All posts by Jill Gough

Learner, Love Questions, Problem-finding, Math w/technology. Interests: Collaborating, PLC, Formative assmt

#LL2LU Mathematical Communication at an early age (TBT Remix)

Continue the pattern:  18, 27, 36, ___, ___, ___, ___

Lots of hands went up.

18, 27, 36, 45, 54, 63, 72
Yes! How did you find the numbers to continue the pattern?

S1:  I added 9.
(Me: That’s what I did.)
S2: I multiplied by 9.
(Me: Uh oh…)
S3: The ones go down by 1 and the tens go up by 1.
(Me: Wow, good connection.)

Arleen and Laura probed and pushed for deeper explanations.

S1: To get to the next number, you always add 9.
(Me: That’s what I did.)
S2: I see 2×9, 3×9, and 4×9, so then you’ll have 5×9, 6×9, 7×9, and 8×9.
(Me: Oh, I see! She is using multiples of 9, not multiplying by 9. Did she mean multiples not multiply?)
S3: It’s always the pattern with 9’s.
(Me: He showed the trick about multiplying by 9 with your hands.)

Without the probing and pushing for explanations, I would have thought some of the children did not understand.  This is where in-the-moment formative assessment can accelerate the speed of learning.

There were several more examples with probing for understanding. Awesome work by this team to push and practice. Arleen and Laura checked in with every child as they worked to coach every learner to success.  Awesome!

24, 30, 36, ___, ___, ___, ___
49, 42, 35, ___, ___, ___, ___
40, 32, 24,  ___, ___, ___, ___

I was so curious about the children’s thinking.  Look at the difference in their work and their communication.

By analyzing their work in the moment, we discovered that they were seeing the patterns, getting the answers, but struggled to explain their thinking.  It got me thinking…How often in math do we communicate to children that a right answer is enough? And the faster the better??? Yikes!  No, no, no! Show what you know, not just the final answer.

My turn to teach.

It is not enough to have the correct numbers in the answer.  It is important to have the correct numbers, but that is not was is most important.  It is critical to learn to describe your thinking to the reader.

How might we explain our thinking? How might we show our work? This is what your teachers are looking for.

The children gave GREAT answers!

We can write a sentence.
We can draw a picture.
We can show a number algorithm. (Seriously, a 4th grader gave this answer. WOW!)

But, telling me what I want to hear is very different than putting it in practice.

It makes me wonder… How can I communicate better to our learners? How can I show a path to successful math communication? What if our learners had a learning progression that offered the opportunity to level up in math communication?

What if it looked like this?

Level 4
I can show more than one way to find a solution to the problem.  I can choose appropriately from writing a complete sentence, drawing a picture, writing a number algorithm, or another creative way.

Level 3
I can find a solution to the problem and describe or illustrate how I arrived at the solution in a way that the reader does not have to talk with me in person to understand my path to the solution.

Level 2
I can find a correct solution to the problem.

Level 1
I can ask questions to help me work toward a solution to the problem.

Learning progression in 4th grade student-friendly language from Kato Nims (@katonims129)
Learning progression developed by #TrinityLearns 2nd Grade community learners-of-all-ages.

What if this became a norm? What if we used this or something similar to help our learners self-assess their mathematical written communication? If we emphasize math communication at this early age, will we ultimately have more confident and communicative math students in middle school and high school?

What if we lead learners to level up in communication of understanding? What if we take up the challenge to make thinking visible? … to show what we know more than one way? … to communicate where the reader doesn’t have to ask questions?

How might we impact the world, their future, our future?

#LL2LU Mathematical Communication at an early age was originally published on October 30, 2013.

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.

#TEDTalkTuesday: designing adjustable seats

How might we design flexible spaces for learning?  This is a current question in education.  I wonder, however, if we are thinking deeply enough about this question.  I hope that we will take up the challenge of thinking about learning spaces in more ways than furniture.

How might we design to leverage simple solutions that will improve the performance of our learners and dramatically expand our talent pool?

Don’t miss this compelling talk from high school dropout turned Harvard professor and consider how we might change and redesign to address the size differences of all learners.

The Myth of Average: Todd Rose at TEDxSonomaCounty

Our learners are not one-dimensional.  How might we design for jagged learning profiles? What if we nurture individual potential and talent in every learner?

How might we “teach to the edges”?

Empower learners to deepen their learning

How might we empower learners to deepen their understanding?

After creating and administering common assessments, the next question is perhaps the most challenging: “Are students learning what we think they are supposed to be learning?” (Ferriter and Parry, 75 pag.)

