Engaging Learners and Realizing the
Development of Mathematical
Practices
ALM Conference
July 15, 2015
Trena L. Wilkerson
Professor, Mathematics Education
Baylor University
Trena_Wilkerson@baylor.edu
NCTM Board of Directors
Session Overview
• What are Mathematical Practices for students and
Mathematical Practices for Teaching
– Principles to Actions: Ensuring Mathematical Success for
All, NCTM 2014
– http://www.nctm.org/PtA/
• Focus on problem solving, perseverance, and
reasoning
• Sample problems, activities, and resources
• Brainstorm & Discuss ways of incorporating MPs and
MTPs-connecting to adult learners and adult
educators
2
Polya
“Mathematics is not a spectator sport. To
understand mathematics means to be able
to do mathematics. And what does it mean
to be doing mathematics? In the first place,
it means to be able to solve mathematical
problems.”
(from a lecture on teaching)
3
CCSS-M Mathematical Practices
Represent what students are doing as they learn mathematics
•
•
•
•
•
•
•
•
•
4
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of
others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Mathematics Teaching Practices
(NCTM, PtA 2014, p. 10)
1. Establish mathematics goals to focus learning.
2. Implement tasks that promote reasoning and problem
solving.
3. Use and connect mathematical representations.
4. Facilitate meaningful mathematical discourse.
5. Pose purposeful questions.
6. Build procedural fluency from conceptual understanding.
7. Support productive struggle in learning mathematics.
8. Elicit and use evidence of student thinking.
5
Questions to Consider
• What are ways that we, as adult leaner educators, can
approach or include the development of MPs in our
programs?
• What are mathematical practices for teaching (MTPs)
that we can include to support teachers and learners?
• What is the role of reasoning and perseverance in
problem solving as related to the development of MPS
and MTPS?
• What are exemplary tasks, tools, and activities we can
include in our programs?
• What are the available resources to support this work?
6
Candy Jar Problem
1. 10 minutes to work individually
2. 10 minutes to share at table (various approaches)
3. 10 minutes to record different strategies to post on wall
7
Principles to Actions: Ensuring Mathematical Success for All, NCTM 201, p. 314
Candy Jar Problem-Sample Student
Answers
•
•
•
•
•
•
•
•
•
8
260
360
40
26
340
50
240
270
I do not
know
•
•
•
•
•
•
•
•
•
2600
65
100
38.14
82.3
30
87
250
I do not
understand
• 108
• 20
• 325
• 36
• 35
• 7R9
• 61
• 50/50
• 6.9
• Less than 78
• 38
but more than
• 160
65
• 13
CCSS-M Mathematical Practices
Represent what students are doing as they learn mathematics
•
•
•
•
•
•
•
•
•
9
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of
others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Mathematics Teaching Practices
(NCTM, PtA 2014, p. 10)
1. Establish mathematics goals to focus learning.
2. Implement tasks that promote reasoning and problem
solving.
3. Use and connect mathematical representations.
4. Facilitate meaningful mathematical discourse.
5. Pose purposeful questions.
6. Build procedural fluency from conceptual understanding.
7. Support productive struggle in learning mathematics.
8. Elicit and use evidence of student thinking.
10
Teaching & Learning Principles
http://www.adultnumeracynetwork.org/
• Curriculum
– include the concepts of number, data, geometry, and algebra at all levels of learning
so that students can develop and connect mathematical ideas.
– weave together all the elements of mathematical proficiency – not only procedural
fluency, but also conceptual understanding, ongoing sense-making, problem solving,
and a positive attitude about learning mathematics.
– feature worthwhile tasks, such as activities that are drawn from the context of real life
experience.
• Learning Environment
– build on what students already know, valuing the various informal and alternative
strategies students use to solve problems.
– include opportunities for students to question, reason, solve problems, define goals
and monitor their own progress by using estimation, mental math, computation, and
technology when appropriate.
11
Teaching & Learning Principles
Continued
• Design
– begin with teachers as mathematics learners and thinkers.
– be a continuing process that is connected to curriculum and assessment standards,
program policy and instruction and current research.
– be welcoming and accessible to all – to literacy and language teachers as well as to
those who primarily teach mathematics.
– be evaluated with respect to its impact on teacher behavior in relation to increased
student learning.
• Content
– establish a deep understanding of the mathematics of the curriculum and its
principles.
– understand how adults’ mathematical knowledge develops, how to recognize previous
misconceptions, and how to assess and engage prior knowledge.
– use a broad range of instructional strategies that utilize a variety of materials to
accomplish learning goals.
12
– understand how research can be used to improve their effectiveness as teachers.
ANN Adult Numeracy Components
NCSALL, 2006
• Context: the use and purpose for which an
adult takes on a task with mathematical
demands
• Content: the mathematical knowledge that is
necessary for the tasks confronted
• Cognitive & affective: the processes that
enable an individual to solve problems, and
thereby, link the content and context
13
The Calling Plans Task
http://www.nctm.org/Conferences-and-Professional-Development/Principles-toActions-Toolkit/The-Case-of-Elizabeth-Brovey-and-the-Calling-Plans-2-Task/
Long-distance company A charges a base rate of $5.00 per
month plus 4 cents for each minute that you are on the
phone. Long-distance company B charges a base rate of only
$2.00 per month but charges you 10 cents for every minute
used.
Part 1: How much time per month would you have to talk on
the phone before subscribing to company A would save you
money?
Part 2: Create a phone plane, Company C, that costs the same
as Companies A and B at 50 minutes, but has a lower monthly
fee than either Company A or B.
14
Video Viewing Focus
Student Mathematical Practices
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and
persevere in solving them.
Reason abstractly and
quantitatively.
Construct viable arguments and
critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of
structure.
Look for and express regularity in
repeated reasoning.
15
Mathematical Teaching Practices
1.
2.
3.
4.
5.
6.
7.
8.
Establish mathematics goals to
focus learning.
Implement tasks that promote
reasoning and problem solving.
Use and connect mathematical
representations.
Facilitate meaningful
mathematical discourse.
Pose purposeful questions.
Build procedural fluency from
conceptual understanding.
Support productive struggle in
learning mathematics.
Elicit and use evidence of
student thinking.
Pose Purposeful Questions
PtA, 2014
Effective Questions should:
•
•
•
Reveal students’ current understandings;
Encourage students to explain, elaborate, or clarify their
thinking; and
Make the mathematics more visible and accessible for
student examination and discussion.
Teachers’ questions are crucial in helping students make
connections and learn important mathematics and science
concepts. Teachers need to know how students typically think
about particular concepts, how to determine what a particular
student or group of students thinks about those ideas, and
how to help students deepen their understanding. (Weiss and
Pasley, 2004)
16
NCTM New PD Resources-MTPs
• http://www.nctm.org/Conferences-and-ProfessionalDevelopment/Professional-Development-Resources/
• http://www.nctm.org/PtAToolkit/
17
NCTM Classroom Resources
http://www.nctm.org/Classroom-Resources/Connect-with-NCTM-Illuminations/
18
Sample NCTM Resources
www.nctm.org
• Implementing the CCSS though Mathematical
Problem Solving: For various grade bands
• Connecting the NCTM Process Standards and
the CCSSM Practices
• 5 Practices for Orchestrating Productive
Mathematics Discussions
• Principles to Action: Ensuring Mathematical
Success for All
19
Trena_Wilkerson@baylor.edu
NCTM Board of Directors
20