Number Talks: developing fluency, flexibility, and conceptual understanding #AuthorAndIllustrate

How might we work on fluency (accuracy, flexibility, efficiency, and understanding) as we continue to teach and learn with students? What if our young learners are supposed to be fluent with their multiplication facts, but… they. ..just…aren’t!?

It really isn’t a surprise, right? Children learn and grow at different rates. We know that because we work with young learners every day.  The question isn’t “Why aren’t they fluent right now?” It isn’t. It just isn’t. The question should be and is:

“What are we going to do, right now, to make this better
for every and each learner in our care?”

In Making Number Talks Matter, Cathy Humphreys and Ruth Parker write:

Multiplication Number Talks are brimming with potential to help students learn the properties of real numbers (although they don’t know it yet), and over time, the properties come to life in students’ own strategies. (Humphreys, 62 p.)

Humphreys and Parker continue:

Students who have experienced Number Talks come to algebra understanding the arithmetic properties because they have used them repeatedly as they reasoned with numbers in ways that made sense to them. This doesn’t happen automatically, though. As students use these properties, one of our jobs as teachers is to help students connect the strategies that make sense to them to the names of properties that are the foundation of our number system. (Humphreys, 77 p.)

So, that is what we will do. We commit to deeper and stronger mathematical understanding. And, we take action.

This week our Wednesday workshop focused on Literacy, Mathematics, and STEAM in grade level bands.  Teachers of our 4th, 5th, and 6th graders gathered to work together, as a teaching team, to take direct action to strengthen and deepen our young students’ mathematical fluency.

We began with the routine How Do You Know? routine from NCTM’s High-Yield Routines for Grades K-8 using this sentence:

81-25=14×4 How do you know?

Here’s how I anticipated the ways learners might think.

Anticipating students’ responses takes place before instruction, during the planning stage of your lesson. This practice involves taking a close look at the task to identify the different strategies you expect students to use and to think about how you want to respond to those strategies during instruction. Anticipating helps prepare you to recognize and make sense of students’ strategies during the lesson and to be able to respond effectively. In other words, by carefully anticipating students’ responses prior to a lesson, you will be better prepared to respond to students during instruction. (Smith, 37 p.)

How many strategies and tools do we use when modeling multiplication in our classroom? It is a matter of inclusion.

It is a matter of inclusion.

Every learner wants and needs to find their own thinking in their community. This belonging, sharing, and learning matters. We make sense of mathematics and persevere. We make sense of others thinking as they learn to construct arguments and show their thinking so that others understand.

Humphreys and Parker note:

They are learning that they have mathematical ideas worth listening to—and so do their classmates. They are learning not to give up when they can’t get an answer right away because they are realizing that speed isn’t important. They are learning about relationships between quantities and what multiplication really means. They are using the properties of the real numbers that will support their understanding of algebra. (Humphreys, 62 p.)

As teachers, we must anticipate the myriad of ways students think and learn. And, as Christine Tondevold (@BuildMathMinds) tells us:

The strategies are already in the room.

Our job is to connect mathematicians and mathematical thinking.

From NCTM’s Principles to Actions:

Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.

And:

Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.

What if we take up the challenge to author and illustrate mathematical understanding with and for our students and teammates?

Let’s work together to use and connect mathematical representations as we build procedural fluency from conceptual understanding.

Humphreys, Cathy. Making Number Talks Matter. Stenhouse Publishers. Kindle Edition.

Leinwand, Steve. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA.: National Council of Teachers of Mathematics, 2014. Print.

Smith, Margaret (Peg) S.. The Five Practices in Practice [Middle School] (Corwin Mathematics Series). SAGE Publications. Kindle Edition.

Embolden Your Inner Mathematician Week 2: Contemplate then Calculate (#CthenC)

For our second session of Embolden Your Inner Mathematician, we focus on Numeracy and Visual Learning: Elicit and use evidence of student thinking.

What is we use powerful tools to elicit student thinking? How might we learn about students to deeply understand them as mathematicians? And then, what actions do we take to ensure mathematical success for all?