What if our learners are grasping the content, but they are struggling to communicate what they know and how they arrive at a conclusion?

How might we make our expectations clear? What if we empower our learners to take action on their own behalf?

What if our culture embraces the three big ideas of a PLC?

Learning is our focus.
Collaboration is our culture.
Results guide our decisions.

Our #TrinityLearns 2nd grade team sat down together last week to analyze the results of the most recent common assessment.  While our young learners are grasping the basic concepts, we want more for them. We want confident, flexible thinkers and problem solvers.  We want our learners to show what they know more than one way, and we want strong clear communication so that the reader can follow the work without to infer understanding.

Teams at this point in the process are typically performing at a high level, taking collective responsibility for the performance of their students rather than responding as individuals. (Ferriter and Parry, 77 pag.)

As a team, these teachers sorted their students’ work into four levels, shared artifacts of levels with each other, and planned a common lesson.

Laurel Martin (@laurel_martin) explained to our children that the artifacts they analyzed were not from their class and that they belonged to a class across the hall.

Here’s the pitch to the students from Sarah Mokotoff’s (@2ndMokotoff) class:

Don’t you just love the messages: Be like scientists. Make observations. Offer feedback on how to improve.

Here’s what it looked like as the children analyzed artifacts from another 2nd grade class:

Once the analysis was complete, our teachers facilitated a discussion where the children developed a learning progression for this work.

From Kerry Coote (@CooteMrs):

We created these together after looking at student work samples that were assigned at each level. Our kids were so engaged in the activity; they were able to compare and give reasons why work was at a level 3 versus a level 4. It was really good to see! I believe this will empower them to be deeper thinkers and gradually move away from giving an answer without showing their thinking and work.

Here’s what the students in Grace Granade’s (@2ndGranade) class developed:


More from Kerry Coote:

After we helped them develop the learning progression, we conferenced with each child looking at their math assessment. They  automatically self-assessed and assigned levels for their thinking. Many scored themselves lower at first, but the activity of crafting the learning progression helped in making sense of explaining their thinking! Today in math a boy asked me – “so Mrs. Coote, what are those levels again? I know the target is Level 3, but I want to use numbers, words, and pictures to get to level 4.”  It is all coming together and making sense more with these experiences!

In their morning meeting the next day, one of Kathy Bruyn’s (@KathyEE96) learners shared the poster she made the night before.

Don’t you love how she explained the near doubles fact and her precise language?  Wow!

Since we focus on learning and results, this team offered learners an opportunity to show growth.

From Samantha Steinberg (@spsteinberg):

This is an example of leveling up after looking at our assessment.  Initially, [he] used the learning progression to rate his work at level 3.  After reading my feedback, he added words to his next attempt to show his additional thinking.

Before the class developed the learning progression:

Screen Shot 2015-10-01 at 8.08.58 PM

After the class developed the learning progression:

Screen Shot 2015-10-01 at 8.09.13 PM

Can you see the difference in this child’s work, understanding, and communication?

A growth mindset isn’t just about effort. Perhaps the most common misconception is simply equating the growth mindset with effort. Certainly, effort is key for students’ achievement, but it’s not the only thing. Students need to try new strategies and seek input from others when they’re stuck. They need this repertoire of approaches—not just sheer effort—to learn and improve. (Dweck, n. pag.)

Kudos to our 2nd grade team for reaching for the top stages of  the seven stages of collaborative teams! Learning is our focus. Collaboration is our culture. Results guide our decisions.

How might we continue to empower learners to deepen their understanding?

Dweck, Carol. “Carol Dweck Revisits the ‘Growth Mindset’” Education Week. Education Week, 22 Sept. 2015. Web. 02 Oct. 2015.

Graham, Parry, and William Ferriter. Building a Professional Learning Community at Work: A Guide to the First Year. Bloomington, IN: Solution Tree, 2010. Print.

Growth mindset = effort + new strategies and feedback

What if we press forward in the face of resistance?

For me, the most frustrating moments happen when a learner says to me I already know how do this, and I can’t learn another way.
Me:  Can’t or don’t want to? Can’t yet?

A growth mindset isn’t just about effort. Perhaps the most common misconception is simply equating the growth mindset with effort. Certainly, effort is key for students’ achievement, but it’s not the only thing. Students need to try new strategies and seek input from others when they’re stuck. They need this repertoire of approaches—not just sheer effort—to learn and improve. (Dweck, n. pag.)

