Designing Math Adventures

For the last several years, Kathy and I have been telling stories associated with our use of human-centered design to support the teaching and learning of meaningful K-8 mathematics. We decided to gather these stories in a book, Designing Math Adventures. After several prototypes, it is finally ready for release. It is available as an ebook or a paperback on Amazon.

Third Time is the Charm

We are also grateful to the Michigan Council of Teachers of Mathematics who invited us to deliver the keynote at their 75th Annual Conference this summer. If you're interested in attending, you can register here.

What will you be teaching on October 20th?

My first real teaching job was at Grant Middle School. I was hired on the Thursday before school started and began teaching the next Monday. I taught 8th grade math in a portable classroom (much like the one shown on the right) and coached the girls and boys basketball teams. It was wonderful but also stressful. I always felt like l was just a few lessons ahead of my students.

At the end of the school year, I vowed that I'd never be so disorganized in my teaching ever again. I spent the entire summer planning my lessons for the next year. I was assigned 8th grade math again and also an Algebra class. By the time school started, I could tell you exactly what I'd be doing during either class any day of the year.

It was my worst year of teaching. I had spent so much time focusing on the content that I had totally forgotten about the students. And I put so much effort into the plans that I was resistant to altering them. My students and I were all miserable.

I share this story with the teachers I work with as a cautionary tale. It's tempting to want to be totally prepared for every lesson. Unfortunately, it's impossible. It's better to have a lesson prototype in mind that can be altered in response to feedback from students. We use a version of this One-Sentence Lesson Plan as a framework for our prototypes for designing math adventures. The incompleteness leaves room for student-voice and student-choice during the lesson experiment.

So, instead of spending the summer planning out every detail of the coming school year, please keep it simple and give yourself some time to recreate and restore. Maybe you could explore some new places. This will hopefully re-energize you and allow you to be more responsive to the students in your classes.

Scampy McScamperson visits Devils Tower

What's the problem we're trying to solving in math education?

The way we ask a question tends to frame our solution. This was one of the most long-lasting lessons that I learned at the d.school's week-long Teaching and Learning Studio. The activity we did early in the week to introduce this idea is one that I still do with teachers. I'll do my best to recreate that experience here.

First, I need you to break into one of two groups. Each group will do a different task. Let's assign your task based on your birthdate. If the day of your birth is odd, then do the first task. If it is even, then do the second one. (e.g. I was born on the 29th, so I'd be in group one.)

To begin, click on the appropriate link provided below. It will open up a copy of a Jamboard with the task's question. You can do your work on the Jamboard (it is yours to keep) or on a separate piece of paper.

Task One (odd birthdates)

Task Two (even birthdates)

Don't read any farther until you've completed your task or you'll ruin the surprise.

Once you've completed your task, click on the other task question and consider how that question might result in different solutions and why.

The first group often comes up with a variety of creative vases. Some have multiple openings at the top. Some hold the flowers sideways. However, because of the way the question is framed, they are all obviously vases.

Very few vases show up in the second group. Instead, flowers grow on the wall or hang from the ceiling. One participant had the flowers floating in a clear container with a fan at the bottom. Sure, it might not actually be feasible, but it is a much more creative way to start seeking solutions.

IBM does a version of this activity in their

Enterprise Design Thinking Course

I ask educators to do these tasks to highlight that most of our efforts to innovate in math classrooms amount to the first task. No matter what we do, we just end up with another vase. This is one reason why Kathy and I wrote Designing Math Adventures - because we want learners to experience something more than another math lesson.

So, what's the question you'll ask to frame your next school year? I'd like to collect them in the comments. Thanks for contributing to our creative problem-solving.

Why write Designing Math Adventures?

In planning the book, Designing Math Adventures, Kathy and I took Simon Sinek's advice and started by identifying our WHY: to support educators' efforts to improve K-8 math learning.

Golden Circle

Based on our interactions with thousands of teachers, we can confirm that Brené Brown is right. Most of us are doing the best we can under the circumstances. This doesn't mean there isn't room for improvement. We wrote this book to help teachers to make the subtle shifts necessary to change their current circumstances and improve the teaching and learning of math in their classrooms. 

