Playing with sentences begins with witnessing writing as performance. It’s a concrete way to reach out and engage our audience’s eyes and ears. (Anderson, 180 pag.)

Intent on learning more about sentence variation, my feedback partner helped me notice that I begin many of my sentences with nouns. Challenged to play more with my writing, I assigned myself the task of writing an 11 sentence paragraph using each of Anderson’s 11 Sentence Pattern Options from Chapter 8, Energy.

As a young learner, I was a memorizer. Doing what was expected of me, I learned the rules required for “the test”. Relieved and exhausted, I promptly forgot them. As concepts became more complex, my workload and anxiety increased. My favorite professor, Allen Smithers, noticed my lack of understanding. Dr. Smithers, patient and determined, challenged me to develop conceptual understanding. He challenged me to learn – not memorize. He expected me to confirm my understanding using drawings, graphs, tables, and equations. I grew as a mathematician, confident and capable. I learned, deeply. I am grateful.

Here’s the breakdown:

I know that I ended my sentence with an adverb instead of an adjective, but I choose to leave it as is.

Playing with sentences and ideas, I tried again.

As a young learner, I was a memorizer. Doing what was expected of me, I learned the rules required for “the test”. Relieved and exhausted, I promptly forgot them. As concepts became more complex, my workload and anxiety increased. Jill Lovorn, mathematician, was lost yet lucky. Success, assumed and shown, was shallow at best. Rote memorization – pages and pages of hidden work – masked missing conceptual understanding. I could use procedures, theorems, techniques, and algorithms. I got the right answers, mysteriously and remarkably. No one knew, sadly. I survived.

Still ending that sentence with an adverb, I enjoyed playing with ideas and with sentences. Here’s the structure with a sentence checkup.

We know young writers will do what feels comfortable. They don’t play with their writing. They don’t try a sentence three different ways when it’s not working. They don’t explore what a varied sentence pattern or length can do for their writing’s rhythm and fluency. (Anderson, 178 pag.)

Blending a little math into writer’s workshop, what if we analyze and visualize our sentence patterns and lengths? Will learners play with their sentences after collecting and graphing a little data as described in 10 Things Every Writer Needs to Know?

Knowing how important visuals are to my learning, I used Google Sheets to “see” the variation in sentence length and to analyze the pattern of my sentence beginning.

Wow! I am not worried about my sentence length. (Are they long? Is there an average number of words in great sentences, or is it about variety and rhythm?) However, I am appalled at the lack of interesting first words. It would have been so easy to write:

“Advance Your Inner Mathematician is a new course we are piloting this semester.”

And, the second sentence could have easily been,

“Anchored in Smith and Sherin’s ‘The 5 Practices in Practice: Successfully Orchestrating Mathematics Discussion in Your Middle School Classroom’, this course supports continued teacher learning after Embolden Your Inner Mathematician.”

Or the two sentences could have been combined into one sentence.

“Advance Your Inner Mathematician, a new course we are piloting this semester is anchored in Smith and Sherin’s ‘The 5 Practices in Practice: Successfully Orchestrating Mathematics Discussion in Your Middle School Classroom’, to support continued teacher learning after Embolden Your Inner Mathematician.”

I notice that this post is chock-full of questions (16 of 18 sentences) – a known trait of my writing. I find the visual of sentence length interesting.

While I chose Google sheets as my tool, students can quickly graph this data by hand (please encourage the use of graph paper so that they attend to precision), and drop it in their writer’s notebook.

Will writers play more with their words and sentences if they see the patterns and frequency?

What looks simple on the surface can be deceptively complex and elegant.

How might we teach our young learners to deepen their algebraic reasoning?

Let’s see what you think…

Unit 8: Cartesian Coordinate Plane, Two-Variable Equations, Graphing, and Regularity in Repeated Reasoning

graph on the Cartesian coordinate plane,

look for and make use of structure,

look for and express regularity in repeated reasoning,

use and connect mathematical representations?

Kristi Story, Trinity’s 6th Grade math teacher, set the above goals for student learning and selected what looks like a simple, yet is actually a deep task that aligns with these goals. Providing opportunities for students to learn important mathematics content and to engage in essential mathematical practices are at the forefront of this planning.

Tasks that provide the richest basis for productive discussions have been referred to as doing-mathematics tasks. Such tasks are nonalgorithmic—no solution path is suggested or implied by the task and students cannot solve them by the simple application of a known rule. (Smith, 16 pag.)

Day 1’s Task is modified from Fawn Nguyen‘s Visual Patterns, pattern #10 puppies. Starting as a simple number talk, how many do you see and how do you see them?

Obviously, 6th graders know that there are three puppies, but how do they see the three? Do they see two puppies in the top row and one puppy in the bottom row? Do they see two puppies in the first column and one puppy in the second column? Either way, they would write 2+1=3. To make their thinking visible, they circle the two and the one. Also, they might see a 2×2 square with one puppy missing and write 2×2-1=3. It is a quick check about attending to precision, making use of structure (making visible what isn’t readily seen), and writing an equation.

