For your second bi-weekly reflection, please make sure you bring in ideas from the **reading** with quotes (don't forget to put in parenthesis the source and the page number), ideas from other media (podcast, video), ideas from your class discussions, and your own experiences.

These questions may guide you, but feel free to expand on other overarching themes:

1) How might teachers need to know the content they are teaching in order to create opportunities for their students that help them (the students) understand important subject specific ideas? Such opportunities to learn may be activities, experiments, excursions, readings, discussions, etc.)

2) Describe the three most important characteristics (in your opinion) of a "good" assessment. Why are these characteristics so important? Support your arguments with evidence.

3) What strategies could a teacher use to make graded assessments more "fair" and equitable?

(1.5-2 pages single spaced)

https://docs.google.com/presentation/d/1r3cX6NiIvVNoM4IGSZvsk85eQcRqZpfY/edit#slide=id.p1

https://docs.google.com/presentation/d/1iXVCWoL9YYdmP0q-D3YOuIE281cTkL7S/edit#slide=id.p1

Copyright © 2021 by The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

DIGITAL OBJECT IDENTIFIER: VOLUME: ISSUE NUMBER:

AUTHOR NAMES:

ARTICLE TITLE:

Mission Statement

The National Council of Teachers of Mathematics advocates for high-quality mathematics teaching and learning for each and every student.

Approved by the NCTM Board of Directors on July 15, 2017.

CONTACT: [email protected]

Mathematics Teacher: Learning and Teaching PK-12, is NCTM’s newest journal that reflects the current practices of mathematics education, as well as maintains a knowledge base of practice and policy in looking at the future of the field. Content is aimed at preschool to 12th grade with peer-reviewed and invited articles. MTLT is published monthly.

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© 2021 by National Council of Teachers of Mathematics, Inc.

Supporting Mathematics Talk in Kindergarten

Kindergartners are capable of engaging in reasoning about mathematics and justifying their thinking using several resources.

Hala Ghousseini, Sarah Lord, and Aimee Cardon

Ms. Sanders’s (all names are pseudonyms) kindergarten students are gathered on the floor for their morning cal- endar activity. They have just counted the total number of days they have been in school. It has been 129 days. The class is particularly focusing on the number 29 and dif- ferent ways to represent it. Students first represent it with bundles of sticks and with base-ten blocks. Then they turn to representing it with tally marks: five groups of five tally marks and four individual tally marks. A student, Jenna, is counting the total number of tally marks, “5, 10, 15, 20, 25, 26, 27, 28, 29.” Ms. Sanders interjects and addresses the class, “So, I notice [Jenna] is counting these last four

by ones. So, when we get over here [pointing to 25], why is Jenna counting by ones instead of counting by fives?”

As several students raise their hands, Ms. Sanders calls on Ryan, who explains, “Because they are, uhm, ones, and you don’t count by fives.” Then, pointing to the representation with bundles of sticks, which had been completed earlier, he continues, “like these!”

Ms. Sanders inquires, “So you are thinking about earlier when we could not count our sticks by tens any- more, and we had to count by ones?”

Ryan nods in agreement and reaches for some base-ten blocks (see figure 1), selecting a tens block and

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a ones block. He says, “This [ones block] is smaller than this one [tens block].” Then pointing to the represen- tation with the tally marks, he signals first to a group of five tally marks and says, “Five,” and then to the

last four ungrouped tally marks and says, “Four.” The teacher asks the class, pointing to the four tally marks, “Do we have a group of five here?” and the students say in chorus, “No.”

Hala Ghousseini, [email protected], Twitter: @hghousseini, is a teacher educator and the John G. Harvey

Professor of mathematics education at the University of Wisconsin. She is interested in classroom mathematics

discourse and studying teaching and teacher learning.

Sarah Lord, [email protected], is a doctoral student in mathematics education at the University of Wisconsin.

She studies children’s mathematical development in number and operations and teaches courses for in-service

teachers aimed at deepening their mathematical knowledge for teaching.

