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For students to develop as mathematicians, it is critical that they actively engage in mathematical thinking and reasoning and take ownership of their learning. While the teacher is key to establishing the classroom culture in which the student develops attitudes, beliefs, knowledge, and skills as a mathematician, the student ultimately chooses whether or not to engage. We argue that formative assessment practices are a vehicle to facilitate students in taking on this role. In this chapter we describe the teacher’s role in developing a student-centered environment through the selection of tasks and use of feedback cycles. Ex-amples are drawn from a diverse third-grade classroom with a teacher committed to using these strategies to help her students become mathematical thinkers.

Clearly, what students are asked to engage in needs to be accessible, relevant, and math-ematically worthwhile. Thinking like a mathematician is more than a set of content stan-dards; it is also a set of broadly useful cognitive skills such as observing patterns, making models, translating among representations, communicating mathematically, and justifying generalizations (NCTM 2000; National Governors Association Center for Best Practices [NGA Center] and Council of Chief State School Officers [CCSSO] 2010). We have a vision of supporting students in becoming mathematicians using a research-based formative assessment cycle involving related aspects of instruction and assessment. Figure 21.1 shows the student’s role in this cycle. The figure highlights two main aspects of the cycle that sup-ports students becoming mathematicians. One aspect is regularly having students engage in mathematical thinking and reasoning; the second is assuring that students own their learn-ing. Bell and Cowie (2001) found that students do this by taking action on the information, expectations, and feedback they receive from the teacher. In this way, formative assessment enhances learning while the learning is taking place.

■ CHAPTER 21

Student Mathematicians Developed through Formative Assessment Cycles

Nicole Rigelman, Portland State University, Portland, OregonKellie Petrick, Hillsboro School District, Hillsboro, Oregon

Using Research to Improve Instruction

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Students Engage in Mathematical Thinking

and Reasoning

• Solve worthwhile mathematical tasks that are cognitively demanding and connected to appropriate standards.

• Engage in mathematical practices such as:

• observe patterns• make use of structure• model mathematics and translate

among various representations• engage in a process of conjectur-

ing and then justifying and generalizing those conjectures

Students Own Their Learning

• Understand the learning goals/intentions and the criteria for success.

• Ask genuine questions to deepen their understanding.

• Critique their thinking and the thinking of others.

• Engage in self- and peer-assessment.• Use feedback as an opportunity to

grow and improve.

Fig. 21.1. The student role in the formative assessment cycle

Students need to engage in metacognitive activities throughout the learning process. Marshall, Horton, and Smart (2009) define metacognition as reflection and self-awareness about one’s learn-ing and suggest that through reflection students determine where their understanding is complete and where it needs more work. When students reflect on their progress toward a learning goal with a thumbs-up, thumbs-sideways, or thumbs-down indication of their level of understanding, this not only provides teachers with important information but also provides information for students as they consider areas for growth. In elementary students, metacognitive reflection addresses “positive self-perception, motivation, and achievement” (Bingham et al. 2010, p. 61) and contributes to the development of higher-order thinking skills (Bingham et al.). Additional components of the student’s role and how teachers can promote such behaviors are in the upcoming sections.

■Assessment for LearningBlack and Wiliam (2009) assert “the teacher is responsible for designing and implementing an effective learning environment, and the learner is responsible for the learning within that environment. . . . The responsibility of learning rests with both the teacher and the learner.” (p. 7). If teachers and students are mutually responsible for learning, then they both must be users of assessment data (Bell and Cowie 2001). As a result of this mutual responsibility, teachers and students have distinct roles in assessment for learning. Figure 21.1 provides a summary of the critical student roles, for the teacher’s role, below are Black and Wiliam’s (2009) five strategies for teachers that make a difference in student achievement. The strategies are—

Student Mathematicians Developed through Formative Assessment Cycles ■ CHAPTER 21

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1. clarifying and sharing learning intentions and criteria for success;

2. engineering effective classroom discussions and other learning tasks that elicit evidence of student understanding;

3. providing feedback that moves learners forward;

4. activating students as instructional resources for one another; and

5. activating students as owners of their own learning (p. 8).

