The Last Day of Class

Another school year has finished (two months ago), and a new one is about to begin.

Our teachers did a lot to promote growth mindset this last year.

Many of us sent our students a poll with statements from Carol Dweck’s book, Mindset, on the first day of class

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and again on the last day of class.

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You can see some change in the way that students responded.

I have wondered whether talking about mindset and promoting growth mindset makes a difference in what and how students learn. I know plenty of teachers are skeptical. I’m convinced that it matters.

Many students participated in Jo Boaler’s How to Learn Math: For Students online course through Stanford University. I heard students talking about making mistakes, their brain growing, and synapses firing on many occasions throughout the year.

What has convinced me more than student responses throughout the class, though, are the voluntary reflections that my students have offered. I received a handwritten letter at the beginning of May from a former student who no longer attends our school. An excerpt follows:

“When you taught my geometry class last year you polled us at the beginning and the end of the year to see if our opinions on innate/static intelligence vs. one’s ability to improve intelligence had changed. I just want to say that though I was doubtful at the time, this idea of an evolving and increasing intelligence through questioning and learning through wrong answers has stuck with me and served me well. I was once pretty insecure in my academic abilities: yes, I made good grades without much trouble, but there’s always someone faster or more confident or more eloquent, and so much of my identity was wrapped up in being a ‘smart’ kid that I was often afraid to speak up and make mistakes. Now, though my grades and academic integrity are still very important to me, I don’t see successes and failures quite so black and white. Rather, I try to see it all as a learning moment, and I thank you for introducing me to some of the ideas of growth mindsets and ‘GRIT’.” – CM

This thoughtful reflection a year after the class ended is coupled with a thoughtful reflection from another student who wrote as the class ended last year. You can see his reflection in this post.

We will start another school year on August 6 … our students are going to hear the message not only that they can be successful in mathematics but that we, their teachers, want them to be successful in mathematics … our students are going to be greeted with open-ended problems that are accessible to all (many of which will come from youcubed’s Week of Inspirational Math) – problems that allow them to realize from the beginning that we don’t all think the same way and that making our thinking visible to others is a good and important learning opportunity for all … our students are going to set norms for how the class will learn together throughout the year … our students are going to hear from Carol Dweck on the power of “Yet” and they might even hear from Sesame Street, too.

What message will your students hear on the first day of class? What will they say about your class when asked how they think classes are going to go this year?

I look forward to school starting again, as the journey continues …

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Posted by on July 25, 2015 in Student Reflection


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Ask, Don’t Tell

I was invited to write a few posts for NCTM’s Mathematics Teacher Blog: Joy and Inspiration in the Mathematics Classroom.

Ask, Don’t Tell (Part 4): The Equation of a Circle

Ask, Don’t Tell (Part 3): Special Right Triangles

Ask, Don’t Tell (Part 2): Pythagorean Relationships

Ask, Don’t Tell (Part 1): Special Segments in Triangles

While you’re there, be sure to catch up on any other posts you haven’t read. There are some great ones by Matt Enlow, Chris Harrow, and Kathy Erickson.

“Ask Don’t Tell” learning opportunities allow the mathematics that we study to unfold through questions, conjectures, and exploration. “Ask Don’t Tell” learning opportunities begin to activate students as owners of their learning.

I haven’t always provided “Ask Don’t Tell” learning opportunities for my students. My coworkers and I spend our common planning time thinking through questions that we can ask to bring out the mathematics. We plan learning episodes so that students can learn to ask questions as well. (Have you read Make Just One Change: Teach Students to Ask Their Own Questions?)

After the Special Right Triangles post, someone commented on NCTM’s fb page something like the following: “Really? You told students the relationships without any explanation?”

I have always used the Pythagorean Theorem to show why the relationship between the legs and hypotenuse in a 45˚-45˚-90˚ is what it is. But I think that’s different from “Ask Don’t Tell”.

I have been teaching high school for over 20 years. And yes. I really used to tell my geometry students the equation of the circle. I told them definitions for special segments in triangles along with drawing a diagram. I told them how to determine whether a triangle was right, acute, or obtuse. And I told them the relationships between the legs and hypotenuse for 45˚-45˚-90˚ and 30˚-60˚-90˚ triangles.

I’ve also been in meetings with teachers who have not thought about decomposing a square into 45˚-45˚-90˚ triangles or an equilateral triangle into 30˚-60˚-90˚ triangles to make sense of the relationships between side lengths.

You can see on the transparency from which I used to teach that I actually did go through an example where an equilateral triangle was decomposed into 30˚-60˚-90˚ triangles; even so, I failed to provide students the opportunity to look for and make use of structure.


Purposefully creating a learning opportunity so that the mathematics unfolds for students through questions, conjectures, and exploration is different from telling students the mathematics, even with an explanation for why.

As you reflect on your previous school year and plan for your upcoming school year, what #AskDontTell opportunities do and can you provide?


