Tag Archives: Jo Boaler

Anticipating @IllustrateMath’s 6.RP Overlapping Squares

To anchor our work in differentiation and mathematical flexibility, we use NCTM’s 5 Practices for Orchestrating Productive Mathematics Discussions by Margaret Smith and Mary Kay Stein.

Kristi Story, Becky Holden, and I worked together during our professional learning time to meet the goals for the session shown below.

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.

The learning goals for students include:

I can use ratio reasoning to solve problems and understand ratio concepts.

I can make sense of tasks and persevere in solving them.

I can look for and make use of structure.

I can notice and note to make my thinking visible.

Kristi selected Illustrative Math’s  6.RP Overlapping Squares task for students. Here are the ways we anticipated how students would approach and engage with the task.

This slideshow requires JavaScript.

Our plan for helping students who are stuck includes providing and encouraging the use of a graphing tool such as graph paper or TI-Nspire software installed on their MacBooks. We also intend to use the following learning progressions.

I can make sense of tasks and persevere in solving them.

I can look for and make use of structure.

Finally, we also want our learners to work on how they show their work.

#ShowYourWork Subtraction

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


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.

Stein, Mary Kay., and Margaret Smith. 5 Practices for Orchestrating Productive Mathematics Discussions. N.p.: n.p., n.d. Print.

Goal work: learn more math, study the Practices

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 learn more math.
  • 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)

productive-struggle-4 productive-struggle-3

3:25 5 min Book Club warm-up

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.

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.

#TEDTalkTuesday: believe and change the future

Many teachers try to be comforting and sympathetic about math, telling girls not to worry, that they can do well in other subjects. We now know such messages are extremely damaging. (Boaler, n. pag.)

What if the messages are different? What if we send the message I believe in you? How might we change our future?

Brittany Wenger: Global neural network cloud service for breast cancer detection

Wenger began studying neural networks when she was in the seventh grade. She attributes her interest in science to her 7th grade science teacher.  As a high school senior, she won the grand prize in the 2012 Google Science Fair for her project, “Global Neural Network Cloud Service for Breast Cancer.”

How might we offer opportunities for integrated studies and human-centered problem solving?

What if we send the message I believe in you? How might we change our future?


Boaler, Jo. “Parents’ Beliefs about Math Change Their Children’s Achievement.” Youcubed. Stanford University, n.d. Web. 20 Sept. 2015.

What we don’t remember about the foundation…

I wonder if, when the house is finished, we forget the foundational infrastructure required for function.  How does water get into and out of my house? Who ran the wires so that our lamps illuminate our space? Who did the work, and what work was done, prior to the slab being poured?

When we recall a basic multiplication fact, it’s like flipping a light switch in our house. The electrical wiring allowing us to turn on the light is linked to sound, safe, and deeply connected infrastructure. (K. Nims, personal communication, August 30, 2015)

Just like the light switch is not part of the foundation, memorization of multiplication facts is also not foundational. It is efficient and functional.  Efficiency must not trump understanding.

We need people who are confident with mathematics, who can develop mathematical models and predictions, and who can justify, reason, communicate, and problem solve. (Boaler, n. pag.)

Screen Shot 2015-08-30 at 7.45.22 PMStudents who rely solely on the memorization of math facts often confuse similar facts. (O’Connell, 4 pag.)

Students must first understand the facts that they are being asked to memorize. (O’Connell, 3 pag.)

What if we have forgotten all the hard work that came prior to the task of memorizing our multiplication facts?

Do we remember learning about multiplication as repeated addition? Have we forgotten the connection between multiplication, arrays, and area?

Conceptual understanding of multiplication lays a foundation for deeper understanding of many mathematical topics.  Memorizing facts denies learners the opportunity to connect ideas, exercise flexibility, and interact with multiple strategies.

The goal is to have confident, competent, critical thinkers. Let’s remember that a strong foundation has many unseen components.  What if we slow down to develop deep understanding of the numeracy of multiplication?

Second, going slow helps the practitioner to develop something even more important: a working perception of the skill’s internal blueprint – the shape and rhythm of the interlocking skill circuits.”  (Coyle, 85 pag.)

How might we serve our learners by expecting them to show what they know more than one way?


