Effective Practices and Shifts
in Teaching and Learning Mathematics
Dr. Amy Roth McDuffie
Washington State University Tri-Cities
Plan for the Session
Focusing on Conceptual Understanding in Teaching and Learning
Mathematics
Discussing the Reading: Principles to Actions
Highlighting the Standards for Mathematical Practice
Doing some math as learners!
Looking into a mathematics classroom – an illustration of effective
practice in action
Wrapping up
An Introduction to Research-Based Foundations for
Teaching and Learning (underlying CCSSM)
Focus on supporting students’ conceptual understanding
first, and then build procedural fluency.
Applications can be used to launch new learning, to solidify
concepts, and to practice (develop fluency).
NCTM has several videos that provide background on
Common Core State Standards for Mathematics (CCSSM)…
http://www.nctm.org/standards-and-positions/common-core-state-standards/teaching-andlearning-mathematics-with-the-common-core/ Select Building a Conceptual Understanding
Discussing the reading: Principles to
Actions: Ensuring Math Success for All
In groups of 2-3, review the Intro and your group’s section. On flip chart, record
3-4 key ideas, with page numbers for reference. Be prepared to briefly share (1-2
minutes per group)
All: Introduction - Effective Teaching and Learning (intro), Pages 7-12
Jigsaw:
1. Establishing Goals & Implementing Tasks, Pages 12-24
2. Mathematical Representations & Mathematical Discourse, Pages 24-35
3. Posing Questions & Building Fluency from Concepts, Pages 35-48
4. Supporting Productive Struggle & Eliciting Student Thinking, Pages 48-57
5. Curriculum, Pages 70-77
Standards for Mathematical Practice –
“Doing” mathematics (See Page 8 in Principles to Actions)
1.
2.
3.
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 (apply math to everyday life, society,
workplace)
5. Use appropriate tools strategically (all technologies, from pencil and
paper to statistical software)
6. Attend to precision (communicate precisely using mathematic
vocabulary, symbols, units, representations, etc.)
7. Look for an make use of structure (find patterns or structures)
8. Look for and express regularity in repeated reasoning (find general
methods, short-cuts).
*Connect practices to Grade Level Math Content Standards
Standards for Mathematical Practice
NCTM video with explanations for each standard…
http://www.nctm.org/standards-and-positions/common-core-state-standards/teaching-andlearning-mathematics-with-the-common-core/ Select Standards for Mathematical Practice
(view first 3).
Applications, Concepts, and Fluency
Learn through Meaningful Applications
Build Conceptual Understandings
Developing Fluency with Facts
When learning new and Procedures
Engage learners and
provide a way to make
sense of math for deeper
learner that is retained
over time. Meaningful
applications are helpful
throughout all phases of
learning.
ideas, students need
opportunities to
understand how and
why, as a first step in
learning math.
After students have
developed conceptual
understandings, they need
to develop fluency with
math facts and with
procedures – but if fluency
is emphasized too early, it
can undermine
understanding math.
Focusing in on an example
Multiplication with multi-digit numbers…
Trajectory for Procedural Fluency with Multi-Digit Multiplication
(after learning about operation of multiplication with single digit numbers)
Conceptual Understandings
Procedural Fluency
Explore and share invented/
alternative strategies and
algorithms for computing
with larger numbers that are
connected to making sense
of the quantities (and place
value). Use models,
pictures, representations,
etc. to show thinking.
Students need experiences
and time to make sense of
what’s happening!
Learn the steps/ methods for
U.S. standard (traditional)
algorithms. Bridge from
invented to traditional with
models, representations, etc.
Justify the steps and prove
that they work. Practice (but
still allow for invented/flexible
strategies - often they are
more efficient!)
*Try 100 x 5 = ____
So, 99 x 5 = _____
This trajectory is reflected in the CCSS for Math!
Buying a Turkey:
Using a (potentially – with caveats) meaningful application/context to
engage students. Try it with a partner…
Lenses activity that connects to place value & alternative/ invented
algorithms (computational methods): Buying a Turkey
Context: 3rd grade class in New York, classroom with diverse students.
Problem: Need a turkey that will feed many guests for Thanksgiving dinner. Teacher wants to
buy a 24 pound turkey. Turkey is on sale for $1.25/pound. What is the cost of the turkey?
Anticipating Strategies:
•Solve the problem as students might. Keep in mind that most of these students do not yet
know the traditional/standard algorithm for multi-digit multiplication (and it is not expected
yet in CCSSM).
•Use as many strategies as you can think of, explain your thinking and listen to your partner’s
thinking.
•Consider what prior knowledge and strategies students might draw on as well as potential
difficulties and/or struggles might arise. (Exploring alternative/invented strategies to the
traditional algorithm)
Questions to keep in mind while watching the
lesson…
◦ How are students’ understandings for number, place value, and
multiplication evident in their work? Why is deep understanding of
place value and multiplication as an operation (not just naming
columns/digits) important to their work?
◦ Why might it be important to experience/develop alternative
algorithms before focusing on fluency with the traditional
algorithm?
◦ How does the context (application) of buying a turkey support
students’ learning and knowledge?
◦ How does the teacher facilitate students’ sharing of strategies?
What types of questions/probes do they ask? How do they keep the
students engaged and involved? Why use these approaches?
Suzanne & Rose’s Solution… How are they solving the problem?
Examining students’ strategies
Look at the examples of student work.
1.
What mathematical reasoning do you see?
2.
What might you want each pair to explain/ share if they were justifying/ defending their
solution to the class?
3.
For Sample E, the solution is not (YET) correct.
a. What evidence of some understanding do you see?
b. What question or prompt might you pose to this pair to encourage them to keep
working WITHOUT telling them what to do or what is incorrect?
c. Why might we want to encourage perseverance and provide space for students to
develop solutions (as compared to teacher correcting their work)?
d. If this pair shared this solution in a whole class discussion, what questions might you
ask and/or how might you sequence this solution with other solutions to be shared?
An Open Array: Another representation for thinking about
multiplication with quantities>10 (included in CCSSM for
Number and Operation)
$1.25 x 24 lbs = ?
$1.00
20 lbs
4 lbs
$.25
The next day…Cooking the Turkey
Going beyond just connecting to money in the math
The teacher looks in her cookbook to see how long she
needs to cook the turkey and plan when to start cooking
and when to serve dinner. The cookbook says that a
turkey needs to be in the over 15 minutes for every
pound. How long does she need to cook the turkey?
Why pose this problem next?
Final Thoughts for Your Work…
Conceptual understandings need to be developed before focusing on procedural
fluency
Applications are important in building conceptual understandings AND in practicing
for fluency – not just “at the end of the chapter” (old methods).
Curriculum materials are important in teaching and learning math as a key resource
and starting point for teachers. Research has established that teaching and learning
is more effective with a high quality curriculum program. A curriculum program is
far more than collection of problems, and some materials can “look like” a
curriculum program – analyze for rigor, coherence, and a progression and sequence
of learning opportunities over time.
In reviewing materials, analyzing content can be easier than analyzing for the
Standards for Mathematical Practice, but the Standards for Mathematical Practice
are a CRITICAL component of learning mathematics and enacting effective,
research-based practices.