A teacher’s role in ensuring assessment is used for learning includes first setting clear goals and success criteria while communicating these to students. Next, the teacher selects cognitively demanding tasks (Stein et al. 2009) and provides students with tools and time to explore these tasks as individuals, with partners, and in larger groups (Hiebert et al. 1997). One way teachers support students’ understanding of the criteria for high-quality work is by collaboratively reviewing and evaluating exemplars with a scoring guide before students are asked to complete their final copy. Finally, teachers press students to own their learning when they ask students to monitor their progress toward the learning goals for the day by writing an exit reflection, self-assessing their work using a scoring guide (Fernandes and Fontana 1996; Higgins Harris and Kuehn 1994), or revising their work based on peer or teacher feedback (Chappius 2009; Hattie and Timperley 2007).

■Developing a Community of LearnersIn the research project described in this chapter, we worked collaboratively, seeking to under-stand more about how teachers develop productive mathematics learning communities. One of us, Nicole Rigelman, is a university professor; the other, Kellie Petrick, is a classroom teacher. We used Petrick’s class for our research.

Currently in the professional literature there are examples of productive classrooms (Hiebert et al. 1997; Kazemi and Stipek 2001) and recommendations for the characteristics of such class-rooms (Stein et al. 2009). Less available to most teachers is guidance on how a teacher achieves such a classroom and supports the development of mathematical thinking over the course of the year. Often, as teachers are new to implementing inquiry-oriented or student-directed instructional approaches, they try out something and do not experience much success because the expectations are so different for both them and their students. This leads them to abandon these approaches before either teacher or students can develop skills in these new roles (Manouchehri and Goodman 2000; Olson and Barrett 2004). We were interested in examining the various sup-ports a group of students needed to become mathematicians who conjecture, justify, and generalize about their mathematical ideas. We were also curious about how long the supports had to be in place before the actions and behaviors were normative in the classroom. We also wondered about challenges that are faced by teachers trying to implement these strategies in diverse settings.

Petrick’s class of thirty learners was diverse in several ways. Forty percent of the students (n = 12) qualified for free or reduced lunch. Seventeen percent of the students were English language learners (n = 5). Seven percent of the students were identified as students with spe-cial needs (response to intervention [RtI] Tier 3; n = 2) with a total of 37% (n = 11) identified as skills intensive (RtI Tier 2). Twenty percent (n = 6) were identified as talented and gifted. Seventeen percent (n = 5) had been diagnosed with attention deficit hyperactivity disorder.

We collected data from classroom observations, student work samples, student interviews, and


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notes from collaborative planning and debriefing conversations. These data were used to confirm observations of students’ thinking and plan next steps for classroom supports.

In the upcoming sections we offer a selection of findings from our ongoing inquiry into our practice. We share the ways in which developing the classroom culture, selecting the tasks, and incorporating feedback cycles influenced these third graders as developing mathematicians.

■Classroom Culture Focused on Developing Mathemati-ciansFrom the first week of school, these students heard about conjectures and the fact that mathema-ticians take time to make observations and see patterns. When students notice things they think are always true, they can make conjectures about ideas. In the same way that the content-focused learning intentions were made explicit to students, so were these cognitive skills-focused goals. This move of making learning goals explicit is critical in the formative-assessment cycle so that students are clear about their role.

In one activity, I (Petrick) reminded students of a conjecture that had been made, “an even plus an odd is always odd.” I told them that today I wanted them to decide if they think this conjecture is always true. I asked for a volunteer to build the number 8 on the magnetic board and another to build the number 9. The model revealed that when 8 and 9 are combined, an odd number, 17, is formed (see fig. 21.2). Not much discussion occurred about how they knew that numbers were defined as odd or even. Students returned to their seats to explore the conjecture with their partners.

Fig 21.2. An example of adding an even number and an odd

One pair discussed adding 27 to 100 and obtained the sum of 127. When asked how they decided 127 was odd they said “because 7 is odd and 27 is odd, but 100 is even.” When asked how they knew 100 is even there was a long pause. Finally, to break the tension I asked, “Is that some-thing you just know?” One student responded, “yes.”

