# Using number lines to build strong, deep academic foundation

Many students struggle with algebraic ideas because they have not developed the conceptual understanding (Hattie, 129 pag.)

Are you a “just the facts ma’am” mathematician, or do you have deep conceptual understanding of mathematics? How did Algebra I, Algebra II, and Calculus go for you? Did you love it,  just survive it, or flat-out hate it?

What if we focus on depth of knowledge at an early age? How might we change the future for our young learners?

Imagine you are back in Algebra I, Algebra II, or Calculus working with polynomials.  Do you have conceptual understanding, procedural fluency, or both?

Learning has to start with fundamental conceptual understanding, skills, and vocabulary. You have to know something before you can do something with it. Then, with appropriate instruction about how to relate and extend ideas, surface learning transforms into deep learning. Deep learning is an important foundation for students to then apply what they’ve learned in new and novel situations, which happens at the transfer phase. (Hattie, 35 pag)

What if, at the elementary school level, deep conceptual numeracy is developed, learned, and transferred?

Our brains are made up of ‘distributed networks’,and when we handle knowledge, different areas of the brain light up and communicate with each other. When we work on mathematics, in particular, brain activity is distributed between many different networks, which include two visual pathways: the ventral and dorsal visual pathways (see fig 1). Neuroimaging has shown that even when people work on a number calculation,such as 12 x 25, with symbolic digits (12 and 25) our mathematical thinking is grounded in visual processing. (Boaler, n pag.)

Using concreteness as a foundation for abstraction is not just good for mathematical instruction; it is a basic principle of understanding. (Heath and Heath, 106 pag.)`

A number line representation of number quantity has been shown in cognitive studies to be particularly important for the development of numerical knowledge and a precursor of children’s academic success. (Boaler, n pag.)

Well, that’s worth repeating, huh?

A number line representation of number quantity has been shown in cognitive studies to be particularly important for the development of numerical knowledge and a precursor of children’s academic success.

Often, we rush to efficiency – to “just the facts ma’am” mathematics. Surface knowledge – memorized facts – is critical to success, but that is not the end goal of learning.  The goal of all learning is transfer.

When we use number lines to support conceptual understanding of number quantity and operations, we deepen and strengthen mathematical foundation.  Our young students are learning that multiplication is repeated addition, that 4 x 5 is 5 four times, which lays the foundation for being able to transfer to the following polynomials.

a + a + a +a = 4a
and
a + 3b +a + 3b = 2a + 6b

Abstraction demands some concrete foundation. Trying to teach an abstract principle without concrete foundations is like trying to start a house by building a roof in the air. (Heath and Heath, 106 pag.)

How might we focus on deep learning and transfer learning by studying and learning visually? What if we embrace seeing as understanding so that we learn to show what we know more than one way?

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.

Hattie, John A. (Allan); Fisher, Douglas B.; Frey, Nancy; Gojak, Linda M.; Moore, Sara Delano; Mellman, William L.. Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning (Corwin Mathematics Series) (p. 35). SAGE Publications. Kindle Edition.

Heath, Chip. Made to Stick: Why Some Ideas Survive and Others Die (p. 106). Random House Publishing Group. 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.

# Embolden Your Inner Mathematician Week 1: Number Talks

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

For our first session of Embolden Your Inner Mathematician, we focus on Subitizing and Number Talks: Elicit and use evidence of student thinking.

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.

Meeting the demands of world-class standards for student learning requires teachers to engage in what as been referred to as “ambitious teaching.” Ambitious teaching stands in sharp contrast to what many teachers experienced themselves as learners of mathematics. (Smith, 3 pag.)

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.

How might we foster curiosity, creativity, and critical reasoning while deepening understanding? What if we listen to what our students notice and wonder?

My daughter (7th grade) and I were walking through our local Walgreens when I hear her say “Wow, I wonder…” I doubled back to take this photo.

To see how we used this image in our session to subitize (in chunks) and to investigate the questions that arose from our wonderings, look through our slide deck for this session.

From  NCTM’s 5 Practices, we know that we should do the math ourselves, predict (anticipate) what students will produce, and brainstorm what will help students most when in productive struggle and when in destructive struggle. What if we build the habit of showing what we know more than one way to add layers of depth to understanding?

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

What if we ask ourselves what other ways can we add layers of depth so that students make sense of this task? How might we better serve our learners if we elicit and use evidence of student thinking to make next instructional decisions?

#ChangeTheFuture

#EmbraceAmbitiousTeaching

#EmboldenYourInnerMathematician

Boaler, Jo, Lang Chen, Cathy Williams, and Montserrat Cordero. “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.

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.

The math committee met this week to work on our goals. We agreed that, for the rest of this school year, we would spend half of our time on learning more math and the other half studying to learn more about the Standards For Mathematical Practice.

We met this week to learn more math and to discuss Chapter 1, Mathematical 1: Make Sense of Problems and Persevere in Solving Them in Beyond Answers: Exploring Mathematical Practices with Young Children by Mike Flynn.

Yearlong Goals:

• We can share work with grade level teams to grow our whole community as teachers of math.
• We can deepen our understanding of the Standards For Mathematical Practice.

Today’s Goals:

• I can make sense of tasks and persevere in solving them.
• I can reason abstractly and quantitatively.
• I can look for and make use of structure.

Resources:

Learning Plan

 3:05 5 min Quick scan of Jo’s YouCubed article (pp. 2, 11) 3:05 20 min Solving equations visually to make sense of the algebra (Learn more math) Pentagon of comprehension (shown below) Challenge to think deeply:  Solving 1-step Equations, 3:25 5 min Book Club warm-up Using #TrinityReads and #BeyondAnswers, tweet ideas from Chapter One that resonate with you or that you have questions about.  Tag Mike Flynn (@MikeFlynn55) 3:30 20 min Use Visible Thinking Routines to guide discussion of Chapter One: Make Sense and Persevere (deepen our understanding of the SMPs.) 3:55 5 min Feedback – “I learned…, “I liked…,”I felt…” Read Chapter 2: Reason Abstractly and Quantitatively

Update on PD (Goal: Scale our work to our teams.)

When we set purposeful team goals, we help each other make progress, and we use our time intentionally.

Flynn, Michael. Beyond Answers: Exploring Mathematical Practices with Young Children. Portland, Maine.: Stenhouse, 2017. Print.

Van de Walle, John. Teaching Student-centered Mathematics: Developmentally Appropriate Instruction for Grades Pre-K-2. Boston: Pearson, 2014. Print.