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 …

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Posted by on September 1, 2015 in Uncategorized


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Team Sorting – Coordinate Geometry

Last year we sorted students into teams using cards at the beginning of each unit.

All cards can be found at this link.

The following are some of the comments that we overheard when students sorted for their Coordinate Geometry unit.

Equations! I can do equations.

If this is algebra, I’m going to do great in this unit.

Are we a team because we both have 18x?

If your slope equals -7, you are here.

Oh! We are doing slopes!

Is your slope -5/3?

Do any of you know how to do this?

Ours all look like 4x-y=2.

2015-02-23 08.27.38  2015-02-23 08.34.43 2015-02-23 08.36.092015-02-23 08.33.25When we asked students at the end of the year what to stop, start, keep, and change, many said that we should keep the Team Sorting Cards. They enjoyed changing teams for each unit and getting to know and work with most of the students in the class.

You can read about previous team sorting here and here.

This year’s class has their first test on Wednesday, and so we look forward to their first team sort on Friday (even though our Tools of Geometry/Construction Unit team sort is lame and needs to be changed before then), as the journey continues …

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Posted by on August 30, 2015 in Coordinate Geometry


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The Best Professional Development Ever

I heard Dylan Wiliam speak in July at the SREB Networking Conference in Atlanta.

What does detoxified-PD look like for you?

In Embedded Formative Assessment, Wiliam says, “Sharing high-quality questions may be the most significant thing we can do to improve the quality of student learning.”

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Our principals are committed to providing common planning time for teachers, and so our teachers meet in their Algebra 1, Geometry, Algebra 2, and Advanced Math teams during a common planning block every other day to work through and plan learning opportunities and questions during this time. This time is, of course, valuable.

But the model that has been most helpful to further our learning as teachers is that three years ago, our administrators incorporated a common geometry class for our teachers to teach. We had 4 teachers and about 25 students in the first block geometry class. The teachers team-taught the class, learning together how to implement inquiry-based (and often technology infused) learning opportunities for students; then they had common planning during second block to make adjustments to the lessons before teaching their own classes the rest of the day.

During the second year, Algebra 2 teachers had common planning during first block and then team-taught Algebra 2 during second block. Last year, Algebra 1 teachers team taught first block and then had common planning during second block. Because of some changes in our schedule, we have another team-taught Algebra 1 class this year.

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This model has been so important in giving every teacher confidence in asking questions, using technology for formative assessment, and allowing students opportunities to explore the mathematics. Note that we don’t have any fewer students taking these courses. We haven’t been able to get an extra teacher unit to incorporate the team-taught classes. Instead, it is a bit of a sacrifice for the other sections to have a few more students each, but we find that sacrifice worth the value of the teachers teaching together. As one of our teachers said, “Team teaching is the best professional development experience I have ever had.” (Then he looked at me, his primary professional development provider, and apologized!)

Another teacher is teaching a class for the first time this year, and his first question was whether his planning period could be when the other teacher teaching the class. He wants to be able to learn in her class first before teaching the class his first time. (Due to the small number of sections, we are unable to let them team-teach this class, but his willingness to spend his planning period observing her class emphasizes the culture of learning from each other that has been created by team-teaching.)

Elham Kazemi’s Shadow Con call to action is for us “to make collective learning opportunities happen”.

How are you already doing this?

How can you make this happen in other ways?

And so the journey continues, together …

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Posted by on August 22, 2015 in Professional Development


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Our First Day Message

What message do your students leave class with on the first day? How do you craft the first day learning episodes to promote that message?

Our students walked into the room with two Which One Doesn’t Belong scenarios.

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Which led to a discussion about working on our math flexibility throughout the year.

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We wondered what one word students would use to describe their feelings about math.

Algebra 1:

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AP Calculus:

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(I wasn’t surprised at the negative feelings towards math in Algebra 1 … but I was surprised at some of the responses from geometry and AP Calculus students.)

That led to the Quick Poll that we’ve sent now for a few years from Mindset.

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We did a short number talk that I saw on Twitter:

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We watched Jo Boaler’s Day 1 Week of Inspirational Math video on mindset and mathematics.

We ran out of time to do our normal opener where students find more than one way to complete a sequence of terms.

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We thought about what it looks like when a team is working well together in math class.

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TeamWork – Calculus (using Navigator for handhelds)

cohesive         1

everyone is building on each other’s idea 1

synchronization         1

organized discussion            1

clockwork       1

coordinated   1

everyone ends up with the correct answesr and understands   1

everyone ends up understanding the problem    1

communication and listening          1

people help each other, in an oreo 1

everyone gives input            1

everyone talking/ explaining at the same time.    1

clear discussion         1

All ideas and all teammates are listened to.           1

a well-oiled machine.            1

lots of words, pointing, and ideas    1

It looks synchronized            1

Like a well-oiled machine.    1

you are productive and no one gets left out          1

A solid conclusion shared by the entire group is reached based upon the well thought out ideas contributed each individual   1

dividing and conquering      1

efficent           1

We are very focused and productive.         1

lots of high fiving and excitement   1

lots of talking and no one hates each other           1

looks like an oreo      1

Organized discussion            1

explaining thought processes to each other          1

success           1

helping each other and an oreo      1

TeamWork – Geometry (using Navigator for Networked Computers)