Since the student had drawn a base-ten model (fig. 21.3), I drew attention to the model he had drawn and asked, “Can you tell by looking at this part of the diagram if 27 is odd or even?” The student responded that the 7 is odd. I clarified by asking if he looked at the 20 to see if it were odd or even and he said he did not. This led me to infer that he had been taught the rule of looking to


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the ones digit to decide if a number is odd or even. So I asked him to look at the 100, wondering if he could look at this 100 model and know if 100 is an even number. The student did not respond. I was not sure if he did not understand the question or if he did not know the answer.

Fig. 21.3. The student’s base-ten representation of 127

Another pair was looking at the following equation and using the rule of looking to the ones digit to determine whether the number was odd or even.

500 + 9999 = 10499 odd

In our debrief after the lesson, we discussed our observations about students’ strategies and their use of the rule versus the model for proving the conjecture. We decided that the next lesson should begin with posing the number sequence arranged as shown in figure 21.4 with a discussion about how odd and even numbers look, given this structure. Then students could move toward generic representation of odd and even numbers versus rules. This move helped students learn what it means to develop more convincing arguments for their conjectures.

Over time, conjecturing became a norm in the classroom. A conjecture box was established for students who were ready to record conjectures. Students would use the conjecture box if it were not yet an appropriate time to share their conjecture so that their thought would not be lost. An example of such a conjecture is shown in figure 21.5. As students were creating fraction number lines as tools that would allow them to compare fractions on a number line showing halves, thirds, fourths, and so on, one student thought the number lines should also continue in the negative direction in increments of sixths.

Another way the classroom culture was reinforced was through the feedback provided to students on their written work. In the following examples, students were solving the S’Mores task shown in figure 21.6. Students were asked to model the situation using a number line and clearly show what happened in this situation. In figures 21.7, 21.8, and 21.9 Petrick used feedback to communicate her expectation for clear labels and precision. At this point in the lesson, students typically either revised their work based on this feedback or set goals for their future work.


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10

10

Fig. 21.4. A visual representation of the odd and even number sequence

Fig. 21.5. A student conjecture about negative fractions during the construction of a fraction number line tool


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I made s’mores with my daughter and had some leftover chocolate for next time! How much choc-olate is left over? [The teacher shows the students the leftover chocolate that was equivalent to 11/2 or 3/2 chocolate bars.]

Fig. 21.6. S’Mores task

Fig. 21.7. A student showing that the left over chocolate is 1/2 + 1/2 + 1/2, 3/2, or 1 whole and 1/2 of a whole. All counts are provided in terms of twelfths on the number line.

Fig. 21.8. A student focused on representing the leftover chocolate but not suggesting what the other part represents. Counts on the number line are twelfths and other names are also given.


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Fig. 21.9. A student showing each part of the situation but labeling points on the number line as opposed to the quantity represented when 3/2 are left over or 1/2 is eaten.

As a final illustration of how these students were becoming mathematicians, we asked what it means to them to be a mathematician. Following are samples of their responses from end-of-the-year interviews.

• I think being a mathematician is when you are pretty good at math and when you really show work when you are doing something. Like if you are doing fractions or something, you use a model or a number line to show your thinking.

• It is someone who knows how to learn math well.

• Being picky about what models you are using, how your jumps are on the number line, when you are measuring, starting at zero, and things like that.

• It is about being confident in yourself and knowing you can do it. It’s not about talent, no matter how smart you are, you can work on it, rely on it more, and do it more.

• It is not about being the smartest kid in class. It is someone who relies on tools, no, uses tools, trying to decide if someone else is right or trying to prove your thinking to them.

These students do not believe that you are either good at mathematics or you are not. They believe everyone can be successful at learning mathematics if they persevere.

■Selection of Tasks to Ensure High-Quality DataPosing tasks that provide insight into students’ mathematical thinking and reasoning is critical to providing data that support the teaching and learning process. While there may be specific task features one may consider when selecting a task, such as providing opportunities to model with mathematics, translate among representations, and so on, what happens during implementation of the task influences the quality of the students’ responses. Hiebert and his colleagues (1997) suggest that reflection and communication are critical to developing deep understanding. Providing students with a scoring guide that communicates expectations for exemplary work can support students by bringing richness to the task (Fernandes and Fontana 1996).

