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?

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.

At the 2018 International T³ Conference in San Antonio, Jennifer Wilson (@jwilson828) and I presented the following 90 minute
session.

#SlowMath: looking for structure and noticing regularity in repeated reasoning

How do we provide opportunities for students to learn to use structure and repeated reasoning? What expressions, equations and diagrams require making what isn’t pictured visible? Let’s engage in tasks where making use of structure and repeated reasoning can provide an advantage and think about how to provide that same opportunity for students.

Here’s my sketch note of our plan:

Dave Johnston (@Johnston_MSMath) recorded his thinking and learning and shared it with us via Twitter.

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.)

What if we teach how to reach? How might we offer targeted struggle for every learner in our care?

Investing time in teaching students how to learn is never wasted; in doing so, you deepen their understanding of the upcoming content and better equip them for future success. (Jackson, 19 pag.)

If we are to harness the power of feedback to increase student learning, then we need to ensure that feedback causes a cognitive rather than an emotional reaction—in other words, feedback should cause thinking. It should be focused; it should relate to the learning goals that have been shared with the students; and it should be more work for the recipient than the donor. (Wiliam, 130 pag.)

When people believe their basic qualities can be developed, failures may still hurt, but failures don’t define them. And if abilities can be expanded – if change and growth are possible – then there are still many paths to success.” (Dweck, 39 pag.)

What pathways to learning are illuminated in order to highlight learning = struggle + perseverance?

Many students would struggle much less in school if, before we presented new material for them to learn, we took the time to help them acquire background knowledge and skills that will help them learn. (Jackson, 18 pag.)

We want every learner in our care to be able to say

But…what if I can’t? What if I have no idea what to look for, notice, take note of, or attempt to generalize?

Investing time in teaching students how to learn is never wasted; in doing so, you deepen their understanding of the upcoming content and better equip them for future success. (Jackson, 19 pag.)

Are we teaching for a solution, or are we teaching strategy to express patterns? What if we facilitate experiences where both are considered essential to learn?

We want more students to experience the burst of energy that comes from asking questions that lead to making new connections, feel a greater sense of urgency to seek answers to questions on their own, and reap the satisfaction of actually understanding more deeply the subject matter as a result of the questions they asked. (Rothstein and Santana, 151 pag.)

What if we collaboratively plan questions that guide learners to think, notice, and question for themselves?

What do you notice? What changes? What stays the same?

Indeed, sharing high-quality questions may be the most significant thing we can do to improve the quality of student learning. (Wiliam, 104 pag.)

How might we design for, expect, and offer feedback on procedural fluency and conceptual understanding?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3 I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

If we are to harness the power of feedback to increase student learning, then we need to ensure that feedback causes a cognitive rather than an emotional reaction—in other words, feedback should cause thinking. It should be focused; it should relate to the learning goals that have been shared with the students; and it should be more work for the recipient than the donor. (Wiliam, 130 pag.)

But what if I can’t look for and express regularity in repeated reasoning yet? What if I need help? How might we make a pathway for success?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3 I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

What do you notice? What changes? What stays the same?

Can we use CAS (computer algebra system) to help our students practice look for and express regularity in repeated reasoning?

What do we need to factor for the result to be (x-4)(x+4)?
What do we need to factor for the result to be (x-9)(x+9)?
What will the result be if we factor x²-121?
What will the result be if we factor x²-a2?

We can also explore over what set of numbers we are factoring using the syntax we have been using. And what happens if we factor x²+1. (And then connect the result to the graph of y=x²+1.)

What happens if we factor over the set of real numbers?

Or over the set of complex numbers?

What about expanding the square of a binomial?

What changes? What stays the same? What will the result be if we expand (x+5)²? Or (x+a)²? Or (x-a)²?

What about expanding the cube of a binomial? Or expanding (x+1)^n, or (x+y)^n?

What if we are looking at powers of i?

We can look for and express regularity in repeated reasoning when factoring the sum or difference of cubes. Or simplifying radicals. Or solving equations.

Through reflection and conversation, students make connections and begin to generalize results. What opportunities are you giving your students to look for and express regularity in repeated reasoning? What content are you teaching this week that you can #AskDontTell?