Category Archives: Reflection

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.

Lesson Study: different teachers, common lesson plan, guaranteed and viable curriculum

What if we share common mission and vision? How might we express our style, individuality, and personality while holding true to a plan and the essentials to learn?

My team, the Academic Leadership Team, includes the Head of School, both Division Heads, the Director of Curriculum, the Director of Technology, and me. We strategically plan using our agreed upon essential learnings.

This week, I had the honor and privilege of observing members of my team launch learning based on our goals and plans.  Can you see our connectedness, themes, and common language?

All School Meeting
with Joe Marshall, Head of School

Upper Elementary Division Meeting
with  Sarah Barton Thomas, Division Head

Early Elementary Division Meeting
with Rhonda Mitchell, Division Head

Instructional Core Meeting
with Jill Gough, Director of Teaching and Learning
and Marsha Harris, Director of Curriculum

Early Elementary Division Meeting
with Rhonda Mitchell, Division Head

Upper Elementary Division Meeting
with  Sarah Barton Thomas, Division Head

How might we team to meet the needs of our diverse learners? What if teaching teams plan common lessons based on guaranteed and viable curriculum? And, what can we learn when we observe each other?

#BeTogetherNotTheSame
#GrowAndLearnTogether

Focus on Instructional Core: establish goals to focus learning

As part of our school’s Pre-Planning, Marsha Harris and I facilitated a faculty-teams workshop to continue our work and learning in the Instructional Core.

Here are my notes from the session.

The agenda, shared ahead of the meeting, looked like this:

The slide deck that accompanies this plan looks like this:

As seen in the slides, we checked in with John Hattie’s research around teacher clarity.

Teacher clarity involves the instructional moves a teacher makes that begin with carefully planning a lesson and making the learning intentions for that lesson or unit clear to herself and her students. 

It extends to consistently evaluating where students are in the learning process and describing the success criteria on which students can assess their own progress and on which the teacher bases her evaluation of a student’s progress with a idea or concept. (Hattie, 38 pag.)

To model teacher clarity, we looked at two drafts for

I can establish goals to focus learning.

First, establish goals:

Then, focus learning:

How might we partner together to establish learning goals? What if we by “do the task as a learner” to notice and note needed prerequisites and anticipate potential learning obstacles? Can we deepen learning experiences by connecting to prior learning standards and strategies?

What if we make learning goals visible so that learners are able to identify what they know and need to know next?  How might we team to anticipate needed questions to assess and advance learning? What if we teach learners to ask more questions to forward and deepen learning? How might we empower learners to level up?

When we focus on learning,
we strengthen the Instructional Core.


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. 38). SAGE Publications. Kindle Edition.

The self-discipline to wait, watch, coach (revised)

My small extended family, there are just 10 of us, blazed through 12 dozen homemade cookies in three afternoons. Home for the holiday, my mother, my daughter, and I bake for pleasure, to help the house smell good, and to pass on important family traditions.  The cookie baking extravaganza has now extended into day 4. The demand for more cookies might be triggered by the smell of chocolate, peanut butter, and sugar wafting throughout the house, back porch, and driveway. Or, it could be gluttony. It’s a holiday; calories don’t count, right?

In her new red and green pjs, AS wakes up raring to go.  Jumping up and down in the kitchen in her new polka dot apron, she asks “How many cookies will we bake today, Mama? How many? How many?”

I hold back a sigh and try not to drop my head; I am tired. I have turkey, dressing, ham, and several casseroles to prepare to carry on our traditions, and I am experiencing cookie overload. I muster my best smile and say, “We need to bake at least 4 dozen cookies. Uncle Jack is coming today, and you know how much he loves your cookies.”

It is our 4th day of cookie baking. Once again, by popular request, we were making Reese’s peanut butter cup cookies.  We make peanut butter cookie dough, roll it into balls, and cook them in mini muffin pans.  As they come out of the oven, we press mini Reese’s peanut butter cups into the center of the cookies.  Delicious.

It is day 4 of this algorithmic work.  The learner is still excited, curious, and engaged.  Am I? Do I feel the same engagement, or am I bored and ready to move on?

For the first 2 dozen, I make the batter, and three generations work together in concert to roll the cookies into balls. The tins come out of the oven holding peanut butter goodness just waiting to receive the Reese’s peanut butter cup candies.  Together, my mother, AS, and I press the candy into the cookies as they come out of the oven. I can still picture my grandmother’s hands doing this work with my mother and me.

Apprenticeship as learning is so important.

I am struck by the lessons my sweet 6-year old, AS, is teaching me about learning with my students. How often do our students watch us do the work to solve the problem or answer the question and pitch in at the last step?   