Aimee Cardon, [email protected], is a doctoral student in mathematics education. Her research interests are in

teacher learning at the preservice and in-service levels.

doi:10.5951/MTLT.2020.0310

Fig. 1

The teacher can support young students’ efforts to communicate mathematical ideas with academic language and by supplying number lines, pictures, number charts, and other representations in the classroom.

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This example reflects how kindergartners are capa- ble of engaging in reasoning about mathematics and justifying their thinking using several resources. Ryan was justifying the claim that after reaching 25, the count continues by ones because there are no more groups of fives. As a young learner still developing his language skills, Ryan may not yet have the proficiency to express his thinking using academic terms. In addition to for- mal and informal linguistic resources, he relies on ges- tures and representations around the classroom, which he uses to argue for the difference between a group of five tally marks and four individual tally marks.

Classroom discourse is integral to mathematics instruction at all levels. The expectation that all stu- dents will engage in mathematics discourse is central to the Common Core State Standards, which empha- size practices like conjecturing, justifying, and recon- ciling different ideas to analyze a problem situation (NGA Center and CCSSO 2010). This work is challenging for learners at any age, but perhaps especially for very young learners who are still developing their general oral language skills and at the same time beginning to acquire academic language. Consequently, the mathe- matical development of young learners is intertwined with the development of their language and communi- cation skills (Greenes, Ginsburg, and Balfanz 2004).

Research suggests that children as early as kinder- garten can consider alternative strategies and are capa- ble of sophisticated mathematical thinking (Carpenter et al. 2014). To engage in this type of mathematical activity, students need to be able to communicate about their mathematical reasoning in ways that others can understand and respond to. This involves their use of both oral language and gestures. In addition, young stu- dents’ efforts to communicate mathematical ideas can be supported in powerful ways by available represen- tations in their classroom (e.g., number lines, pictures, and number charts). The classroom teacher plays a vital role in providing opportunities for students to commu- nicate their mathematical reasoning with academic language and supporting their efforts to represent their emerging ideas using available classroom resources.

We draw on data from a larger study of teachers’ facilitation of classroom discourse in elementary class- rooms to highlight this vital role of the teacher in sup- porting students’ take up of academic language and encouraging them to use classroom resources in support of their communication. We elaborate three import- ant practices that a kindergarten teacher used to sup- port students’ mathematics discourse. We focus on the

work of a kindergarten teacher, Ms. Sanders, because we saw in her classroom consistent evidence of kindergart- ners participating in socially and intellectually demand- ing whole-class mathematics discussions. We observed and videotaped five of Ms. Sanders’s mathematics les- sons over a period of four months and interviewed her after each lesson. As we analyzed videos and transcripts of class discussions and interviews, we attended to the practices Ms. Sanders used to facilitate mathematical discourse and develop intellectual community among her students. Our analysis yielded three important prac- tices that we highlight in this article with related sen- tence frames for teachers (see figure 2): (1) establishing expectations that support mathematical discourse, (2) eliciting student thinking, and (3) narrating stu- dent thinking. In what follows, we use vignettes from Ms. Sanders’s kindergarten classroom to elaborate the nature of these practices and to illustrate how the prac- tices supported the efforts of young learners to commu- nicate about mathematical ideas in powerful ways.

ESTABLISHING EXPECTATIONS FOR MATHEMATICAL TALK Ms. Sanders set expectations in two different ways, which directly contributed to the success of the mathe- matical discussions in her classroom. The first expecta- tion established the norm of explaining one’s thinking. Using phrases like, “Be ready to share with us how you knew that your answer was correct,” or “We are going to look closely at Katie’s work and talk together about her thinking,” Ms. Sanders often reminded her students well ahead of a discussion that they should prepare to share their ideas. These reminders occurred throughout the lesson, including when she was giving directions for an independent task and when she was consulting with stu- dents working independently or with partners.

