How might we learn to show our work so that a reader understanding without having to ask questions? As we work with our young learners, we want them to grow as mathematicians and as communicators.

We ask students to show their work so that a reader understands without having to ask them questions. What details should we add so that our thinking is visible to others?

To show (and to assess) comprehension, we are looking for mathematical flexibility.

Learning goals:

I can use ratio and rate reasoning to solve real-world and mathematical problems.

I can show my work so that a reader can understanding without having to ask questions.

Level 4:
I can demonstrate mathematical flexibility with ratio and rate reasoning to show what I know more than one way using tables, equations, double number lines, etc..

Level 3: I can use ratio and rate reasoning to solve real-world and mathematical problems.

Level 2:
I can make tables of equivalent ratios relating quantities with whole-number measurements, and I can use tables to compare ratios.

Level 1:
I can use guess and check to solve real-world and mathematical problems.

I wonder if any of our learners would think to use a spreadsheet. How might technology help with algebraic reasoning? What is we teach students to create tables based on formulas?

How might we experiment with enough tools to display confidence when explaining my reasoning, choices, and solutions?

Choice of an appropriate tool is learner based. How will we know how different tools help unless we experiment too?

How might we learn to show our work so that a reader understanding without having to ask questions? As we work with our young learners, we want them to grow as mathematicians and as communicators.

We ask students to show their work so that a reader understands without having to ask them questions. What details should we add so that our thinking is visible to others?

To show (and to assess) comprehension, we are looking for mathematical flexibility.

I taught 6th grade math today while Kristi and her team attended ASCD. She asked me to work with our students on showing their work. Here’s the plan:

Learning goals:

I can use ratio and rate reasoning to solve real-world and mathematical problems.

I can show my work so that a reader can understanding without having to ask questions.

Level 4:
I can demonstrate mathematical flexibility with ratio and rate reasoning to show what I know more than one way using tables, equations, double number lines, etc..

Level 3: I can use ratio and rate reasoning to solve real-world and mathematical problems.

Level 2:
I can make tables of equivalent ratios relating quantities with whole-number measurements, and I can use tables to compare ratios.

Level 1:
I can use guess and check to solve real-world and mathematical problems.

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?

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?

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!

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.

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?

A doodler is connecting neurological pathways with previously disconnected pathways. A doodler is concentrating intently, sifting though information, conscious, and otherwise, and – much more often than we realize – generating massive insights. (Brown, 11 pag.)

How might we practice, experience, and engage in a different way of connecting with information? What if we exercise our own creativity to create visuals of what we are learning?

Rather than diverting our attention away from a topic (what our culture believe is happening when people doodle), doodling can serve as an anchoring task – a task that can occur simultaneously with another task – and act as a preemptive measure to keep us from losing focus on [a] topic. (Brown, 18 pag.)

It seems counterintuitive, but I can attest to my own improvement in focus, attention, and engagement.

People using even rudimentary visual langue to understand or express something are stirring the neurological pathways of the mind to see a topic in a new light. (Brown, 71 pag.)

Yes, it takes practice. Yes, it is difficult at first.

How might we foster a community of learners where everyone bravely and fiercely seeks feedback?

I was at EduCon in Philadelphia when this tweet arrived last week.

Am I showing enough work? How do I know? What if we partner, students and teachers, to seek feedback, clarity, and guidance?

Success inspires success.

Yesterday, I dropped by Kato‘s classroom to work on the next math assessment and found our learners working together to apply math and to improve communication.

Now, I was just sneaking in to drop off and pick up papers. But, how could I turn down requests for feedback?

Here’s the #showyourwork #LL2LU progression in the classroom:

Grade 4

Level 4
I can show more than one way to find a solution to the problem.

Level 3 I can describe or illustrate how I arrived at a solution in a way that the reader understands without talking to me.

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.

And here’s one child and her work. “Ms. Gough, will you look at my work? Can you understand it without asking me questions? Is is clear to you?”

I see connected words, pictures, and numbers. I like the color coding for the different size bags. I appreciate reading the sentences that explain the numbers and her thinking. I also witnessed this young learner improve her work and her thinking while watching me read her work. She knew what she wanted to add, because she wished I knew why she made the final choice. I’d call this Level 4 work.

What if we foster a community of learners who bravely and fiercely seek feedback?

What if content isn’t the essential learning? What if content is just the vehicle to learn process?

Imagine these essential learnings:

Show your work: I can describe and illustrate my thinking so that a reader understands without having to talking with me.

Mathematical flexibility: I can apply mathematical flexibility to show what I know using more than one method.

Our 5th graders just started a unit on fractions. What if we use fractions to teach our young learners to show their work and demonstrate flexibility of thought?

60 seconds of pair-share to improve answers to what is a fraction?

Regularly return to the essentials to learn and the associated learning progressions. How are we doing? At what level are we right now? What is a next step?

60 seconds of pair-share to improve answers to what are equivalent fractions?

How might we use technology for learning and investigation?

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How might we put process and product on even ground? What if we emphasize communication and flexibility and use content and skill as vehicles to show what you know and how you can communicate?

Level 4
I can analyze different pathways to success, find connections, between pathways, and add new strategies to my thinking.

Level 3 I can apply mathematical flexibility to show what I know using more than one method.

Level 2
I can show my work to document one successful method.

Level 1
I can find and state a correct solution.

Level 4
I can show what I know using words, numbers, and pictures.

Level 3 I can describe and illustrate how I arrived at a solution so that a reader understands without having to talking with me.

Level 2
I can describe or illustrate how I arrived at a solution so that a reader understands without having to talking with me.