Baking the second 2 dozen is a very different story.  Thanks to my mother, AS her very own measuring spoons, spatula, and mini muffin pan that bakes 1 dozen muffins.  Empowered now that she has her own pan, she takes charge. It would have been so much faster for me to roll the cookies.  But, no…her pan; her cookies. Her mantra: “I can do it myself!”

So, I watch, wait, and coach.  I try not to cringe. I hold my comments so that I do not undermine her independence and confidence. Too small, the balls will be difficult to press candy into after baking in the oven.  Too big and they will blob out on the pan during baking. Patiently, I ask, “I wonder, honey, if the peanut butter cup will fit into that ball once baked. What do you think?” She fixes most of these problems with a little coaching from me.

Isn’t this happening in our classrooms?  It is so much faster and more efficient for the teacher to present the material.  We can get so much more done in the short amount of time we have. But, how much does the learner “get done” or learn?  When efficiency trumps learning, does anyone really have success? How do we encourage “I can do it myself!”? How do we find the self-discipline to watch, wait, and coach?

As she demands more independence, her confidence grows.  Can you believe that she would alter my recipe for the first 2 dozen cookies?  As our second dozen bakes, I press the peanut butter cups into my cookies. Miss I-Can-Do-It-Myself decides that Hershey kisses will be just as good or better.  With no prompting (or permission) she creates a new (to her) cookie. She has Hershey Kisses, and she wants to use them.

Worth repeating: “As she demands more independence, her confidence grows.” When we intervene too soon, are we stripping learners of their confidence and independence? Are we promoting productive struggle? Do we let them grapple enough?  

Does it really matter which method a learner uses to solve a problem or answer a question?  Isn’t it okay if they use the distributive property or an area model to multiply? Does it really matter which method is used to find the solution to a system of equations?  Shouldn’t they first find success? Don’t we want our learners to understand more than one way? Is our way always the best way?

Is AS pleased with herself and her creativity?  You bet. Are her cookies just as good as the original recipe?  Sure! How can you go wrong combining chocolate and peanut butter?

We must applaud the process that learners use to solve a problem or respond to a question.  We must praise them when they try something different. We must promote and encourage risk-taking, creativity, and problem-solving.

We must find the self-discipline to be patient while learning is in progress, to watch, wait, and coach.  We must embrace and promote the “I can do it myself!” attitude.

We must.


The self-discipline to wait, watch, coach was originally published on Dec 26, 2010.  This revision is inspired by what we are learning in Embolden Your Inner Writer.

I am grateful for the thoughtful, challenging, advancing feedback from Marsha Harris, Amanda Thomas, Kate Burton, Becky Holden, Cathrine Halliburton, and Lauren Kinnard.

High-purpose environment. Teacher clarity. Touchpoints of praise.

I wait patiently for my turn.

Carrots. Beep. Doritos. Beep. Milk. Beep.

Donned in her green Publix smock, she makes eye contact and small talk with the customer ahead of me as she swipes items across the reader.

Hamburger. Beep. Kale. Beep. Beep. Beep.

She says, “That will be forty-two. twenty-eight” Wincing, she shook her head and said, “No, no wait! It is twenty-eight forty-two.” Smiling sheepishly, she blushes and says “Ugh! I just hate numbers.” The customer, patient and kind, concludes her business at the register and goes on about her way.

I cannot stop myself. Why can’t I stop myself from attempting to put salve on the raw wound that someone else – knowingly or unknowingly – has inflected on this poor young woman? I hear my internal voice say, “You don’t have to fix this. You really can’t fix this. You did not do this.”

I know I should stop myself. I cannot. I softly say, “So I’m a math teacher. It is easy to mix numbers up. Don’t worry.”

And then it happens… again. It breaks my heart a little more every time. Though it is not unexpected, I brace myself for what is coming.

She takes a deep breath. In a painful blurt, she replies, “I did so many posters just so I could pass.  She decided that was never going to ‘do’ math well, so she let me create bulletin boards and cut out letters in order to pass. I just hate it. Math was never my thing. Early, we knew that I could not do it, and we created workarounds so I could pass and graduate.”

So then, as always, I apologize for her terrible experience.

I am so sorry.

I am so sorry that any child is led to believe they cannot be successful at math – the language, art, and communication tool that is my love and passion.

I am so sorry that any child is led to believe they cannot be successful.

I seethe inside that any teacher would “extra credit” a child out of learning.

High-purpose environments are filled with small, vivid signals designed to create a link between the present moment and a future ideal. They provide the two simple locators that every navigation process requires: Here is where we are and Here is where we want to go. The surprising thing, from a scientific point of view, is how responsive we are to this pattern of signaling. (Coyle, 180 pag.)

