Math 753 • Session 5


While proportional reasoning will factor into some of what we do in the future, we’re shifting ahead to one of the second themes of the course: arithmetic to algebra. Aside from a few of the usual activities (Estimation 180, Visual Patterns, Graphing Stories, etc.) and a few rounds of sharing our own creations in those categories, the major new piece tonight is an AIMS activity written (by Dave Youngs) called The Fascinating Triangle.



Two flavors: PDF, Keynote


Warm-Up 4. Official handbook is here.

Estimation 180

Onward to Days 41-50! I couldn’t resist the Giant Wheel (Day 41) or Little Man Stadel standing in front of a giant tire (Day 46). Here’s a look at the rest of the challenges from 41-50:


After completing two of Mr. Stadel’s challenges, teachers will share the estimation challenges they created for homework.

Course Themes

Arithmetic to algebra (#1) and four representations (#5).

(Turns out I wasn’t a liar!)

The Fascinating Triangle

Three years ago when I first taught this class (when I had even less of a clue as to what I was doing than I do now), I taught half of the content and Dave Youngs taught the other half. During his portion he developed the idea of algebra as generalized arithmetic with the use of a series of activities/explorations. The Fascinating Triangle was one of those activities, and there are more patterns and connections in this little problem than I thought possible.

I love the “topic” listed on the teacher resource page:

Screen Shot 2013-09-24 at 9.59.23 PM

Graphing Stories

As with today’s Estimation 180 challenges, we’ll explore two “official” challenges (Elevation, since nothing beats watching kids endure intense dizziness for the sake of mathematics; Distance from Camera, for the wonderful and stark contrast with Distance from Center of Carousel). Afterwards, teachers will share and discuss their creations in small groups.

Visual Patterns

A second week of sharing our own Visual Patterns, followed by a journey into two (basic) quadratic patterns:



In the weeks ahead, as we transition to course goal #3 (Expressions and Equations) we’ll begin watering the seeds planted by all of these visual patterns to simplify/expand/factor expressions, identify equivalent expressions, and solve equations.

Big Ideas in Algebra

We ran out of time last week and didn’t have an opportunity to discuss the Session 3 reading assignment, or the various comments teachers in the class left on the Session 3 post. We’ll carve out some time for that discussion in Session 5.

Reading Assignment

Calling it a “reading assignment” isn’t entirely accurate. However, you’ll do some reading to get started, and I imagine some of the “missions” will involve reading as well.

Whatever it should be called, the assignment is this:

Step 1
Go to

Step 2
Sign up

Step 3
Get excited!

“Create Your Own” Assignments

Estimation 180 (#2)

Another round of create-your-own (one or two) estimation challenges. Again, they may be inspired by what we’ve done in class, or what you see over at Estimation 180, but they must be your own invention. A few more details:

  • Take a photo of something that can be estimated
  • Use presentation software (Apple Keynote, Google Presentation, or Microsoft PowerPoint) to create a “question” slide and an “answer” slide
  • Make sure the answer is not revealed in the “question” slide
  • Share your slides with me via email no later than 11:59 pm on Monday, September 23

Visual Patterns (#3)

Do it again! That is, create one or two more of your own visual patterns. They may be inspired by what we’ve done in class, or what you see over at Visual Patterns, but they must be your own invention. A few more details:

  • Create a visual for Step 1, Step 2, and Step 3. (you may include Step 4 if you find it helpful or necessary.)
  • Create each step by carefully drawing, using a computer, or taking photographs of patterns you see or build in the physical world. (I would love to see the latter.)
  • Put Steps 1-3 (or 1-4) on a single sheet of paper (physical, digital, or both). Bring at least four physical copies of your visual pattern to class next week.
  • Ideally, your pattern will fit the linear growth theme we’ve explored in the first few sessions.

Math 753 • Session 4

Personal challenge for this post: Brevity. You’ll know by the end if I achieved my goal. 🙂


A few of our “usual” activities,  annotating graphing stories, expansion of the “create your own” theme into new territory, and a fresh attempt at a proportional graphing challenge.