This week’s session began with a gallery walk using Amy Lucenta and Grace Kelemanik’s first five Contemplate then Calculate (#CthenC) lessons found on at Fostering Math Practices.

From Ruth Parker and Cathy Humphreys in Making Number Talks Matter:

No matter what grade you teach, even high school, so-called “dot” cards (which may not have dots) are a great way to start your students on the path to mathematical reasoning. We say this because, from experience, we have realized that with dot cards, students only need to describe what they see— and people have many different ways of seeing! Arithmetic problems, on the other hand, tend to be emotionally loaded for many students. Both of us have found that doing several dot talks before we introduce Number Talks (with numbers) helps establish the following norms:

• There are many ways to see, or do, any problem.

• Everyone is responsible for communicating his or her thinking clearly so that others can understand.

• Everyone is responsible for trying to understand other people’s thinking.

To embolden mathematicians and to prepare to elicit and use evidence of student thinking, teaching teams must practice to develop the habits put forth in 5 Practices for Orchestrating Productive Mathematics Discussions.

You can see our teacher-learner-leaders working to deepen their understanding of and commitment to the Making Number Talks Matter: norms, Smith and Stein’s 5 Practices for Orchestrating Productive Mathematics Discussions, and NCTM’s Principles to Actions: Ensuring Mathematical Success for All.

How might we continue to deepen our understanding of NCTM’s teaching practices? What if we team to learn and practice?

Elicit and use evidence of student thinking.
Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.

In ambitious teaching, the teacher engages students in challenging tasks and collaborative inquiry, and then observes and listens as students work so that she or he can provide an appropriate level of support to diverse learners.  The goal is to ensure that each and every student succeeds in doing meaningful, high-quality work, not simply executing procedures with speed and accuracy. (Smith, 4 pag.)

Worth repeating:

The goal is to ensure that each and every student succeeds in doing meaningful, high-quality work, not simply executing procedures with speed and accuracy.

We continue to foster creativity, visual and algebraic representation to strengthen our mathematical flexibility as we learn together.

When mathematics classrooms focus on numbers, status differences between students often emerge, to the detriment of classroom culture and learning, with some students stating that work is “easy” or “hard” or announcing they have “finished” after racing through a worksheet. But when the same content is taught visually, it is our experience that the status differences that so often beleaguer mathematics classrooms, disappear.  – Jo Boaler

#ChangeTheFuture

#EmbraceAmbitiousTeaching

#EmboldenYourInnerMathematician

Seeing as Understanding: The Importance of Visual Mathematics for Our Brain and Learning.” Journal of Applied & Computational Mathematics 05.05 (2016): n. pag. Youcubed. Standford University, 12 May. 2016. Web. 18 Mar. 2017.

Humphreys, Cathy; Parker, Ruth. Making Number Talks Matter (Kindle Locations 339-346). Stenhouse Publishers. Kindle Edition.

Kelemanik, Grace, and Amy Lucent. “Starting the Year with Contemplate Then Calculate.” Fostering Math Practices.

Leinwand, Steve. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA.: National Council of Teachers of Mathematics, 2014. (p. 46) Print.

Smith, Margaret Schwan., et al. Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K-5. The National Council of Teachers of Mathematics, 2017.

PD Planning: Number Talks and Number Strings

As we begin the second part of our school year and as the calendar changes from 2016 to 2017, we review our goals.

The leaders of our math committee set the following goals for this school year.

Goals:

• Continue our work on vertical alignment.
• Expand our knowledge of best practices and their role in our current program.
• Share work with grade level teams to grow our whole community as teachers of math.
• Raise the level of teacher confidence in math.
• Deepen, differentiate, and extend learning for the students in our classrooms.

Our latest action step works on scaling these goals in our community. The following shows our plan to build common understanding and language as we expand our knowledge of numeracy.  Over the course of two days, each math teacher (1st-6th grade) participated in 3-hours of professional learning.

Our intentions and purpose:

We started with a number talk and a number string from Kristin Gray‘s NCTM Philadelphia presentation. We challenged ourselves to anticipate the ways our learners answer the following.

