Thursday, February 20, 2020

Math Language Routine: Stronger and Clearer Each Time


We spend a lot of time perfecting how we support students in the classroom during the learning of mathematics, but helping students develop independence in using those supports and their benefits can be tricky. In thinking about why this happens, I have realized that some of the supports I have used actually lower the cognitive demand of tasks, so when students are faced with a non-routine problem, they can struggle to apply the tenets of those supports in a new context.

The Mathematics Language Routines maintain the rigor of the language and amplify it so that students are practicing in class what they will see on those measures. They are also more deeply learning the language, as they spend time refining their responses and then reflecting on how communication helped them more clearly articulate their thinking.

My school adopted Illustrative Mathematics last year and that was the first place I learned about them. I reached out to the literacy coach in my building who worked with me to develop the protocols with my students for "Three Reads" and "Stronger and Clearer Each Time." I'll focus on the latter in this post.
The goal of "Stronger and Clearer Each Time" is to help students revisit and refine their thinking to present a well-developed idea either in response to a prompt, as a solution to a problem, or any other opportunity for communication where precision is a focus. It also provides a chance for participants to reflect on how communication strengthens our thinking when we learn from others.

The routine begins with independent think time, where students write in response to the prompt on their own. I typically set an amount of time for them to “write everything you can about ________” I encourage the use of drawings, diagrams, words, and phrases. Before students move on to the next part, I have them pause, read what they have written, and jot down 3-4 key words from it to remind them what they want to share with their partner. 

Next, students go through 2-3 rounds of collaboration, where they spend a couple of minutes with a partner to share their own thinking and get feedback, and then do the same for their partners. At the end of each pairing, students quickly jot down something new they heard from their partner. 

After we have 2-3 rounds of partner talk, students head back to their home bases to write a revised draft of their thinking, incorporating the new information they heard. I have found that sharing out and thanking each other is a great closing to the collaboration piece; and it also reinforces social skills and gives students a chance to practice friendly language. We use sentence stems or closing statements that they choose from and high five before heading back to their seats.

Finally, we close and reflect on the process. This also launches what is perhaps the most important part of the routine; the reflection. I make sure to save time for us to think about and discuss how much our responses were improved and strengthened as a result of speaking and listening with others. Students often marvel at the change in their responses, and this piece is not something to skip.



One of the ways I’ve used this routine is to reinforce key terms or big ideas during a unit. This example is from a week ago when my students were exploring theoretical probability. There was some confusion between it and experimental probability, so I started our class with Stronger and Clearer, simply asking, “What is theoretical probability?” Below, you can see the initial draft and revised draft of one student. Being able to talk helped add detail and specific language to her response. The right image shows the note-taking tool students used to jot things down that their partners said during collaboration time. They use this information to revise their initial drafts. My favorite moment was when one student said, “Wow, I knew way more than I wrote down at first!”




Stronger and Clearer Each Time can also be used when students are solving problems, where the prompt becomes the problem and the discussions are around solution methods and justifications.

How will you use Stronger and Clearer in your classroom?

Friday, January 24, 2020

Math + Retrieval Practice = Sustained Learning

Retrieval practice is a hot topic in the Twitterverse right now, and for good reason; it is a high-yield and easily implemented strategy that works.

The idea behind retrieval is that bringing information to mind (getting information OUT of students' heads rather than cramming more IN) boosts memory and helps to disrupt the forgetting curve. A team of cognitive scientists has compiled an amazing set of research and tools at https://www.retrievalpractice.org/. I highly recommend spending a day reading and learning there, and checking out the books Make it Stick and Powerful Teaching to understand current thinking in cognitive science about this strategy, as well as awesome, practical strategies.

Two non-negotiables for effective retrieval practice:

  • Retrieval practice is practice; a formative experience that is not graded.
  • Feedback is an indispensable part of the process. Close the loop.

I've fallen in love with several strategies over the past 1.5 years of trying this in my classes.
Here are 3 of them that I feel are particularly powerful for my mostly emergent bilingual mathematicians (examples follow after the table):


This is a Retrieval Grid I used recently with my Algebra I class to ease back in after winter break:

Here are some picture prompt responses:




(I am working on being more systematic about collecting work samples, and will store them here when I have them, and I also Tweet them out from @anneagost).

For those of you in the Chicago area, I hope to see you at MMC next weekend to talk more about these and other tools for retrieval practice in math class!

Wednesday, August 1, 2018

Take that Math Talk, Flip it and Reverse It (Backward Math Talks)

Despite the title, this post is not an ode to Missy Elliott (though maybe I should write one of those sometime).

How many times have you participated in a math talk at a professional development session with other adults and seen many awesome representations and strategies come out? Have you then gone back to your school, excited to try that same prompt with your students, only to find that the number of strategies is 1 or 2...or none, mentally?