Two flavors: PDF, Keynote


Warm-Up 3. Official handbook is here.

Estimation 180

I was tempted to do some diaper estimation (this is a regular part of my life at home these days), but I opted instead for Red Vines (Day 36 and Day 37). Here’s a look at what we’re skipping:

753 Session 4.012

Course Themes

Ratios and proportional reasoning (#2) and four representations (#5).

I think I won’t be a liar if I promise that we’re moving into new territory next week (probably “arithmetic to algebra”).

Graphing Stories

I’m moving this section of the class forward in the evening so we’re sure to have enough time to discuss. (In recent sessions I’ve run short on time and had to cancel this activity.)

In addition to reviewing the first four graphing stories, we’ll annotate solution graphs using ThingLink (HT: John Stevens). Here’s a sample created (i.e., annotated) by yours truly (image credit goes to Dan Meyer, video credit to Adam Poetzel).

More details on this assignment below.

Desmos Proportional Graphing Challenge v2.0

Heading into a class or a workshop, my excitement level for a particular activity is often directly proportional to the amount of time I spent creating/preparing/tweaking. In line with that model, I was very excited to see how the first version of the Proportional Graphing Challenge would play out.

In a nutshell: It was terrible.

I’ll share my thoughts on how any why things went wrong sometime in the future, but for now I have a second approach that (I hope) deals with some of the weaknesses of v1.0.

One major shift: No more browser-tab juggling. Instead, students will have paper copies of the challenges as they use Desmos to match various graphs.

Here are the handouts: One, TwoThree

Visual Patterns

Last session I assigned our first “Create Your Own” visual pattern. I’m excited to see what everyone brings to class. We’ll do a pair/trade/solve/discuss model for one, two, or three rounds, as time allows.

I’m hopeful that this mixing of ideas will lead to even more creative patterns in future sessions.

Big Ideas in Algebra

Last session’s reading assignment included a handful of blog post responses to the previous week’s reading assignment (Grant Wiggins’ “bashing algebra” post). No new reading assignment for this week, but we’ll carve out a space to discuss how last week’s reading is shaping our own “big ideas” list.

Proportion Play

I originally thought we wouldn’t work through every Running Game challenge as last week’s solving session culminated in a great look at a variety of solution approaches to the second half of the Day 5 Challenge.

variety of solution approaches

However, after trying the Day 7 Challenge with my own students, I’m convinced that there is still some untouched, discussion-rich territory to explore in the upcoming challenges. We may still jump ahead to one of the super challenges in the next couple of weeks, but for now I’m curious to see how teachers attack Day 7 and Day 8.

Reading Assignment

Nothing official this week. However, I invite you to explore one of the blogs you’ve come across in recent weeks if you find yourself with 30 minutes to spare one day this week. Or try one of these amazing blogs:

“Create Your Own” Assignments

Estimation 180 (#1)

Create one or two of your own estimation challenges. They may be inspired by what we’ve done in class, or what you see over at Estimation 180, but they must be your own invention. A few more details:

  • Take a photo of something that can be estimated
  • Use presentation software (Apple Keynote, Google Presentation, or Microsoft PowerPoint) to create a “question” slide and an “answer” slide
  • Make sure the answer is not revealed in the “question” slide
  • Share your slides with me via email no later than 11:59 pm on Monday, September 23

Annotate a Graphing Story (#1, #2)

The first annotation should be completed in class. The steps for each annotation are the same.

