Indicators of Teacher Instructional Practices That Elicit Student Mathematical Thinking

Indicator / Description / ExampleS
Classroom Discourse / Classroom discourse consists of any of a variety of exchanges in which ideas about mathematics are communicated within a classroom. As an instructional strategy, this may include dialog between teacher and students, dialog among students, presentations, use of images or diagrams, or any of a host of communicative methods to share mathematical ideas. /
  1. Teacher communicates expectations for students to explain their thinking process(es) and defend them to one another.
  2. Teacher takes a neutral stance, encouraging students to discuss and argue about mathematical ideas.
  3. Teacher poses a problem to students, and leaves students to their own facility and collaboration to find a solution to the problem.
  4. Teacher allows for student-driven discourse in groups.

Open-Ended Questions/problems / Asking open-ended questions is a key strategy teachers use to stimulate mathematical thinking. Open-ended questions can embody the demand for student exploration, sense-making, application of ideas, extension of ideas, construction of new ideas, and struggling with the unknown. Teacher provides for multiple entry points that can lead to one solution or multiple solutions. / Teacher poses questions such as:
  • Can you explain what you are thinking?
  • How can we use what we know about working with slope to help us understand the patterns of in/out data tables
Teacher presents a problem such as:
  • The slope of a straight line changes from 2 to 3. How does that affect its data table of (1,4), (2,6), (3,8)?

“How” and “why” Questions / Asking ‘how” and “why” questions gives students an opportunity to describe what they did and why they did it. The students have to be able to give reasons for the choices they made. Reasoning through problems and offering evidence in support of ideas are essential components of mathematical thinking. This leads to the development of student metacognition. / Teacher requires students to support their answers with evidence.
  • How do you know if your answer is correct?
Teachers ask students to describe how they reasoned through a task.
  • What did you do?
  • Why did you do it?

Mathematical Representation / Mathematical representation refers to multiple ways in which mathematical concepts are presented. Because one specific idea can often be presented in many different forms – in number, in the English language, in diagram – it is important to the development and expression of students’ mathematical thinking for them to understand how they can create mathematical representations to express their mathematical ideas. / 1.Teacher records multiple student representations and facilitates discussion about the connections among students’ representations.
  • What is the relationship between the multiple ways we can represent functions?
2. Teacher models her own mathematical thinking through the questions she asks herself.
3. Teacher encourages the use of visuals and manipulatives so that students can think about a problem or concept.
Use of Contextual Problems / A central rationale for teaching mathematical thinking is that it is necessary in the world beyond the classroom. The world beyond the classroom includes real life situations that present mathematical problems as well as natural occurrences of mathematics in the world. / Marissa went to a party at her friend’s house. She promised her strict father she would get home by 11PM. He told her she would be grounded for a month if she comes home one second passed 11. Was she grounded?
Here is some information to help figure this out.
  • She left the party at 10:05 PM.
  • She drove an old Buick that made a lot of noise.
  • She lived 18 miles from where the party was held.
  • She drove at the same rate the whole way home.
  • Marissa was 12 miles from home after 20 minutes.

Indicators of Student Demonstration of Mathematical Thinking

Indicator / Description / ExampleS
Reasoning and Evidence / Reasoning is thecapacity for logical thought, explanation, and justification. Students reason through problems and offer evidence in support of ideas.
An aspect of reasoning is reflection. Students reflect on their own thinking, looking at whether their approach is sensible or if there is a need for self-correction. / Examples of students demonstrating reasoning and evidence-giving behaviors includes explaining their reasoning process, giving numerical, algebraic or geometric proof, and giving visual evidence of a correct answer.
Examples of student reflection behaviors include students justifying a claim, refuting a claim of a classmate, correcting their own claims, and investigating a new claim proposed by a classmate.
Observing, Conjecturing, and Generalizing / Thinking mathematically consists of making observations of available information, making conjectures about possible solutions to problems, and generalizing from a set of particular cases to formulate new concepts for future application. / Students observe different patterns within a set of data, look for relationships within the data and attempt to conjecture a possible general idea. They test their conjecture for its validity and work to prove the conjecture leading to a generalization.
Making Connections / New mathematical concepts make sense to students as they are connected to the other ideas students have already learned. Students make connections around big ideas in mathematics and construct new knowledge by making connections to prior knowledge. / Students observe and explain coherence between mathematical ideas and procedures.
Students observe and explain a connection between the task and past experience, past problems, or other subjects.
Students use knowledge or other mathematical understandings to develop a new understanding.
Strategizing / Students with strategic competence not only come up with several approaches to a non-routine problem…but also choose flexibly among guess and check, algebraic approaches, or other methods to suit the demands presented by the problem and the situation in which it was posed. / Students choose a correct and efficient strategy based on the mathematical situation in the task.
Students show evidence of applying prior knowledge to the problem-solving situation.
Students show flexibility of mind by being able to rethinkstrategies while solving problem.
COMMUNICATION / Students organize and consolidate their mathematical thinking through communicating their mathematical thinking coherently and clearly to peers, teachers, and others; analyze and evaluate the mathematical thinking and strategies of others; use the language of mathematics to express mathematical ideas precisely. / Students communicate through verbal/written accounts using both familiar, everyday language and formal mathematical language.
Students communicate their thinking and reasoning through use of diagrams or mathematical symbols.
REPRESENTATION / Students create and use representations to organize, record, and communicate mathematical ideas; select, apply, and translate among mathematical representations to solve problems; and
use representations to model and interpret physical, social, and mathematical phenomena. / Studentsrepresent mathematical concepts and ideas using diagrams, tables, and mathematical symbols. Students are able to translate from one type of representation to another.