HOW can teachers change their circumstances? Sometimes it's as simple as doing more of what’s working and less of what’s not. The level of intentionality we are talking about requires an honest examination of our teaching practice. This type of reflection isn't always easy.

WHAT Designing Math Adventures offers is a design thinking tool that supports teachers in creating meaningful math lessons. By considering five straightforward questions, teachers can write a lesson that responds to the mathematical brilliance of their students in thirty minutes or less. We know teachers are incredibly busy, so we want to ensure that the approach is both sustainable and satisfying.

Where have I been?

I haven't posted on this blog for nearly two years and nine months. It's not that I lost confidence in the power of blogging as a way to share and reflect on my thinking. I just got busy.

First, I took a position as director of the Design Thinking Academy [DTA] at Grand Valley State University. The goal of the academy is to support the use of design thinking methods and mindsets across the campus community. My work entailed visiting classes, scheduling pop-up courses, facilitating semester-long design challenges, managing creativity kiosks, and organizing the GVSU Design Thinking Speaker Series.

Ela Ben-Ur was our first speaker. She held a series of workshops where she introduced participants to a design thinking tool she created - the Innovators' Compass.

Ela Ben-Ur: Design Thinking and You (GVSU DT Speaker Series)

I learned about Ela's work on this episode of the Design Thinking 101 podcast.

This introduction to the Compass lead to the second project that has been occupying my time - writing a book. Kathy and I have been wanting to write about the Teaching & Learning Cycle for some time, but we always felt like something was lacking. When we learned about the Innovators' Compass and how it can help people to get unstuck or explore uncharted territory, we thought it would be a good resource for teachers engaging in the Cycle.

Innovators' Compass - design thinking cycle (hexagons) - Teaching & Learning Cycle (purple)

Now that a draft of the book is done (#DesigningMathAdventures) and sent to some publishers, I'm not as busy. Because I had set aside an hour each morning to engage in creative writing and didn't want to lose that momentum, I decided to fire up the old blog. I figured it would not only give me something to do while we wait to hear from the publishers but also allow me to share some parts from our writing that didn't make the latest cut.

Thanks for indulging me in this exercise of creativity. As always, if you have any questions, please post them below or reach out on Twitter (@delta_dc). The comments are open.

How might we deal with the mismatch?

Last week I heard Kat Holmes talk about her new book, Mismatch. I learned how designing for "normal" can miss the mark. As a result, many of us encounter mismatched experiences in our physical and virtual spaces. Effectively dealing with these mismatches requires inclusive design principles: 1) recognizing exclusion; 2) learning from diversity; and 3) solving for one - extending to many. In order to illustrate the second principle, Kat introduced us to Victor Pineda who made the following point:

There's a triangle of three different things that have to come together to really unlock human accomplishments for people with disabilities. And those involve assistive technology, personal assistance—somebody that's aware, understanding how to support you, and three is coping strategies. And so these three things sort of create a variety of tools.

Of course, I started connecting these ideas to teaching.

A curriculum is often designed for the normal/average student.

In reality, a student and the curriculum are typically mismatched.

To help the student to connect with the curriculum and be a contributor, the teacher might need to offer personal assistance, assistive technology, and/or coping strategies. And we can ask the student (learning from diversity, or as Dr. Emdin writes - co-teaching) to participate in the design.

Having planned for one student, the teacher must consider how to extend the design to many students.

After Kat's talk, I had the opportunity to watch a student-teacher deal with the mismatch between her students' experiences solving problems involving scientific notation [8.EE.A.4] and the curriculum used in her school. We talked about using a think-aloud (personal assistance) to create an anchor chart (assistive technology) that students could refer to (coping strategy) while solving 8.EE.A.4 problems. 

In the next lesson, she tried these ideas out. The lesson started with her making her thinking visible while solving an 8.EE.A.4 problem. Next, she asked students what they noticed in her thinking and added it to an anchor chat. She happened to put the anchor chart in the back of the room so it was obvious later in the lesson how many of the students were using this tool as they turned in their seats to see it. Because of her efforts, students were able to successfully connect to the curriculum.

