## #NCTMLive and #T3Learns webinar: Implement tasks that promote reasoning and problem solving, and use and connect mathematical representations.

On Wednesday, May 2, 2018, Jill Gough (@jgough) and I co-facilitated the second webinar in a four-part series on the Eight Mathematics Teaching Practices from NCTM’s Principles to Actions: Ensuring Mathematical Success for All.

Implement tasks that promote reasoning and problem solving,
and Use and connect mathematical representations.

Effective teaching of mathematics facilitates discourse among learners to build shared understanding of mathematical ideas by analyzing and comparing approaches and arguments.

• How might we implement and facilitate tasks that promote productive discussions to strengthen the teaching and learning of mathematics in all our teaching settings – teaching students and teaching teachers?
• What types of tasks encourage mathematical flexibility to show what we know in more than one way?

Our slide deck:

View this document on Scribd

Our agenda:

 7:00 Jill/Jennifer’s Opening remarks Share your name and grade level(s) or course(s). Norm setting and Purpose 7:05 Number Talk: 81 x 25 Your natural way and Illustrate Decompose into two or more addends (show it) Show your work so a reader understands without asking questions Share work via Twitter using #NCTMLive or bit.ly/nctmlive52 7:10 #LL2LU Use and connect mathematical representations Self-assess where you are Self-assessment effect size Think back to a lesson you taught or observed in the past month. At what level did you or the teacher show evidence of using mathematical representations? 7:15 Task:  (x+1)^2 does/doesn’t equal x^2+1 7:25 Taking Action (DEI quote) 7:30 #LL2LU Implement Tasks That Promote Reasoning and Problem Solving 7:35 Graham Fletcher’s Open Middle Finding Equivalent Ratios 7:45 Illustrative Mathematics: Jim and Jesse’s Money 7:55 Close and preview next in the series

Some reflections from the chat window:

I learned to pay attention to multiple representations that my students will create when they are allowed the chance to think on their own. I learned to ask myself how am I fostering this environment with my questioning.

I learned to pay attention to the diversity of representations that different students bring to the classroom and to wait to everyone have time to think

I learned to pay attention (more) to illustrating work instead of focusing so much on algebraic reasoning in my approach to teaching Algebra I. I learned to ask myself how could I model multiple representations to my students.

I learned to pay attention to multiple representations because students all think and see things differently.

I learned to make sure to give a pause for students to make the connections between different ways of representing a problem, rather than just accepting the first right answer and moving on.

I learned to pay attention to the ways that I present information and concepts to children… I need to include more visual representations when I working with algebraic reasoning activities.

Cross-posted on Experiments in Learning by Doing

## I can elicit and use evidence of student thinking

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.

Elicit and use evidence of student thinking.

Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.

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 elicit and use evidence of student thinking, we refer to 5 Practices for Orchestrating Productive Mathematics Discussions by Peg Smith and Mary Kay Stein and Dylan Wiliam’s Embedding Formative Assessment: Practical Techniques for K-12 Classrooms along with Principles to Actions: Ensuring Mathematical Success for All by Steve Leinwand.

To deepen our understanding around eliciting evidence of student thinking, we anticipate multiple ways learners might approach a task, empower learners to make their thinking visible, celebrate mistakes as opportunities to learn, and ask for more than one voice to contribute.

From  NCTM’s 5 Practices for Orchestrating Productive Mathematics Discussions, we know that we should do the math ourselves, anticipate what learners will produce, and brainstorm how we might select, sequence, and connect learners’ ideas.

How will classroom culture grow as we focus on the five key strategies we studied in Embedding Formative Assessment: Practical Techniques for F-12 Classrooms by Dylan Wiliam and Siobhan Leahy?

• Clarify, share, and understand learning intentions and success criteria
• Engineer effective discussions, tasks, and activities that elicit evidence of learning
• Provide feedback that moves learning forward
• Activate students as learning resources for one another
• Activate students as owners of their own learning

To strengthen our understanding of using evidence of student thinking, we plan our hinge questions in advance, predict how we might sequence and connect, adjust instruction based on what we learn – in the moment and in the next team meeting – to advance learning for every student. We share data within our team to plan how we might differentiate to meet the needs of all learners.

How might we team to strengthen and deepen our commitment to ensuring mathematical success for all?

What if we anticipate, monitor, select, sequence, and connect student thinking?

