Tag Archives: LL2LU

I can establish mathematics goals to focus learning

We strive to grow in our understanding of the Eight Mathematics Teaching Practices from NCTM’s Principles to Actions: Ensuring Mathematical Success for All. This research-informed framework of teaching and learning reflects a core set of high leverage practices and essential teaching skills necessary to promote deep learning of mathematics.

Establish mathematics goals to focus learning.

Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions.

In order to support our teaching teams as they stretch to learn more, we drafted the following learning progressions. We choose to provide a couple of pathways to focus teacher effort, understanding, and action.

When working with teacher teams to establish mathematics goals to focus learning, we refer to 5 Practices for Orchestrating Productive Mathematics Discussions by Peg Smith and Mary Kay Stein and Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning by John Hattie, Douglas Fisher, and Nancy Frey along with Principles to Actions: Ensuring Mathematical Success for All by Steve Leinwand.

To deepen our understanding around establishing mathematics goals, we anticipate, connect to prior knowledge, explain the mathematics goals to learners, and teach learners to use these goals to self-assess and level up.

From  NCTM’s 5 Practices for Orchestrating Productive Mathematics Discussions, we know that we should do the math ourselves, predict (anticipate) what students will produce, and brainstorm what will help students most when in productive struggle and when in destructive struggle.

Once prior knowledge is activated, students can make connections between their knowledge and the lesson’s learning intentions. (Hattie, 44 pag.)

To strengthen our understanding of using mathematics goals to focus learning, we make the learning goals visible to learners, ask assessing and advancing questions to empower students, and listen and respond to support learning and leveling up.

Excellent teachers think hard about when they will present the learning intention. They don’t just set the learning intentions early in the lesson and then forget about them. They refer to these intentions throughout instruction, keeping students focused on what it is they’re supposed to learn. (Hattie, 55-56 pag.)

How might we continue to deepen and strengthen our ability to advance learning for every learner?

What if we establish mathematics learning goals to focus learning?

Cross posted on Experiments in Learning by Doing

Hattie, John A. (Allan); Fisher, Douglas B.; Frey, Nancy; Gojak, Linda M.; Moore, Sara Delano; Mellman, William L.. Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning (Corwin Mathematics Series). SAGE Publications. Kindle Edition.

Leinwand, Steve. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA.: National Council of Teachers of Mathematics, 2014. (p. 21) Print.

Stein, Mary Kay., and Margaret Smith. 5 Practices for Orchestrating Productive Mathematics Discussions. N.p.: n.p., n.d. Print.


Posted by on March 27, 2018 in Professional Learning & Pedagogy


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Sneak Peek: Leading Mathematics Education in the Digital Age

Leading Mathematics Education in the Digital Age


How can leaders effectively lead mathematics education in the era of the digital age? There are many ways to contribute in our community and the global community, but we have to be willing to offer our voices. How might we take advantage of instructional tools to purposefully ensure that all students and teachers have voice: voice to share what we know and what we don’t know yet; voice to wonder what if and why; voice to lead and to question.

Sneak peek for our session includes:

How might we empower our learners to own their learning? How might we provide opportunities for our learners to level up to the learning target, knowing what they know and what they don’t know yet? How might we encourage our learners to add to the learning of their classmates?

Interested? Here’s a sneak peek at a subset of our slides as they exist today. Disclaimer: Since this is a draft, they may change before we see you in San Antonio.

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Here is Jill’s sneak peek, in case you missed it.



Posted by on March 16, 2017 in Professional Learning & Pedagogy


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Using Technology Alongside #SlowMath to Promote Productive Struggle

Using technology alongside #SlowMath to promote productive struggle
2017 T³™ International Conference
Sunday, March 12, 8:30 – 10 a.m.
Columbus AB, East Tower, Ballroom Level
Jennifer Wilson
Jill Gough

One of the Mathematics Teaching Practices from the National Council of Teachers of Mathematics’ (NCTM) “Principles to Actions” is to support productive struggle in learning mathematics.

  • How does technology promote productive struggle?
  • How might we provide #SlowMath opportunities for all students to notice and question?
  • How do activities that provide for visualization and conceptual development of mathematics help students think deeply about mathematical ideas and relationships?