I'm still processing a lot of this and would appreciate you sharing your thoughts in the comments.

[This blog post was written with the help of the Innovators' Compass. Check out my planning.]

Can we have five more minutes?

Excuse the pun, but it was like clockwork. I would assign a group project (something like making a concept map), give them 20 minutes, and set the classroom timer. The timer would go off and nearly all the groups would ask for more time - usually about five minutes. It got to the point where I would just add five minutes into the plan but then they'd still want more time. 

It wasn't as if they hadn't been working the entire time; they were just really invested in getting it perfect. Even the smallest detail, like the use of colors, had to be debated. I explained that these details didn't matter as much as the connections they were making, but somewhere they got the notion that the presentation of their ideas was paramount.

Then I was introduced to Design Thinking and the principles of bias to action and prototyping to a solution. I decided to apply these principles to the problem of students attempting to create the perfect poster. I went back to giving groups only twenty minutes but I broke it up into smaller intervals - each with a defined purpose.

First three minutes: organize the concepts in a way that reflects how you see them related and glue them on the paper.

I knew if I simply moved on to the next phase nothing would change; they would get stuck in the same old debates. Therefore, I had the groups rotate clockwise around the room. They were now looking at another group's vision of how the concepts might be arranged. 

Next three minutes: consider the previous group's arrangement, talk through what the arrangement might represent, and add connections (nothing more).

During this time, I kept reminding them that the previous group had only spent three minutes coming up with the configuration of concepts upon which they were working. There was nothing they could do to ruin it. This was simply a prototype and they had limited time to add their contributions. After three minutes, they rotated clockwise to a new group's poster.

Third interval of three minutes: consider the work of the previous groups, talk through what the work represents, and add descriptions to the connections.

I reminded them that only six minutes of work had gone into poster. The previous groups didn't really have anything invested in what was already done. So they shouldn't worry about doing anything that might change the poster. I even encouraged them to add new connections if they thought it made sense. After three minutes, they rotated clockwise to another new poster.

Fourth, fifth, and sixth intervals of three minutes: repeated the previous intervals - add more concepts, add more connections, and add more descriptions.

Each time, I repeated the mantra: "The other groups only spent three minutes on the poster. You can't ruin it. Just get to work." 

After 18 minutes, each group returned to their original poster. There were some audible gasps and laughs. Rarely had the poster turned out as expected but each group could infer the intent behind the decisions other groups had made. They spent the last two minutes creating an artist statement for their concept map - something they thought an observer ought to notice.

None of the groups asked for more time. They were satisfied that the posters were prototypes - works in progress that allowed viewers to add their own perspective. We used the "extra" five minutes to do a gallery walk and see how our work turned out.

We're really going to get to do it, aren't we?

One of the projects the pre-service elementary teachers (math majors) that I teach worked on this semester was designing a 4th-grade statistics lesson to address 4.MD.B.4.

The teachers went through a design cycle to make the lesson. They ... 

• Built empathy by observing two fourth-grade classes;
• Defined the problem by developing a User/Needs/Insight statement;
• Brainstormed a variety of possible activities;
• Developed a prototype SAFARI Lesson;

• Tested the lesson by sharing it with the classroom teacher; and
• Revised it based on her feedback.

Three teachers co-taught the lesson in two different fourth-grade STEM classes. They made adjustment between the lessons based on what worked and what didn't. Afterwards, they reflected on the experience and shared the lesson with me. The lesson was so cool, I decided to make a few adjustments and use it in another class for pre-service elementary teachers (mostly non-math majors) that I teach. Here is the SAFARI Lesson that I taught.

Schema Activation - Prediction

Directions: "You have two sticky notes. On the green sticky, I want you to predict the number of seconds you think it would take you to write the alphabet from A to Z. On the purple sticky, I want you to predict how long it would take you to write the alphabet in reverse order from Z to A. You have a quarter of a minute. Go!"

Focus - 5.MD.B.2

Share lesson target: "We are going to make line plots to display data sets of measurements in fractions of a unit."