The Brownie Sharing task, shown in figure 21.10, provided an opportunity for students to revisit an idea they had explored earlier in the year, congruence, but in the context of fair shares. Figures 21.11, 21.12, and 21.13 illustrate the range of student thinking, from focusing only on


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the fact that the brownies were divided into equal sized parts, so they must be equal, to showing through cutting and rearranging how the pieces are equal. Once students had almost completed their initial work on this task, they shared their work with a partner and gathered feedback regard-ing potential revisions to make their justifications clearer. Their work on this day closed with an exit reflection in response to the prompt, “What would you say to help Gabrielle understand your answer?” These moves supported students in finalizing a high-quality response while pressing them to be sure that they had convinced Gabrielle as they reflected on their solutions.

Gabrielle’s Brownie Isabella’s Brownie

Brownie Sharing TaskThe sisters decided to each order a different brownie so they could share. Gabrielle ordered a white choc-olate brownie and Isabella ordered one with peanut butter chips. Gabrielle said that the piece Isabella gave her was not half of her brownie. Their cuts are shown below. Did they each get a fair share of each other’s brownie? How do you know?

Fig. 21.10. Brownie Sharing task

Fig. 21.11. A student considering only the fact that the brownies were divided

into equal-size parts


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Fig. 21.12. A student considering that each girl received one part from each brownie and that two pieces from each original brownie made one whole

Fig. 21.13. A student transforming the brownie pieces so that the triangle forms an L shape

■Feedback and Revise CyclesWhile the basic premise of all formative assessment is that the teacher and students collect, inter-pret, and act on assessment data, not all formative assessment cycles are successfully completed.


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The collection of formative-assessment data alone is not enough to positively affect learning; therefore, teacher feedback to students, student reflection, and data-driven instructional decisions are all important components of assessment for learning (Black and Wiliam 1998). With this in mind, we share a structure for students to receive feedback from one another, samples of the teacher’s comment-only feedback, and the students’ responses to learning in a classroom where they had opportunities to receive feedback and revise based on that feedback.

To further the goal of putting learning in my students’ hands, I (Petrick) used explain-evaluate pairs where one student took on the explainer role, explaining her thinking and solution strategy to her partner while her partner actively listened. The partner was the evaluator, asking genuine questions to make sense of her partner’s thinking and letting the partner know if there were parts that were unclear. Once she completed this process, then the pair could continue work on the task either independently or as partners. The pairs were purposefully partnered so that the explainer was a student who could use additional practice communicating her mathematical thinking while the evaluator may have needed to work on listening and questioning skills. When the explain-evaluate protocol was new to the students, sentence frames and sample questions were provided to support students’ productive interactions.

Knowing that students often respond better to feedback when it is comments only rather than comments accompanied with an evaluation such as points or a grade (Black and Wiliam 2009), I worked hard to provide comments to students that connected to the learning intentions or to the habits of a mathematician. Below are some examples of comments on a recent set of fraction problems.

• Nice to show the number bonds [i.e., addition combinations represented pictorially to show the part-part relationship] that link to fractional units.

• Redo number bonds; a few changes are needed.

• Nice to show your skip counting on the number line.

• Does your number line need to show equal jumps? Why or why not?

• How many minutes would 1/4 of 60 minutes be?

• Label your model to match the information in your story problem.

• I like the way you match the model to a new model and use your labels to truly convince others.

• Fantastic skip counting . . . now prove the big idea that you need the same size whole to compare fractions.

Finally, we wanted to hear what the students thought about receiving feedback on their work and having a chance to revise. This was of interest to us as it would help us continue to refine our practice. Students’ statements regarding feedback included the following:

• Sometimes I feel kind of sad about that, like when she writes you a question. Sometimes it’s like “Wow, you did it” and sometimes it’s like “You need to rethink this.” It reminds me of being behind in my work because I have to do it the next day.


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• I think it is pretty good. She doesn’t say that we’re wrong, she says rethink. I might need to be more precise or to prove a little bit more. Maybe I need to match my models.

• I like getting feedback because when you get it you get to fix what you did wrong. Well not wrong, you just need to rethink. I like it because you get to know how other people think about your work.

• It is like having a conversation with the teacher. If I don’t understand something, I can ask a question and she will write me a note or a question that will help me think about it some more. Or we will have a conversation about it.

• I think it helps because when you go to a test she does not give you feedback but by then you kind of know a lot. When you have a chance to fix your stuff you notice what you are missing and I think that helps a lot.