[Cross posted at Experiments in Learning by Doing]


Posted by on March 12, 2017 in Professional Learning & Pedagogy


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Read, apply, learn

Read, apply, learn
2017 T³™ International Conference
Saturday, March 11, 8:30 – 10 a.m.
Columbus H, East Tower, Ballroom Level
Jennifer Wilson
Jill Gough

How might we take action on current best practices and research in learning and assessment? What if we make sense of new ideas and learn how to apply them in our own practice? Let’s learn together; deepen our understanding of formative assessment; make our thinking visible; push ourselves to be more flexible; and more. We will explore some of the actions taken while tinkering with ideas from Tim Kanold, Dylan Wiliam, Jo Boaler and others, and we will discuss and share their impact on learning.

[Cross posted at Experiments in Learning by Doing]

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Posted by on March 11, 2017 in Professional Learning & Pedagogy


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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 Experiments in Learning by Doing]

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Posted by on March 10, 2017 in Professional Learning & Pedagogy


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Short Cycle Formative Assessment: 45-45-90 Triangles

I am enjoying our slow book chat on Dylan Wiliam’s Embedding Formative Assessment. (You can download the first chapter here, if you are interested.)

Chapter 1 is Why Formative Assessment Should Be a Priority for Every Teacher. Wiliam convinced me of this in Embedded Formative Assessment, but I still learned plenty from this chapter. My sentence/phrase/word reflection was actually a paragraph:

Formative assessment emphasizes decision-driven data collection instead of data-driven decision making.

As I planned our Special Right Triangles lesson for Wednesday, I decided what questions to ask based on what was essential to learn.

Level 4: I can use the Pythagorean Theorem & special right triangle relationships to solve right triangles in applied problems.

Level 3: I can solve special right triangles.

Level 2: I can use the Pythagorean Theorem.

Level 1: I can perform calculations with squaring and square rooting.

We started class with a Quick Poll.

I was surprised at how long it took students to get started. I hadn’t planned it purposefully, but the way the triangle was given forced them to make more connections than if the two legs had been marked congruent.


Eventually, everyone got a correct answer (and the opportunity to learn more about using the square root template) using the Pythagorean Theorem.

I asked them to determine the hypotenuse of a 45˚-45˚-90˚ triangle with a leg of 10 next. As soon as they got their answer, they announced “there’s a pattern”.

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They conjectured what would happen for legs of 12 and 7.

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I asked them to select a number between 20 and 100 for the leg and convince themselves that the pattern worked for that number, too.

I loved, though, that the first student whose work I saw had to convince himself that it worked for a side length of x before he tried a number between 20 and 100. I took a picture of his work and let him share it later in the class.

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Students shared their results with the whole class, and then I sent another poll.

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Which led us to reverse the question using the incorrect answer. If √6 is the hypotenuse, what is the leg?

And then a poll to determine the leg given the hypotenuse.

And another poll to determine the leg given the hypotenuse.

I set the timer for 2 minutes and asked students to Doodle what they had learned, using words, pictures, and numbers. And I was pleased that more than the majority took their doodles with them when class was over.

Wiliam says, “But the biggest impact happens with ‘short-cycle’ formative assessment, which takes place not every six to ten weeks but every six to ten minutes, or even every six to ten seconds.” (page 9)

I sent this poll first thing on Friday.

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Students gave these responses after working alone for 1-2 minutes.

I didn’t show the results, and got these responses after students collaborated with a partner for next minute or two.

When I gave a similar question a previous year, allowing collaboration, the success rate was informative but abysmal.

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And so the journey continues … focusing on decision-driven data collection, giving my students and me the opportunity to decide what do next based on “short-cycle” formative assessment.


Wiliam, Dylan, and Siobhán Leahy. Embedding Formative Assessment: Practical Techniques for F-12 Classrooms. West Palm Beach, FL: Learning Sciences, 2015. Print.


Posted by on January 9, 2016 in Geometry, Right Triangles


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How Do Your Tests Make the Grade?

For several years now, I’ve been learning from Jill Gough about #LL2LU (Leading Learners to Level Up). It’s taken a while, and it is always a work in progress, but we are definitely convinced that writing learning goals for our students in language that they can comprehend is important and makes a difference in their learning.

Jill and I have worked on leveling the Standards for Mathematical Practice:


And for each lesson, our team levels the content standard:

Level 4: I can determine the congruence of two figures using rigid motions.

Level 3: I can map a figure onto itself using rotations.

Level 2: I can identify and define rotations.

Level 1: I can apply and perform rotations.

