# Deep practice: building conceptual understanding in the middle grades

Deep practice:
building conceptual understanding in the middle grades
2017 T³™ International Conference
Friday, March 10, 10:00 – 11:30 a.m.
Dusable, West Tower, Third Floor
Jill Gough
Jennifer Wilson

How might we attend to comprehension, accuracy, flexibility and then efficiency? What if we leverage technology to enhance our learners’ visual literacy and make connections between words, pictures and numbers? We will look at new ways of using technology to help learners visualize, think about, connect and discuss mathematics. Let’s explore how we might help young learners productively struggle instead of thrashing around blindly.

[Cross posted at Easing The Hurry Syndrome]

# What is a Fraction? – visual lesson

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

Imagine these essential learnings:

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?

Lesson tic-toc

• Introduce the two essential learnings.
• 60 second quick write to recall what is a fraction?
• 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 second quick write to recall what are equivalent fractions?
• 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.

Level 1
I can find a correct solution to a task.

# Common denominators – “Let’s see why”

Everybody knows that you must have common denominators to add fractions, right?  Do we know why? If asked to construct a viable argument, could we? Can we draw it (i.e., communicate why visually)?  How mathematically flexible are we when it comes to fractions? From Jo Boaler’s How to Learn Math: for Students:

…we know that what separates high achievers from low achievers is not that high achievers know more math, it is that they interact with numbers flexibly and low achievers don’t.

Today’s Building Concepts lesson: Adding and Subtracting of Fractions with Unlike Denominators, had our young learners working to show their understanding of adding and subtracting fractions in multiple ways.

Kristi Story (@kstorysquared) used a phrase today that has really stuck with me is “Let’s see why…”  It immediately reminded me of Simon Sinek’s How great leaders inspire action.

And it’s those who start with “why” that have the ability to inspire those around them or find others who inspire them.

I wonder if, when young learners struggle with numeracy, it is because they do not see why.  Have they been so concerned with “getting the right answer” that they have missed the theory, reasoning, and geometry?

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What if we  leverage appropriate tools and use them strategically? What if we use technology to personalize learning and offer every learner the opportunity to see why?

#LL2LU draft for use equivalent fractions as a strategy to add and subtract fractions.

Level 4:
I can solve real-world and mathematical problems involving the four operations with rational numbers.

Level 3:
I can solve word problems involving addition and subtraction of fractions by using visual fraction models or equations to represent the problem.

Level 2:
I can add and subtract fractions with unlike denominators, including mixed numbers, by replacing given fractions with equivalent fractions.

Level 1:
I can understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

I can recognize and generate simple equivalent fractions, and I can explain why the fractions are equivalent using a visual fraction model.

#LL2LU for I can apply mathematical flexibility.

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.

#LL2LU for I can construct a viable argument and critique the reasoning of others.

Level 4: I can build on the viable arguments of others and use their critique and feedback to improve my understanding of the solutions to a task.

Level 3: I can construct viable arguments and critique the reasoning of others.

Level 2: I can communicate my thinking for why a conjecture must be true to others, and I can listen to and read the work of others and offer actionable, growth-oriented feedback using I like…, I wonder…, and What if… to help clarify or improve the work.

Level 1: I can recognize given information, definitions, and established results that will contribute to a sound argument for a conjecture.

# What is a Fraction? … be flexible, use appropriate tools strategically

What if we use technology to visualize new concepts and interact with math to investigate and learn? What if we pair a process learning progression with a content learning progression?

By the end of this lesson, we want every learner to be able to say:

I can explain and illustrate that a fraction a/b is the quantity formed by a parts of size 1/b, and I can represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0.

AND

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

We have completed Jo Boaler’s two courses – How to Learn Math: For Students, and How to Learn Math: For Teachers and Parents.  As a team we are working on our math flexibility with math learners of all ages.  We challenge ourselves to offer more visuals and additional pathways for success. How might we leverage appropriate tools and use them strategically?

Enter: Building Concepts lessons from Texas Instruments.  Kristi Story (@kstorysquared) used What is a Fraction? to review and assess what is already known with our 6th graders.

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To offer a glimpse of the learning experience, a copy of my raw notes from this lesson are below.

Kristi starts with The Number of the Day to chalk talk a number talk.

It is obvious that our students have an understanding of fractions, decimals and percents.  Kristi encourages students to and modeled making connections between different representations of 2 1/5, the number of the day.  Many students answered aloud and enthusiastically moved to the board to draw or write a different representation.  By using the chalk talk method, this number talk encouraged number flexibility and creativity and the number talk offered all learners the opportunity to expand their understanding and fluency.

Kristi launches the TI-Nspire software and the lesson What is a Fraction? and encourages our students to explore and investigate what the software will do and interpret the results.  This led to a side conversation about 1.5/3 and complex fractions.

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Kristi introduces the vocabulary of unit fraction. Interesting discussion and another chance for mathematical flexibility happens when students are asked to describe/illustrate what happens when the value of the denominator increases.  How does the number of equal parts in the interval from 0 to 1 change? What happens to the length of those parts?

Students clearly possess background knowledge of fractions, and Kristi challenges them to become more flexible in representing fractions.  Note: Many students are drawing circles to represent fractions.  In addition, we want them to draw number lines  and rectangles.

The discussion transitions to compare 3/5 to 7/5. Student answers included

3/5 is 3 copies of 1/5.
3/5 is a little more than 1/2
3/5 is 60% of the way between 0 and 1
3/5 is 2/5 back from 1
7/5 is 2/5 more than 1
7/5 is 3/5 less than 2
Both are 2/5 away from 1 but in different directions.

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Kristi and students use Think-Pair-Share to describe how they decided to explain their answer to the question Is 11/8 closer to 1 or 2? Kristi asks everyone improve their answer based on partner feedback. Kristi asks for volunteers to read their partner’s idea.

From me to Kristi:

I thought today was great! I love how you facilitated a discussion encouraging all learners to talk about math. My notes are attached.  Thank you for your willingness to pilot this software with our students.  I was glad to hear that you have enjoyed this start with fractions.

From Kristi:

Thank you for all the feedback. As I said yesterday, it was exciting to present fractions in a way that I think will make a difference in their understanding of fractions. I’m looking forward to continuing this series.

What if we use technology to visualize new concepts and interact with math to investigate and learn?

#LL2LU for What is a Fraction?

Level 4:
I can decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation, and I can justify decompositions by using a visual fraction model.

Level 3:
I can explain and illustrate that a fraction a/b is the quantity formed by a parts of size 1/b, and I can represent a fraction a/b on a number line diagram by marking off a lengths 1/from 0.

Level 2:
I can represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts.

Level 1:
I can explain and illustrate that a fraction 1/b is the quantity formed by 1 part when a whole is partitioned into b equal parts.

I can compare fractions by reasoning about their size.

Level 3:

#LL2LU for Mathematical Flexibility

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.

What if we pair a process learning progression with a content learning progression?