We also referred to Making Number Talks Matter to find Humphreys and Parker’s four strategies for multiplication.  We pressed ourselves to anticipate more than one way for each multiplication strategy to align with Smith and Stein’s 5 Practices for Orchestrating Productive Mathematics Discussions.

From our earlier work with Lisa Eickholdt, we know that our ability to talk about a strategy directly impacts our ability to teach the strategy.  What can be learned if we show what we know more than one way? How might we learn from each other if we make our thinking visible?

After working through Humphreys and Parker’s strategies (and learning new strategies), we transitioned to the number string from Kristin‘s presentation.

The goal for the next part of the learning session offered teaching teams the opportunity to select a number string from one of the Minilessons books shown below.  Each team selected a number string and worked to anticipate according to Smith and Stein’s 5 Practices for Orchestrating Productive Mathematics Discussions.

To practice, each team practiced their number string and the other grade-level teams served as learners.  When we share and learn together, we strengthen our understanding of how to differentiate and learn deeply.

Deep learning focuses on recognizing relationships among ideas. During deep learning, students engage more actively and deliberately with information in order to discover and understand the underlying mathematical structure.
—John Hattie, Doug Fisher, Nancy Frey

As we begin the second part of our school year and as the calendar changes from 2016 to 2017, what action steps are needed to reach our goals?

Hattie, John A. (Allan); Fisher, Douglas B.; Frey, Nancy; Gojak, Linda M.; Moore, Sara Delano; Mellman, William L. (2016-09-16). Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning (Corwin Mathematics Series) (p. 136). SAGE Publications. Kindle Edition.

Humphreys, Cathy; Parker, Ruth (2015-04-21). Making Number Talks Matter (Kindle Locations 1265-1266). Stenhouse Publishers. Kindle Edition.

Norris, Kit; Schuhl, Sarah (2016-02-16). Engage in the Mathematical Practices: Strategies to Build Numeracy and Literacy With K-5 Learners (Kindle Locations 4113-4115). Solution Tree Press. Kindle Edition.

Smith, Margaret Schwan., and Mary Kay. Stein. 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA: National Council of Teachers of Mathematics, 2011. Print.

Fluency: comprehension, accuracy, flexibility, and efficiency

No strategy is efficient for a student who does not yet understand it. (Humphreys & Parker, 27 pag.)

If both sense and meaning are present, the likelihood of the new information getting encoded into longterm memory is very high. (Sousa, 28 pag.)

When we teach for understanding we want comprehension, accuracy, fluency, and efficiency. If we are efficient but have no firm understanding or foundation, is learning – encoding into longterm memory – happening?

We don’t mean to imply that efficiency is not important. Together with accuracy and flexibility, efficiency is a hallmark of numerical fluency. (Humphreys & Parker, 28 pag.)

What if we make I can make sense of problems and persevere in solving them and I can demonstrate flexibility essential to learn?

If we go straight for efficiency in multiplication, how will our learners overcome following commonly known misconception?

common misconception: (a+b)²=a² +b²

correct understanding: (a+b)²=a² +2ab+b²

The strategies we teach, the numeracy that we are building, impacts future understanding.  We teach for understanding. We want comprehension, accuracy, fluency, and efficiency.

How might we learn to show what we know more than one way? What if we learn to understand using words, pictures, and numbers?

What if we design learning episodes for sense making and flexibility?

Humphreys, Cathy, and Ruth E. Parker. Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4-10. Portland, ME: Steinhouse Publishers, 2015. Print.

Sousa, David A. Brain-Friendly Assessments: What They Are and How to Use Them. West Palm Beach, FL: Learning Sciences, 2014. Print.

Number Talks: how AND why

Listening informs questioning. (Berger, 98 pag.)

How do we know learning has occurred? How do we know how learning has happened? What if we pause and listen to learn?

If both sense and meaning are present, the likelihood of the new information getting encoded into longterm memory is very high. (Sousa, 28 pag.)