Last year, Jill Gough wrote several blog posts about assessing the quality of the assessments that we give, which led me to Beyond the Common Core: A Handbook for Mathematics in a PLC at Work, High School, by Mona Toncheff and Tim Kanold. Kanold and Matt Larson have written a Leader’s Guide for the handbook in which they offer an Assessment Instrument Quality – Evaluation Tool and a High-Quality Assessment Diagnostic and Discussion Tool. (If needed, you can access the reproducibles through the Leader’s Guide page – Figures 1.11 and 1.16) I also had the opportunity to attend Mona’s session on these tools at either NCSM or NCTM (the sessions run together).

Jill’s posts and Mona’s session made me think that while the assessments we give might not need a complete overhaul, they definitely needed some overhaul if we agreed with the Level 4 Descriptors in the Assessment Instrument Quality Evaluation Tool.

How do your assessments measure up on the following indicators?

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(From Assessment Instrument Quality – Evaluation Tool)

While our learning standards throughout the unit are clear, we don’t typically include them on the assessment. And in fact, I wasn’t sure that I wanted to include all of them on the assessment. Wouldn’t the learning goal give away what to do on some of the test items?

While our assessment is neat, organized, and easy to read, we typically don’t allow students to write on the test (because that would require more time standing at the copy machine). So they aren’t well-spaced and there is no room for teacher feedback.

We changed the last geometry test of last year and the first calculus test of this year to meet the indicators on the assessment evaluation tool.

Here are some comments from our students:

  • I like how our goals for the unit were on the test to remind/help us in what we are looking for.
  • I love the new format. It allows me to go to the sections I can do quickly immediately and save the most difficult problems for last. Because of this, I was able to have enough time to complete every problem.
  • I like that is shows our goals for the unit. This made me feel like the work I was putting in meant something.
  • The formatting/competencies let me know which skills I needed to use. It kept me from getting confused like I usually do.
  • I like the formatting because it keeps similar questions next to each other. This way we can focus on one thing at a time.
  • I like how it starts with basic skills then gets harder. It’s like a warm-up for the end.
  • I like this new formatting because it gives me more space for my work and it won’t be so hard to notice which work goes with which problem.
  • I feel like the new format for the test. It is a lot more organized and easier to read through. On previous tests, the pages felt crammed and a little disorganized. This is an improvement.

As for the learning goal giving away what to do to solve the problem, we decided that we are okay with that on some of the items. And, at Jill’s suggestion, we include some culminating items at the end of the assessment with the leveled learning progression of a practice learning goal, such as I can look for and make use of structure:

Level 4: I can integrate geometric and algebraic representations to confirm structure and patterning.

Level 3: I can look for and make use of structure.

Level 2: I can rewrite an expression into an equivalent form, draw an auxiliary line to support an argument, or identify a pattern to make what isn’t pictured visible.

Level 1: I can compose and decompose numbers, expressions, and figures to make sense of the parts and of the whole.

Or I can show my work:

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.

Thank you, Jill, Mona, Tim, and Matt, for making us rethink what our assessments look like, as the journey continues …


Posted by on September 1, 2015 in Professional Learning & Pedagogy


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SMP6: Attend to Precision #LL2LU

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

I can attend to precision.


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But what if I can’t attend to precision yet? What if I need help? How might we make a pathway for success?


Level 4:
I can distinguish between necessary and sufficient conditions for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.


How many times have you seen a misused equals sign? Or mathematical statements that are fragments?

A student was writing the equation of a tangent line to linearize a curve at the point (2,-4).

He had written


And then he wrote:

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He absolutely knows what he means: y=-4+3(x-2).

But that’s not what he wrote.


Which student responses show attention to precision for the domain and range of y=(x-3)2+4? Are there others that you and your students would accept?

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How often do our students notice that we model attend to precision? How often to our students notice when we don’t model attend to precision?

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Attend to precision isn’t just about numerical precision. Attend to precision is also about the language that we use to communicate mathematically: the distance between a point and a line isn’t just “straight” – it’s the length of the segment that is perpendicular from the point to the line. (How many times have you told your Euclidean geometry students “all lines are straight”?)

But it’s also about learning to communicate mathematically together – and not just expecting students to read and record the correct vocabulary from a textbook.

[Cross posted on Experiments in Learning by Doing]


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Special Right Triangles: 45-45-90

I gave my students our learning progression for SMP 8 a few weeks ago as we started a unit on Right Triangles and had a lesson specifically on 45-45-90 Special Right Triangles.


