Effective Practices and Shifts in Teaching and Learning Mathematics

Dr. Amy Roth McDuffieWashington State University Tri-Cities

Plan for the SessionFocusing on Conceptual Understanding in Teaching and Learning Mathematics Discussing the Reading: Principles to ActionsHighlighting the Standards for Mathematical PracticeDoing some math as learners!Looking into a mathematics classroom – an illustration of effective practice in actionWrapping 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-and-learning-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

1. Make sense of problems and persevere in solving them2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning of others4. 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 –“Doing” mathematics (See Page 8 in

Principles to Actions)

Standards for Mathematical Practice

NCTM video with explanations for each standard…

http://www.nctm.org/standards-and-positions/common-core-state-standards/teaching-and-learning-mathematics-with-the-common-core/ Select Standards for Mathematical Practice (view first 3).

Applications, Concepts, and Fluency

Learn through Meaningful Applications

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.

Build Conceptual Understandings

When learning new ideas, students need opportunities to understand how and why, as a first step in learning math.

Developing Fluency with Facts and Procedures

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

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!

Procedural Fluency

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 $.25

20 lbs

4 lbs

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 fluencyApplications 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.