[Anticipated learner responses are in brackets.]

"Who thinks they can write the alphabet forward the fastest? [13 seconds or 2 letters per second] Who thinks they will take the most time to write the alphabet forward? [52 seconds or 1 letter every 2 seconds] Alright, let's get up and stand in order from fastest predicted time to slowest predicted time."

Learners order themselves

"As I listened in, it became apparent that several of you made similar predictions. It would be interesting to see how the predictions cluster. But we could potentially have a lot of unique guesses. In order to gather those guesses, let's round our predictions to the nearest quarter-of-a-minute. For example, Sam guessed 20 seconds forward and 55 seconds backwards. He would round to 1/4 of a minute for forward and one minute, four-fourths, backwards. Work with your neighbors to round your predictions to the nearest quarter-of-a-minute and then post them on the board - forward at the front and backwards at the back."

Learners post rounded predictions

"What do you notice about the data sets?" [The writing in reverse predictions are "higher" and more spread out than the forward predictions.] 

"Why?" [We are familiar with writing the alphabet forwards so we think we can do it faster and know more what to expect.]

Activity - Writing the Alphabet in Reverse Order

Directions: "I am going to give you three pieces of paper."

At this point in my lesson, one of the pre-service teachers asked, "We're really going to get to do it, aren't we? We're really going to find out how long it takes us to write the alphabet from Z to A? Is it weird that I am so excited about this?" I reassured her that it wasn't weird - that my other pre-service teachers had designed a pretty cool lesson.

Directions continued: "You have a choice. On the yellow paper, you may write the alphabet forward on one side and use it to help you to write it from Z to A on the other side. You'll see the second sheet has the alphabet already on the back in the form of classic blocks, like the ones my grandson plays with. If you choose that one, you will incur a 1/2 minute penalty, which means you will add 30 seconds to your time. The last piece is simply scrap paper; use it if you want to try to write the alphabet from Z to A without any other support.

"A few more things: 

• You must start at Z and write the letters in reverse order to A. You can't cheat and start at A on the right-side of your paper. 
• The letters must be legible. Your table-mates will decide if they can read your letters, and you will earn a 5 second penalty for each letter they can't read.
• When you finish, check the timer on the front board, record your time, and round it to the nearest quarter-of-a-minute.

At this point, a student rose his hand to ask a question. The girl who was so excited blurted out, "I just want to get started!" The other student asked if the letters had to be upper or lower case. I said it didn't matter to me.

Set the online stopwatch and say, "Go!"

When everyone is finish, have learners trade papers check letters for legibility.

Reflection - Noticing and Naming

Directions: "If you used the yellow paper (wrote A to Z on the back), write your result, to the nearest quarter-of-a-minute, on the yellow sticky note. If you used the blocks, and added 30 seconds to your time, write your rounded result on the blue sticky note. If you did it without any support, write your rounded result on the pink sticky note. Your rounded results go on the line plot on the back board underneath your predictions."

"What do you notice?" [Look for opportunities to introduce terminology related to measures of center and spread, like median, mode, and range]

I want to ... - Choice

Directions: "What do you want to do now? Here are some ideas:

• Try it again using a different level of support and add it to the line plot;
• See if there is a difference between writing in upper and lower case;
• Try it forward and compare it with your prediction;
• Gather more data from your friends and family over Thanksgiving;
• Consider other activities that ask people to do familiar things in unfamiliar ways and what the data might show; or
• Come up with your own idea to extend your learning."

What's the hurry?

The moment you (some of you) have been waiting for [insert drumroll] ... the Carousel Lesson Design process. Previously, we learned about SAFARI lessons and prototyping. In this post, I share how to encourage teachers to embrace creativity and connectivity while collaborating on a week long unit design.

First, you need some ingredients. It's best if you have: 

Investigations Curriculum

• 5 willing teachers;
• 1 set of targets;
• 1 rich curriculum;
• 5 pieces of easel paper;
• Various scented (optional), colored markers;
• Multiple sticky notes;
• 1 lesson design framework; and
• 1 timer

Each teacher is assigned one of five sequential lessons and given 5 minutes (no more, no less) to look through the lesson in order to determine what is important.  At the end of this time, they use another 2 minutes to set up the SAFARI lesson framework on their easel paper and write down some of the most important ideas from the lesson they were assigned.