Another expectation that Ms. Sanders promoted was the use of available classroom resources to support math- ematical explanations, including charts and number lines, pictorial representations, and manipulatives. Students knew that they could draw on these resources as they endeavored to communicate their thinking. In our anal- ysis, we noted three specific ways Ms. Sanders supported her students’ mathematics discourse with resources:

1. She made resources available around the class- room and physically within the students’ reach.

2. She encouraged students to move freely around the room to seek resources to support their thinking.

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3. She frequently oriented students to resources they could use in supporting their explanations.

In our data, we saw examples of students’ use of resources both as a direct response to the teacher’s sug- gestion and as a result of their own spontaneous initia- tive. As the example in the introduction to this article illustrates, when Ms. Sanders asked Ryan why Jenna was counting by ones instead of counting by fives when she reached 25, he independently reached for bundles of sticks and base-ten blocks to explain his thinking. He used these classroom resources to argue that when one gets to 25, enough ones are not present to make a group of five. He stated, “Because they are, uhm, ones, and you don’t count by fives.” His actions reflected the routine ways the students interacted with resources in Ms. Sanders’s class- room, which we will further illustrate in the next section.

ELICITING STUDENT THINKING In Ms. Sanders’s classroom, students’ participation in mathematics discourse was structured around several mathematical tasks that engaged them in noticing shapes and patterns and making sense of magnitudes and rela- tionships among numbers. After posing a task, Ms.

Sanders consistently elicited student thinking using two forms of questioning. Her elicitation regularly started with open-ended questions that solicited students’ ini- tial ideas and explanations and surfaced the represen- tations they drew on, such as “How do you know? What do you notice?” She then followed with more probing questions that targeted mathematical ideas (e.g., Why did you start with 9?), language precision (e.g., What do you mean by “drew them the same way”?), and representa- tions (e.g., What do those dots mean here?). Our analysis revealed that starting with open-ended questions allowed the kinder gartners to choose several representations as contexts to support their explanations. The probing ques- tions, in turn, pressed students to further describe and articulate the mathematical ideas they were attempting to convey using several representations. This sequenc- ing of open-ended and more probing questions is an approach that researchers have also shown to support the development of mathematics discourse among emergent bilinguals (Banse et al. 2016).

To illustrate this process, we return to the lesson we featured in the introduction, in which the class was fig- uring out the number of days they had been in school. It had been 129 school days. We rewind to the part where the class was representing this number on a place-value

Fig. 2

These three practices and possible sentence stems can support mathematics talk.

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pocket chart with bundled sticks, grouped in hundreds and tens. The class had just collectively counted nine sticks in the ones pocket (listen to audio 1). Ms. Sanders asked, “How many days until we make another group of tens?”

In this example, the open-ended nature of Ms. Sanders’s first question provided an opportunity for Aiden to identify and draw on resources that he appeared to view as relevant to his sense making: He reached out to the hundred chart and used it to elabo- rate his reasoning, using language that suggested that he was treating the chart as a number line where “one more day” was represented as a “hop” from 9 to make a group of 10. The series of probing questions that Ms. Sanders posed in response to Aiden’s initial state- ment supported him in clarifying his reasoning and making visible the ways in which he was mentally act- ing on numerical objects. In this process, Aiden began by treating 9 and 10 as two points on the number line. As he responded to Ms. Sanders’s questions, his lan- guage became more precise, referring explicitly to the place value of the digits in 9 and 10. After establishing that the number 9 meant 9 ones, Aiden’s reference to the hop suggested that he was adding an additional 1 to make a group of 10—using the expression “a group of 10” in his response.

Additionally, this exchange demonstrates how stu- dents responded to Ms. Sanders’s consecutive prompts by appealing to several different representations to which they had access. Aiden and Ryan relied on both the pocket chart with bundles of tens and ones and the hundred chart to share and clarify their ideas about place value. The mathematical activity demonstrated by Aiden and Ryan in this exchange—well-grounded in representations—is important for young children’s mathematical development as they learn to make deci- sions about how to express their reasoning (Russell et al. 2017). It also benefits children’s language develop- ment as students use representations as language prox- ies and take up the teacher’s more academic language as they gain experience communicating their mathemati- cal ideas.