So, what is the disconnect? Well, for one thing, students have varied prior experiences and so the models and strategies in their toolboxes also vary. More than that, though, is that once we learn algorithms, it is hard to go back to truly experience the building of conceptual models. Sometimes well-meaning parents teach their children to solve problems the way they learned, using the algorithm. Sometimes the pressure of high-stakes testing makes us feel like we need to introduce it so that students can tackle those questions independently. Whatever the reason, early introduction of algorithms happens, and it is harmful. 

Algorithms are still a goal, eventually, but when we get there we want students to be able to think about what they're doing. Quantity gets lost in algorithms. Kids need to be doing mental checks and asking themselves, "Does this make sense?"

Fluency is bigger than answers.

(Based on research from Add it Up)

So, if students do not use the models or strategies I am hoping to see during my math talk, it is up to me to get them into our conversation. This is where my girl Missy comes in.


If students aren't familiar with the various strategies and models I want to see in a math talk, no amount of wait time is going to magically transfer them into their heads. Why not provide the models and have students work backward to make sense of why those models represent the problems we pose?

Take, for example, 16 x 25, as in the example at the top of this post. Rather than ask, "I want you to find the product of 16 x 25 in as many ways as you can," we can ask:


In this particular example, my goal is to help students see how we can break apart factors and use the associative property to make the problem easier to solve mentally.

Pam Harris has some great examples at her site, https://www.mathisfigureoutable.com/ that show lots of different visual models for various operations, like the one below, as well as problem strings to get those SMP juices flowing.


I wondered more about the idea of "learning backward" and came across a TED Talk by GM Maurice Ashley that really resonated. You can check it out here.

Toward the end of the talk, he mentions the adage of youth being wasted on the young, which made me think:

So, instead of being frustrated when students don't have the strategies and models we wish they did, let's work to "flip it and reverse it" by taking ownership of the solution with backward math talks to promote sense-making and rich discussion about mathematics.

I have found this planning template useful. You can access it here in Word form.


Happy planning! Please share any that you try in your classroom with me @anneagost!




Saturday, April 28, 2018

My Reflections from NCTM Annual 2018

I'll admit it: I have had some amazingly good and some amazingly "meh" experiences attending NCTM conferences in the past. It is a big investment of both time and money, and I sometimes felt as though I couldn't justify it based on what I felt I got out of the conference.

This year, however, that changed. This was, by far, the best NCTM conference I have attended, which has caused me to reflect on why. Here's what I have come up with so far:

1.  I put myself out there.

Tracy Zager put a call out for volunteers for NCTM Game Night and I responded saying I would be happy to help. In typical Tracy form, she was inclusive and thoughtful and started a Twitter group message involving those who voiced interest. I got to meet people from all over the U.S. and Canada, and felt part of the community that had previously felt a bit too exclusive to want to break into. I also volunteered at the #MTBoS booth one morning (more about that below).

I also made an effort to talk to people in every session, beyond the basic sharing of name/location/role, but asking real questions and having deep discussions about teaching and learning mathematics. What an opportunity that I have neglected to take advantage of in the past!

2. Twitter and all its magic.

Since my last #NCTMannual experience, I have become much more active on Twitter. This enabled me to choose sessions in a more informed manner (Have I already heard what this person has to say via social media? Is this someone I've only briefly encountered and want to know more?) and I also finally got to attend a session with Sara VanDerWerf, whose blog and energy have inspired me for years (her session did not disappoint).


I volunteered to work the #MTBoS booth and got to meet some amazing people, and also spread the word about how Twitter in general and #MTBoS specifically have helped me grow as an educator. SO many people said, "I don't have time for Twitter." This is such a common thing that I hear from colleagues, friends, and professional connections. Is this something you struggle with, too?

3.  I took time to reflect on my learning each day.

I wrote in a journal about the ideas that resonated with me and followed up with people's blogs and websites after hearing from them during sessions. This was a way to solidify some of the ideas I Tweeted about during the conference and marinate in them after the "conference high" to bring them back to my context.

I learned a lot at #NCTMannual, and one of the biggest takeaways is a set of strategies for making the most of conference experiences. Please comment with your ideas, too!

Thursday, March 15, 2018

Math Talks are Awesome: Share Them!

So, I know I talk about math talks a lot, but there is good reason! They are such a versatile and accessible tool for promoting problem solving, discourse, precision, pattern-sniffing, and so much more.

