Category Archives: Connecting Ideas

Deep Practice, Leveling, and Communication (TBT Remix)

Does a student know that they are confused and can they express that to their teacher? We need formative assessment and self-assessment to go hand-in-hand.

I agree that formative self-assessment is the key. Often, I think students don’t take the time to assess if they understand or are confused. I think that it is routine and “easy” in class. The student is practicing just like they’ve been coached in real time. When they get home, do they “practice like they play” or do they just get through the assignment? I think that is where deep practice comes into play. If they practice without assessing (checking for success) will they promote their confusion?  I tell my students that it is like practicing shooting free throws with your feet perpendicular to each other. Terrible form does not promote success. Zero practice is better than incorrect practice.

With that being said, I think that teachers must have realistic expectations about time and quality of assignments. If we expect students to engage in deep practice (to embrace the struggle) then we have to shorten our assignments to accommodate the additional time it will take to engage in the struggle.  We now ask students to complete anywhere from 1/3 to 1/2 as many problems as in the past with the understanding that these problems will be attempted using the method of deep practice.

Our version of deep practice homework:
“We have significantly shortened this assignment from years past in order to allow you time to work these questions correctly. We want you do work with deep practice.

  • Please work each problem slowly and accurately.
  • Check the answer to the question immediately.
  • If correct, go to the next problem.
  • If not correct, mark through your work – don’t eraseleave evidence of your effort and thinking.
    • Try again.
    • If you make three attempts and can not get the correct answer, go on to the next problem. “

I also think that the formative assessments with “leveling” encourage the willingness to struggle. How many times has a student responded to you “I don’t get it”? Perhaps it is not a lack of effort. Perhaps it is a lack of connected vocabulary. It is not only that they don’t know how, is it that they don’t know what it is called either. It is hard to struggle through when you lack vocabulary, skill, and efficacy all at the same time. How might we help our learners attend to precision, to communicate in the language of our disciplines?

Now is the time to guide our young learners to develop voice, confidence (and trust), and a safe place to struggle.

Deep Practice, Leveling, and Communication was originally published on November 20, 2010


#ShowYourWork: words, pictures, numbers

In her Colorful Learning post, Learning: Do our students know we care about that?, Kato shared the following learning progression for showing your work.

What if we guide our learners to

I can describe or illustrate ow I arrived at a solution so that the reader understands without talking to me?

Isn’t this really about making thinking visible and clear communication?  Anyone who has taught learners who take an AP exam can attest to the importance of organized, clear pathways of thinking. It is not about watching the teacher show work, it is about practicing, getting feedback, and revising.

Compare the following:

What if a learner submits the following work?

Screen Shot 2015-11-09 at 8.43.46 AM

Can the reader understand how the writer arrived at this solution without asking any questions?

What if the learner shared more thinking? Would it be clearer to the reader? What do you think?


How often do we tell learners that they need to show their work? What if they need to show more work? What if they don’t know how?

How might we communicate and collaborate creatively to show and tell how to level up in showing work and making thinking visible?

How might we grow in the areas of comprehension, accuracy, flexibility, and deeper understanding if we learn to communicate clearly using words, pictures, and numbers?

Master of them all: make sense of problems and persevere

In his #CMCS15 session, Michael Serra challenges us to consider:

“Of all the Mathematical Practices, there is one that stands above the others: Make sense of problems and persevere in solving them.”

If our learners cannot make sense of tasks and persevere in solving them, will they even find opportunities to experience the other Standards for Mathematical Practices?

What actions do we take to develop and grow a collaborative culture of perseverance?  How might we leverage gaming to foster perseverance, inspire struggle, and promote flexible thinking?


Can we demonstrate enough self-regulation to hold our solution long enough for our learners to persevere, productively struggle, and find a solution pathway for themselves?

How might we develop a community of learners that, when asked if they’d like a hint, say a loud, resounding NOWe can persevere; we can do it ourselves!

HMW walk the walk: 1st draft doesn’t equal final draft

In her #CMCS15  session, Jessica Balli (@JessicaMurk13) challenges us to consider how we might redefine mathematical proficiency for teachers and students. Are our actions reflecting a current definition or are we holding on to the past?

How might we engage with the Standards for Mathematical Practice to help all redefine what it means to be ‘good at math’?

Do we value process and product? Are we offering opportunities to our learners that cause them to struggle, to grapple with big ideas, to make sense and persevere?


Do we value our learners’ previous knowledge or do we mistakenly assume that they are blank slates? What if we offer our learners opportunity to show what they know first?  How might we use examples and non-examples to notice and note and then revise?

What if we take up the challenge to walk the walk to prove to our learners (and ourselves) that a first draft is not the same as a final draft?

Mistakes: it’s what you do next…

Mistakes: everybody makes them; the key is what happens next.

In his #CMCS15  session, Making Math Mistakes and Error Analysis: Diamonds in the Rough session, Andrew Stadel (@mr_stadelDivisible by 3, and Estimation 180) challenges us to make our thinking visible and to seize opportunities to deepen understanding.

  • Math mistakes are a valuable window into student thinking
  • Analysis of mistakes can help drive instruction, curb student misconceptions, and strengthen formative assessment.

How might we strength formative assessment to spur action?  Knowing is not enough.  What if we bright spot work found in the mistake to show something was going well?


Do we practice?  How often do we reflect on our struggles? Knowing what went well and where we struggled, how might we consider taking new tack in what we do next?

Do something different… It’s what happens next.



#TEDTalkTuesday: Brave space a.k.a. turning collisions into connections

Yesterday, I had the privilege of attending a session at GISA on Implicit Bias facilitated by Trinity’s very own Gina Quiñones () and Lauren Kinnard ().

Lauren and Gina began the session by setting norms, challenging us to level up from a safe space to a brave space. How might we dare to be brave enough to express what we think and feel? What if we listen to others to learn?

Screen Shot 2015-11-03 at 6.53.47 PM

They challenged us to consider how might we turn our own cultural collisions into more meaningful connections and shared the following TED talk.

Turning cultural collisions into cultural connections: Nadia Younes at TEDxMontrealWomen

I am grateful to work and learn with brave leaders, and I am thankful for all who trust enough to share brave space.

Fluency: comprehension, accuracy, flexibility, and efficiency

No strategy is efficient for a student who does not yet understand it. (Humphreys & Parker, 27 pag.)

If both sense and meaning are present, the likelihood of the new information getting encoded into longterm memory is very high. (Sousa, 28 pag.)

When we teach for understanding we want comprehension, accuracy, fluency, and efficiency. If we are efficient but have no firm understanding or foundation, is learning – encoding into longterm memory – happening?

We don’t mean to imply that efficiency is not important. Together with accuracy and flexibility, efficiency is a hallmark of numerical fluency. (Humphreys & Parker, 28 pag.)

What if we make I can make sense of problems and persevere in solving them and I can demonstrate flexibility essential to learn?


Flexibility #LL2LU

If we go straight for efficiency in multiplication, how will our learners overcome following commonly known misconception?

common misconception: (a+b)²=a² +b²


correct understanding: (a+b)²=a² +2ab+b²

The strategies we teach, the numeracy that we are building, impacts future understanding.  We teach for understanding. We want comprehension, accuracy, fluency, and efficiency.

How might we learn to show what we know more than one way? What if we learn to understand using words, pictures, and numbers?

What if we design learning episodes for sense making and flexibility?

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.