How might we elicit and use evidence of student thinking to advance learning for every learner?

Cross posted on Experiments in Learning by Doing

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.

Wiliam, Dylan; Leahy, Siobhan. Embedding Formative Assessment: Practical Techniques for F-12 Classrooms. (Kindle Locations 2191-2195). Learning Sciences International. Kindle Edition.

Posted by on March 29, 2018 in Professional Learning & Pedagogy

## Webinar: Establish Mathematics Goals to Focus Learning, and Elicit and Use Evidence of Student Thinking.

On Wednesday, March 28, 2018, Jill Gough (@jgough) and I co-facilitated the first webinar in a four-part series on the Eight Mathematics Teaching Practices from NCTM’s Principles to Actions: Ensuring Mathematical Success for All.

Establish Mathematics Goals to Focus Learning, and Elicit and Use Evidence of Student Thinking.

Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.

• How might we communicate with clarity to ensure that learners are focused on high quality mathematical goals?
• What types of tasks provide opportunities for learners to notice, note, wonder, and take action as agents of their own learning?
Our slide deck:
Agenda:

 7:00 Opening remarks Share your name and grade level(s) or course(s). – Maybe a poll? Norm setting and Purpose 7:05 Establish mathematics goals to focus learning #LL2LU 7:10 Task:  Illustrative Math – Fruit Salad? Demonstrate flexibility Look for and make use of structure Share your work via Twitter using #NCTMLive or http://bit.ly/IMfruitsalad-dropyourwork 7:25 Quotes from Taking Action 7:30 Elicit and use evidence of student thinking #LL2LU Dylan Wiliam James Popham #LL2LU 7:35 Let’s Do Some Math Robert Kaplinsky: Open Middle Biggest Product ? 7:45 Talking Points – Elizabeth Statmore Suggestion One talking about learning math Suggestion Two talking about mathematics Geometry Talking Points example Ratios and Proportional reasoning Talking Points example 7:55 Close and preview next in the series
Some reflections from the chat window:
• I learned to pay attention to how my students may first solve the problem or think about it prior to me teaching it to try and see connections that are made or how I can meet them. ~C Heikkila
• I learned how to pay attention to how I introduce tasks to students. Sometimes I place limits on their responses by telling them what I expect to see in their responses as it relates to content topics. I will be more mindful about task introduction. ~M Roland
• I learned to pay more attention to mathematical operations, and to look for more solutions that can satisfy the given problem. ~B Hakmi
•  I also learned the importance of productive struggle and to be patient with my students. ~M James
• I’m thinking about how to encourage my teachers to intentionally teach the mathematical practices. ~M Hite
• I learned to pay attention to the learning progressions so I can think of the work as a process and journey. ~B Holden
• A new mathematical connection for me was the idea of graphing values for the product example. ~A Warden
• I learned to pay attention to peer discussions to discover how well students are learning the concepts. ~M Grech
• Am I anticipating the roadblocks to learning? ~L Hendry

An audio recording of the webinar and the chat transcript can be viewed at NCTM’s Partnership Series.

Cross posted at Experiments in Learning by Doing

## 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

## What I Learned Today: Scale Drawings and Maps

I asked my 15-year-old what she learned today at school. She paused for a moment and then answered my question by asking me what I learned at school today.

It took me a while to think about what I had learned [which will make me more patient when I ask her the question again tomorrow], and then I remembered and shared with her:

We are working with some teachers who are using the Illustrative Mathematics 6–8 Math curriculum. The 7th grade teachers are in Unit 1, Scale Drawings. They are working with Scale Drawings and Maps. Today I learned to look more closely at the scale given for a map.

Look at the following for a moment. What’s the same? What’s different?

What’s different about the scales on the last two?

Attend to precision, MP6, says, “Mathematically proficient students try to communicate precisely to others. … They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately.”

I’m not sure that we would have noticed a difference, except that we were trying to find some assessment items from another source and saw that many aligned to 7.G.A.1 included a scale in the form of “1 cm = 100 miles”. I’ve looked at lots of maps and I never noticed the incongruity of saying that 1 cm equals 100 miles. We don’t really mean that 1 cm equals 100 miles, right? Not in the same sense that we say 4 quarters equals \$1 or 3+4=7. Is there any wonder that our students misuse the equal sign?

