A few months ago, Levi Patrick recommended that I read “5 Practices for Orchestrating Productive Mathematics Discussions.” I’m so glad he did because the book has completely changed my approach to teaching through problems. Thanks Levi!

So, I’m going to start blogging about the implementation of these practices with some of the problems I’ll be using in class this year. If you haven’t read the book, then this may be a little foreign. Therefore, I provided a few quotes to help provide a background. For those who have read the book or are familiar with the practices, feel free to offer suggestions for improvement!

# Learning Goal

“To ensure that a discussion will be productive, teachers need to have clear learning goals for what they are trying to accomplish in the lesson, and they must select a task that has the potential to help students achieve those goals…The key is to specify a goal that clearly identifies what students are to know and understand about mathematics as a result of their engagement in a particular lesson.” (Smith & Stein, 2011)

For my first lesson, I wanted to use a good intro task to provide a context for solving equations. My official learning goal was to, “create a purpose for solving equations and connect student strategies to formal equations.” To accomplish this, I decided to use Jon Orr’s “Filling it Up!” 3 act lesson. It’s the most intriguing task I’ve found so far for accomplishing the learning goal. Also, I ended up recording a video to save on materials and created a lesson page to work from. Check it out to know what I’ll be writing about for the rest of this post.

# Anticipating

“Anticipating students’ responses involves developing considered expectations about how students might mathematically interpret a problem, the array of strategies—both correct and incorrect—that they might use to tackle it, and how those strategies and interpretations might relate to the mathematical concepts, representations, procedures, and practices that the teacher would like his or her students to learn.” (Smith & Stein, 2011)

For the anticipation stage, I created the following chart based on something similar in the book, and I worked the problem out in the ways that I thought the students might approach the problem.

# Monitoring

“One way to facilitate the monitoring process is for the teacher, before beginning the lesson, to create a list of solutions that he or she anticipates that students will produce and that will help in accomplishing his or her mathematical goals for the lesson.” (Smith & Stein, 2011)

After creating the list of possible solutions in the anticipation stage, I actively looked for these strategies during the group work session in order to take note of which groups were using what strategies. I used the following chart to keep track.

As the classes worked, I quickly realized that everyone was using the first strategy anticipated. No one used a table or equation. However, I also was on the lookout for the possible mistakes listed in the anticipating stage. Most of the mistakes actually did occur, and the most common was adding the can volume instead of subtracting. This led to helpful conversations.

# Selecting

“The selection of particular students and their solutions is guided by the mathematical goal for the lesson and the teacher’s assessment of how each contribution will contribute to that goal. Thus, the teacher selects certain students to present because of the mathematics in their responses.” (Smith & Stein, 2011)

Since everyone used the same strategy, I decided to pick an outgoing student to present his or her group’s work. I did this because it was the first week of school, and I didn’t want to put a less confident student in front of the class until I build more trust with the students during the first few weeks.

# Sequencing

“Having selected particular students to present, the teacher can then make decisions regarding how to sequence the student presentations. By making purposeful choices about the order in which students’ work is shared, teachers can maximize the chances of achieving their mathematical goals for the discussion.” (Smith & Stein, 2011)

I’m excited about this phase of the 5 practices, but since everyone used the same strategy, I only had one student per class present. I didn’t want to have the same presentation over and over.

# Connecting

“Rather than having mathematical discussions consist of separate presentations of different ways to solve a particular problem, the goal is to have student presentations build on one another to develop powerful mathematical ideas.” (Smith & Stein, 2011)

For this phase, I decided to create a connection handout for the class to get strategies on paper and have a page for future reference. We walked through the strategy that the class used to solve the problem, and we connected it to a formal equation to solve. Next, I took Kyle Pearce’s advice and connected the overall activity to traditional word problems that can be found on standardized tests.

Click here for the handout, and let me know how it can be improved!

# Materials:

Here are the links to all of the materials mentioned in this post and used during the lesson.

## Featured Comment:

Ryan Dent mentioned a great strategy for improving the presentation process:

I’d love to start a discussion specific to this thinking…after we’ve anticipated, monitored, selected and then prepared a sequence of work, I strongly believe that all kids should first have the opportunity to analyze the representation, and turn-and-talk about what they think that kid was thinking before the kid who built the representation explains what he/she did. I see a lot of SMP 1, 2 and 3 when this happens. If you take that part out (the turn-and-talk on interpreting the representation first), and have the “author” of each representation present what he/she did, that “author” of the representation becomes the lecturer in the room.

