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!

Monday, December 18, 2017

Math Fun for a Crazy Week

It is so close I can taste it!

Here are some of my favorite math activities for those unexpected times when you have a group in front of you that you had not planned to see, or to see for that extra 20 minutes.

1. SolveMe Mobiles from EDC: https://solveme.edc.org/Mobiles.html
These puzzles are set up like the mobiles hanging above your infant's crib, and they present "multiple balanced collections of objects whose weights must be determined by the puzzler." They gradually increase in difficulty, and there is the option to build your own puzzles, too. Lots of algebraic reasoning going! They also have other amazing puzzles (my other faves are Mystery Number and Who-Am-I and a paper handout of some samples.

2. Central Park by Desmos: https://teacher.desmos.com/centralpark
Students move from guesses to algebraic rules to design parking lots that place barrier in the appropriate places. This is a great tool for transitioning students from arithmetic to algebra, and fosters the "Guess-Check-Generalize" thinking that helps them make that move.

3. Open Middle Problems: http://www.openmiddle.com/
Searchable by grade level and content strand, these have been my jam this semester, and kids absolutely love them. I have found that setting up some templates for them to use for the problems I have given them helps them understand the structure at first. This support can be gradually taken away as they get used to the way these amazing problems work.

4. Graphing Practice: http://www.math-aids.com/Graphing/Four_Quadrant_Graphing_Characters.html
Perhaps not as sexy as Desmos, but great for places with low or no tech. Put on some festive tunes and allow students to practice plotting points on the Coordinate Plane. The end result could be mittens, a present, or other holiday images.

5. Math = Love Activities: https://mathequalslove.blogspot.com/p/algebra-1.html
Sarah Carter is a super creative teacher, and generously shares her activities and ideas on her blog. Head on over to see her activities and you are sure to get inspired to try something new. Three of my favorites are Four FoursRolling Dice for Point-Slope Form, and Snowball Fight.

This time of year is definitely crazy, and I know that I am certainly looking forward to the opportunity to unplug and unwind. However, for many of our students, we should remember that breaks can bring uncertainty, changes in routine, and missing teachers, friends, heat and warm meals. So, in the midst of celebrating our impending, hard-earned break, keep in mind that some may not be looking forward to it and need some reassurance that we will be back soon better than ever.

Merry everything, and happy break!

Tuesday, November 14, 2017

Math Talk Feature: WODB

https://wodb.ca/ is brilliant.

I was enlightened to this resource's existence a little over a year ago via the magic of the #MTBoS. I have used it with students and teachers, both of whom love it as much as I do.

The biggest thing I have learned is that the prompt, "Which one doesn't belong?" was leading my students/participants to think there was one right answer, which I didn't want. Rather, I wanted to see what they knew about the numbers/shapes/graphs we were analyzing, and their relationships to each other.

So, the prompt shifted to "Find a reason why each one doesn't belong," and the reasoning and open communication followed; students began making observations and noticing things that I didn't even think of, because they were trying to find reasons for all of the quadrants rather than focusing on one "correct" answer.


The sample above is my favorite to use as a first experience with WODB for both teachers and middle school students. You can see the bullet points which are reasons 6th grade students shared for each number. My favorite is the second reason for 9: "only number whose digits don't add to 7." Awesome.

Beyond fun, this allows teachers to assess students' prior knowledge of numbers - vocabulary like perfect square, prime number, factor, multiple, and digit all came up naturally in this class.

One of the teachers I work with this year is developing many of his own with graphs and equations for an Algebra I class, and we are seeing students make sense of the features of both as well as the relationships among representations. We were inspired by images like these from the site:


I am excited about this resource, and hope we can all continue to contribute ideas to grow it more and more. I will leave you with some of my favorite shapes sets:




Friday, October 13, 2017

The Way We Say the Things We Say

I have been thinking a lot lately about language, and becoming more mindful of the way I phrase things during coaching visits and broader education conversations.

During a recent coaching visit, a teacher stopped me during our post-conference and said, "I just have to tell you, when you say 'our students' it makes me feel like you really care about us, and I get why my students are so comfortable with you." Aside from the warm fuzzies I felt from her generous words, I was hit with a realization: we all need a lot more "we," "our," and "us" in our lives.

Life is hard. People are fighting battles everyday; be it anxiety, lack of sleep, relationship issues, dealing with grief, or a myriad of other issues, many of which are battles we fight alone. We try to keep private our struggles to keep up with the pressures of "being ok".

In so many contexts, there is this focus on the individual, and I think it is because we are all working so hard with so little formal acknowledgement of what it takes to do all the things. This is especially true in my experiences in education, and I am finding that is a common challenge for many of us.

Often, in classrooms where management issues prevail, I pay attention to language and there is often a focus on "MY classroom" or "MY time." We've all been there. But I think that simple shifts in language can not only build more cohesive communities in our classrooms, but also get us back to our roots of teaching as teamwork and not a competition.

Life is tough enough. Let's bring joy and togetherness to the forefront of our work.



Thursday, August 24, 2017

Building Community in Mathematics Class

The best way to build a math community is through doing mathematics together.

I'm sure we all remember the first day of school when we woke up excited and ripe with anticipation about how our year would play out, only to be hit with a day full of rules, syllabi, and boredom that didn't actually look like a typical school day at all.

