Indicators of Teacher Instructional Practices That Elicit Student Mathematical
Thinking
INDICATOR
DESCRIPTION
EXAMPLES
1.
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.
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, sensemaking, 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.
“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.
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.
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
Teacher
poses questions such as:
problem.
4. Can
you allows
explainfor
what
you are thinking?
Teacher
student-driven

How
can we
use what we know about
discourse
in groups.
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)?
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?
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
REASONING AND
EVIDENCE
OBSERVING,
CONJECTURING, AND
GENERALIZING
MAKING CONNECTIONS
STRATEGIZING
COMMUNICATION
DESCRIPTION
EXAMPLES
Reasoning is the capacity for logical thought,
explanation, and justification. Students reason
through problems and offer evidence in support
of ideas.
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.
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 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.
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.
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.
Students with strategic competence not only
come up with several approaches to a nonroutine 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 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 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 rethink strategies while solving problem.
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.
Students represent mathematical concepts and
ideas using diagrams, tables, and
mathematical symbols. Students are able to
translate from one type of representation to
another.