Something to do once:

  • Create a ThingLink account (look for “Join Now!” in the top right)

Things to do each time:

  • Select a graphing story from here
  • Watch one of the videos
  • Carefully sketch your solution by hand (consider using pens, colored pencils, and/or markers)
  • Take a photo with a smartphone
  • Upload your image to ThingLink (start with the “Create” button in the top right corner)
  • Add annotations to help tell the story of the graph
  • Share your annotated image with me via email (copy any paste the URL into an email)
  • Here’s the URL of my example:

Visual Patterns (#2)

Create one or two more of your own visual patterns. They may be inspired by what we’ve done in class, or what you see over at Visual Patterns, but they must be your own invention. A few more details:

  • Create a visual for Step 1, Step 2, and Step 3. (you may include Step 4 if you find it helpful or necessary.)
  • Create each step by carefully drawing, using a computer, or taking photographs of patterns you see or build in the physical world. (I would love to see the latter.)
  • Put Steps 1-3 (or 1-4) on a single sheet of paper (physical, digital, or both). Bring at least four physical copies of your visual pattern to class next week.
  • Ideally, your pattern will fit the linear growth theme we’ve explored in the first few sessions.

Math 753 • Session 3

New here? Check out the background to this series before you dive in.


Once again we’ll get things rolling with some MATHCOUNTS problem solving and Estimation 180… er, well… estimation.

For at least one more week the primary focus will be on the following two course themes:

#2: Ratios and Proportional Relationships

#5: Four Representations: Numerical, Graphical, Algebraic, Verbal

We’ll also continue the “Four Big Ideas in Algebra” conversation.



Two flavors: PDF, Keynote


After two weeks of Warm-Ups we’re ready for our first Workout. Still need a copy of the handbook? It’s right here, it’s awesome, and it’s free.

Estimation 180

Nothing says mathematical reasoning like a couple rolls of toilet paper. In Session 3 we’ll work on Day 28 and Day 29 on the site. Here’s a look at the rest of what Days 21-30 have to offer:

Days 21-30

Four Big Ideas in Algebra

We’ll spend a few minutes discussing our reactions to the Session 2 reading assignment (a Grant Wiggins blog post), including any affirmation, surprises, disagreement, new insights, or new questions that came up in the course of reading the post and its comments, and writing a comment of our own.

For now, we’ll leave the larger “Four Big Ideas” conversation alone until a later session.

The Running Game

I’ve tweaked the new handout a bit based on a comment from the class in the Session 2 Feedback Form. Here’s the latest and greatest version.

Our scheduled challenges: Day 5 and Day 6

Visual Patterns

No time for a new style of visual patterns (we’ll continue pressing forward in a week or two). For now I want to camp on linear growth a bit longer. Instead of giving a few new challenges from the website (or of my own invention), I have a “create your own” assignment for each member of the class. Details below.

Automatic Change Dispenser

On my way through the checkout at a local supermarket, I saw this:

Act 1

I think the MTBoS is sharpening (or further twisting?) my brain, because I instantly wondered…

What’s the total value of those coins?

So I asked this on Twitter:

I’ll add a full length blog post about how my thoughts grew from “Hey, this is a cool estimation task” to “Whoa, this could be a pretty sweet three act task if I don’t blow the presentation (like I usually do).” For now, check out the resources I’ve posted for this task over at 101qs.

Desmos Proportion Graphing Challenges

Earlier this year Dan Anderson, Justin Lanier, and I launched a website called Daily Desmos. Each day we (or someone else from our awesome and growing authoring team) creates one basic (er, well, not so basic) and one advanced graphing challenge. They’re fun to make and fun to solve.

(For more background, go here. For details about what’s next for Daily Desmos, go here and here.)

We recently had several discussions about how to make Desmos more useful to classroom teachers. One idea: Create a series of related challenges, intentional in sequence, progressing from simple to more challenging, and in doing so provide students with a sandbox for developing their graphing skills in an enjoyable, dynamic, exploratory (yet still somewhat structured) environment. (Sorry about that last sentence; it got a bit out of control.)

At any rate, because one of the major themes in Math 753 is proportional reasoning, and because we’ve been discussing creating a sequence of linear graphing challenges, I created a sequence of proportional graphing challenges (some might use the term direct variation). I’m waiting for feedback on the quality (or lackthereof) of this sequence. Once we’re happy with the quality and format, we’ll turn our attention to a linear sequence, then (probably) quadratic, and so on.


