
The impetus of this inquiry came from the SWST observing the students in the grade 2 class playing a game of 'Subtraction Bingo'. The students were solving the subtraction problems on their card and one student was "stuck" on the problem of '20-10'. The host teacher asked the student how he would solve this problem and the student replied that he would need to use counters. The student retrieved some of the math manipulatives and began counting them out in order to solve the problem.
At this point the SWST initiated a conversation with the host teacher about Number Talks, and the idea that students as young as kindergarten are able to make doubles and count by tens, which is one of several strategies for addition, subtraction, and multiplication that Number Talks foster: “Beginning as early as Kindergarten, children are able to recall sums for many doubles. This strategy (Doubles/Near Doubles) capitalizes on this strength by adjusting one or both numbers to make a double or near-doubles combination.” (Parrish, 2010)
Also, that number talks could help her students build their overall mathematical thinking, discussion, and reasoning while “creating understanding of the numerical relationships that provide the foundation for the mathematical rules and procedures” (Parrish, 2010).
The host teacher was quite interested and thus began the journey…
What are Number Talks
Number talks arose from a situation where it was noted that: “classrooms are filled with students and adults who think of mathematics as rules and procedures to memorize without understanding the numerical relationships that provide the foundation for these rules.” (Parrish, 2010)
Simply defined; number talks are five- to fifteen-minute classroom oral sharing of mental math strategies used to solve purposefully crafted equations posed by the educator, who takes the role of facilitator. Number Talks are not intended to replace curriculum work, or to take up the majority of time spent in a mathematics class.
Further, a number talk is a brief, daily routine which gives students the opportunity to practice with computational fluency, to explore number relationships within addition, subtraction, multiplication and division, and to share and learn with their classmates.
“The National Council of Teachers of Mathematics (NCTM, 2000) recommends that students explore different strategies to solve problems and understand how different strategies are used in different contexts...Instead of providing the answers and listing the steps to follow, we become facilitators and questioners, guiding discussions and helping students make meaning of the mathematics.” (Krpan, 2013)
In Number Talks, it is the students who discover the strategies.
Importance of oral communication/strategy sharing in mathematics
We recognize that literacy in our classroom means going beyond simple reading and writing. We recognize mathematical literacy, goes beyond pencil and paper tasks, and the use of mathematical discourse, with tools such as number talks, fosters that mathematical literacy.
“When we encourage students to talk about mathematics, we provide them with opportunities to rehearse their thinking and link ideas together...Student achievement and engagement are improved when students have opportunities to rehearse their thinking and link ideas together.” (Krpan, 2013).
The Grade Threes
For the grade three team, 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 what we were seeing and hearing from this student work. Some of the student samples that were discussed that day led to the following observations and discussion:
The consensus amongst the team was that students, anecdotally, were successful when they were given the opportunity to discuss and share with their classmates their thinking during math problem solving. The student work study teacher, who had already been working in the grade 2 class using Number Talks, then made a presentation on the work and progress that had been going on in that space, that students were able to create, utilize and identify multiple strategies of addition when given the chance to share with each other, orally, the mental math strategies that they were using.
Theory of Action
If we implement 'Number Talks' in our classroom as a means of sharing student thinking, then students will become more adept at communicating and justifying their strategies.
The hope was that this communication and justification would become present in their writing, following a given amount of time practicing orally.
This theory of action became important, not only in the grade three class, but in the grade two class where the SWS had spent a longer period of time, as such the students had greater exposure to the number talks. The host and SWS teacher were able to apply this theory of action, and concluding results, to the work being done in the grade two space.
- apply developing reasoning skills to make and investigate conjectures (e.g., through discussion with others);
- communicate mathematical thinking orally, visually, and in writing, using everyday language, a developing mathematical vocabulary, and a variety of representations.
Resources and Research Referenced
Helping Children Build Mental Math and Computation Strategies
By Sherry Parrish
Developing Student Thinking and Problem Solving Through Communication
By Dr. Cathy Marks Krpan
for Orchestrating Productive Mathematics Discussions
By Margaret S. Smith, Mary Kay Stein
By Dr. Cathy Marks Krpan
The Ontario Curriculum
Monographs: Capacity Building Series
Collaborative Inquiry in Ontario: What Have We Learned and Where Are We Now?, September 2014
Where to start?