Boaler, Jo. “The Stereotypes That Distort How Americans Teach and Learn Math.” The Atlantic. Atlantic Media Company, 12 Nov. 2013. Web. 30 Aug. 2015.

Coyle, Daniel (2009-04-16). The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. Random House, Inc.. Kindle Edition.

O’Connell, Susan, and John SanGiovanni. Mastering the Basic Math Facts in Multiplication and Division: Strategies, Activities & Interventions to Move Students beyond Memorization. Portsmouth, NH: Heinemann, 2011. Print.

struggle + perseverance = learning

How are we facilitating experience where learners can risk and grow in sense making and perseverance?  We want every learner to be able to say:

I can make sense of tasks and persevere in solving them.

An important and powerful aspect of teachers’ practice concerns the ways in which they treat mistakes in mathematics classrooms. Research has shown that mistakes are important opportunities for learning and growth, but students routinely regard mistakes as indicators of their own low ability. (Boaler, n. pag.)

Do we teach mistakes as opportunities to learn? What if we slow down – pause – to reflect on what didn’t work well and plan a new tact?

In analyzing a series of setbacks, a key question to ask is Am I failing differently each time? “If you keep making the same mistakes again and again,” the IDEO founder David Kelley has observed, “you aren’t learning anything. If you keep making new and different mistakes, that means you are doing new things and learning new things.”(Berger, 124 pag.)

How might we take up the challenge to focus on learning? What if we teach the importance of struggle?

Struggle is not optional—it’s neurologically required: in order to get your skill circuit to fire optimally, you must by definition fire the circuit suboptimally; you must make mistakes and pay attention to those mistakes; you must slowly teach your circuit. You must also keep firing that circuit—i.e., practicing—in order to keep myelin functioning properly. After all, myelin is living tissue. (Coyle, 43-44 pag.)

I can make sense of tasks and persevere in solving them.

How might we amplify the important practice of how we treat mistakes? What if we teach and learn how to pay attention to mistakes and how to change based on what we learn?

What pathways to learning are illuminated in order to highlight learning = struggle + perseverance?

What if we slow down to focus on learning?


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

Boaler, Jo. “Ability and Mathematics: The Mindset Revolution That Is Reshaping Education.” Forum 55.1 (2013): 143. FORUM: For Promoting 3-19 Comprehensive Education. SYMPOSIUM BOOKS Ltd, 2013. Web. 2015.

Coyle, Daniel (2009-04-16). The Talent Code: Greatness Isn’t Born. It’s Grown. Here’s How. Random House, Inc.. Kindle Edition.

Time, accuracy, speed, & precision (TBT Remix)

It is critical that we take a moment to review the emerging evidence on the impact of timed testing and the ways in which it transforms children’s brains, leading to an inevitable path of math anxiety and low math achievement. (Boaler, Jo)

Her name was Mrs. Hughes.  I can still hear her:

F … F … J … J … F … F … J … J.

Time, accuracy, speed, and precision were ultimately important in the typing class I took my sophomore year of high school.  I am glad that I touch-type.  At typingtest.com, you can assess your typing speed and accuracy.  Here are my latest results:

Screen Shot 2014-12-06 at 11.31.17 AM

And, later in the day…

Screen Shot 2014-12-06 at 6.02.14 PM

There are plenty of people that do not touch type, some hunt-and-peck.  Is their work some how diminished because they may need more time than a touch typist?  If not witnessing the time and effort, would the reader of the product even know whether the author was speedy or not?

Are accuracy and precision when typing more important than speed and time?  Wouldn’t it be better to take more time and have an accurate product than to be quick with errors?

This has me thinking about assessment, testing, and time.  In a perfect world, we want both speed and accuracy.  What if we can’t have both?  What if a learner needs more time to demonstrate what they know?  Do we really expect all children to perform and produce at the same speed?  Are we sacrificing accuracy and precision for the sake of time?  Should it be the other way around? Are we assessing what our learners know and can show or how fast they can think and work?

How important is it to complete an assessment
within a fixed, pre-determined period of time?

How might we offer learners more time to demonstrate what they know and have learned?

Time is the variable; learning is the constant.


Time, accuracy, speed, & precision was first posted on April 30, 2012

Boaler, Jo. “Timed Tests and the Development of Math Anxiety.” Web log post. Jo Boaler. N.p., 06 July 2012. Web. 09 Nov. 2014.