Ms. Sanders’s elicitation and pressing for student thinking in the context of Aiden’s strategy also engaged more than one student. Her invitation to students to apply their thinking to Aiden’s reasoning positioned the work of responding to his thinking—interpreting it and

making it visible—as a collective responsibility of the class. In the process, Ms. Sanders attended to students’ understanding of important mathematical ideas and to shared language and representations that can sup- port this work. She oriented students to Aiden’s idea by asking them how he knew that he has a group of tens, a focal, and complex, mathematical idea for students’ mathematical development at this stage (Carpenter et al. 2014). She also infused their exchanges with pre- cise mathematical language, “Can someone add on, using our words, ‘the tens place’ and ‘the ones place’?”

NARRATING STUDENT THINKING Ms. Sanders also fostered a rich discourse community within her kindergarten classroom by narrating student thinking. Narration involves providing a running com- mentary that describes what a student says and does (or may have said or done prior to speaking). Young learners draw on their knowledge of oral language (Strickland 2006) and multimodal forms of communi- cation to support their attempts to convey coherent accounts of their thinking (Ball 1993; Dunphy 2015). In this context, narration is a particularly important type of support that teachers can offer. Beyond revoicing, narration serves to integrate a student’s verbal and non- verbal attempts at communication into a holistic narra- tive that promotes their conceptual understanding.

Ms. Sanders’s practice of narration is evident in the preceding example. When Ms. Sanders asked Aiden to clarify what he meant by “this is where we start,” Aiden mainly pointed to the place-value pocket chart and said, “That’s our one.” At this moment, Ms. Sanders engaged in an act of narration in which she not only repeated Aiden’s words but also illustrated the con- nections he was making between the hundred chart and the place-value chart. She started by pointing to the place-value chart, “So, Aiden is looking at the ones place, and he found that we have nine ones today.” Then she pointed to the hundred chart to connect what Aiden said to his initial claim, “So, he’s starting at nine, and then what did you say next, Aiden?” Simultaneously, through her gestures to the different representations that the Aiden used, Ms. Sanders integrated into her narration his nonverbal embodied sense making (Alibali and DiRusso 1999) and provided language support through her connection to representations as visual cues (Cady, Hodges, and Brown 2010).

Additional instances were observed in which Ms. Sanders used narration to highlight mathematical

Audio 1: Listen to Students Explaining How Many Days until They Make Another Group of Tens.

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processes evident in student work. In another lesson, for example, Ms. Sanders was helping students build an understanding of the term equals to mean “the same amount.” As part of this lesson, students completed a task in which they were to first give an equal number of snowballs to both a penguin and a polar bear and then represent the equality with an equals sign.

After students completed their work on this task, Ms. Sanders regathered the students for a whole-group discussion in which several students shared their work using a document camera. One of these students, Kara, shared how she gave eight snowballs to the bear and eight to the penguin. Evident on her notebook were semivisible dots representing snowballs she had erased (see figure 3).

Ms. Sanders used narration to highlight the nature of these semivisible dots. She started by noting that they represented snowballs that Kara had erased. Then she continued.

Ms. Sanders: OK, so we see the 8 there. Now, I want to show you something that Kara did. I was thinking, “Oh, my goodness, Kara is really think- ing hard about her math today!” At first, when Kara was working, the polar bear had 8 [she points to the polar bear’s snowballs] and the pen- guin had 10 [pointing to the penguin’s snowballs]. And, we were talking, and we said, “Hmm, well, if the polar bear has 8 and the penguin has 10, do they have the same amount?”

Fig. 3

Kara’s snowballs representation showed snowballs she had erased.

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Students: No! [in chorus] Teacher: And [Kara] said, “Oh! I need to fix that. I

need to take some away.” And that’s why we see, when she’s thinking hard about her math, and she said, “Oops!” and took those away [pointing to the semi-erased dots], and she double checked: They both have 8.