5 Small Steps
In her book ‘Number Talks’ Parrish suggests starting with five small steps:
1. Start with smaller problems to elicit thinking from multiple perspectives.
2. Be prepared to offer a strategy from a previous student.
3. It is all right to put a student’s strategy on the back burner.
4. As a rule, limit your number talks to five to fifteen minutes.
5. Be patient with yourself and your students as you incorporate number talks into your regular math time. (Parrish p.28)
The host and SWS teacher decided to focus on a strategy of ‘making tens’ for the first number talk. This involves having students break numbers apart in order to quickly make ten, being able to take numbers apart with ease, or fluency, is the key to using this strategy.
Initial objectives for our first session included establishing the following norms:
- Use of the ‘silent thumb’. When students have solved the equation on the board using their chosen strategy, instead of raising a hand or calling out an answer they are to place their thumb on their chest. They may raise an additional finger for each strategy they have used to solve the equation.
- To let students know that the activity was about using ‘mental math’, and that they would be given appropriate time to solve the equations, in their head.
- Further, that a key component of Number Talks occurs when students are called upon to ‘defend their solutions’, this is the time when students will share strategies and learn from each other.
- Number Talks are a ‘safe space’ and if after hearing students share solutions a student who has provided an incorrect answer wishes to change it they may do so
- A 5-15 minute time goal for each session
Number Talks took place on a large movable white board. The students were seated on a carpet in front of the board, but within visual contact of our (later developed) strategy wall. During subsequent Number Talks students would begin to transition automatically to the ‘Number Talks’ area. “One way to elevate the status of number talk time and signal its importance is to have a specific location in the classroom where the students gather for this purpose.” (Parrish p.17)
The number talks for the grade twos took place in an 'open concept' pod, and the noise from the adjoining classrooms added some difficulty to following and recording the strategies, as the student voices were difficult to register.
Initial Number Talks Observations
The majority of students were found to use the strategy of ‘counting all’ (starting at 1, counting up to the first addend, then adding the second addend, again by ones) or ‘counting on’ (starting at the first or larger addend and adding the second by ones). Students did not seem to be ‘jumping’ to our chosen strategy of making tens. Although Parrish offers that teachers should be prepared to “offer a strategy from a previous student” as a way to “help to know how to enter the conversation”. Discussions were had between the host and the SWS teacher on the idea of introducing strategies to students however, it was decided that there was importance and power to the students discovering the the strategy on their own, and not being directed to it.
The host and SWS teacher engaged in many conversations, such as the one above, during the collaborative inquiry process, often times during the conducting of the Number Talks, to the point where the students became accepting, even expecting, of this co-learning approach.
On the last equation, one student, A, had a more complex sounding method for arriving at his answer. The host teacher and the SWST attempted to dissect his answer with the class, but it was found to be too complex for whole class discussion. The rest of the class was sent to continue their inquires, while the SWST sat with the student to have him further explain his thinking. Putting this strategy on the ‘back burner’ allowed for the class discussion to continue, while still honouring his individual answer.
The results of that conversation are highlighted in the following picture.
Student used a complex (and less efficient) method of decomposing the equation, he did not need to add 6+3 initially, but appeared to have greater flexibility with numbers than some of his classmates.
He was able to explain his method out several times and never varied.
Here, students' requested to do their own number talk during a 'class meeting':
Student engagement was quite high throughout the process in the grade two space. The students were eager to share their solutions as well as their strategies. Initially students names were recorded on the board next to their answer, a method which was later stopped, as it seemed to encourage students to share their answer even if it was the same which had already been shared.
Regardless, the facilitator of the Number Talk still continued to provide students with the opportunity to voice a a 'same answer' as it was felt this would assist in keeping the students involved and engaged.
On a day that the SWS was not in the school, the following email was received from the host teacher:
The kids requested a "number talk" for one of our class meeting blocks. So great to see this love of learning for math. Thank you for inspiring that and making it contagious.
Shown here is an example of the collaborative anticipation that took place between the host and SWS teacher.
In their book “5 Practices” Smith and Stein introduce anticipation as the first practice “...make an effort to actively envision how students might mathematically approach the instructional task that they will work on...how students might mathematically interpret a problem, the array of strategies – both correct and incorrect.” Further they state that “Sometimes teachers find that it is helpful to expand on what they might be thinking individually by working on the task with colleagues...” (Smith and Stein, 2011). This is supported as well by the Ministry of Education monograph on ‘Collaborative Inquiry’, where as part of professional discourse it encourages teachers to “Engage with colleagues and knowledgeable others to implement actions responsive to learners in the classroom.” (Collaborative Inquiry in Ontario What We Have Learned and Where We Are Now).