And so the journey continues, grateful for the authors of this curriculum who make me pay closer attention to attending to precision and grateful for my daughter who makes me think and share about what I’m learning, too …

## Blending Technology with Paper and Pencil

My geometry class is 1:1 this year; each student has her own MacBook Air. Students share responses to questions digitally in class using TI-Nspire Navigator for Networked Computers. Students explore mathematics using TI-Nspire dynamic graphs and geometry software. Students explore mathematics and share responses digitally using Demos Activity Builder. We use Canvas, an online learning management system, for assignments. We use Google Drive for sharing electronic documents with each other, and we use MathXL, online homework with built-in learning help, to practice mathematics. What place does pencil and paper have in my students’ learning and understanding of mathematics?

Even though many of the tasks that my students do for geometry take place digitally, I am convinced that pencil and paper plays an important role in how much mathematics my students not only learn but also remember. In a Wall Street Journal article, “Can Handwriting Make You Smarter?“, Robert Lee Hotz reports that students who take notes by hand usually outperform students who type notes when assessed more than one day after the class period. Students who type notes quickly type everything the professor says, but students who handwrite notes have to process the information while they are hearing it to select what is important to remember (Hotz 2016).

Hotz cites the work of Mueller and Oppenheimer published in Pyschological Science. Their research studies showed that students who took notes by hand performed better on conceptual questions than those who took notes on a laptop. Students performed about the same on factual questions. Their hypothesis for why is that students who take notes by hand choose which information is important to include in their notes, and so they are able to study “more efficiently” than those who are reviewing an entire typed lecture (Mueller and Oppenheimer 2014). Note: These studies are on college students; I have found little research on grade school students.

For several years now, my students and I have been learning how to learn mathematics using the Standards for Mathematical Practice. MP8, “look for and express regularity in repeated reasoning”, has pushed me to think about having students record what they see instead of just noticing and discussing it.

One of the ways that I’ve learned to talk about “look for and express regularity in repeated reasoning” is to ask students to notice what changes and what stays the same as we take a dynamic action on a geometric figure. Consider a recent learning episode from my classroom.

Students were told that our learning intention was “I can look for and express regularity in repeating reasoning”. The content was conceptual development of the equation of a circle in a coordinate plane using the Pythagorean Theorem. I did not share that specific content with students up front, however, because I wanted it to be revealed as the lesson progressed. I showed them a dynamic right triangle in the coordinate plane.

What changes? What stays the same?

I could have let them simply discuss what they noticed. But instead I asked them to “Notice & Note”, using words, pictures, and numbers to write and sketch what they saw.

Then I asked them to share what they noticed with a partner and add to their own notes as desired.

Our classroom discussion revealed that the equation of the circle formed by tracing point P was x2 + y2 = 52.

Students continued to “Notice & Note” as they moved a circle around in the coordinate plane. What changes? What stays the same?

As we moved P around in the coordinate plane, and then as they later moved the circle around in the coordinate plane students noted what they saw. Eventually, students generalized the center-radius form of an equation.

Notice & Note, by Kylene Beers and Robert E. Probst, is a guide of signposts (strategies) for close reading of text. Students are taught signposts to notice while they are reading, and they are asked to stop reading and note what the signpost might imply. “Again & Again” is one signpost. Do you notice an event in the text that keeps happening again and again? Do you notice a phrase in the text that is repeated again and again? Stop reading, note it, and think about what that might mean (Beers and Probst 2013). How might we take advantage of the ways that students are learning to read text in their English Language Arts (ELA) classes to guide students in inquiry based exploration of mathematics?

In her online course, Sunni Brown, author of The Doodle Revolution, states that “Tracking content using imagery, color, word pictures and typography can change the way you understand information and also dramatically increase your level of knowledge and retention” (Brown 2016). How do we make tracking content using words, pictures, and numbers a reality in the 1:1 classroom? My experience is that it doesn’t happen without deliberate emphasis on its importance.

In Reading Nonfiction, Beers and Probst write “When students recognize that nonfiction ought to challenge us, ought to slow us down and make us think, then they’re more likely to become close readers” (Beers and Probst 2016). Our ELA counterparts are on to something. Effective classroom instruction is not just about creating learning episodes for our students to experience the mathematics using the Math Practices. Effective classroom instruction incorporates practices that will help students remember what they are learning longer than for the next test.