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The Primacy-Recency Effect: a conversation with Elizabeth (episode II)

Dear Elizabeth,

I was so glad to know of others out there familiar with Primacy-Recency Effect. I first learned about it when Jill encouraged me to read How the Brain Learns Mathematics. I still love her blog post about her school’s Social Media Experiment for practicing primacy-recency.

My colleagues and I have been thinking a lot about #AskDontTell learning episodes, but we also recognize that the mathematics does eventually need to be revealed and we need to provide a balance between conceptual development, fluency, and application.

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When I think about how most math classes I attended were set up (and some that I still visit), we were definitely going over homework during Prime-time-1, learning new material during Down-time, and starting our homework/practice during Prime-time-2.

Doesn’t that have to be different than providing students an opportunity for productive struggle during Prime-time-1, even if some of the framing moves into Down-time?

I wonder how these ideas are connected to the deep practice that Daniel Coyle emphasizes in The Talent Code, and in particular, to his experiment about struggle.

We have the luxury of 95-minute classes, and so our takeaway from learning about the Primacy-Recency Effect and thinking about when and how students encounter a new idea has been to create a series of smaller learning episodes (usually 4) for the block, maximizing the amount of Prime-time. (This doesn’t work for every single class period, but our collaboration in creating lessons makes it happen more often than if we worked in isolation.)

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I still start my classes with an opener that gives students some team practice on the mathematics that we have been doing and pushes them a bit towards the mathematics for the day’s lesson … Learning Episode 1. Every time I read this article, though, I wonder whether I should continue that practice? Would the opener be better as a closer … Learning Episode 4?

Thank you for thinking through these questions with me.


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Posted by on July 16, 2015 in Professional Development


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#AskDontTell: Pythagorean Relationships

I have been invited to write a few posts for NCTM’s Mathematics Teacher Blog: Joy and Inspiration in the Mathematics Classroom. You can read my second post here. While you’re there, be sure to catch up on any other posts you haven’t read. There are some great ones by Matt Enlow, Chris Harrow, and Kathy Erickson.

My post starts with a quote by one of my students: A few weeks ago, I overheard one student telling another, “Will you help me figure this out? Don’t just tell me how to do it.” How many of the students in our care are thinking the same thing? How often do we tell them how to do mathematics? How often do we provide them with “Ask, Don’t Tell” opportunities to learn mathematics?

After reading the post, John Golden tweeted the following:

John had no idea that I happen to be reading Creating Cultures of Thinking by Ron Ritchart (you can preview the first chapter at the link), and so the language that he chose to use was timely. I’m deep in the midst of thinking about how we teach our students to learn … about the cultures that we are creating with our students.

Ritchart quotes Lev Vygotsky: “Children grow into the intellectual life of those around them.” And then says himself, “… learning to learn is an apprenticeship in which we don’t so much learn from others as we learn with others in the midst of authentic activities.” [p. 20]

Ritchart later asks, “What difference does it make if a teachers asks, ‘Is your work done?’ or ‘Where are you in your learning?’” [p. 44]

I wonder what you think. Does it matter whether we ask our students whether they are finished with their work or where they are in their learning? I think it might. Focusing on the learning instead of the work creates a culture of thinking. Focusing on the learning instead of the work causes students to say, “Will you help me figure this out? Don’t just tell me how to do it.”

And so the journey of creating cultures of thinking continues …


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#AskDontTell: Special Segments in Triangles

I have been invited to write a few posts for NCTM’s Mathematics Teacher Blog: Joy and Inspiration in the Mathematics Classroom. You can read my first post here. While you’re there, be sure to catch up on any other posts you haven’t read. There are some great ones by Matt Enlow, Chris Harrow, and Kathy Erickson.

Oh – and here’s Kate’s hot chocolate picture. As you can imagine, paring down a blog post to include only two pictures was a challenge for me!

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Posted by on May 29, 2015 in Angles & Triangles


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0.9 Repeating

I got to teach one of my favorite lessons in a Precalculus class this week, which I developed several years ago from a paper by Thomas Osler, Fun with 0.999…

We started with a Quick Poll. Students could select as many or as few choices as they wanted.

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I shared their responses separated

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and grouped together.

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In the first class, one student selected all three choices.

In the second class, 5 students selected all three choices.

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I set the timer for a few minutes and asked students to think individually about how they could argue their selection(s).

Then I asked them to talk together about their ideas.

I walked around and listened. These are the conversations I heard:

A: 1/3 is 0.3 repeating. 2/3 is 0.6 repeating. If we add 1/3 and 2/3, we get 1. If we add 0.3 repeating and 0.6 repeating, we get 0.9 repeating.

B: 1/9 is 0.1 repeating. If we multiply 1/9 by 1, we get 1. If we multiply 0.1 repeating by 9, we get 0.9 repeating.

C: 1/3 is 0.3 repeating. If we add 1/3 three times, we get 1. If we add 0.3 repeating three times, we get 0.9 repeating.