Teachers need to determine the gap between students’ current level of performance or understanding and the expected level of mastery. (Hattie, 66 pag.)

If someone received just three or more touchpoints, or instances, of praise in a single quarter, their performance score in the next review period significantly increased. If they received four or more touchpoints of praise or recognition in a quarter, the retention rate increased to 96 percent over the next year. (Achor, Kindle Locations 1766-1768.)

How might we create more classrooms that are high-purpose environments where teacher clarity empowers learners to close gaps between what is known and what is needed?  What if we highlight what is going well to create touch points of praise to embolden learners to reach for a next level?

CULTURE: from the Latin cultus, which means care.


Achor, Shawn. Big Potential: How Transforming the Pursuit of Success Raises Our Achievement, Happiness, and Well-Being (Kindle Locations 1766-1768). The Crown Publishing Group. Kindle Edition.

Coyle, Daniel. The Culture Code: The Secrets of Highly Successful Groups (Kindle Locations 2378-2380). Random House Publishing Group. Kindle Edition.

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.

I know something!

A motley band of brothers traveling as a pack ambled in at 7:58 for Pre-Cal. Just in time. From a distance they are handsome and well dressed in coats and ties. Up close, they are a little disheveled, like they stepped out of their clothes last night and into them again this morning.  My favorite part is the peacock-ness of their hair.  They slept hard, and it shows. It helps me remember how young they are when they look so grown up. I wonder how many of them brushed their teeth and decided not to go there.

They live together, play together, and learn together. Boarding school builds family and familiarity and deepens connectedness.  Yesterday’s test results require a preface. I prep them by explaining what I learned about them and what we need to do now.  Learning is our focus, and this moment in time tells us that we need to do more, go deeper, and practice harder. This milestone marker says that we need to back up and try again.

I made a 42!” he exclaimed with genuine jubilation. It’s what he said next that confirmed his glee.  “I know something!

“I know something!”

I am in awe of what happens next.  Every man, child, boy learner, in their coats and ties, stands and high-fives him as if he as won a gold medal, sunk the winning shot, made the field goal in the final 30 seconds.

And then, just as quickly as it started, they were all seated as he said “Bring it, Ms. Gough. What do we do now?”

Group culture is one of the most powerful forces on the planet. We sense its presence inside successful businesses, championship teams, and thriving families, and we sense when it’s absent or toxic. (Coyle, xvii pag.)

Culture is a set of living relationships working toward a shared goal. It’s not something you are. It’s something you do. (Coyle, xx pag.)

Live, learn, work, and serve in a community that highlights what is going well. Focus on learning. Build living relationships. Work toward shared goals.

CULTURE: from the Latin cultus, which means care.


Coyle, Daniel. The Culture Code: The Secrets of Highly Successful Groups. Random House Publishing Group. Kindle Edition.

Available for learning

I feel it coming.

Have you seen surfers catch big waves? They paddle out, wait and watch.  Some waves pass by. Others crash over them. Sometimes the surfer gets up, punches through and rides it out.  Other times they wipe out. It is fascinating that they get back up and go back out. They persevere. I wonder how I can do that.

I feel it coming, but I cannot predict how I’m going to handle it. It is different every time. Sometimes I cry. Sometimes I turn and fight an internal battle to stand my ground to punch through the wave. Sometimes I can ask for help, but not always.

I am afraid it won’t be perfect.

I am afraid I can’t do it.

I am afraid others will laugh at me.

I am afraid I’ll disappoint my parents.

I am afraid I’m not good enough.

I am afraid I won’t be accepted.

I am afraid I don’t belong.

I am afraid I am not enough.

I am afraid.

I am afraid.

Labels make it easy to attribute student disengagement to a lack of ability or motivation, when disengagement often results from a lack of confidence. (Hassan and Lennard, 70 pag.)

The key to creating psychological safety, as Pentland and Edmondson emphasize, is to recognize how deeply obsessed our unconscious brains are with it. A mere hint of belonging is not enough; one or two signals are not enough. We are built to require lots of signaling, over and over. This is why a sense of belonging is easy to destroy and hard to build. (Coyle, 13 pag.)

How might we learn more about our learners? What actions are needed so that learners know they are psychologically safe?

What conditions must be set so that more learners punch through and rides out the wave?

CULTURE: from the Latin cultus, which means care.


Coyle, Daniel. The Culture Code: The Secrets of Highly Successful Groups. Random House Publishing Group. Kindle Edition.

Hasson, Julie, and Missy Lennard. Unmapped Potential: an Educator’s Guide to Lasting Change. Dave Burgess Consulting, Inc., 2017.