The Geometry Nspired Activity Special Right Triangles contains an Action-Consequence document that focuses students attention on what changes and what stays the same. The big idea is this: students take some kind of action on an object (like grabbing and dragging a point or a graph). Then they pay attention to what happens. What changes? What stays the same? Through reflection and conversation, students make connections between multiple representations of the mathematics to make sense of the mathematics.

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Students start with what they know – the Pythagorean Theorem.

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Looking at the side lengths in a chart helps students notice and note what changes and what stays the same:

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The legs of the triangle are always the same length.

As the legs increase, the hypotenuse increases.

The hypotenuse is always the longest side.


Students begin to identify and describe patterns and regularities:

All of the hypotenuses have √2.

The ratio of the hypotenuse to the leg is √2.


Students practice look for and express regularity in repeated reasoning as they generalize what is true:

To get from the leg to the hypotenuse, multiply by √2.

To get from the hypotenuse to the leg, divide by √2.

hypotenuse = leg * √2

Teachers and students have to be careful with look for and express regularity in repeated reasoning. Are we providing students an opportunity to work with diagrams and measurements that make us attend to precision as we express the regularity in repeated reasoning that we notice?

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In a Math Practice journal, Kaci writes about “look for regularity in repeated reasoning”. We figured out that half of a square is a 45-45-90 triangle, and students were trying to determine the other two sides of the triangle given one side length of the triangle. She says “To find the length of the hypotenuse, you take the length of a side and multiply by √2. The √2 will always be in the hypotenuse even though it may not be seen like √2. In her examples, the triangle to the left has √2 shown in the hypotenuse, but the triangle to the right has √2 in the answer even though it isn’t shown, since 3√2√2 is not in lowest form. She says, “I looked for regularity in repeated reasoning and found an interesting answer.”

What opportunities can we provide our students this week to look for and express regularity in repeated reasoning and find out something interesting?


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Piecewise Functions

Piecewise Functions

We started a unit on piecewise functions in Algebra 1 with the following leveled learning progression:

Level 4: I can sketch a graph of a piecewise-defined function given a verbal description of the relationship between two quantities.

Level 3: I can interpret key features of a piecewise-defined function in terms of its context.

Level 2: I can determine the domain and range of a function given a context.

Level 1: Using any representation of a function, I can evaluate a function at a given value of x, and I can determine the value of x for a given value of f(x).

We started with an opener to ensure that students were successful with Levels 1 and 2 so that we could reach our target (Level 3) during the lesson.

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Questions 1-4 gave us evidence that most students could evaluate a function at a given value of x and determine the value of x for a given value of f(x) using any representation of a function.

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Questions 5-6 gave us evidence that our students needed more support determining the domain and range of a function given a context.

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Without showing the results from the opener that are pictured above, we talked all together about the context, reading the graph, but not explicitly discussing the domain and range. When we sent the question as a Quick Poll, we saw evidence that more students could determine the domain and range of a function given a context.

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We probably could have spent more time on Level 2 in class. But we didn’t. Instead, we had to provide additional support for Level 2 outside of class, through homework practice, zero block, and after school help.

To open our discussion of piecewise functions, we showed this picture from the front of the Jackson airport parking garage.

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What do you notice? What do you wonder?

Students wrote down a few observations individually, then shared their thoughts with a partner. We selected some for our whole class discussion. In particular, it was helpful that one student specifically said, “pay depends on time”.

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How much would you pay for a certain number of hours of parking?

Similar to an idea from the Internet Plans Makeover, we asked students to choose a number between 0 and 24. If you park that many hours, how much will you pay?

We asked students to check work with a partner before submitting. The result wasn’t quite as disastrous as when we tried the Internet Plans Makeover.

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Showing the grid helps some.

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And having a whole class discussion about the stipulations of the sign helped even more.

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Did anyone choose a non-whole number?

What would happen if you parked for 1.5 hours?

Or 2 hours and 20 minutes?

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We also used the Math Nspired lesson Dog Days or Dog Years with good success. What we are still trying to decide is which comes first … the structure from the Dog Days or Dog Years lesson about creating piecewise functions? Or the less structured conceptual introduction from the cost of parking at the airport? I’m not sure it’s wrong (or even better) to start with either one. But we still wonder, as the journey continues …


Posted by on March 30, 2015 in Algebra 1


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