#LL2LU Mathematical Communication at an early age

When my daughter was learning to talk, we coached her in the moment.  When she exclaimed “Mommy, I eat-ed the whole thing!,” we countered with “you ate the whole thing!” Right there in the moment, we offered feedback and correction.  When someone gave her an unexpected gift, we would way “what do you say?” “Thank you.” Right there in the moment. 

In the moment… Hmm… As a high school and junior high math teacher, I often lamented and worried about my learner’s inability to successfully communicate what they knew.  Did I coach them in the moment – in the learning episode? Or did I correct them only on “test” day?

This week I’ve been helping in 4th grade math – well, I hope I’ve been helping.  Arleen Honick and Laura McRae allowed me to join their math sessions for Unit 3 as they work with our young learners on multiplication, division, number sentences, & algebra.

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.

What if this became the norm in our elementary math learning? 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 challenge them to make their thinking visible?

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


    • Thanks Jeff. I agree. What if we made our expectations clear enough that learners could self-assess their communication?

      As you can see, Kato helped me revise the language to make it simpler for 4th graders. I’d love to know if you use it and how you might adjust it for your students.


  1. Jill, I would love to experiment with these levels in Fourth Grade. I like the levels about showing your work, and that they never say “show your work.” I find that that phrase overwhelms Fourth Graders (of all abilities) because they don’t really know what it means. Level 3 and 4 are good. I wonder if they are too wordy or have too many action steps to follow. I would love to chat more about the wording and hear what you have to say. Great post!!


    • Really, really good feedback, Kato! Thanks. (I promise not to work on this any more this weekend.) I’d love to design a unit 4 lesson (or 2) with you. Yay!

      How about this simpler version?

      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.


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