Continuing the number talk, how many do you see and how do you see them?

6th graders immediately know that there are five puppies, but how do they see the five? Do they see three puppies in the top row and two puppies in the bottom row and write 3+2=5? Do they see two puppies in the first two columns and one puppy in the third column and write 2+2+1=5 or 2×2+1=5? Do they see a 2×3 rectangle with one puppy missing and write 2×3-1=5? Additional practice attending to precision, making use of structure (making visible what isn’t readily seen), and writing an equation.

6th graders immediately know that there are seven puppies, but how do they see the seven? Do they see four puppies in the top row and three puppies in the bottom row and write 4+3=7? Do they see two puppies in the first three columns and one puppy in the fourth column and write 2+2+2+1=7 or 2×3+1=7? They might also see a 2×4 rectangle with one puppy missing and write 2×4-1=7.

The important reflection question is: Did I use the same structure for each of the figures, or did I make use of different structures with each figure?

Using previously discovered structures, students predicted the number of puppies in Figure 4 and in Figure 10. Connecting to the algebra in their previous unit, they wrote a generalization for any figure number using their structure and reasoning. We found the following different expressions.

(n+1)+n. where n is the figure number
2(n+1)-1, where n is the figure number
1+2n, where n is the figure number

“These all represent the same pattern. Are they equivalent expressions?” asked Kristi. Using the distributive property, and combining like terms, they proved equivalence.

Committed to deep understanding for our young learners, Kristi asked students to graph (Figure Number, Number of Puppies) on the coordinate plane.

Trained to notice and note, our students were surprised to discover a linear pattern.

JH said, “Hey, to go from one point to the next, all you have to do is go up 2 and over 1.”
When asked, CJ interpreted the point (6, 13) saying “that means that there will be 13 puppies in Figure 6.”

My #ObserveMe notes illustrate more of the details and flexibility.

Our students graphed points and a line on the Cartesian coordinate plane, made use of structure, expressed regularity in repeated reasoning, used and connected mathematical representations, and deepened algebraic reasoning.

That’s a lot of Algebra I for a 6th grader, don’t you think?

Deep learning. Empowered learners.

Never underestimate the power of a motivated learner.

Smith, Margaret (Peg) S.. The Five Practices in Practice [Middle School] (Corwin Mathematics Series). SAGE Publications. Kindle Edition.

It really isn’t a surprise, right? Children learn and grow at different rates. We know that because we work with young learners every day. The question isn’t “Why aren’t they fluent right now?” It isn’t. It just isn’t. The question should be and is:

“What are we going to do, right now, to make this better
for every and each learner in our care?”

Multiplication Number Talks are brimming with potential to help students learn the properties of real numbers (although they don’t know it yet), and over time, the properties come to life in students’ own strategies. (Humphreys, 62 p.)

Humphreys and Parker continue:

Students who have experienced Number Talks come to algebra understanding the arithmetic properties because they have used them repeatedly as they reasoned with numbers in ways that made sense to them. This doesn’t happen automatically, though. As students use these properties, one of our jobs as teachers is to help students connect the strategies that make sense to them to the names of properties that are the foundation of our number system. (Humphreys, 77 p.)

So, that is what we will do. We commit to deeper and stronger mathematical understanding. And, we take action.

This week our Wednesday workshop focused on Literacy, Mathematics, and STEAM in grade level bands. Teachers of our 4th, 5th, and 6th graders gathered to work together, as a teaching team, to take direct action to strengthen and deepen our young students’ mathematical fluency.

Anticipating students’ responses takes place before instruction, during the planning stage of your lesson. This practice involves taking a close look at the task to identify the different strategies you expect students to use and to think about how you want to respond to those strategies during instruction. Anticipating helps prepare you to recognize and make sense of students’ strategies during the lesson and to be able to respond effectively. In other words, by carefully anticipating students’ responses prior to a lesson, you will be better prepared to respond to students during instruction. (Smith, 37 p.)

How many strategies and tools do we use when modeling multiplication in our classroom? It is a matter of inclusion.

It is a matter of inclusion.

Every learner wants and needs to find their own thinking in their community. This belonging, sharing, and learning matters. We make sense of mathematics and persevere. We make sense of others thinking as they learn to construct arguments and show their thinking so that others understand.

Humphreys and Parker note:

They are learning that they have mathematical ideas worth listening to—and so do their classmates. They are learning not to give up when they can’t get an answer right away because they are realizing that speed isn’t important. They are learning about relationships between quantities and what multiplication really means. They are using the properties of the real numbers that will support their understanding of algebra. (Humphreys, 62 p.)

Our job is to connect mathematicians and mathematical thinking.

From NCTM’s Principles to Actions:

Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.

And:

Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.

What if we take up the challenge to author and illustrate mathematical understanding with and for our students and teammates?

Let’s work together to use and connect mathematical representations as we build procedural fluency from conceptual understanding.