People are building off of other ideas and are able to make an educated descision using everyone’s input.           1

When a team is working well together they may not always agree on the answer or how to get to it; they always work to find the right one no matter how difficult or stressful it may be.            1

When a team works well together they are listening to each other and trying to trying all methods (to solve). Everyone listens as much as they talk.     1

When a team is working well together, there will be a lot of communication and listening to others’ ideas and meathods.          1

The team looks uniform and united, and if you fail, then your team can figure out what to do better next time.   1

People bouncing ideas off of each other and creating new ideas or improving old ones. Even if some people are wrong, people correct them in a kind way and tell them how to correctly solve a problem. All team members are putting in effort and carrying their weight, not just leaving others to do all the work. The team finds multiple ways to solve the problem and chooses the best one.    1

All team members working together and solving problems at the same time individually then comparing answers and learning from different views.  1

When a team is working well together, each individual student listens to one another and actually thinks about each teammate’s idea and sees it as a viable solution. A team is working well together when they get along and are respectful to one another.        1

When a team works well together, they work easily and doesn’t argue when someone has a different answer than the other person. And they get the answer right. They should be able to recieve and give feedback from the other.       1

When a team, in math, is working well together, I think of a deep conversation of different ideas. I see different solutions and ways a proplem could be solved coming from all sides of the table… if you know what I mean. 1

When no one argues and everyone considers others solutions to a problem.  1

It looks very good and fluent when your team is working well together. When you have a team to help bounce ideas off of each other and to help each other reach the goal they need it is very useful. Everybody is going to make mistakes, and when your team knows that and will help you to find what you did wrong, you will have a larger success rate.    1

When your team is working well together, every member is sharing their thoughts and ideas, right or wrong. The team members aren’t embarrassed about getting the wrong answers because they know that their other teammates will help them to understand their mistakes and learn from them. Each member comes up with different ideas when the team works together, so that way every person benefits wih a new way of thinking.     1

The team is able to get more done, and do it quicker.      1

When a team is working well the group closer together and they’re all listening to other team members input and are also giving their own input.  1

It looks productive and focused. We are all concentrating on the problem that the team has to solved.      1

It looks like we all know what we are doing. It looks like we have more ideas and know more ways to solve our problem. It makes us look as if we understand the problem more and in most cases we probably do, when we work together well.     1

You are makeadvancements and improve one another and also agree upon an answer.       1

When the team is working well together it helps other people of the team to increase their knowledge because each person may see things in a different way.            1

Everyone is not arguing. People are using teamwork and getting the right answer while teaching others on the team how to do it differently.            1

Everyone is learning and helping each other when they may not know the answer to something or need help figuring out how to do something. No one makes fun of anyone if they get a wrong answer because we all need to learn and grow.    1

Somebody will suggest something and people will get excited or say “”yes!””. Then another person will suggest something and everybody will enjoy making progress. People that originally disagreed will change their minds because of something another person explained. They will keep working together until the project is finished.      1

The team is working well when they all have corresponding ideas that come together to get a problem correct.   1

When a team is working well together the team everything flows and everyone is participating. Everyone is helping, ideas are exchanged, and people are learning. Everyone is set on one goal and everyone is headed to achive that goal.           1

It is when everyone in that team is listening to and coming up with ideas. The whole group is cooperating and completing the task given. That is when a group is working well together.     1

Everyone understands the objective and is comprehending well. They understand why and how the team got the answer. No one is confused and everyone feels like they are making a contribution to the work     1

We talked about Popham’s four levels of formative assessment.

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Our goal is for all students to reach the learning goals and not just for the “smartest” and “fastest” to do so. Which means that we will have to help each other. Which also gets into Wiliam’s Five Key Strategies for formative assessment, in particular, activating students as learning resources for one another.

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The bell rang before I could have my students reflect on what they learned and what they will do in geometry this year, so I sent them a Google form to complete outside of class.

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Algebra 1 students answered a Quick Poll before leaving about what they learned.

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today, I learned that anyone can learn math

This year i am going to improve in math   1

About my teachers to be better at math.   1

I have learned If I practice I can become better.

In math this year, I am going to practice more.     1

be able to learn more about math and I will try harder. 1

I have learned that I have three very cool teacher that will make math fun this year and I also learned that math isnt everybodys fav but evryone a math person

In math this year i am going to get better at it      1

You can’t be a math person everyone is good at math it you pratice, so i am going to pratice.           1

I have learned that all three of my math teachers went to NWRHS and in math class this year i will not get bored in class         1

i have learned to think outside of the box. i am going to learn to thinkoutside the box          1

that we have three teachers.

try to work out everything better. 1

that math will be exciting this year and every one is good at math       1

That everyone is good in math in some way. In math this year, I am going to have an A.       1

I have learned that my teachers are cool

in math im going to inprove.            1

I have learned how to think outter box .

