# 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.

# 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.

We will never know our reach unless we stretch. (Heath, 131 pag.)

When students don’t make errors, it’s probably because they already know the content and didn’t really need the lesson. (Hattie, 17 pag.)

Whack! One second everything was fine, then, for a fraction of a second, black. Kah-whop! I could see my phone, which used to be in my back pocket, hit the ice and slide about 8 feet in front of me.  Searing, hot pain surfaced in my left knee. It’s like I have a view from the ceiling. I can see myself face down on the ice. Cold. Wet.

I return to my eye’s view. I am really not sure what to do as I watch a nice soul skate over to my phone and bring it to me. While it was only a few seconds, it felt like 5 minutes of slow motion.  I was upright by then; no longer spread eagle face down on the ice.

A sweet young thing glided up and laughed at me. “Ouch!” I heard myself say, “Don’t laugh! I’m hurt, and I don’t think I know how I’m gonna get up.” I saw her flinch but not leave me. My eyes confirmed that I was in a crowd and no one seemed to know what to do but stare.  #NotGood

The music teacher—“a woman with a beehive-ish hairdo and a seemingly permanent frown on her face”—led the choir in a familiar song, using a pointer to click the rhythm of the song on a music stand. Then, Sloop remembered, “She started walking over toward me. Listening, leaning in closer. Suddenly she stopped the song and addressed me directly: ‘You there. Your voice sounds . . . different . . . and it’s not blending in with the other girls at all. Just pretend to sing.’ ” The comment crushed her: “The rest of the class snickered, and I wished the floor would open and swallow me up.” For the rest of the year, whenever the choir sang, she mouthed the words. (Heath, 141 pag.)

Whatever momentary lapse in concentration caused me to fall – splat – did not feel good.  And the laugh, while meant to make light of an awkward situation, was crushing.  It was a mistake and a painful one at that.

We hear it at school. We want our learners to be risk takers, to work on the edge of their ability, to fail faster, fail up, fail forward.  Right?

Get out there! Try something different! Turn over a new leaf! Take a risk! In general, this seems like sound advice, especially for people who feel stuck. But one note of caution: The advice often seems to carry a whispered promise of success. Take a risk and you’ll succeed! Take a risk and you’ll like the New You better!  That’s not quite right. A risk is a risk. (Heath, 131 pag.)

Errors help teachers understand students’ thinking and address it. Errors should be celebrated because they provide an opportunity for instruction, and thus learning. (Hattie, 16 pag.)

And just like that, she arrived.  An angel on the ice.  As she stretched out her hands, palms up, she said “Just take my hands.” I could get one foot square on the ice, though I felt like I was buried in a foot of snow, and then the other. Patiently she said “Now look at me and just press down.” I was up; shaken, but not broken.  Her beautiful brown eyes connected with mine and she smiled warmly as she said firmly “you are up and you are fine.” Just as quickly and elegantly as she arrived, she floated away.

Then, in the summer after her seventh-grade year, she attended a camp for gifted kids in North Carolina called the Cullowhee Experience. She surprised herself by signing up to participate in chorus. During practice, she mouthed the words, but the teacher noticed what she was doing and asked Sloop to stick around after class. The teacher was short and thin, with hair down to her waist—a “lovely flower child,” said Sloop. She invited Sloop to sit next to her on the piano bench, and they began to sing together in the empty room. Sloop was hesitant at first but eventually lowered her guard. She said, “We sang scale after scale, song after song, harmonizing and improvising, until we were hoarse.” Then the teacher took Sloop’s face in her hands and looked her in the eyes and said: “You have a distinctive, expressive, and beautiful voice. You could have been the love child of Bob Dylan and Joan Baez.” As she left the room that day, she felt as if she’d shed a ton of weight. “I was on top of the world,” she said. Then she went to the library to find out who Joan Baez was. “For the rest of that magical summer,” Sloop said, she experienced a metamorphosis, “shedding my cocoon and emerging as a butterfly looking for light.” (Heath, 142 pag.)

My knee still throbbed and most of me was shaking.  I limped over to the edge of the rink until I could steady my nerves.  I’m not sure which hurt worse, my knee or my pride.  In either case, it hurt. But, I was up and I was fine.

The words, tones, facial expression, and body language we use with our learners matters.

Memorizing facts, passing tests, and moving on to the next grade level or course is not the true purpose of school, although sadly, many students think it is. School is a time to apprentice students into the act of becoming their own teachers. We want them to be self-directed, have the dispositions needed to formulate their own questions, and possess the tools to pursue them. (Hattie, 32 pag.)

How might we highlight what is going well for our young learners, accent the positive, and gently guide them to stretch, risk, and reach? What if we craft our feedback so our learners know we believe in their ability and expect great things even when they stumble, fall, and hurt? What if we guide their apprentice work to learn to use needed tools and hone their skills.

Our hopes and dreams for learning don’t include pretending – just stand there and mouth the words. Our learners must emerge as butterflies.

What type of feedback are we practicing? Laughter to make light of a stumble? Calm, “take my hand and push; you are fine?”

The promise of stretching is not success, it’s learning. (Heath, 131 pag.)

What great mentors do is add two more elements: direction and support. I have high expectations for you and I know you can meet them. So try this new challenge and if you fail, I’ll help you recover. (Heath, 123 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.