Choosing Tasks that Encourage
Students to Use Mathematical
Practices
When you were working did you…
1. make sense of a
problem and
persevere in solving?
2. reason abstractly and
quantitatively?
3. construct a viable
argument and critique
the reasoning of
others?
4. model with
mathematics?
5. use appropriate tools
strategically?
6. attend to precision?
7. look for and make use
of structure?
8. look for and express
repeated reasoning?
Memorization – The task/solution

Involves reproducing previously learned facts, rules,
formulas or definitions or committing these to memory.

Cannot be solved using procedures because a procedure
does not exist or because the time frame in which the
task is being completed is too short to use a procedure.

Is not ambiguous. Such tasks involve the exact
reproduction of previously seen material, and what is to
be reproduced is clearly and directly stated.

Has no connection to the concepts or meaning that
underlie the facts, rules, formulas, or definitions being
learned.
High Cognitive Demand Tasks
Procedures with Connections – The task/solution

Focuses students’ attention on the use of procedures for
the purpose of developing deeper understanding of
mathematical concepts and ideas.

Suggests explicit and/or implicit pathways to follow
that involve the use of broad general procedures that
have close connections to underlying conceptual ideas
as opposed to narrow algorithms.

Can usually be represented in multiple ways, including
the use of manipulative materials, diagrams, and
symbols. Making connections among the
representations helps students develop meaning.

Requires some degree of cognitive effort. Although
general procedures may be followed, they cannot be
followed mindlessly. Students are engaged in
conceptual ideas that underlie the procedure and
develop understanding.
Procedures without Connections – The task/solution
Doing Mathematics – The task/solution

Is algorithmic. The use of a procedure either is

Requires complex, non-algorithmic thinking
specifically called for or is evident from prior

Requires students to explore and understand the nature
instruction and/or experience.
of mathematical concepts, processes, or relationships.

Requires limited cognitive demand for successful

Demands students do some type of self-monitoring or
completion. Little ambiguity exists about what needs to
self-regulation of their own cognitive processes.
be done and how to do it.

Requires students to access relevant knowledge and
a. Is not connected to the concepts or meaning that
experiences and make appropriate uses of them in
underlie the procedure being used.
working through the task
b. Is focused on producing correct answers.

Requires students to analyze task constraints that may
c. Requires no explanation or explanations focus solely on
limit possible solution strategies or solutions.
describing the procedure that was used.

Requires considerable cognitive effort and may cause
some level of anxiety for the students as they are
working through the problem.
Stein, Smith, Henningsen, & Silver (2009). Implementing standards-based mathematics instruction.
A casebook for professional development. New York: Teachers College Press
Low Cognitive Demand Tasks
http://classroom.westsidehsfaculty.org/webs/jschroe1/upload/worksheet_12.5_volume_of_pyramids_new.pdf
Blocks measure 1 1/2 inches on each edge. A cube one
foot high, one foot wide, and one foot deep is made with
these cubes. How many little blocks are in the large cube
of blocks?
The area of the floor of a rectangular room is 315 sq. ft.
The area of one wall is 120 sq. ft. and the area of another
is 168 sq. ft. The floor and ceiling are parallel. What is the
volume of the room?
Source: Wheatley, G. & Abshire, A. (2007). Developing mathematical fluency:
Activities for grades 5-8. Tallahassee, FL: Mathematics Learning.
A
To create a cone,
Cut out the circle below.
Cut through to the center on the
radius line.
You will slide the radius line to
different points on the circle to
create cones.
Try the different cones. Predict the
cone you believe will have the
greatest volume.
O
B
N
C
M
L
D
K
E
F
J
G
I
H
http://middlemathccss.wordpress.com/8th-grade-math/8th-d4-geometry/volume-of-cylinderscones-spheres/
Complete the table and then plot your Radius and Volume on the graph
below. Label your points with the appropriate letter. Which point has the
maximum volume? _______ How does the data below help you better
estimate the position for the maximum volume?
Letter
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
Radius
Height
Volume
Make a second prediction for maximum volume.
Use your prior knowledge to solve for the height using the length (l) and the
radius (r). Show all work.
h = ___________________________
Using the formula for volume of a cone, find your maximum cone volume
in cubic centimeters. Show all work. V = ______________________
What happens to the volume of a rectangular
prism if one, two, or all dimensions are
doubled? (begin with a 2 x 3 x 1)
Source:
Chapin, S. H. & Johnson, A. (2006) Math Matters: Understanding
the math you teach, Grades K-8 (Second Edition). Sausalito, CA:
Math Solutions.
Use a centimeter ruler to measure the l, w, and h of a
block. Build larger and larger cubes using blocks.
Record the number of blocks on any edge and the
volume for each of these larger cubes. What patterns
do you observe? Generalize to a formula for volume of
a cube.
Source:
Hatfield, Edwards, Bitter, & Morrow (2008). Mathematics
Methods for Elementary & Middle School Teachers
A swimming pool is 8 m long, 6 m wide and 1.5
m deep. The water resistant paint needed for
the pool costs $6 per square meter.
How much will it cost to paint the interior
surfaces of the pool?
http://www.vitutor.com/geometry/solid/volume_problems.html
Making tasks more challenging…
• Open Tasks – framed in such a way that a
variety of strategies or solutions are possible
• Parallel Tasks – sets of related tasks at
different levels that are close enough in
context that they can be discussed
simultaneously
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.
New York: Teachers College Press
Open Tasks
A shape has a triangle for a cross-section. What
could the shape be?
One vertex of a triangle is at the point (1,2).
After a reflection, one vertex is at the point
(5,8). Name all three vertices of the original and
final triangles.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.
New York: Teachers College Press
Parallel Tasks
Option 1: A and B are the location of Arjun’s and Bob’s
homes. They want to meet equally far from both
homes. Where could they meet?
Option 2: Carol is joining Arjun and Bob. Locate a
meeting place equally far from all three homes.
A
B
C
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.
New York: Teachers College Press
Changing Textbook Tasks
http://classroom.westsidehsfaculty.org/webs/jschroe1/upload/worksheet_12.5_volume_of_pyramids_new.pdf