Prior to the initial number talk this anticipation did not take place. As such, the facilitator found it difficult to track and record student strategies as they were sharing. In all subsequent number talks anticipation was done, which made the facilitating more manageable.
The Collaborative Environment
In her book ‘Math Expressions’, Dr. Cathy Marks Krpan states “Before students begin to work collaboratively on mathematical activities, we need to ensure that they develop an understanding of group norms, or behaviours that guide cooperative work.” (Krpan, 2013).
At the beginning of the number talk on day 2, and in all subsequent experiences, we briefly reviewed the collaborative norms and expectations that we were establishing throughout the process, these included: choosing wisely where to sit when coming to the carpet (and what this means), using a silent thumb to indicate that you have reached a solution, we are here to engage in mental math, to learn from our peers by sharing strategies, and that mistakes would be valued (and students who offered an incorrect response would be given the chance to defend their incorrect response or to change it).
Utilizing Incorrect Answers
Following the strategy sharing the teacher conducting the number talks returns to the students who have provided incorrect answers and ask whether they wish to defend, or change their incorrect answer.
Of note, the same student provided an incorrect answer for each question, but would then change his answer given the opportunity. This was discussed with host teacher the possibility that this was attention seeking. However, he did choose to defend his incorrect answer for the last question, and while doing so named the strategy that he used as: 'counting up' (which was seen as a positive that he was using the math language). NB: When counting up he arrived at the correct answer, not his original one. This was also seen as a positive.
The above illustrates two conditions discovered when doing Number Talks firstly, that allowing for incorrect answers helped to foster a safe collaborative learning environment; “A first step toward establishing a respectful classroom learning community is acceptance of all ideas and answers – regardless of any obvious errors” (Parrish 2010).
Secondly, allowing students the opportunity to explore their incorrect answers could often lead them to self-correction, particularly after being witness to other students sharing correct strategies. “Mistakes are necessary for learning. It is not enough to say that mistakes are “celebrated.” The collective examination and understanding of mistakes as a regular part of mathematics instruction can shift students’ erroneous views of mathematics-as-answer-getting and help them learn to analyze the validity of procedures.” (Humphrey’s and Parker, 2015).
Host Teacher Takes the Reins/Collaborative Communication
The host teacher reported that the number talk was successful. The students were able to identify the strategy of 'making tens'. The host teacher was able to identify and illustrate the strategies that the students were using. The students now have three addition strategies (counting all, counting up, and making tens) from which to draw from.
Communication from host teacher following her initial number talk:
Good news, one of the students discovered the making tens strategy!
I myself called "counting on" counting up instead. I also used the number line to show "counting all" when it should only be used to show "counting on".
Even something like colour coding while teaching using the white board gave me lots to keep track of.
Definitely lots of learning and more to come.
Thanks for sharing.
The above email illustrates the ongoing communication that occurred between the host and SWS teacher.
Teacher and SWST conversation led to the feeling that the students had grasped the concept of making tens, and that given these problems it was the most efficient strategy. We decided to do one (or two) more making tens number talk, to create anchor charts with the strategies used thus far, and to select a new strategy to move on to.
Photo from SWST led number talk on day 6.
To be noted, students are no longer using (or at least defending) counting all. SWST and host teacher discussed that observing students counting on their fingers might indicate that they are still "counting all" and that those students would benefit from a small group number talk.
As can be seen here, the students arrived at the 'decomposition' strategy. They were decomposing to make tens.
Collaborative Communication in Mathematics and the ELL
Part way through the inquiry a new student joined the class who had emigrated to Canada recently and was an English Language Learner (ELL). She was included directly in the Number Talks when she arrived, and the SWST received the following email from the host teacher about her participation:
I can't remember if I mentioned the new student in our class just arrived to Canada with no English language and participated for the first time during your math talk session with the kids and again today. It was neat to see the rest of the kids so surprised and trying to figure out how she was able to participate when she only spoke Urdu in our class all week. How exciting to be able to share that with the parent when mom asked how her daughter did this week.
In her book ‘Math Expressions’ Krpan discusses supporting ELL learners in the mathematics classroom, she notes that some educators may decide to not include ELL learners in math discourse, but may instead have them engage in a separate activity, a practice which she cautions against: “By excluding ELLs from experiencing mathematical discourse, you may be limiting their opportunities to develop an understanding of how mathematical vocabulary is used in natural learning contexts.” (Krpan, 2013)
Midpoint Interviews and Opportunities
At this stage the host and SWS decided to focus on three students (designated as “low”, “medium” and “high”) in order to collect some data on the impact that the number talks were having in the classroom. It was observable, anecdotally, that math discourse and learning were improving, but were there any deeper understandings? The SWS teacher withdrew the three selected students, separately, and prompted them with: "Tell me about number talks."