As classes implement them more and more, they see the payoff in increase confidence and robust number sense. Many of my schools this year have asked me to help K-5 teachers get started using math/number talks in their classes, so I thought I would share my process for introducing them:
  1. I start with a rationale for math talks, which is tailored to the school's perceived need (often this is building number sense in students, increasing engagement, and/or facilitating discourse). 
  2. We experience 4 types of talks: Which One Doesn't Belong, Dot Talk, Number of the Day, and a computational number talk. I ask the participants to play the role of student and to suspend "teacher talk" until they have experienced the full routine. 
    • Along the way, the participants have a template they use to jot down what they notice about teacher moves and students moves, any questions they want to ask after we experience the talks, and connections they see to the SMPs.
    • After each talk we pause for them to think and ink, then we discuss noticings and wonderings they capture on their templates. 
  3. Once we complete the first 3 types, I introduce the full protocol and do a number talk, stressing that this type comes after a safe classroom environment and community has been established. 
  4. We pause for any additional noticings and wonderings and discuss, then they think-pair-share about these questions:
    - How might we use math talks to assist students in developing their understandings of the big
      ideas in our courses? How does this relate to number sense?
    - What are the big ideas in our courses that might lend themselves to math talks? 
  5. I introduce my resource list and invite participants to play around with the sites on it.
  6. Teams form to plan a math talk using this template. If there is time, I encourage them to practice with a partner or small group.
    • Follow up is key, so either by passing that off to a leader or coming back, I make sure to have sharing of the one they tried in their classes at the next gathering. 
Here is a link to my slides: https://tinyurl.com/MathTalkIntroSlides

Here is a document where I am trying to compile websites that have "math talk-ish" prompts available: https://tinyurl.com/MathTalkResources


What am I missing? Please comment and let's make these working documents work for us.




Thursday, February 15, 2018

Math Talk Progressions: Making Distributed Practice Meaningful

'Tis the season for anxiety around upcoming standardized tests. Woof.

Many of us feel that there is so much content over the course of a year that we may not have time to even get to every topic, let alone revisit it to practice or reinforce skills along the way.

Something I am noticing this year is that we, the adult math teachers, spend a lot of time thinking about scope and sequence and, if we are lucky, having vertical conversations to make sure the flow of our units and the big mathematical ideas makes sense over the course of our students' careers with us. But when do these messages get shared with our students? When do they get the opportunity to see the beauty of mathematics and how the big ideas evolve and connect?

NCTM's definition of procedural fluency points out an important truth:


Being in love with math and number talks as I am, I see a solid opportunity to merge some of the ideas I have been thinking about to build meaningful distributed practice. I wrote about my own progression of using math talks last April. As I have gone through this school year thinking about students' understanding of big ideas with teachers, purposeful series of math talks seem to be the answer to a lot of the areas students struggle to make sense of.

I think that sharing that overarching beauty of the big ideas would help them retain and connect and, therefore, remember the processes and procedures, too.  We can't accomplish this through isolated experiences.

Math Talk Progressions (#MathTalkProgs? Yes. Let's do this.) are series of related math talks that build toward an idea, representation, strategy, or understanding that we want students to take away. Inspiration can come from current learning topics, areas students are struggling with (i.e. fraction sense or place value), or just some fun topic you wouldn't otherwise make space for.

The Progression is meant to take place over the course of several days, and the goal is to illuminate patterns,

I designed this planning template, which has been helpful when first starting out designing these and still serves as a structured brainstorming place as I anticipate student moves and my questions.

Now, we have transitioned into jumping onto slides pretty quickly during the design phase of writing these. You can find some on my Math Talk Progressions page. I will continue to add as more are developed, and would love for you to contribute, too!

What other ways are you incorporating meaningful practice into your classes over time?

Wednesday, January 17, 2018

Navigating Numberless Word Problems

Do you struggle with students who have aversions to word problems? Do they just scan for numbers and perform any old operation on them without considering important context or relationships? Brian Bushart's blog is always an awesome read, but when I read about Numberless Word Problems last year, it was a total game changer for me in tackling this issue.

Numberless word problems are just what they sound like; word problems with the numbers (and question stems) removed, initially, to provide an opportunity for students to make sense of the relationships in the problem before rushing into computation. They are empowered to access the problem situation and employ their own wonderings, contexts, and questions to make sense of it. 

I have been working on NWP with some of the teachers I coach, and we have seen students ask lots of interesting and insightful questions about the situations presented, and many are even applying this idea of slowing down and working to understand what is happening in problems before rushing to compute.

Recently, we have been adapting questions from the standardized test they use to get at the structure of those items through NWP math talks. 

This is the general structure we use to plan (it can certainly be modified depending on the problem, like in this geometry example). We also brainstorm questions we anticipate students might come up with and choose one for a final slide where we actually ask them to solve.



Here is an example from a test prep resource:

So where can you find Numberless Word Problems to use with your students?

  • Start with Brian Bushart's blog: https://bstockus.wordpress.com/numberless-word-problems/
  • Develop them from existing problems found in your textbooks or other resources
  • Write them based on real school or classroom scenarios (i.e. planning for a field trip)
  • Get in on the Twitter action with #numberlesswp and share your amazing ideas!

Math Language Routine: Stronger and Clearer Each Time

We spend a lot of time perfecting how we support students in the classroom during the learning of mathematics, but helping stude...