How would you add 39 to 67? Would you use the traditional algorithm? Would you need paper? How might we teach flexibility, sense making, and numeracy to build fluency and confidence?

Number talks are about students making sense of their own mathematical ideas. (Humphrey & Parker, 13 pag.)

How might we seize the opportunity to confer with our learners to see if they are making sense of what is being taught?

This is the challenge – and joy – of teaching by listening to students. (Humphrey & Parker, 13 pag.)

If interested in additional examples of number talks, both the how and the why, listen to Jo Boaler and her students from the Stanford Online MOOC How to Learn Math: For Teachers and Parents.

Do we believe our learners – every one of them – are capable of developing proficiency in mathematics?

How might we show what we know more than one way?

How might we continue to send the message I believe in you and mean it?

What if we listen to learn?

I am grateful to Kristin Gray (@MathMinds) and Crystal Morey (@themathdancer) for their leadership and facilitation as a dozen #TrinityLearns faculty participate in an online book club (#mNTmTch) for Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding Grades 4-10 along with over 600 educators across the globe.

Berger, Warren (2014-03-04). A More Beautiful Question: The Power of Inquiry to Spark Breakthrough Ideas . BLOOMSBURY PUBLISHING. Kindle Edition.

Humphreys, Cathy, and Ruth E. Parker. Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4-10. Portland, ME: Steinhouse Publishers, 2015. Print.

Sousa, David A. Brain-Friendly Assessments: What They Are and How to Use Them. West Palm Beach, FL: Learning Sciences, 2014. Print.

Flexibility and sense-making to build confidence and long-term memory

In his TEDxSonomaCounty talk, The Myth of Average, Todd Rose (@ltoddrose) challenges us to consider and act to leverage simple solutions that will improve the performance of our learners and dramatically expand our talent pool.  (If you’ve not seen his talk, it is worth stopping to  watch the 18.5 minute message before reading on.)

There are far too many students who feel like they are no good at math because they aren’t quick to get right answers. (Humphreys & Parker, 9 pag.)

Efficiency must not trump understanding.  How often do we remember the foundation once we’ve mastered “the short cut?” Were we ever taught the foundation – the why – or were we only taught to memorize procedures that got to an answer quickly?

Of course, students must be able to compute flexibly, efficiently, and accurately. But they also need to explain their reasoning and determine if the ideas they’re using and the results they’re getting make sense. (Humphreys & Parker, 8 pag.)

How might we design and implement practices that help our young learners make sense of what they are learning?  In Brain-Friendly Assessments: What They Are and How to Use Them, David Sousa explains how necessary sense-making and meaning are to transfer information from working memory into long-term memory.

The brain is more likely to store information if it makes sense and has meaning. (Sousa, 28 pag.)

Dr. Sousa continues:

We should not be measuring just content acquisition. Rather, we should be discovering the ways students can process and manipulate their knowledge and skills to deal with new problems and issues associated with what they have learned.  (Sousa, 28 pag.)

The first chapter of Making Number Talks Matter highlights the importance of number talks.  We want our young learners to develop flexibility and confidence working with numbers.

Listen to Ruth Parker and Cathy Humphreys discuss Number talks:

From Jo Boaler’s How to Learn Math: for Students:

…we know that what separates high achievers from low achievers is not that high achievers know more math, it is that they interact with numbers flexibly and low achievers don’t.

What if we take action on behalf of our young learners?  What if we offer multiple pathways for success?

How might we dramatically expand our talent pool?

I am grateful to Kristin Gray (@MathMinds) and Crystal Morey (@themathdancer) for their leadership and facilitation as a dozen #TrinityLearns faculty participate in an online book club (#mNTmTch) for Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding Grades 4-10 along with over 600 educators across the globe.

Humphreys, Cathy, and Ruth E. Parker. Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4-10. Portland, ME: Steinhouse Publishers, 2015. Print.

Sousa, David A. Brain-Friendly Assessments: What They Are and How to Use Them. West Palm Beach, FL: Learning Sciences, 2014. Print.