As you and your students engage with the problem events in Data Modeling, you add to the narrative, writing yourselves into our developing narrative about probability and chance. Welcome to the story!

Formative Assessments

Each unit includes a formative assessment and a scoring rubric that helps teachers relate student responses to levels of one, and sometimes, two constructs. The rubric guides selection of student responses for further class discussion—we call these discussions assessment conversations. The aim of the assessment conversation is to bring out different ways of student thinking about a statistical idea and to help students make progress toward more powerful ways of thinking. More powerful ways of thinking are those that help students reason more generally about a variety of situations in which data and chance play important roles.Instruction is often extended and modified in light of student responses to formative assessments. For example, teachers often modify a formative assessment item slightly and ask students to use what they have learned from the conversation to respond again to the item.

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Classroom Community

Using Data Modeling as a tool for learning depends on the prior existence or concurrent development of classroom discussion and teacher questioning directed toward student thinking. For our purposes, classroom discussion is a conversation among students and between students and teacher about important mathematical ideas. It is a dialogue among the students and teacheras they attempt to make meaning together via an exchange of ideas. This view of discussion is often at odds with what most teachers mean by discussion, in which the teacher asks a question, calls on a student, the student answers, teacher evaluates the response (right/wrong), then asks another question. This pattern is sometimes referred to as IRE (initiate-respond-evaluate) cycle and is a traditional teacher-questioning model, because only the teacher asks the questions and evaluates the responses. In our view of classroom discussion, students’ talk dominates, as they build ideas with each other. Students are likely to ask questions of themselves and each other and to share ideas. To emphasize the contrast, we often use the term "kid talk,"rather than classroom discussion, because that’s the part that is most important. It tells us what the children do and don’t understand. Teachers still ask questions, but those questions are designed to elicit and elaborate on student ideas.

In order for students to share their ideas and risk "being wrong" in public, classroom norms that define, actualize, and constrain student talk must be in place, in both large-and small-group work. Examples of helpful constraints include no "dissing" by word or gesture, and we discuss ideas, not people. The teacher plays an important role in modeling, reminding, and enforcing these norms so that students feel safe to talk. Teacher questioning provokes and guides classroom discussion. Teachers formulate queries that prompt, clarify, amplify/enhance, and extend student thinking about the Data Modeling concepts. The Data Modeling curriculum units each include many examples of teacher questions, when to ask them, and likely responses from students. In addition,to guide culminating conversations about important mathematical ideas, Talk Moves for each unit are available on this web site. Talk Moves are a core of questions that teachers can use to start a conversation, keep it going, and ensure that at its end, students have had the opportunity to learn about important mathematical ideas. An excellent resource for learning more about classroom discussions and teacher talk is:

Chapin, S., O’Connor, C. and Anderson, N. 2009. Classroom discussions: Using math talk to help students learn, 2nd edition. Sausalito, CA: Math Solutions.

To support sharing of ideas, students should keep a written record, a learning journal, of their thinking to which they can refer during discussions. This record may include words, numbers, graphics, and/or symbols. Students use the record to document their ideas, interpret observations, develop arguments, and provide evidence for their thinking about concepts they encounter. As students progress through the units, the record allows them to revisit and reconsider their changing ideas. It also allows the teacher to review the development of students’ thinking and make appropriate adjustments to instruction in response. Keeping all of their thinking in a math notebook simplifies storage and retrieval of students’ work.

Academic Language

Exchanging ideas in the math classroom obliges teachers to model, and students to engage with, the academic language of the discipline. Educators have recently heard a great deal about Academic Language and its importance to student success in school. According to Goldenberg (2008, p. 9), academic language"refers to more abstract, complex, and challenging language that will eventually permit you to participate successfully in mainstream classroom instruction. (It) involves such things as...possessing and using content-specific vocabulary and modes of expression in different academic disciplines such as mathematics and social studies." Academic language is much more than a list of pertinent vocabulary; it is, instead, how knowledgeable people communicate in a particular field of study.

In classroom discussions, students should receive multiple opportunities to try out both the generic and content-specific components of academic language in mathematics. By generic, we mean school words that are used across subject areas, such as compare, define, and synthesize. Content-specific language refers to words and phrases that help people in disciplines communicate and share ideas.

To support the development of academic language, teachers may create a Word Wall of relevant terms that allows students to consult the list whenever necessary in the communication process. However, the terms are developed with students on an as-needed basis and recorded on the Word Wall only after students have constructed the ideas to which they refer. In other words, we do not advocate pre-teaching vocabulary. Students first encounter concepts within activity structures. After engaging with, talking about, and clarifying their understanding of the idea, the teacher may introduce the appropriate term, telling students that mathematicians agree to call this ___.

A second support of academic language is a posted chart of often-used phrases, summary statements, and argument structures that help children engage in the social practices and conventions of mathematics. These may include: I noticed that ___; I agree/disagree with ____ because ____; I used to think ____, but now I think ____; I learned that ____; I know that ____ because ____; I conclude that ____; In summary, I think ___; among many others. If teachers model these structures and explicitly teach and repeatedly encourage children to use them in conversation they will become the norm for students during listening, speaking, and recording/representing. As with the Word Wall, students contribute to this resource and then rely upon it.

A third support of academic language is the math notebook. Here students can record descriptions and graphics, try out ideas, notice patterns, and build arguments with the evidence they have compiled. They can revisit their thinking to see both its origin and its progress. Teacherscan review the notebooks to gauge student understanding and revise teaching in response.

Finally, establishing a question wall that records student-generated questions invites students to participate in the practice of mathematics because their classroom tasks become investigations into the novel using the familiar. This is what mathematicians do.

Goldenberg, C. (2008) Teaching English language learners: What the research does—and does not—say. American Educator, Summer (pp. 7-44).

Academic Language by Unit

With each Data Modeling Unit are tables of Academic Language. Some terms that maintain their common meanings (collection, measure, lightweight) are also included for consideration with English Language Learners. Others are included for students who may be unfamiliar with a context (batting average, fouled). Teachers need to be aware that not all children will understand the meaning of all of the words used.

With this list, teachers will know the context of terms (where they were introduced and their frame of reference). In addition, only the first reference to the term is recorded. It is our experience that teachers tend to use the language of the discipline within the classroom and eventually many students adopt it. However, teachers will wish to name concepts that have been developed so that eventually everyone is using conventional terms and referents (what the term means). By Unit 5, the terms are used in conjunction with each other to form phrases representing concepts. For example, sample variability is introduced in Unit 3, explored more thoroughly in Unit 4 and extended in Unit 5 to include sample-to-sample variability to form yet another concept. Terms are also combined to form definitions or describe processes and conditions, and definitions are combined to explain other concepts. Therefore, students will be expected to understand and utilize most of the terms, phrases, and structures below to name learned concepts.