What if we offer a pathway for learners to help others learn, and at the same time, learn new strategies?

What if we deem the following as essential to learn?

I can demonstrate flexibility by showing what I know more than one way.

I can construct a viable argument, and I can critique the reasoning of other.

The trick is to choose a goal just beyond your present abilities; to target the struggle. Thrashing blindly doesn’t help. Reaching does. (Coyle, 19 pag.)

How might we provide pathways to target the struggles to learn new strategies, to construct a viable argument, and to critique the reasoning of others?

MathFlexibility #LL2LU


What if we press forward in the face of resistance and offer our learners who already know how to do this pathways to grow and learn?

How might we lead learners to level up?

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

Dweck, Carol. “Carol Dweck Revisits the ‘Growth Mindset’” Education Week. Education Week, 22 Sept. 2015. Web. 02 Oct. 2015.

Could it be as simple as adding rather than subtracting? (TBT Remix)

I prefer to think of myself as their coach.  “I coach kids to learn algebra” says that I am dedicated to my kids.  “I teach 8th grade algebra” indicates that my dedication may be to the content.  Being their coach does not make me less of an evaluator.  Their athletic coaches evaluate them all the time.  The coach decides which kids make the team and which kids are cut.  The coach decides who starts and who rides the bench.  The coach decides how much playing time, if any, each player has.

There are some things I just have to do as their teacher.  Yes, I mean grading.  (Remember, our grade books are sparse; we have very few grades.  We assess quite often; we grade little.)  We’ve just finished our semester exams.  My team grades together in the same room using the same scoring guide.  Prior to our exam day, we agreed on the questions as well as the solutions, predicted student errors, and completed the exercise of negotiating partial credit.  Some say that is good enough; there is no reason to grade in the same room when everyone understands the scoring guide.  Really?  Would we say that there is no reason to play on the same court or field since everyone knows and agrees upon the plays?  Don’t we expect the other team to have a plan of their own?

Are our learners the opponents in the exam process?
Are we trying to keep them from scoring?
Do they feel that we are? 

Are we still considered their coach?
Are we trying to help them compete?
Do they feel that we are?

How are we thinking about scoring items on the summative assessment?  Do our scoring guides assign points for good work or do they document how we will subtract points for errors?  Are we grading in team?  Do we take our issues to our teammates or our table-leader when we have a question about work that is out of the norm or unexpected?  (Or, is the amount of partial credit awarded based on how nice, sweet, cooperative, participative -or not – a child is? YIKES!)

Could we alter everyone’s mindset about this stressful event by changing our approach and attitude about how we mark, score, and grade each item?  What if we add points for what is done well instead of subtracting points when an error occurs?  Could our scoring guides be more about assigning credit and less about docking points?  What if we chose to add points for bright spots in the work instead of appearing to play “gotcha” by subtracting points?  Would our grades be closer to representing a true score of what has been learned?

How would a learner respond if we handed them a paper that was filled with +4, +2, +3 and so on rather than -2, -4, -3?

Let’s try adding up the good things we find
rather than playing “gotcha”
by subtracting when an error is found.

Could the self-reflection prompts during the exam analysis process, similar to the post-game film analysis, ask the learner to identify why they earned the points that were scored?  Could we get them to write about what they did well?  Could they work in team to identify what others did well that they wish they had done too?  Could they work in team to identify what others did that they find different or unusual and explain why it worked?  Would this process motivate them to improve their understanding and help each other learn?

Would this help us all learn to blend the 4C’s (critical thinking and problem solving; communication, collaboration; and creativity and innovation) with the 3R’s?

Can we use this type of process to add to our learning?  Could it be as simple as adding rather than subtracting?  Are we willing to experiment?

Could it be as simple as adding rather than subtracting? was originally posted on December 23, 2010.

#TEDTalkTuesday: Ideas can spark a movement

How might we uncover passions and connect ideas? What if we listen to learn?

Maya Penn: Meet a young entrepreneur, cartoonist, designer, activist …

Ideas can spark a movement. Ideas are opportunities and innovation. Ideas truly are what make the world go round. If it wasn’t for ideas, we wouldn’t be where we are now with technology, medicine, art, culture, and how we even live our lives.

We live in a big, diverse and beautiful world, and that makes me even more passionate to save it. But it’s never enough to just to get it through your heads about the things that are happening in our world. It takes to get it through your hearts, because when you get it through your heart, that is when movements are sparked. That is when opportunities and innovation are created, and that is why ideas come to life.