Early elementary teachers can do various things to facilitate opportunities for their students to use their own pictorial representations, make more meaning of their written mathematical systems, and improve their mathematical thinking. To develop a culture in which children represent their mathematical thinking, certain components must be present in terms of the physical environment, the classroom community, the styles of interactions, instructional practices, routines, and professional communities. Simply telling students to "draw what they are thinking" is not enough.
First of all, the physical environment of the classroom should be rich in mathematical content. In a rich mathematical environment, a variety of manipulatives, whether commercial or found, are provided as concrete learning tools. Throughout the classroom, one can see number lines, calendars, graphs and charts on display at a level that is accessible to children. As Carruthers and Worthington note, providing a "graphics area" that includes not only literacy-based, but also math-related resources is critical to the math-rich environment. This area should include writing and drawing tools, paper of varying sizes and shapes, clipboards, graph paper, forms, lists of children's names, rulers, calculators, measuring tape, check books, shapes, clocks, etc.
In addition, in (most likely preschool and kindergarten) classrooms that have role-play areas, strategic placement of notepads by telephones, clipboards by doctor/police/ firefighter areas, calendars on the refrigerator, and shopping lists and coupons in the store are just a few more ways to create a math-rich environment. Of course, simply providing math-related objects and realia does very little to develop students' mathematical thinking if a nurturing community that embraces children's mathematical thinking and representations is not established and maintained.
To begin, if the physical resources noted above are in a teacher's classroom, he or she must pay attention to make sure students can use them to help build their understanding of mathematical concepts. For example, the math manipulatives should be readily accessible to students, and limitations should not be put on which manipulatives students can use for which math problems. Rather, a teacher can facilitate discussions as to how students use different resources and how effective and efficient each of them are. The displays and data placed on the walls should be meaningful to students and caution should be taken to ensure that students' representations carry as much weight as conventional representations. The "graphics area" described above should be made available not only during math time, but also during any sort of "free time." As "free time" dwindles across school districts, we have to get creative as teachers and create "mobile graphic areas" and provide access to these resources at any spare moment that we can find: at arrival and dismissal, when students get done early with their breakfast or lunch, and at recess or during "Fun Friday." Finally, and perhaps most importantly, when these moments are provided, be sure to pay attention to what students develop during this time.
Interacting with Students
When we listen and carefully observe what students are thinking, we not only show that we value children's thinking and graphical marks, but we also get significant insight into what students' mathematical understandings are and how they can be valued and further developed within our classrooms. While allowing students to use their own pictorial representations is very meaningful for them, if we are not listening carefully to what students are saying and what their drawings are representing, we can easily make inaccurate assumptions as to what a child knows or doesn't know. A Primary Teacher Associate with NRICH (a team of teachers supported by the University of Cambridge aiming to enrich the mathematical experiences of all learners), Bernard Bagnall, pleads for caution on making assumptions when looking at children's representations of mathematics:
We need to consider the child's ability to communicate (and the opportunities we
provide for communication), the child's understanding of his or her own
mathematics and to bear in mind that this may not necessarily, at this stage, be
totally in line with 'school maths.'
By paying close attention to students' "self-talk" and talk among peers as they developed their mathematical representations, Woleck noted how the language attached to these rich pictures provides significant insight into the students' mathematical thinking and understanding. Through these informal and spontaneous conversations, "children came to question, debate, defend, clarify, and refine their mathematical understandings."
Of course, in order to nurture these types of interactions, a teacher must be willing to allow students to talk to one another and to think through their process out loud. Also, circulating and listening in on these conversations, focusing on students' mathematical thinking, will play a valuable role in assessing where students are and providing additional opportunities for discussion.
In addition to providing opportunities for these student-initiated discussions, teachers should carefully plan and adapt lessons to facilitate discussion that takes advantage of the mathematical knowledge and experiences children bring to the classroom. A "Capacity Building Series" publication distributed by the Ontario Ministry of Education gives several suggestions about how to facilitate "math talk" that will encourage and foster children's growth in the understanding of mathematical concepts in the early elementary classroom. One very important component is allowing students to talk about their mathematical thinking after they have worked through solving a problem. During this time, the teacher encourages students to share out their variety of strategies, justify their solutions, and make connections between them, while working to facilitate the development of generalizations of mathematical concepts.
This same article proposes the use of Suzanne Chapin's "Five Productive Talk Moves" to further facilitate meaningful discussions around mathematical thinking:
1) Revoicing—Repeating what students have said and then asking for clarification
So you're saying it's an odd number?