As I think about our district’s continued implementation of 1:1 technology, I am convinced that we need to pay attention to when we are asking, encouraging, and requiring students to use pencil and paper to create a record of what they are learning. I am interested in thinking more about how we might blend the use of dynamic graphs and geometry software with Notice & Note – using words, pictures, and numbers, along with color, so that students not only have a record of what they are learning but also have a better chance of remembering it later. And so, the journey continues …

References

Beers, G. Kylene, and Robert E. Probst. Notice & note: Strategies for close reading. Portsmouth: Heinemann, 2013. Print.

Beers, G. K., & Probst, R. E. (2016). Reading nonfiction: Notice & note stances, signposts, and strategies. Portsmouth: Heinemann.

Brown, S. (n.d.). Visual Note-Taking 101 / Personal Infodoodling™. Retrieved April 25, 2016, from http://sunnibrown.com/visualtraining

Hotz, Robert Lee. “Can handwriting make you smarter?” The Wall Street Journal. 04 Apr. 2016. Web. 25 Apr. 2016.

Mueller, P. A., and D. M. Oppenheimer. “The pen is mightier than the keyboard: Advantages of longhand over laptop note taking.” Psychological Science 25.6 (2014): 1159-168. Web.

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Posted by on August 3, 2017 in Circles, Coordinate Geometry, Geometry

## Is This a Rectangle?

One of our learning intentions in our Coordinate Geometry unit is for students to be able to say I can use slope, distance, and midpoint along with properties of geometric objects to verify claims about the objects.

G-GPE. Expressing Geometric Properties with Equations

B. Use coordinates to prove simple geometric theorems algebraically

1. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

We recently used the Illustrative Mathematics Task Is This a Rectangle to provide students the opportunity to practice.

We also used Jill Gough’s and Kato Nims’ visual #ShowYourWork learning progression to frame how to write a solution to the task.

How often do we tell our students Show Your Work only to get papers on which work isn’t shown? How often do we write Show Your Work next to a student answer for which the student thought she had shown her work? How often do our students wonder what we mean when we say Show Your Work?

The Show Your Work learning progression begins to help students understand what we mean when we say Show Your Work. I have seen it empower students to ask each other for feedback on their work: Can you read this and understand it without asking me any questions? It has been transformative for my AP Calculus students as they write Free Response questions that will be scored by readers who can’t ask them questions and don’t know what math they can do in their heads.

We set the timer for 5 minutes of quiet think time. Most students began by sketching the graph on paper or creating it using their dynamic graphs software. [Some students painfully and slowly drew every tick mark on a grid, making me realize I should have graph paper more readily available for them.]

They began to look for and make use of structure. Some sketched in right triangles to see the slope or length of the sides. Some used slope and distance formulas to calculate the slope or length of the sides.

I saw several who were showing necessary but not sufficient information to verify that the figure is a rectangle. I wondered how I could steer them towards a solution without telling them they weren’t there yet.

I decided to summarize a few of the solutions I was seeing and send them in a Quick Poll, asking students to decide which reasoning was sufficient for verifying that the figure is a rectangle.

Students discussed and used what they learned to improve their work.

It occurred to me that it might be helpful for them to determine the Show Your Work level for some sample student work. And so I showed a sample and asked the level.

But I didn’t plan ahead for that, and so I hurriedly selected two pieces of student work from last year to display. I was pleased with the response to the first piece of work. Most students recognized that the solution is correct and that the work could be improved so that the reader knows what the student means.

I wish that I hadn’t chosen the second piece of work. Did students say that this work was at level 3 because there are lots of words in the explanation and plenty of numbers on the diagram? Unfortunately, the logic is lacking: adjacent sides perpendicular is not a result of parallel opposite sides. Learning to pay close enough attention to whether an argument is valid is good, hard work.

Tasks like this often take longer than I expect. I’m not sure whether that is because I am now well practiced at easing the hurry syndrome or whether that is because learning to Show Your Work just takes longer than copying the teacher’s work. And so the journey continues …

Posted by on April 11, 2017 in Coordinate Geometry, Geometry, Polygons

## Leading Mathematics Education in the Digital Age

Leading Mathematics Education in the Digital Age
2017 NCSM Annual Conference
Pre-Conference Sessions
Jennifer Wilson
Jill Gough

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.

[Cross-posted at Experiments in Learning by Doing]

## 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.

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

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

## 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]