Category Archives: Algebra

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.

Paper.Productive Struggle.195

From The 5 Practices in Practice: Successfully Orchestrating Mathematical Discussion in your Middle School Classroom:

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.

Agenda: Embolden Your Inner Mathematician (10.17.18) Week 6

Week Six of Embolden Your Inner Mathematician

We commit to curation of best practices, connections between mathematical ideas, and communication to learn and share with a broad audience.

Course Goals:
At the end of the semester, teacher-learners should be able to say:

  • I can work within NCTM’s Eight Mathematical Teaching Practices for strengthening the teaching and learning of mathematics.
  • I can exercise mathematical flexibility to show what I know in more than one way.
  • I can make sense of tasks and persevere in solving them.

Today’s Goals

At the end of this session, teacher-learners should be able to say:

  • I can use and connect mathematical representations. (#NCTMP2A)
  • I can make sense of tasks and persevere in solving them. (#SMP-1)
  • I can show my work so that a reader understands without have to ask me questions.

From Principles to Actions: Ensuring Mathematical Success for All

Use and connect mathematical representations: 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.

From Taking Action: Implementing Effective Mathematics Teaching Practices in Grades K-5

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.)

Learning Progressions for today’s goals:

  • I can use and connect mathematical representations. (#NCTMP2A)

  • I can show my work so that a reader understands without have to ask me questions.

Tasks:

  • Visual representation of multiplication, exponents, subtraction. (Connect 2nd-5th grade with Algebra I and II.)
  • Apples and Bananas task (see slide deck)

What the research says:

Not only should students be able to understand and translate between modes of representations but they should also translate within a specific type of representation. [Smith, pag. 139] 

Equitable teaching of mathematics includes a focus on multiple representations. This includes giving students choice in selecting representations and allocating substantial instructional time and space for students to explore, construct, and discuss external representations of mathematical ideas. [Smith, pag. 141]

Too often students see mathematics as isolated facts and rules to be memorized. [Smith, pag. 141]

\Anticipated work and thinking:

Slide deck:

[Cross posted at Sum it up and Multiply it out]


Gough, Jill, and Jennifer Wilson. “#LL2LU Learning Progressions.” Experiments in Learning by Doingor Easing the Hurry Syndrome.WordPress, 04 Aug. 2014. Web. 11 Mar. 2017.

Leinwand, Steve. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA.: National Council of Teachers of Mathematics, 2014. (p. 21) 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.

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.)

Screen Shot 2018-08-26 at 6.50.50 PM

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.

Notice success, celebrate multiple milestones, level up

Learning intentions are more than just statements to convey to students what the learning is composed of; they are a means for building positive relationships with students. (Hattie, 48 pag.)

It is what I didn’t notice.  The bell rang. As always, I heard a chorus of “Thank you, Ms. Gough. Bye, Ms. Gough.” It was normal practice – and a much appreciated practice – for my students to say thank you and goodbye as they left for their next class.

I thought to myself “what a great class, everything went well, and they are so nice.” I busied myself straightening my desk, organizing paper, and mentally listing off the things I needed to do before my next class rolled in.  Eat lunch was at the top of the list.

Then, I sensed it. I was not alone.  It is what I didn’t notice.  There she sat, so still, except for the river of tears falling out of her beautiful, sad, green eyes. The river ran off the desk and pooled on the floor. “What is wrong?” I asked as I sat down beside her.

As I gently placed my hand on her arm, her shoulders began to shake as she said “I f..f..f..failed!” Whoosh, another flood of tears.

Now, she had not failed from my point of view. Her test score, damp as her test was now, showed a grade of 92 – an A.  And yet, she deeply felt a sense of failure.  As we sat together and looked at her work, we discovered that there was one key essential learning – in fact, a prerequisite skill – that caused her to stubble.

Tears, still streaming down her face, she said “I don’t know where I’m going wrong. I don’t miss this in class, but on the test, I fall apart.”

The point is to get learners ready to learn the new content by giving their brains something to which to connect their new skill or understanding. (Hattie, 44 pag.)

So, of course, the stumbling block for this sweet child is a known pain point for learners who master procedures without conceptual understanding.  Consistently, she expanded a squared binomial by “distributing” the exponent – a known pitfall. #petpeeve

When our learners do not know what to do, how do we respond? What actions can we take – will we take – to deepen learning, empower learners, and to make learning personal?

Kamb’s insight was that, in our lives, we tend to declare goals without intervening levels. We declare that we’re going to “learn to play the guitar.” We take a lesson or two, buy a cheap guitar, futz around with simple chords for a few weeks. Then life gets busy, and seven years later, we find the guitar in the attic and think, I should take up the guitar again. There are no levels. Kamb had always loved Irish music and had fantasized about learning to play the fiddle. So he co-opted gaming strategy and figured out a way to “level up” toward his goal:

Level 1: Commit to one violin lesson per week, and practice 15 minutes per day for six months.

Level 2: Relearn how to read sheet music and complete Celtic Fiddle Tunes by Craig Duncan.

Level 3: Learn to play “Concerning Hobbits” from The Fellowship of the Ring on the violin.

Level 4: Sit and play the fiddle for 30 minutes with other musicians.

Level 5: Learn to play “Promontory” from The Last of the Mohicans on the violin.

BOSS BATTLE: Sit and play the fiddle for 30 minutes in a pub in Ireland.

Isn’t that ingenious? He’s taken an ambiguous goal—learning to play the fiddle—and defined an appealing destination: playing in an Irish pub. Better yet, he invented five milestones en route to the destination, each worthy of celebration. Note that, as with a game, if he stopped the quest after Level 3, he’d still have several moments of pride to remember. (Heath, 163-164 pgs.)