2) Repeating—Asking students to restate someone else's reasoning (
Can you repeat what he just said in your own words?
3) Reasoning—Asking students to apply their own reasoning to someone else's reasoning (
Do you agree or disagree and why?
4) Adding on—Prompting students for further participation (
Would someone like to add something more to that?
5) Waiting—Using wait time (
Take your time...We'll wait…
Regardless of the curriculum a district is using, facilitating discussions using some of the above guidelines can be extremely valuable in helping students share their mathematical thinking and more deeply understand mathematical concepts by seeing and hearing about how their peers have solved similar problems. Of course, if and when students have a challenging time articulating their mathematical thinking, using their pictorial representation can help them communicate their ideas and provide others with visual aids to understand their process.
Bansho (Board Writing)
When students are sharing their mathematical thinking, the classroom becomes an environment in which student learning is consolidated and both individual and collective mathematical thinking is occurring. One instructional strategy that is used in Japan and, more recently, in Ontario, is referred to as "bansho" or board writing. The Ontario Ministry of Education has adapted the Japanese bansho and made suggestions for implementation to further facilitate children sharing their mathematical ideas in their classroom community. To plan a "bansho" lesson, teachers should use what they know about their students and the grade-level standards to develop an objective, choose a problem that is aligned with this objective, and anticipate each part of the lesson. The problem-solving lesson itself consists of the three parts: before, during, and after students solve the problem presented. Before working on solving the problem, the teacher activates students' mathematical knowledge and presents the problem to the students, as well as records it on the board or chart paper. Students identify information provided and needed to solve the problem, and this information is recorded on the board. Next, students are provided with paper on which they can represent how they solve the problem and record their solution. During this time the teacher observes and records students' solutions, as well as facilitates discussions among students in anticipation of the final parts of the lesson. After students have solved the problem, the teacher guides a class analysis and discussion of solutions, during which time students' work can be sorted and reorganized on the "bansho" in order to draw attention to commonalities and progression towards the objective. During this discussion, the teacher should annotate on or around the students' work in order to explicitly expose students to conventional, abstract representations. Following this discussion, the class collectively develops and records a summary or highlights the key ideas, strategies and representations as they relate to the objective. Finally, the teacher provides students with some additional practice problems to which they can apply their developing mathematical understandings.
While this is not a format that would necessarily be used for every math lesson, I believe the bansho instructional strategy is an ideal framework for solving story problems. It allows students to represent their own strategies and solutions, promotes a mathematical community that values children's thinking, facilitates rich discussion and conversation around mathematical thinking, and provides opportunities for the teacher to effectively and appropriately model alternative ways of representing mathematics.
Modeling vs. Examples
With the above "bansho" instructional strategy, the teacher does not provide examples of how to record a solution using an equation before students work on solving the problem. Rather, the teacher waits until
students have solved the problem to annotate additional models of representation that can be used. This format and the timing of introducing abstract symbols to students are very much in line with the philosophy and suggestions presented by Carruthers and Worthington. They highlight the confusion that often arises between the terms
and caution that modeling should not be interpreted as providing direct examples:
We knew that we needed gradually to introduce children to standard symbols and
various labels but when we provided an example at the beginning of a lesson –
intending to offer one possible way – the children copied exactly what we had done
with limited understanding.
When providing direct models that may be increasingly abstract for children, teachers should display models along-side with and in reference to children's own representations in order to demonstrate that they value what students are bringing to the classroom as well as connect this new type of representation to something that is meaningful to the students. As students get older, it will become more and more necessary to provide representations that can communicate with a broader audience, but particularly when approaching new (and potentially more complex) problems, students need to understand their own process and style of representation first.
