Category Archives: Reflection

#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?

Flexibility and sense-making to build confidence and long-term memory

In his TEDxSonomaCounty talk, The Myth of Average, Todd Rose (@ltoddrose) challenges us to consider and act to leverage simple solutions that will improve the performance of our learners and dramatically expand our talent pool.  (If you’ve not seen his talk, it is worth stopping to  watch the 18.5 minute message before reading on.)

There are far too many students who feel like they are no good at math because they aren’t quick to get right answers. (Humphreys & Parker, 9 pag.)

Efficiency must not trump understanding.  How often do we remember the foundation once we’ve mastered “the short cut?” Were we ever taught the foundation – the why – or were we only taught to memorize procedures that got to an answer quickly?

Of course, students must be able to compute flexibly, efficiently, and accurately. But they also need to explain their reasoning and determine if the ideas they’re using and the results they’re getting make sense. (Humphreys & Parker, 8 pag.)

How might we design and implement practices that help our young learners make sense of what they are learning?  In Brain-Friendly Assessments: What They Are and How to Use Them, David Sousa explains how necessary sense-making and meaning are to transfer information from working memory into long-term memory.

The brain is more likely to store information if it makes sense and has meaning. (Sousa, 28 pag.)

Dr. Sousa continues:

We should not be measuring just content acquisition. Rather, we should be discovering the ways students can process and manipulate their knowledge and skills to deal with new problems and issues associated with what they have learned.  (Sousa, 28 pag.)

The first chapter of Making Number Talks Matter highlights the importance of number talks.  We want our young learners to develop flexibility and confidence working with numbers.

Listen to Ruth Parker and Cathy Humphreys discuss Number talks:

From Jo Boaler’s How to Learn Math: for Students:

…we know that what separates high achievers from low achievers is not that high achievers know more math, it is that they interact with numbers flexibly and low achievers don’t.

What if we take action on behalf of our young learners?  What if we offer multiple pathways for success?

How might we dramatically expand our talent pool?

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.

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.

Grading and feedback: what we do matters

Thinking about feedback and marking papers… How should we mark our learners’ work? Do we offer the opportunity to learn through mistakes and corrections?

And, I wonder if we are unintentionally incorrectly using ratios and proportional reasoning when we then put a score on the paper.

Consider the following student’s work from a recent assessment.


Do you see the error?  Is it a big error? Does this young learner understand the task and how to solve it? What feedback should this learner receive?

This child was told that there was a multiplication error in the work. Do you agree?  Is it a matter of close reading on the teacher’s part? What feedback do we hope for to accompany the arrows shown below?


What if we exercise the art of questioning in our feedback? Compare What if you think about what happened here? to You have a multiplication error here. Which feedback will cause more action?

The score for this question was marked as 3.5/4.  Losing 1/2 point for this error seems reasonable.  Would losing 12.5 points also seem reasonable?

If we scale this out of 100 rather than 4,  that 1/2 point become 12.5 points.  Is that what we intend to do, and is it the message that we want to send?

Now, as it happened, this was a 4 question assessment.  This young learner’s questions were marked 4/4, 4/4, 3/4, and 3.5/4.  In question 4, there was the addition error described above. In question 3, the learner multiplied in the first step when division should have been used.  All of these points seem reasonable as long as the items each garner 4 points.  However, proportionally scaled up to 100 points, the 1-point error is now a 25 point error.

How might we rethink grading and scaling? What does research tell us about translating scores between scales?

If learning is our focus and results guide our decisions, what steps do we take now?

And, how are these results guiding the decisions of our young learners?

Listeners: evaluative, interpretive, generative

What type of listener are am I right now? Do I know what modes of listening I use? How might I improve as a listener? What if I actively choose to practice?

Listening informs questioning. Paul Bennett says that one of the keys to being a good questioner is to stop reflexively asking so many thoughtless questions and pay attention— eventually, a truly interesting question may come to mind. (Berger, 98 pag.)

I’ve been studying a paper Gail Burrill (@GailBurrill) shared with us a couple of weekends ago.  The paper, Mathematicians’ Mathematical Thinking for Teaching: Responding to Students’ Conjectures by Estrella Johnson, Sean Larsen, Faith Rutherford of Portland State University, discusses three types of listening: evaluative, interpretive, and generative.

The term evaluative listening is characterized by Davis (1997) as one that “is used to suggest that the primary reason for listening in such mathematical classrooms tends to be rather limited and limiting” (p. 359). When a teacher engages in evaluative listening the goal of the listening is to compare student responses to the “correct” answer that the teacher already has in mind. Furthermore, in this case, the student responses are largely ignored and have “virtually no effect on the pre-specified trajectory of the lesson” (p. 360).

When a teacher engages in interpretive listening, the teacher is no longer “trying simply to assess the correctness of student responses” instead they are “now interested in ‘making sense of the sense they are making’” (Davis, 1997, p. 365). However, while the teacher is now actively trying to understand student contribution, the teacher is unlikely to change the lesson in response.