Please click on the following link to access the student interviews:
The SWS and host teacher discussed the interviews, and drew the following conclusions: Students were able to identify the process involved with number talks, one student giving a specific example, and they were all able to name the strategies that we had covered in class. However, there did not seem to be any “deeper” connection, saying that it makes math easier, more efficient etc. This was possibly because little time had been spent in the number talks discussing and comparing efficient strategies (time constraints) or perhaps the issue was that the learning goals were not made clear and reinforced throughout the number talks.
Establishing Learning Goals
Discussion with the host teacher about the mid-point interviews led to the decision to have a meeting with the class in order to discuss WHY we have number talks and to introduce and establish the learning goals. Efficiency. Flexibility. Confidence. Fun.
From the ASSESSPEEL monograph: A learning goal is a statement that clearly articulates the learning that will take place during a lesson or a series of lessons.
A defined focus for our learning goals became about accuracy, efficiency and flexibility (and primarily the later two): “Accuracy denotes the ability to produce an accurate answer; efficiency refers to the ability to choose an appropriate, expedient strategy for a specific computation problem; and flexibility means the ability to use number relationships with ease in computation.” (Parrish 2010)
Scaffolding was done in order to facilitate learning goal conversation.
The writing in blue are specific examples of learning goals that students had encountered in literacy (the host teacher noted that they primarily discussed learning goals in reading and writing - and less often in math), the writing in red represents generalized information about WHAT a learning goal was, with the word 'learning' being written in green. Students were given time to "turn and talk" with an elbow partner prior to contributing their ideas. This was found to be beneficial.
Ideas that the students had on what specific goals would be for number talks; WHY we do number talks.
The 'general' information about learning goals was left on the board. Ideas that the students had for what number talks goals are are written in blue, the final goal, written in green, was provided by the SWST at the end.
The next step was to create an anchor chart of the number talks learning goals, in pedagogical language, to be referred to by students during the number talks.
Of note, the students were able to easily arrive at specific examples of learning goals, but the majority of them struggled to achieve an understanding of more abstract, what are learning goals.
Anchor charts were created.
The student's words are displayed on the right, and the "teacher words" on the left. By connecting the ideas on the right to those on the left, using string or some other material, it was felt would provide students easier access to the vocabulary.
The host teacher noted that as an added benefit to this is that students are being introduced to new vocabulary.
Assessment and Accountability
Six Suggestions for Accountability
From the beginning, host and SWS teacher discussed how to assess the learning that was going on with the number talks in the classroom
Parrish suggests six ways to develop accountability with students:
1. Ask them to use finger signals to indicate the most efficient strategy.
2. Keep records of problems posed and the corresponding student strategies.
3. Hold small-group number talks throughout each week
4. Create and post class strategy charts.
5. Require students to solve an exit problem using the discussed strategies.
6. Give a weekly computation assessment.
Initial Adoption of Strategies
After engaging in eight number talk sessions, students were given a computation assessment (paper and pencil), using a series of equations from the ‘Making Tens’ strategy. There were 14 samples that were returned by students, all of whom used strategies from number talks. Of those 14, 9 used counting on or counting all as their strategy, the simplest strategy. Of note though, even at that time, the students who were using the simple strategies were also including an illustration to support their solution.
The host teacher also assigned explaining number talks to parents as homework, and sent this communication to the SWS: “Explaining numbers talks to their parents was their weekend commitment. It will be great when they share and reflect on explaining the math strategies used with their parents.”
Assessing the Attainment of “Efficiency”
Throughout the process students were polled to identify which strategy were the most efficient:
Students were able to distinguish (based on silent vote - with an almost majority) that compensation was a more efficient strategy than counting on.
Students were posed the question (after first equation) of whether or not the two methods of using compensation were the same strategy or a different strategy, they were given time to turn and talk with a partner, and we then had a class discussion. They voted (overwhelmingly) that they are different strategies. The implications or importance of this are not known. When asked to explain their opinions most students pointed to the fact that different numbers were used to compensate.
SWS and host teacher debriefed after that the reason that the students felt the strategies were different was because the numbers were different.