After 2 minutes the teachers rotate (like a Carousel) to the next lesson. Day 1 goes to Day 2 ... and Day 5 goes to Day 1. They use what they know from their own lesson and the important points the previous teacher wrote down to inform them about the lesson. They also have exactly 2 minutes to add to the lesson. I am constantly reminding them, "Don't worry about designing it perfectly. You don't even know for sure what the lesson is about. Don't worry about offending the teacher that started the lesson. They spent all of 2 minutes on it so far."

The teachers aren't always crazy about the artificial time crunch. However, it helps to contribute to their creativity (think MacGyverMath) while ensuring progress. It keeps them from letting their perfectionism get in the way.

Rotate! And repeat ... three more times (Note: only two interactions shown below) at 2 minutes a piece.

The teachers are now back at their original lessons. They take 1 minute to read through what has been added to their initial ideas. The sticky notes are used to identify questions for the author or indicate likes (thumbs up). The teachers can also continue to add new ideas based on what they have seen in the other lessons. After 1 minute the teachers rotate again and again and again and again and again. At each lesson they answer questions, add stickies, or contribute ideas.

At the end, the teachers have spent 20 minutes to design a five-day unit.

Yes, there is still some work to do to sift out the essential elements of the lesson. These will be written in the SAFARI format and then shared with their peers for feedback. Finally, the lessons are tested out in the classroom. The next post is about one of those lessons.

Where are we in the SA F A R I?

I might have gotten a bit ahead of myself in the last post (or perhaps I am building suspense - you decide). Before I introduce you to the Carousel Lesson Design process, let me explain a bit more about the SAFARI prototypes. This should put the framework into a clearer context.

As I said, SAFARI is based on an instructional approach called the workshop model. SAFARI is an acronym for the components of the model [Schema Activation, Focus, Activity, Reflection, and "I want to ..."], and in Swahili it means journey. So in designing a lesson, we are thinking about it as a journey from the known to the new.

One thing that design thinking has shown me, is that this journey also reflects the flair and focus necessary for innovative thinking.

The Schema Activation begins with a flair. Sometimes referred to as the anticipatory set, it ought to be an open-ended invitation for everyone to join the journey. Next, the lesson quite literally Focuses the learners' attention on what to look for during the journey; this might be a "think aloud" in typical workshop lessons. Entering the Activity portion, the lesson once again flairs - allowing each learner room to roam. Here you might encounter what Dr. Jo Boaler refers to as a "low floor, high ceiling" task. After a set amount of time exploring, we refocus by Reflecting on what was important during the activity. We should not wait until everyone is finished before making time to consolidate our thinking. In fact, we want students to cry out for more time; it's what Ellin Keene calls fostering learning lust. Finally, we ask learners to consider what comes next by brainstorming "I want to ..." statements related to the lesson. Perhaps they want to spend more time exploring the task they didn't finish. This represents the final flair until tomorrow when the journey starts all over again.

After explaining the components to my teachers, I give them 10 minutes to develop three different SAFARI lessons related to some topic.

At the end of the 10 minutes, I ask the teachers to share their prototypes with a peer for feedback. The teachers are often resistant to share unfinished products because of the implicit need for perfection usually associated with typical lesson plans. I remind them, however, that they had, on average, three-and-a-third minutes on each prototype. So no one expects their lessons to be perfect.

Afterwards, the teachers express appreciation for the process. Not only have they outlined three possible lessons that they could use in their classrooms, but because the lessons were incomplete their peers were able to add innovative ideas in the blank-spaces. "I wouldn' have thought of doing it that way," one teacher admitted. She continued, "And if I had created a full-blown lesson plan, I don't think [the other teacher] would have been able to see and share this amazing idea."