Graphing Stories

Since we ran short on time in the first two sessions, I want to revisit the first four stories in Session 3. Soon we’ll continue moving forward, and soon after that we’ll have an assignment where each of us (myself included) has to create our own 15 second graphing story (or two).

Reading Assignment

We’ll continue with our “Big Ideas in Algebra” discussion by reading a few responses to Grant Wiggins’ original post. Read the following (including all comments):

Your task (if you’re in Math 753, or following along at home):

  1. Read the four posts above, including all of the comments.
  2. Spend at least 24 hours with the ideas bouncing around your brain.
  3. Add your own voice to the conversation by posting a thoughtful comment, either on one of the blogs above, or on this post.
  4. Things to discuss might include (a) your latest list of four big ideas in algebra, (b) ways in which your list is being reshaped as a result of reading several more perspectives, (c) new questions or confusion you have about the big ideas in algebra, (d) new clarity you have about anything related to this discussion, and (e) anything else that comes to mind as a result of the reading assignment.

Visual Patterns Assignment #1

Create one or two of your own visual patterns. They may be inspired by what we’ve done in class, or what you see over at Visual Patterns, but they must be your own invention. A few more details:

  • Create a visual for Step 1, Step 2, and Step 3. (you may include Step 4 if you find it helpful or necessary.)
  • Create each step by carefully drawing, using a computer, or taking photographs of patterns you see or build in the physical world. (I would love to see the latter.)
  • Put Steps 1-3 (or 1-4) on a single sheet of paper (physical, digital, or both). Bring at least four physical copies of your visual pattern to class next week.
  • Ideally, your pattern will fit the linear growth theme we’ve explored in the first few sessions.

Math 753 • Session 2

If you’re just joining us, check out the background to this whole experiment, as well as the Session 1 post.


We’ll begin with some problem solving and estimation to warm up our brains (complements of MATHCOUNTS and Estimation 180). The majority of our activities in this session will focus on the following two course themes:

#2: Ratios and Proportional Relationships)

#5: Four Representations: Numerical, Graphical, Algebraic, Verbal

Additionally, we’ll begin digging into the CCSSM Standards for Mathematical Content (what I’ll refer to from now on as the “CCSSM Content Standards”).



If you want ’em, get ’em here: PDF, Keynote


We got our wheels turning by working through problems from Warm-Up 2. Need the handbook? Get it here.

Estimation 180

I’ve really enjoyed working through every challenge in my fourth period class this year (we’re on Day 16 on the 16th day of school!). We don’t have enough sessions to do the same thing in Math 753, but I want the teachers to have a sense of the types of challenges we’re skipping over. I’ve provided a two-slide preview of the Day 1-10 and Day 11-20 challenges in the hopes that they’ll be drawn back to them later (either on their own or with their students).

753 Session 2.005 753 Session 2.013

Our challenges for Session 2: Day 13 and Day 14

CCSSM Content Standards

Our first real venture into the content standards for CCSSM. After briefly discussing the Four Big Ideas in Algebra conversation that started with Grant Wiggins’ 100th blog post, teachers will work on this:

753 Session 2.029

The Running Game

I’m excited to bring the next pair of Running Game challenges to the class, partially because the challenges increase slightly in difficulty with each pair of days, but also (and primarily) because I have a shiny new handout.

Our scheduled challenges: Day 3 and Day 4

Visual Patterns

In the first session we explored several proportional relationships. (Check the Session 1 slides for specifics.) In this second session we’ll branch out to look at patterns involving a steady rate of increase with a slight shift away from simple multiples. For example, instead of 3, 6, 9, 12, etc., we might look at 4, 7, 10, 13, etc.

If that makes no sense, check out the Session 2 slides.

Soda Fountain Task, v2.0

In last week’s session I introduced a half-baked task based on caloric content of beverages at the In N Out soda fountain. The task was mediocre, but the context (in my opinion) had some merit. With that in mind, I’ve revamped the task. The focus now is on making connections among multiple representations.

The slide deck contains a few potentially useful images, but the real goods are here:

Students will work in small groups (2 to 4, ideally) to cut out and then match the various representations contained in each packet of “ingredients.”