“Number Talks offer an opportunity to glean information about each student’s thinking and approach to the problem. Having the children in close proximity allows you to observe whether the students are using or developing an over reliance on certain tools during mental computation. Possible situations to note for second-through fifth grade students are – use of fingers for calculations, especially basic facts.” (Parrish, 2010)
Moving Students Forward
The SWS and host teacher noted that a number of students were still using their fingers when given time to solve the equations, further, there were a number of students who were choosing to use and defend the ‘counting on’ strategy, even after numerous exposures to more efficient strategies that were being identified and adopted by other students in the class. The host and SWS teacher further discussed how to move students along to more efficient strategies, but not 'banning' them from using a less efficient one.
"J" – A Success Story
It was decided to focus efforts on one student “J”. J had been previously identified as a mid to high level math student, but was noted to be continually relying on the strategy of ‘counting on’ in our number talks. The SWST had a conversation with J prior to a number talk and challenged her to, after using counting on, solve the equation using another method. She was promised that appropriate time would be given (which led to increased wait time).
As an additional form of accountability the host and SWS teacher decided to create anchor charts for the students to use. The strategies were not posted until the students were making functional use of them, versus waiting until they had been mastered, the thinking being that they would still provide meaning and be accessed by students during the number talks.
The host teacher noted the following, based on student use of the anchor charts: “They noticed your anchor chart and also commented on how "counting all" no longer seemed effective as it just "takes too long".
Collaborative Inquiry Key Learning
Through continued practice with Number Talks student 'G' became one of the greatest contributors to the strategy sharing. In following video, student 'G' makes use of compensation, effectively, and more importantly, with confidence.
At the conclusion of the collaborative inquiry, the grade two students were given a post assessment (paper and pencil) to complete. The assessment consisted of four addition problems from the Number Talks resource. The instructions on the assessment were: “Answer the following questions, in as many ways as you can, explain the strategies that you used to solve it.”
- 100% of the students used, identified, and illustrated a Number Talk strategy in their problem solving
- 43% of the students used multiple strategies to solve at least one problem
- 0% of the students used the standard algorithm to solve a problem
- 57% of the students used counting all or counting on (base strategies), correctly, but of those students 100% ALSO included a visual (a number line) to support their answer
Impact of the use of Number Talks
- Greater chance for students to have their voice heard in the classroom
- Increased focus and growth towards a more collaborative learning environment
- Increased engagement and excitement towards mathematics, coupled with an increased student confidence level
- Teachers were able to gather more information on student thinking, versus strictly paper and pencil tasks
- Teachers able to identify, and respond to, students lagging in understanding in a more timely fashion
- Number talks can be used as a "springboard" for richer mathematical discussions, such as: comparing and discussing the efficiency of strategies
- When you learn about number talks you could know the other answers by using new strategies you can get easier ways to do the hard questions
- Number talks are something that can help you learn mental math and help you learn faster in math, or equations that are hard for you
- I learn very new strategies that I didn't know. And I learned that it's a faster way to help you in math
- I want to learn new things from number talks and I want to solve equations that are hard for me so I can get better at them.
- Prompt => Are Number Talks helping you get better at math? Yes they are. Because whenever my mom gave me harder equations my brain was able to solve it. P-> When you are solving a question at home, in your book, do you now try and solve it in your head first? Yes.
- “Builds confidence...leading to success...leads to experimenting with numbers”
- “Provides for a common language to be used among students and teachers”
- “It was a real eye opener when students were asked what their math goals were. It was much easier for them to describe their language goals in reading and writing but not so much in math...why? We need to have more discussion with students around setting math goals and being efficient.”
- “Instills confidence in students, especially those who may struggle with the “standard algorithm” method of thinking”
- “Labeling the strategies allows students to better explain what they were already thinking”
Further Areas of Exploration
2. How will exposure to Number Talks effect students in their future years mathematics classrooms?
3. Exploring that multiple uses of the same strategy (on the same equation) does not make it a different strategy.
4. Getting student's to deeper explore the strategies, their commonalities, and uses in real world contextual problems.
5. How can we use common incorrect answers to deepen understanding of the class?
6. In regards to differentiation of instruction, where do Number Talks fit in?
Suggestions to Guide Further Discussion
In their book 'The 5 Practices' Smith and Stein argue that as classroom teachers we must go beyond the simple "show-and-tells" of strategies in mathematics classes. "Learning how to follow up on the ideas that surface in such student presentations is much more difficult". They go on to suggest certain moves that can be employed to go deeper with the learning. These could be applied to Number Talks and are:
1. Revoicing (repeating).
2. Asking students to restate someone else's reasoning.
3. Asking students to apply their own reasoning to someone else's reasoning.
4. Prompting further participation
5. Using wait time.
Peel District School Board