Smith, M., & Stein, M. (2011). *5 Practices for Orchestrating Productive Mathematics Discussions*. Reston, VA: National Council of Teachers of Mathematics

Fantastic post Dane and thanks for laying everything out for us to follow along. Love how purposeful you’ve made every move.

I especially like the little connection you’ve thrown in on the handout “What it could look like on a test”. I’ll be stealing that for sure but giving credit where’s it due! Thanks for making your understanding of math education accessible to all teachers.

Cheers my friend!

Thanks Graham! This really means a lot coming from you. Your work is incredible!

I definitely have to give credit to Kyle Pearce for the connection piece you mentioned. His 4 part math lesson post is really insightful.

Thanks again for the kind words! Looking forward to seeing more from you soon.

Awesome post here Dane! You’ve really kicked it up a notch with the videos and pictures! Love the tie in to the book!

Something I’ve been battling with lately is: Should we let students “experience” the math? What I mean is, letting students actually see, measure the jug, cup, etc. I think there is value there.

Or is it not really worth the time?

I’m not sure which is the right answer yet. Thoughts?

Thanks Jon! Totally stole everything from you obviously. I appreciate you sharing!

Definitely hear you and agree with the experience the math aspect. I was really torn between making a video and actually letting the kids do the measurements. I think I prefer for the kids to get the hands on experience, but I didn’t do that this time because I was worried about getting behind the pacing calendar for my district (I know that’s not always the best reasoning). I’m in a new district this year, and I didn’t want to get behind so early in the year. Therefore, I figured I’d save time by making the video and preventing clean up time from adding too much to the day. I also wanted to get to the connection handout in the same day so I could jump into solving equations the following day.

I don’t know if there is a true right answer, but I know it’s more fun for the kids to do hands on. I’m all for more fun.

I definitely love that you made a video for this! I didn’t run this lesson last semester because of time too. I should do a study… Two groups 1 hands on and 1 video. See if better understanding.

That sounds like a great idea. Let me know how it goes if/when you do that. I think it’s an important question with all of the great 3 Act Lessons out there that could potentially be turned into hands on experiences.

Dane,

I love the way you’ve step-by-stepped the way through your process. Taking the time to provide a detailed explanation of the what and why of your teaching is so helpful, especially for those who haven’t quite wrapped their minds around how this kind of learning can look. I’ll be sending folks here for insight.

Thank you!

Thank you for the kind words and sharing! I really appreciate it. I know I’ve struggled a lot (and still struggle) through my teaching journey, so it’d be awesome to potentially help others. Thanks again for the positive feedback!

What a great idea to expose the process that you’ll go through. We have many teachers that have been implementing the 5 Practices for a second year now that might be able to share some insights and collaborate, as well as learn together with you that span all the way down to Kindergarten.

Dr. Smith’s work appears to be foundation in NCTM’s Principles to Actions, which Phil Daro himself speaks to the validity of here…https://www.youtube.com/watch?v=C-GMYg4Lrcw

I’d love to start a discussion specific to this thinking…after we’ve anticipated, monitored, selected and then prepared a sequence of work, I strongly believe that all kids should first have the opportunity to analyze the representation, and turn-and-talk about what they think that kid was thinking before the kid who built the representation explains what he/she did. I see a lot of SMP 1, 2 and 3 when this happens. If you take that part out (the turn-and-talk on interpreting the representation first), and have the “author” of each representation present what he/she did, that “author” of the representation becomes the lecturer in the room.

Can we start a discussion on that idea? I’d love for somebody to critique my reasoning on that so I can make my understanding of what’s best more robust.

Thanks for the kind words, Ryan! I really like the turn-and-talk idea. Presentations aren’t always as useful as I’d like, so maybe using that strategy could help improve the effectiveness. It’s hard to get the kids to really focus on the strategies used and how they connect. I try to use connection handouts in order to somehow force the students to analyze the different representations.

I’ll add your comment into the post! Hopefully we can start the discussion you mentioned and others can maybe chime in about their experiences.