The narration support that is evident in this excerpt suggests the way Ms. Sanders used it to position students’ strategies and the processes evident in their work as resources for the classroom community. By highlighting the way Kara revised her work, Ms. Sanders was normal- izing the process of revising as an aspect of doing mathe- matics (Lampert 2001). In her narration, she framed this work as an aspect of “thinking hard” about the mathe- matics in relation to the meaning of “Do they have the same amount?” Ms. Sanders assigned agency to Kara in the process, centering Kara as the learner who made particular choices as she worked on this task.

CONCLUSION We know from prior research in early childhood contexts that children have a natural inclination to engage in a wide range of mathematical activities such as counting, patterning, and developing spatial relationships. Our study shows that when teachers offer appropriate opportunity and support, children of this age can also engage in rich classroom dis- course that involves communication about emerg- ing mathematical concepts and ways of reasoning. Whether young children are involved in play-based learning or are learning in a more “academic” setting, we believe that teachers’ practices can give essential support for children’s sense making and understand- ing (Pyle and Danniels 2017). By using the teaching practices we highlight here, teachers can foster the development of young learners—as mathematical thinkers, as communicators, and as members of an intellectual community—in a holistic and integrated way.

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Track.” Cognitive Development 14, no. 1 (January): 37–56. Ball, Deborah Loewenberg. 1993. “With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School

Mathematics.” The Elementary School Journal 93, no. 4 (March): 373–97. Banse, Holland W., Natalia A. Palacios, Eileen G. Merritt, and Sara E. Rimm-Kaufman. 2016. “5 Strategies for Scaffolding Math

Discourse with ELLs.” Teaching Children Mathematics 23, no. 2 (September): 100–108. Cady, Jo Anne, Thomas E. Hodges, and Clara Brown. 2010. “Supporting Language Learners.” Teaching Children Mathematics 16,

no. 8 (April): 476–83. Carpenter, Thomas P., Elizabeth Fennema, Megan L. Franke, Linda Levi, and Susan Empson. 2014. Children’s Mathematics:

Cognitively Guided Instruction. 2nd ed. New Hampshire: Heinemann. Dunphy, Liz. 2015. “Transition to School: Supporting Children’s Engagement in Mathematical Thinking Processes.” In Mathematics

and Transition to School, edited by Ann Gervasoni, Amy MacDonald, and Bob Perry, pp. 295–312. Singapore: Springer. Greenes, Carole, Herbert P. Ginsburg, and Robert Balfanz. 2004. “Big Math for Little Kids.” Early Childhood Research Quarterly

19, no. 1 (April): 159–66. Lampert, Magdalene. 2001. Teaching Problems and the Problems of Teaching. New Haven, CT: Yale University Press. National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO).

2010. Common Core State Standards for Mathematics. Washington, DC: NGA Center and CCSSO. http://www.corestandards.org. Pyle, Angela, and Erica Danniels. 2017. “A Continuum of Play-Based Learning: The Role of the Teacher in Play-Based Pedagogy

and the Fear of Hijacking Play.” Early Education and Development 28, no. 3 (September): 274–89. Russell, Susan Jo, Deborah Schifter, Virginia Bastable, Traci Higgins, and Reva Kasman. 2017. But Why Does It Work: Mathematical

Argument in the Elementary Classroom. New Hampshire: Heinemann. Strickland, Dorothy S. 2006. “Language and Literacy in Kindergarten.” In K Today: Teaching and Learning in the Kindergarten

Year, edited by Dominic F. Gullo, pp. 73–84. Washington, DC: National Association for the Education of Young Children.

ACKNOWLEDGMENT

The research reported in the article was supported by a grant from the Spencer Foundation.

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- ARTICLE TITLE: Supporting Mathematics Talk in Kindergarten
- AUTHOR NAMES: Ghousseini, Hala; Lord, Sarah; and Cardon, Aimee
- DOI: 10.5951/MTLT.2020.0310
- VOLUME: 114
- ISSUE #: 5