D: If x=0.9 repeating, then 10x=9.9 repeating. (It was clear that a few students had seen Vi Hart talk about 0.9 repeating. Even so, this was all they had for now.)

E: I think this is like Zeno’s Paradox. To walk across the room, you have to walk halfway, and halfway again, and halfway again.

This was the perfect opportunity to deliberately sequence the students’ thinking and let them make connections between their arguments (5 Practices style). With which conversation would you start?

We started with argument C. More than one person shook their head in disbelief, even though they agreed that the argument was convincing.

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Next we moved to argument A, which was very similar to argument C.

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Next we moved to argument B.

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I had a few suggestions of what to do, based on the article from the AMATYC Review. We went to one of those next that the students hadn’t thought of: If x=0.9 repeating, what happens when you divide the equation by 3?

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A student shared their work differently in each class, showing that x=1.

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We moved next to argument D. Again, students shared their thinking differently in each class.

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No one thought about Zeno’s Paradox in the first class. So I asked them how we could express 0.9 repeating as a sum.

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And then I sent a Quick Poll to collect their responses.

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In the second class, I asked the students with argument E to share their thoughts. They got at the infinite sum idea, so without decomposing 0.9 repeating as a class, I sent the Quick Poll. Lots of students came up with a sum that equaled 1. Only one of those was clearly 0.9+0.99 +0.999+…

(I didn’t show them the responses equal to 1 in green when I showed them their results.)

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So we practiced look for and make use of structure together. How can we decompose 0.9 repeating into a sum?

I sent the poll again.

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We concluded the lesson by polling the first question again. In the first class, 4 additional students believed only that 0.9 repeating = 1 at the end.

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In the second class the number of students selecting only choice A changed from 6 to 13.

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Our #AskDontTell journey continues, one lesson at a time …


Posted by on May 2, 2015 in Precalculus


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Connecting Factors and Zeros

NCTM’s Principles to Actions suggests Mathematics Teaching Practices for teachers. Two of those are the following.

MTP 1 Establish mathematics goals to focus learning

MTP 6 Build procedural fluency from conceptual understanding

If the goal for students is to use the factors of a quadratic function to determine its zeros, what concepts must students understand to meet that learning goal?

Our team wrote this leveled learning progression for our lesson.

Level 4: I can factor a quadratic function.

Level 3: I can use the factors of a quadratic function to determine its zeros.

Level 2: I can expand the product of two binomials.

Level 1: I can solve an equation in one variable.

Level 1: I can determine the zero(s) of a function from the graph of a function.

We decided to first ensure that students know what a zero is, and we checked this is more than one way on the opener for the day. (See this source for similar Level 1 problems.)

Students had to place a point at the zero of the function.

Almost all students were able to note that the point of interest is where the graph intersects the x-axis.

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Students had to name the coordinates of the zero of the function, which about half could do.

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And then students had to answer a question about a zero in context. A few more than half could do this.

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We decided that students also need to be able to solve an equation in one variable.

Which they could easily do.

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And we also decided that if students are going to meet the learning goal, they are also going to have to be able to multiply binomials. Which you can tell from the results that they could not easily do (Q8 and Q9).

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In the lesson, we started with the zeros of a linear function.

What do you notice?

If I give you a similar equation, can you tell me the zero?

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What do you notice on this page?

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If I give you a similar equation, can you tell me the zero?

We checked in with students using some Quick Polls.

What do you notice about the answers for this first poll?

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(I noticed that not all are x-intercepts.)

Students showed some improvement as we continued.

The answers are all x-intercepts.

We asked questions like …

How can we tell that (-6,0) is the correct choice using the equation?

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We spent a long time on linear functions. Some might think we spent too long.

Then we looked at a quadratic function.

And we related the linear factors to the quadratic visually.

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This is part of a Math Nspired activity called Zeros of a Quadratic Function, where there is a lot more flexibility in changing the factors.

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Our leveled learning progression for the second lesson changed a little:

Level 4: I can factor a quadratic function.

Level 3: I can use the factors of a quadratic function to determine its zeros, and I can use the zeros of a quadratic function to determine its factors.

Level 2: I can rewrite a quadratic function given in factored form to standard form.

Level 1: I can determine the zero(s) of a quadratic function from the graph of a function.

When we checked for student understanding during the opener of the second lesson, we saw that students were able to determine the zero(s) of a quadratic function from the graph of a function.

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Lots of students were at Level 1, determine the zeros when given the graph and the equation.

Not as many were at the target – but definitely more than had reached it the day before.

We have worked to build procedural knowledge from conceptual knowledge in our unit on Zeros and Factors. Our standards say that we want students to “Factor a quadratic expression to reveal the zeros of the function it defines”. The standards don’t say that we want students to factor a quadratic expression just for the sake of factoring.

What opportunities are you providing your students to concentrate on relationships rather than just results?


Posted by on April 27, 2015 in Algebra 1


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