Choral Counting gets to the heart of what we want for our mathematical communities. This activity creates space for all students to notice, to wonder, and to pursue interesting ideas. Students and teachers alike wonder together about patterns, and why and how numbers change or stay the same. [Franke, Kindle Locations1526-1528}
I wonder what can be learned from using a number line or ten-frames to shed more light on the patterns naturally found from members of the chorus.
Beginning with 6 and counting by 5s, we counted. Learners began adding “because…” to what they noticed. #Awesome
Choral Counting is an invitation; it provides an opportunity for each student to generate important mathematical ideas and for teachers to be curious about their students’ thinking. [Franke, Kindle Location 2057]
One learner said, “To move from one row to the next row, you add 30 because 6×5 is 30.” It is a regularity that repeats. Using the number line shows that to move from 6 to 36 there are 6 hops of 5 or a distance of 30.
The next comment was, “Each term on the diagonal going from the top left to the bottom right increases by 35 because 7×5 is 35.” Another regularity that repeats. Again, the number line shows 7 hops of 5 from 6 to 41, 11 to 46, 41 to 76, and so on.
Awesome that one “I notice…” that includes “because” inspires additional ones. Facilitating meaningful mathematical discourse invites students to develop and share important mathematical ideas.
What tools are within reach of learners as they deepen their numeracy and understanding? What is to be gained when we both author and illustrate mathematical understanding?
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.”
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.
As part of our Embolden Your Inner Writer course, Marsha and I drafted a learning progression for each chapter to help our writers when they feel stuck or need a push. However, these are just drafts. In order to feel confident, to have the courage to use them, we must use them ourselves, share them with learners, and seek feedback.
I’m trying out the following learning progression for Anderson’s chapter on Models, Chapter 2.
I can strengthen my craft, word choice, and mechanics by applying techniques from models and mentor texts.
Enamored with Daniel Coyle’s writing, I picked up my copy of The Talent Code, and found the following sentence.
The goal is always the same: to break a skill into its component pieces (circuits), memorize those pieces individually, then link them together in progressively larger groupings (new, interconnected circuits). [Coyle, 84 pag.]
Noticing the colon, I wondered if I am skilled at using them, knowing when to use them, and using them correctly. (Ok…I’m not, but what can I learn?)
Students need to know the truth: writing is cumulative. [Anderson, 9 pag.]
If I read and observe how these authors use a colon, I think I can use it myself to imitate the great writers.
Perseverance calls for action: show an attempt to think and question, ask and seek clarifying questions, try again with new information and actions.
What do you think?
I’m not sure I “read like a writer” as stated in Level 1, but I annotated well. I could find sentences that helped me think about using a colon. Maybe I read more like a writer than I thought. Hey, that’s one of the tips! Then, I collected and recorded examples to imitate as suggested in Level 2. Curiosity caused me to want to know more. I have asked questions, and I love how Jeff Anderson, in Mechanically Inclined, offers notes and a visual.
And, then…boom! I was struggling with a sentence in my previous post when it dawned on me: Use a colon! Here’s what I wrote:
The editor in my head – no, not the editor – the critic in my head convinces me to wait: wait until I know, wait for someone else, wait.
While I think I’m currently at Level 3 (maybe Level 4 when I press publish), I have more to learn and more work to do to be confident that “I can strengthen my craft, word choice, and mechanics by applying techniques from models and mentor texts.”
Do you know any learner’s that are stuck? Are they convinced that they can’t?
“Fear of imperfection keeps us perched on the edge, afraid to dive in and start writing. If we sit and wait for the perfect words, they don’t come. Inertia sets in. Our mind halts. The clock slows. Much like hesitating at the edge of the ocean, afraid of the shock of cold, we wait. And in waiting, our anxiety spins.” (Anderson, 9 pag.)
Hesitating at the edge, afraid, we wait. How might we develop brave, bold learners who wonder – on paper – what they are thinking so that they might see it? What do we do to overcome the fear of the blank page? This fear, as real as it seems, is just a doodle away from getting your feet wet, right? The editor in my head – no, not the editor; the critic in my head convinces me to wait: wait until I know, wait for someone else, wait. What force is needed to overcome inertia? Is it just as simple as a doodle?
Are math and writing this closely related? Wow! Far too many students will not write the first step in math because they are not sure if they are going to be right? If they are going to be right, are they learning anything?
In Daniel Coyle’s “The Talent Code,” he writes about deep practice, working at the edge of your ability so that you make mistakes, learn, and repeat.
Deep practice is built on a paradox: struggling in certain targeted ways — operating at the edges of your ability, where you make mistakes — makes you smarter. (Coyle, 18 pag.)
The second reason deep practice is a strange concept is that it takes events that we normally strive to avoid —namely, mistakes— and turns them into skills. (Coyle, 20 pag.)
In SMP-1, “I can make sense of tasks and persevere in solving them,” the first level asks for a visible attempt to think and reason into the task.
Are our young mathematicians and writers stuck due to inertia? Is it blank page fright? Is there space in class to draft and redraft, making revisions as you go? Are missteps celebrated and seen as opportunities to learn?
The goal is always the same: to break a skill into its component pieces (circuits), memorize those pieces individually, then link them together in progressively larger groupings (new, interconnected circuits) [The Talent Code]
Salva’s younger brother, Kuol, was taking care of just one cow; like his brothers before him, he would be in charge of more cows every year. [A Long Walk to Water]