Graphing Stories

Water volume (by Esteban Diaz-Ibarra) and Distance from center of carousel (by Adam Poetzel).

Compliments of course to Dan Meyer and BuzzMath for the excellent resource.

Reading Assignment

Grant Wiggins (author of Understanding by Design) recently started a conversation, in his 100th blog post, no less, about the big ideas in algebra. The key passage:

Here is a thought experiment: can you identify 4 big ideas in algebra, ideas that not only provide a powerful set of intellectual priorities for the course but that have rich connections to other fields? Doubt it. Because algebra courses, as designed, have no big ideas, as taught, just a list of topics. Look at any textbook: each chapter is just a new tool. There is no throughline to the course nor are their priority ideas that recur and go deeper, by design. In fact, no problems ever require work from many chapters simultaneously, just learning and being quizzed on each topic – a telling sign.

Your task (if you’re in Math 753, or following along at home):

  1. Read the post and all of the comments. (Get a beverage and a snack ready; there are quite a few.)
  2. Spend at least 24 hours with the ideas jostling around in your brain.
  3. Add your own voice to the conversation by posting a comment, either on Wiggins blog, or here.
  4. Things to discuss might include (a) your own list of 4 big ideas, (b) ways in which your list is being reshaped as a result of our class and the discussion started by Grant Wiggins, (c) questions you’ve always had about the big ideas in algebra, (d) questions you never knew you had until now, and (e) anything else that comes to mind as a result of the reading assignment.

Bonus task (this kind of bonus, not the “points” kind):

  • Never used Twitter? Get your toes wet by exploring Grant Wiggins’ timeline. Keep your eyes out for new threads and clarifying comments in the “big ideas in algebra” conversation.
  • Don’t worry if you get distracted and fall down a few unrelated rabbit holes. Part of the beauty in the conversations on Twitter is that you can find millions of different topics, discussed at varying levels of intensity, and they’re often just a click or two away.

Math 753 • Session 1

For background, this.


Introductions, establishing course routines, discussing course themes, first round of recurring tasks/problems, proportional reasoning, graphing as storytelling, MTBoS shout-out.


Slide Deck

In recent workshops and classes I’ve been experimenting will pulling more and more of the resources I’ll using during a given session into a Keynote slide deck. It’s pretty easy to navigate around the Internet, but it’s even easier to progress through a set of slides. Also, preparing the slide deck in this semi-comprehensive way forces me to think more thoroughly through transitions from one activity to another and connections between our tasks each evening.

At any rate, the latest effort in this experiment is here, in two formats: PDF, Keynote

Math Counts

We’ll use MathCounts as a resource for our problem solving sets this semester. General information on this national middle school mathematics competition is here. Direct access to this year’s handbook (which contains over 300 problems) is here. (Note: I pulled our first problem solving set from the 2012-2013 handbook, but the new handbook is now available, along with a fresh site redesign.)

Estimation 180

For extended commentary on why I think Estimation 180 is awesome, check out my guest post over at Andrew Stadel’s Estimation 180 site. (For further evidence of my love for Estimation 180—imitation, sincerity, flattery, etc.—check out The Running Game and Proportion Play (details below).

Common Core State Standards for Mathematics (CCSSM)

We’ll dig into the common core standards more next session.

The Running Game

A series of bite-sized proportional reasoning challenges, with some new additions since we talked last week. More details coming later, but I think I’m done with “Running Game” challenges. In the near future I hope to add a second set of challenges, “Partial Produce.”

And I almost forgot! I have a new draft for the Running Game handout.

Visual Patterns

I love Fawn Nguyen’s Visual Patterns website. Her blog and Twitter are worth your time as well.

First, to bring Visual Patterns into my classroom, I’ve created a series of introductory challenges. They begin with simply proportional patterns, move into linear, then quadratic, advanced quadratic, and finally oblong and triangular numbers. I’m finding that with this set of sequenced training patterns in hand I can lead students of varying experience, age, and ability into the world of visual patterns. My ultimate goal is to turn students loose on the full set of challenges available on Visual Patterns, and with a few carefully chosen (or newly created) patterns I can do just that.