These students, for the most part, experienced clear benefits from the self-reflection needed to read and respond to the feedback provided on their assignments. A few suggested that the feed-back even influenced their work on future assignments and tests, although it is difficult to know if this is related to their deepened content knowledge or well-developed confidence.

■ConclusionBoth teachers and students are important in the formative-assessment process so it can more fully contribute to learning. Formative assessment is not simply the tasks or questions used to prompt student thinking or the teacher using information from assessments to guide instructional decisions. It is also students engaging in a formative assessment cycle—a cycle empowering them as mathematicians. We found in the examples that as students understood the classroom goals, they embraced them and became intrinsically motivated to persevere in problem solving. We also saw evidence that the various forms of feedback provided for students—verbal or written work, self-reflection, or peer assessment—supported students in both understanding how they were doing and what they should do to improve their performance.

Both the teacher and the students serve important roles in the formative assessment cycle—we highlighted a few components that seem most critical to developing students who are becoming mathematicians in their own right. We have witnessed a group of students who are now much more willing to engage with tasks where there is not an obvious solution and trust that they will receive feedback and guidance that will support them as they navigate their learning.

■ReferencesBell, Beverly, and Bronwen Cowie. “The Characteristics of Formative Assessment in Science Education.”

Science Education 85, no. 5 (2001): 536–53.

Bingham, Gary, Teri Holbrook, and Laura E. Meyers. “Using Self-Assessments in Elementary Classrooms.” Phi Delta Kappan 95, no. 5 (2010); 59–61.

Black, Paul, and Dylan Wiliam. “Inside the Black Box: Raising Standards through Classroom Assessment.” Phi Delta Kappan 80, no. 2 (1998): 139–48.

Black, Paul, and Dylan Wiliam. “Developing the Theory of Formative Assessment.” Educational Assessment, Evaluation and Accountability 21, no. 1 (2009): 5–31.


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Chappius, Jan. Seven Strategies of Assessment for Learning. Upper Saddle River, N.J.: Pearson Education, 2009.

Fernandes, Margarida, and Fontana, David. “Changes in Control Beliefs in Portuguese Primary School Pupils as a Consequence of the Employment of Self-Assessment Strategies.” British Journal of Educational Psychology 66 (1996): 301–13.

Hattie, John, and Helen Timperley. “The Power of Feedback.” Review of Educational Research 77, no. 1 (2007): 81–112.

Hiebert, James, Thomas P. Carpenter, Elizabeth Fennema, Karen C. Fuson, Diana Wearne, Hanlie Murray, Alwyn Olivier, and Piet Human. Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, N.H.: Heinemann, 1997.

Higgins, Karen M., Nancy A. Harris, and Laura L. Kuehn. “Placing Assessment into the Hands of Young Children: A Study of Student-Generated Criteria and Self-Assessment.” Educational Assessment 2, no. 4 (1994): 309–24.

Kazemi, Elham, and Deborah Stipek. “Promoting Conceptual Thinking in Four Upper Elementary Mathematics Classrooms.” The Elementary School Journal 102, no. 1 (2001): 59–80.

Manouchehri, Azita, and Terry Goodman. “Implementing Mathematics Reform: The Challenge Within.” Educational Students in Mathematics 42, no. 1 (2000): 1–34.

Marshall, Jeff C., Bob Horton, and Julie Smart. “4E x 2 Instructional Model: Uniting Three Learning Constructs to Improve Praxis in Science and Mathematics Classrooms.” Journal of Science Teacher Education 20 (2009): 501–16.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000.

National Governors Association Center for Best Practices and Council of Chief State School Officers (NGA Center and CCSSO). Common Core State Standards for Mathematics. Washington, D.C.: NGA Center and CCSSO, 2010. http://www.corestandards.org.

Olson, Jo, and Jeffrey Barrett. “Coaching Teachers to Implement Mathematics Reform Recommendations.” Mathematics Teacher Education and Development 6 (2004): 63–78.

Stein, Mary Kay, Margaret Schwan Smith, Marjorie A. Henningsen, and Edward A. Silver. Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development. 2nd ed. New York: Teachers College Press, 2009.

Student Mathematicians Developed through Formative Assessment Cycles

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