It's a good reminder that the creative process is often about finding and cultivating the cracks that allow new ideas to grow. So how does this get any better? Here's what you've been waiting for - the Carousel Lesson Design ... in the next post.

Wanna go on a SA F A R I?

Recently, I have been considering how to apply principles from Design Thinking to education. For example, I think creating lesson prototypes instead of lesson plans can help teachers in several ways. 

d.school

1. We make prototypes faster than plans, which means more time for other teachery-things; 
2. Because less time is spent developing a lesson prototype, the teacher is usually less invested in sticking to it when things go poorly;
3. Multiple prototypes are often made, so if the teacher wants to shift gears, there's another idea available; and 
4. A prototype has more blank-space (less teacher control), which leaves more room for interesting interpretations of the lesson by students.

When it came time to develop a framework for the lesson prototype, I returned to a familiar instructional approach: Math Workshop. I wrote about this approach with some GVSU math colleagues in a Mathematics in the Middle School article. Table 1 from the article, shown below, gives a good overview of the Math Workshop Model that we use.

I tend to use "Schema activation" instead of  "Making connections" for the first part of the lesson because of its use in reading workshop - where I was introduced to the approach.

Last year, I shared this method with Kristin Frang. We were working on a project that required designing lessons. I wrote the framework on the board using only the initials:

Kristin said, "If only you had an "I" it could be safari."

Having been on an actual safari (and being a bit obsessive about acronyms), I couldn't believe I had never seen this before. All the framework needed was an "I" to complete the word, safari. 

The "I" became "I want to ..." as a way to recognize the importance of students making their own choice about what to explore in their learning journey. And that's the coolest thing about SAFARI lesson design - in Swahili, safari literally means travel or journey.

With the framework developed, the next thing to do was create a process for using the framework to design the lesson prototypes. I wanted to stay true to Design Thinking and use a process that would foster innovation. So in the next post, we will explore Carousel Lesson Design.

What is the purpose of pre-assessments?

In our course, Probability and Statistics for K-8 Teachers, we are trying to apply Design Thinking to a project involving teaching in a local 6th-grade math class. We will be focusing on 6.SP from the Common Core State Standards. Before we begin planning our lessons, we want to know what students can already do in order to build on their strengths. Therefore, we decided to design a pre-assessment.

We recently went through the Design Thinking process and tested out our pre-assessment ideas with the 6th-grade teacher. Design Thinking is an iterative process, and the feedback we received from the teacher reinforced this idea. It was clear that we had not spent enough time defining the project, which resulted in a lot of disconnected pre-assessment ideas. So tomorrow we will return to the Define step using the Project Priority Puzzle shown below.

Instructions:

• Select a phrase from each row (what, why, how, and when) in the table below that you feel ought to define our 6.SP pre-assessment. If you think a phrase is missing, write it in one of the blank spaces provided; 
• Use scissors to cut out each of your selections, along with the top phrase; and 
• Combine the phrases in order from top to bottom using tape to create your “Define” artifact.

Project Priority Puzzle

You could help us out by providing your definition of pre-assessment. Use the puzzle pieces above or create your own. Please add your definitions to the comments. Thank you in advance for your support of our future teachers.

Are you ready for "something completely different?"

My wife gave me The Complete Monty Python's Flying Circus for my 48th birthday. I just got around to watching the first episode and saw the following sketch. It demonstrates perfectly what happens when we fail to use formative assessment to inform planning and instruction.

It is important to understand that formative assessment does not need to be in the form of a test. We can gather data simply by observing. Imagine how this Italian class might have looked if the teacher had really seen the students during the lesson and adjusted his instruction accordingly. In the Design Thinking methodology, this is a part of the first phase - Empathy.

This reminds me of an article by Dr. Rheta Rubenstein titled, "The Learning of Mathematics: Perspectives of Different Researchers." (It comes from our text, Teaching and Learning Middle Grades Mathematics.) Under the section on Vygotsky, she includes the following diagram attributed to Dr. Lauren Resnick.

Vygotsky in Venn Diagram Form

As Terry Jones's character discovers, when teachers use scripted lessons that do not take learners' current level of understanding into account very little learning actually occurs.