So let's flip that switch and start the year with mathematics, modeling the classroom we want to have all year. Norms are not set and learned in a day; they are nurtured and develop through discussion throughout the entire course of the year. By starting the year with an accessible, engaging, mathematics task and reflecting on what went well and what could be improved, we set the tone for this conversation and mindfulness about how our classes function to continue throughout the year.

I like to use this form to help gather students' reflections on the day.

Lots of folks in Twitter-land have great activities to use that they have generously shared. Here are some of my favorites:

  • Sara Van Der Werf's "100 Numbers to Get Students Talking" has been fun for me to use with both middle school students and adults in professional development sessions. Establishing what group work looks like is achieved through purposeful scaffolding and photographs that she describes in her blog post.
  • Annie Forest uses "Me In Numbers" to share some information about herself while also giving students a chance to showcase their number sense by providing a bank of numbers for them to reasonably match to a statement. I love how she then gives the students a chance to make their own that she attempts that night. 
  • Sarah Carter uses "Broken Circles" to help establish group norms and roles. I love that it is a little variation from the wordy, algebra-heavy problems I tend to use, giving another access point to engage more learners. (Sarah also shares her awesome posters for classroom set-up).
  • I also love these two books, which have card sets with clues to help kids work together to solve a math riddle. Get It Together has lots of levels and categories of problems, and even has a sample on its site to whet your palette. United We Solve is similar, and has a sample in the preview at this link.
  • Get started with your daily number sense routines right away! Number talks are my jam, and I love using WODB at the beginning of the year since they are so open-ended, but there are many, many more.
Regardless of the task, focus on access, engagement, and reflection built around mathematics!






Thursday, April 27, 2017

An Ode to Card Sorts

Why are card sorts so dang magical?

Any chance we get to shift students' thinking is great, but resources like Formative Assessment Lessons (Classroom Challenges) from the Shell Center take it to a new level. I have implemented many of these as a classroom teacher and now as an instructional coach, and each and every time kids make new connections, experience "a-ha" moments, have meaningful mathematical discussions, and are highly engaged.

I have used card sorts to review and reinforce concepts, to make connections among representations, and establish group work norms. Some of them have even helped me allow students to discover new ideas, including vertex form of quadratic equations through this gem!

Today I was in a classroom where the teacher was helping students make connections among linear tables, graphs, and equations. She used Mrs. Math's sort to give her class time to explore. Her launch was to ask them what they noticed and wondered about the three representations in purple. Students' thinking is in green.


Next, students explored only the graph and equation cards to try to make matches. They were overwhelmed at first, but then started to create subcategories like "positive slopes" and "goes through the origin."


And thus began the magic. Some of their observations and strategies were noted on the board as they discovered them. The scribing doesn't do the conversations justice, but you get the idea of some of the connections they were making.


The Shell Center was my first introduction to these powerful tools, but since then I have discovered several other sources.
I'm sure many of you have come across some great resources, too. Please share in the comments so that we can all build our arsenals of these amazing tools! 

Thursday, April 13, 2017

Begin again.

I started a blog a while back, and had every intention of writing faithfully. Then my world was shaken when my dad, who was also my closest friend and most enthusiastic soundboard for all things life and education, lost his battle with acute myeloid leukemia. That was three months ago today. Suddenly, all of my efforts were needed simply to function and the blog ended as quickly as it had begun.

But this space will not be used for wallowing in that enormous loss. I can't call my dad to share my excitement, challenges, revelations, or random thoughts anymore, but I hope that writing them here will honor those conversations and make navigating the tumultuous world of education a bit more mindful, for myself and for others. 

Losing my father made me come to a full emotional stop. Slowly, I am shifting toward a new normal. My dad’s favorite number was 53, and I felt it was an apt number to describe the turn I have made in the grieving process; significant, but not impressive, not a clear mathematical benchmark. It also seemed relevant to what I am starting to shift in my practice leading professional development.

It is common practice to throw things away in education quickly and with reckless abandon in our pursuit of the latest trend. Often, the replacements for those original things are not fundamentally different, but they are presented as if a whole new philosophy has emerged. This presentation often demotivates those doing the work, and leads to poor buy-in, or worse, viewing all PD or all new initiatives as useless.

So what is a gal to do when she is responsible for leading PD and knows that we humans do not learn something deeply the first time we work with it?

Rather than dwelling on the frustration I feel when sharing information that people think they have heard before, I try to focus on the shift that has been made to get to the “new” tool or idea.

Take, for example, number talks. This year with my quarterly PD groups I have tried to illustrate how my own practice with number talks evolved over time to help them reflect on best practices. Here are my stages:
1.    Just do them.
·       When I learned about number talks, I just tried a bunch of them.
·       The CCSS-M Standards for Mathematical Practice (SMPs) 1, 3, and 6 were my goals.
·       I learned that the power was in the routine; the more I led them, the better my students got at them.
2.    Try to illuminate structure.
·       The discovery of number strings helped me think about how I could foster SMPs 2, 7, and 8.
·       I learned about them here and worked to build effective strings to help my students recognize and use structure to compute mentally.
3.    Purposefully plan toward a big idea.
·       I was sold. I wanted to incorporate them into my daily instruction to address needs my students had.
·       I started to plan a set of 3-5 related prompts that built toward a big idea.
·       This helped me be faithful to implementation because I was prepared, and it also helped me dissect the standards to understand how big ideas developed conceptually.


So yes, we have all heard number talks and many of us are using them with our students. But if we make just a 53-degree shift in our thinking about them, we might just find something new to enhance an already solid practice.

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 p...