Finally, generative listening can “generate or transform one’s own mathematical understanding and it can generate a new space of instructional activities” (Yackel et al., 2003, p. 117) and is “intended to reflect the negotiated and participatory nature of listening to students mathematics” (p. 117). So, when a teacher is generatively listening to their students, the student contributions guide the direction of the lesson. Rasmussen’s notion of generative listening draws on Davis’ (1997) description of hermeneutic listening, which is consistent with instruction that is “more a matter of flexible response to ever-changing circumstances than of unyielding progress towards imposed goals” (p. 369).

If you’d like to read about these three types of listening the authors continue their paper with a case study.

Evaluative listeners seek correct answers, and all answers are compared to the one deemed correct from a single point of view.

Interpretive listeners seek sense making.  How are learners processing to produce solutions to tasks? What does the explanation show us about understanding?

Generative listeners seek next steps and questions themselves. In light of what was just heard, what should we do next? And, then they act.

For assessment to function formatively, the results have to be used to adjust teaching and learning; thus a significant aspect of any program will be the ways in which teachers make these adjustments. (William and Black, n. pag)

“Great teachers focus on what the student is saying or doing,” he says, “and are able, by being so focused and by their deep knowledge of the subject matter, to see and recognize the inarticulate stumbling, fumbling effort of the student who’s reaching toward mastery, and then connect to them with a targeted message.” (Coyle, 177 pag.)

What if we empower and embolden learners to ask the questions they need to ask by improving the way we listen and question?

Unless you ask questions, nobody knows what you are thinking or what you want to know.” (Rothstein and Santana, 135 pag.)

How might we practice generative listening to level up in the art of questioning? What is we listen to inform our questioning?

How might we collaborate to learn and grow as listeners and questioners?

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

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

Davis, B. (1997). Listening for difference: An evolving conception of mathematics teaching. Journal for Research in Mathematics Education. 28(3). 355–376.

Johnson, E., Larsen, S., Rutherford (2010). Mathematicians’ Mathematicians’ Mathematical Thinking for Teaching: Responding to Students’ Conjectures. Thirteenth Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education Conference on Research in 
Undergraduate Mathematics Education. Raleigh, NC. Retrieved from on September 12, 2015.

Rothstein, Dan, and Luz Santana. Make Just One Change: Teach Students to Ask Their Own Questions. Cambridge, MA: Harvard Education, 2011. Print.

Wiliam, Dylan, and Paul Black. “Inside the Black Box: Raising Standards Through Classroom Assessment.” The College Cost Disease (2011): n. pag. WEA Education Blog. Web. 13 Sept. 2015.

Yackel, E., Stephan, M., Rasmussen, C., Underwood, D. (2003). Didactising: Continuing the work of Leen Streefland. Educational Studies in Mathematics. 54. 101–126.

#TEDTalkTuesday: the power of yet

How might we teach the power of yet? Is it in the culture of our classroom and our school? What if we include a norm that gives permission to add “yet” to any sentence that has “cannot” in it?

Carol Dweck: The power of believing that you can improve

How are we raising our children? Are we raising them for now instead of yet? Are we raising kids who are obsessed with getting A’s? Are we raising kids who don’t know how to dream big dreams?

Here are some things we can do. First of all, we can praise wisely, not praising intelligence or talent. That has failed. Don’t do that anymore. But praising the process that kids engage in: their effort, their strategies, their focus, their perseverance, their improvement. This process praise creates kids who are hardy and resilient.

What if it is as simple as adding the word yet? How might we change our future?


Bringing differences to the same essential-to-learn

One fear we encounter while forwarding the tenets of professional learning communities is the perceived loss of autonomy. Yesterday my team used their strengths to teach and facilitate learning of essentials without such loss.

I have the privilege of attending multiple division meetings, and yesterday was such a day.  We met as an entire community to start conversation about our common goals for this year.  Then, each division met to  learn from and with their Division Head.

The through-lines and essentials to learn are clear to me. I appreciate seeing these examples of how we facilitate essentials learning by leveraging our unique strengths and talents.  There was no cookie-cutter lesson everyone delivered.  Yet, the essentials were taught, experienced, and practiced in each session.

Below are my notes from all three session. I hope you can see the essentials, through-lines, and goals that I see.

Joe Marshall – Whole School Meeting, 8:30 a.m.

Rhonda Mitchell – EED meeting 10:30 a.m.


Maryellen Berry – UED meeting 1:00 p.m.


Awareness of self and others. Giving our strengths and talents to our team. Honoring different points of view.

How might we keep open, unbroken circles?

#TEDTalkTuesday from A More Beautiful Question

One of the choices for summer reading in our community is A More Beautiful Question: The Power of Inquiry to Spark Breakthrough Idea by Warren Berger (@GlimmerGuy).

The following three are featured in the book, and I thought we might want to hear from them to add depth to the reading.

Mick Ebeling: The invention that unlocked a locked-in artist

Jack Andraka: A promising test for pancreatic cancer … from a teenager

Dan Meyer: Math class needs a makeover