Cognitively Guided Instruction and Solving Problems Using "Just-Right" Numbers
As students develop their mathematical understandings at different rates, differentiation is needed in order to ensure that students are able to approach solving the problems provided, but that they are also adequately challenged so that new learning can take place. One such differentiation method often used by teachers is to provide different students with completely different story problems. While this method can be rather time-consuming, challenging to manage, and ineffective in helping students deal with the variety of problems as outlined by CCSS (such as adding to, taking from, putting together, taking apart, and comparing),
there is another potential way to differentiate story problems. As part of the Cognitively Guided Instruction approach
(which would likely support many of the strategies outlined above), this differentiation strategy is known as "Using 'Just-Right' Numbers." Once established in the classroom setting, it can be a very appropriate instructional strategy for students with a variety of mathematical understandings, and it is feasible to facilitate. With this strategy, teachers determine what type of story problem they would like to have students work with and write a story problem frame. A basic example of a "joining"
frame would be: "Raúl had___ apples. Jonathan gave him ___ more apples. How many apples does Raúl have in all?" Then the teacher develops sets of numbers that will meet the needs of his or her variety of students. For example, one student may need to work on the concept of simply putting together two quantities, so the set of 2 and 3 would be manageable for him or her. Meanwhile, a group of students may be working on combining sets that are greater than 10 and determining how to recycle their fingers, so a set of 8 and 5 might be appropriate for them, or 5 and 8 if an emphasis is needed on adding more efficiently by counting on starting with the greater number. More advanced students may be working with understanding place value, in which case a set of 18 and 30 might be appropriate for them. By using the same story line, this story problem can be introduced to the whole class, using the basic set [2, 3] as a placeholder. This way everyone can hear an example of the story problem, visualize what is going on, and identify what is known and unknown before moving forward with developing a strategy to solve the problem. Depending on the needs of the students, all students can start with the basic set and work their way through progressively more challenging sets of numbers, the teacher can help guide students to the set that is most appropriate for them to start with, or the students can select which set of numbers is "just-right" for them. Most importantly, with the implementation of this differentiation strategy, the teacher engages all students in the problem-solving process, and they are able to use strategies and represent them in ways that are most meaningful to them.
Establish a Routine for Solving Story Problems
As with all frequently used procedures, it is very beneficial to develop a routine for approaching and solving story problems. The anchor chart shown in Figure 3 illustrates the steps that I have taught my first-grade students to follow when solving story problems, and they can be adapted to accommodate other grade-levels and classes. The research that I have done in preparation of writing this unit has both confirmed the method that I have used, and revealed new ideas as to how it can be revised and improved upon. It is best to spend at least one lesson on each "step" of the problem-solving process towards the beginning of the year and re-visit individual steps as needed. While focus may be on one step or another based on the lesson objective or needs of students, be sure to go through all steps to reinforce what students should do when they are solving problems independently.
Figure 3: Anchor chart used to help establish a problem-solving routine
At the beginning of the year, as many first-grade students are still emergent readers, the first step is to listen to the story problem. The focus during this step is to visualize what they hear or read (which is also practiced during literacy) in order to make more sense of the story problem. Next, we work on retelling the story problem using our own words. I encourage the students to keep track of what they already know from the story and identify what it is that they need to find. Since I have been using
, I have attempted to use the "number bond" as more of a graphic organizer to help students organize their information. When beginning a story problem with first graders, we often do the first two or three steps as a whole group. As students become more independent, I release them to do the third step and even the first and second independently. However, steps four through six are always done independently or with a partner. A significant number of lessons are spent identifying different strategies that students are using, as students continue to develop more strategies and more efficient strategies throughout the year. While the fifth step on my anchor chart is listed as "record," based on my expectation that students represent the thinking they used to solve the problem, it should be labeled as "represent." If students are using "drawing" as a strategy, the fourth and fifth steps may be combined. Because students already have been exposed to a significant number of worksheets and been requested to record things that are not necessarily meaningful to them, representing their own thinking has proven to be challenging for them. By very mindfully observing and listening to students (as discussed above), I have been able to model different ways that individual students can communicate their thinking and strategies. I usually do this both with spoken words and through pictorial representations, and I would encourage other teachers to do the same as a means of helping students become comfortable with representing their thinking independently. One common question that comes up is from students who "just know" the solution to a problem. I encourage them to try to represent that on their paper, as well as represent how they can
that they are correct. This leads us to the final step, which is checking their work. As their peers share more strategies, students often will check their work by using a different strategy to see if they arrive to the same solution. While this routine may feel lengthy, it helps to ensure that students are, indeed, solving problems in ways that are meaningful to them and, thus, improving their mathematical thinking.
Professional Learning Communities
Finally, one critical part of maintaining a mathematical community is ensuring that all educators (paraprofessionals, teachers, parents, administrators) have the opportunity to discuss and develop their own understandings of mathematical processes, as well as the mathematical thinking that our students are representing. This sort of engagement can take place in a variety of professional learning communities, including grade- or building-level data teams, vertical (across grade-level) teams, teacher- or leadership-led professional development, parent-teacher associations, parent-teacher conferences, or one of the many social networks available. Ideally, we, as educators, will be able to share what we are discovering about children's mathematical thinking, get insight and ideas from each other, and develop even better strategies to meet the needs of our students.