What if I’d made my thinking visible?

What if I’d connected this learning to how 3rd graders are taught multiplication of two digit numbers by decomposing into tens and ones.  What if I’d connected this learning to how 3rd graders are also taught to draw area models to visualize the distributive property?

What if I’d shared my thinking and intentionally connected prior learning in levels?

By using Kamb’s level-up strategy, we multiply the number of motivating milestones we encounter en route to a goal. That’s a forward-looking strategy: We’re anticipating moments of pride ahead. But the opposite is also possible: to surface those milestones you’ve already met but might not have noticed. (Heath, 165 pag.)

How might we help our learners level up, experience success at several motivating milestones, and notice successes that might otherwise go unnoticed?

By multiplying milestones, we transform a long, amorphous race into one with many intermediate “finish lines.” As we push through each one, we experience a burst of pride as well as a jolt of energy to charge toward the next one. (Heath, 176 pag.)

Taken together, these practices make learning visible to students who understand they are under the guidance of a caring and knowledgeable teacher who is invested in their success. (Hattie, 48 pag.)


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). SAGE Publications. Kindle Edition.

Heath, Chip. The Power of Moments: Why Certain Experiences Have Extraordinary Impact. Simon & Schuster. Kindle Edition.

Embolden Your Inner Mathematician: Week 2 agenda

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.
  Principles to Actions: Ensuring Mathematical Success for All

Slide deck

7:15 Establishing Intent, Purpose, Norm Setting

8:00 Continuing Talking Points – Elizabeth Statmore (@chessemonkeysf)

8:15 Number SplatsSteve Wyborney (@SteveWyborney)
8:25 Fraction SplatsSteve Wyborney (@SteveWyborney)
8:45 Planning for Splats

9:00 Closure and Reflection

  • I learned to pay attention to…
  • I learned to ask myself…
  • A new mathematical connection is…
9:15 End of session

Homework:

  • Elicit and use evidence of student thinking using Splats. What will/did you learn?
  • Write to describe your quest for Closest to One using Open Middle worksheet with I can show my work so a reader understands without asking me questions.
  • Deeply Read pp. 207-211 from TAKING ACTION: Implementing Effective Mathematics Teaching Practices in K-Grade 5
    • What the Research says: Elicit and Use Evidence of Student Thinking
    • Promoting Equity by Eliciting and Using Evidence of Student Thinking
  • Read one of the following from TAKING ACTION: Implementing Effective Mathematics Teaching Practices in K-Grade 5
    • pp.183-188 Make a Ten
    • pp.189-195 The Odd and Even Task
    • pp. 198-207 The Pencil Task

 


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

Kaplinsky, Robert, and Peter Morris. “Closest to One.” Open Middle.

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.

Statmore, Elizabeth. “Cheesemonkey Wonders.” #TMC14 GWWG: Talking Points Activity – Cultivating Exploratory Talk through a Growth Mindset Activity, 1 Jan. 1970.

Wyborney, Steve. “The Fraction Splat! Series.” Steve Wyborney’s Blog: I’m on a Learning Mission., 26 Mar. 2017.

Sketch notes from #TMC17 (a.k.a. Twitter Math Camp)

Becky Holden (@bholden86) and I attended Twitter Math Camp (#TMC17) at Holy Innocents Episcopal School in Atlanta, GA from Thursday, July 27 to Sunday, July 30.

This conference is by teachers, for teachers. The structure of TMC contains the following lengths of presentations:

  • Morning sessions (One session that meets Thursday, Friday and Saturday mornings for 2 hours each morning)
  • Afternoon sessions (Individual 1/2 hour sessions on Thursday)
  • Afternoon sessions (Individual 1 hour sessions Thursday, Friday and Saturday)

To honor Carl Oliver‘s (@carloliwitter) #PushSend request/challenge, here are my sketch notes from the sessions I attended.

Differentiating CCSS Algebra 1
— from drab to fab using Exeter Math 1 & Exploratory Talk
Elizabeth Statmore (@cheesemonkeysf)

The Politics(?) of Mathematics Teaching
Grace Chen (@graceachen)

What does it mean to say that mathematics teaching is political, and what does that mean for our moral and ethical responsibility as mathematics teachers?

Bridging elementary skills & concepts to high school & beyond
Glenn Waddell, Jr. (@gwaddellnvhs)

Micro-decisions in Questioning
David Petersen (@calcdave)

All I Really Need To Know I Learned From The MTBoS
…Not Really, But Close
Graham Fletcher (@gfletchy)

Hitting The Darn ‘Send’ Button
Carl Oliver (@carloliwitter)

Practical Ideas on the Kind of Coaching
We Need to Provide and Demand
Steve Leinwand (@steve_leinwand)

What is not captured in my notes is play: game night, trivia, crocheting, and tons of fun.

How might we grow, learn, and play in community when together and when apart?

Read, apply, learn

Read, apply, learn
`2017 T³™ International Conference
Saturday, March 11, 8:30 – 10 a.m.
Columbus H, East Tower, Ballroom Level
Jennifer Wilson
Jill Gough

How might we take action on current best practices and research in learning and assessment? What if we make sense of new ideas and learn how to apply them in our own practice? Let’s learn together; deepen our understanding of formative assessment; make our thinking visible; push ourselves to be more flexible; and more. We will explore some of the actions taken while tinkering with ideas from Tim Kanold, Dylan Wiliam, Jo Boaler and others, and we will discuss and share their impact on learning.

[Cross posted at Easing The Hurry Syndrome]