I am going to try to do better than the year before in math.       1

I have learned this year to be more excited to more about math.

In math this year, I am going to try my hardest.   1

That I have 3 math teachers. Also math wont be so bad. 1

pay attention 1

nothing and this year I’m going to start fresh       1

I have learned about my classmates and teachers.

In math this year, I am going to try harder.           1

I learned my other classmate’s names and how to do Four 4’s.

In this year of Math I plan on working harder and to advance in my knowledge of Math.     1

I have learned that you need to thing hard to get some of the questions right.

In math this year, I am going to make my math skills better and i want to work math fast.   1

I learned that everyone can do math. I math this year, I will work really hard and do the best I can.          1

I have learned how everybody can do math even if you are not a math person and your brain grow if you work it out. In math this year i am going to try to learn more and get a better grade in math.   1

I have learned in this class period to hold a more open mindset towards math and the many different answers to a mathmatical situation. Different methods can be accepted, and not every person will see math in the same light. I am going to learn to find an excitement in the subject of math this year, hopefully.           1

i have learned peoples names in my class in math this year, i am going to make good grades           1

I have learned that anybody can learn algebra and be good at it. This year I’m going to payattention and do my best at algebra.         1

I have learned that everyone is smart at math, people just need to embrace it.

In math this year, I am going to become smarter in math.          1

I have learned… that you can get differnet answers just by using 4 4s, also I learned how everyone is well in Math, it’s just kids with more experience are more better in it.

In math this year, I am going to.. learn more about math and hopefully actually start to like it.        1

i have learned everybody is good at math and i am going to try my best to make good grades         1

Everyone can learn about math very well, and no one is a math person. I am goin to learn about everything that i can and try my best at it.I am also gonna try and figure out problems the best of my ability.   1

I have learned that the more you practice and work your brain the more it grows.

In math this year, I am going to practice and work my brain so I can learn and get better.   1

Today I have learned that practicing something, even if you don’t fully understand it, still helps. This year I am going to try my hardest to achieve high grades in math.           1

I have learned today in class, everytime you have to find something out there can always be more than one way to it.

In math this year, I am going to try and learn new things that I didnt understand last year.            1

Today I learned that no matter how much i get frustrated that i will always have someone to leaN ON THIS CLASS TO HELP ME WITH MY SITUATION … In math thjis year imgoing to succeed in all things all I do my grades will also be also better.           1

i have learned that anyone can be good at math with good practice.

in math this year im going to learn different things and hopefully get better at math.           1

practice math everyday and that it’s okay to make mistakes because we learn from them.   1

that we can learn things easier by practice so in math this year, i am going to practice if i have troubles until i fully understand it.       1

I have learned that the brain can grow with what you learn.

In math this year, I am going to study more.         1

I have learned that your brain is always changing and to get better at something you have to keep practicing.

In math this year, I am going to practice what I’ve learned in order to get better at it.           1

I have learned that your brain can grow just by learning new things.

I’m going to practice more on math because, if you practice more on something you can get better at it.    1

That math doesnt always have t0 be boring, it cab be reall        1

i am going to try my best to past     1

I have learned that everyone is capable of being in the highest math class there is. There is no such thing as a “”math person””. In math this year, I am going to try my hardest to maintain the highest grade I can, and pay attention in class in order to make good grades.   1

practice and do my best to make a good grade even thooe me and math REALLY dont like eachother. but im going to try to do my best            1

IF you practice you can get better at it       1

I learned that anyone can do math well and in math this year, I am going to try.        1

that your brian can grow the more you practice something , this year i am going to pay more attention.    1

I have learned that anybody can be good at math, you just have to work for it.

In this year of math, I am going to pay more attention than I did last year so I can get better and so I can get good grades.        1

I have learned if I keep practicing and going to different levels I will become better.

This year, I am going to study and practice math on different levels so, I will become better at math.         1

We didn’t really have a lesson, but I have learned about how the brain works with how good you are at something. In math this year, I am going to have no tardys, have straight A’s, and not disrespect the teacher in any manner.    1

that i dont have to dread coming to math. i can grow and learn from it. im going to try to make all a’s        1

You can get better at something if you practice at it.

I am going to try harder. And give up if I don’t understand it.   1

I have learned that as long as I continue to practice math or algebra I will slowly get better at it.

In math this year, I am going to attept to be more optemistic about the work and try harder to get better grades.           1

be able to be good in math by the end of the year. and that i will be able to succeced in anything that i do by just practicing.    1

I learned that when you have a difficult time with a question that your brain is growing at the same time.           1

We went from math is “complicated, hard, frustrating, …” to “I can do this” in 95 minutes. I believe our students left hearing our first day message:

Everyone can learn math.

Our brains are growing when we struggle to solve a problem.

There isn’t just one way to solve a problem.

Learning more than one way to solve a problem grows our mathematical flexibility.

Working with a team is an important part of how we learn mathematics.

And so the journey continues as a new school year begins …

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Posted by on August 17, 2015 in Student Reflection


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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.)

Screen Shot 2015-07-16 at 2.51.54 PM

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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