Second, I’ve tweaked Fawn’s excellent handout to provide students with more room to draw, sketch, etc., and I’ve added a fifth task (working backward from a given number of items to find an unknown step number). My handout is here in three formats: PDF, Pages, Word

(The file was made using Apple’s Pages, so I make no promises that the Word file preserved the formatting perfectly.)

In N Out Soda Fountain Task

In class we worked through an early version of this task, based on this photo taken a few days ago at an In N Out soda fountain. I know there’s a worthwhile task on proportional reasoning in there, and you can see my rough thoughts by looking through the slide deck. I expect I’ll revise this task later in the course (based in part on comments and brainstorming from our first session) and bring a new version back for further discussion.

For now, have a look at the slide deck to review what we explored.


It’s free, it’s amazing. Did I mention it’s amazing? Oh, and it keeps getter better.

  • Head over to the Main page
  • Click through to the calculator
  • Create a free account (you might want a Google Account while you’re at it)
  • Experiment and explore!
  • Not sure where to begin? Just start by clicking on everything you see.
  • If you’d prefer a more structured introduction, download Quick Start Guide or the full length User Guide

Graphing Stories

Our fifth theme for the course (if you take my list in the slide deck over FPU’s list in the course catalog) will be exploring algebra through (and making connections between) four representations:

  • Numerical
  • Graphical
  • Algebraic
  • Verbal

We’ll explore graphs as data rich, contextualized storytellers, and Graphing Stories will serve us well in that effort.

Reason and Wonder

If you’re reading this, you’ve already made your way to my blog. Have a look around!

Reading Assignment

Nothing this week.

Math 753 Background

In 2010 I team taught a class in the grad math/science program at Fresno Pacific University. My teaching partner (Dave Youngs) was more a mentor than a colleague at that point, as I had only recently finished my journey through the masters in education program at FPU. I benefitted greatly from the opportunity to work side by side with someone who knew the ropes, and I enjoyed that sort of partnership (first with Dave, and later with another mentor, Richard Thiessen) for three or four semesters. Afterwards, I tried my hand flying solo through a couple of courses.

When my wife and I had twins in November 2012, I took a break from the adjunct instructor gig. The girls are almost a year old now (and diapers are expensive!) so with my wife’s full blessing and encouragement I’m back in the classroom. This semester I’m teaching Math 753, Concepts in Algebra, to a small (but amazing) group of teachers whose positions range second grade to middle school. Our first session was this past Wednesday (August 28).

This is actually the same course I taught in my first semester as an adjunct instructor, working side by side with Dave, but I feel like it’s brand new for two reasons. (1) I’m no longer team teaching. All of Dave’s expertise is now an email or phone call away, rather than right there in the room while I’m teaching. (2) In the three years since I first taught this course, my philosophy and practice (as a teacher of adults, as well as a middle and high school teacher of mathematics) have shifted more than a little bit.

Why Post on the Blog?

“Great Mike, thanks for sharing. Um… why are you sharing this?”

After each class session I’ll post a brief summary or reflection, a small collection of links to resources used or discussed in the class, and—more often than not—a reading assignment (in the form of links to articles and/or blog posts). My purpose for posting these sessions (and hence this background) is threefold.

  1. I want the participants in the class to dip their toes into the mathtwitterblogosphere. I could easily share resources with my students another way (Moodle, Dropbox, Edmodo, Piazza, etc.) but by posting them here, I’m hoping to use my blog to draw them into reading more widely and exploring more deeply the strange and amazing community that I discovered back in March.
  2. By making the course goings-on somewhat public, I’m motivated to design a better course than I might if everything we did in Math 753 remained hidden in our own little corner of the world. It’s not that I would just phone it in, but in sharing publicly I’m putting a bit more pressure on myself to create an even more meaningful course.
  3. In the off chance that someone not in the class is interested in exploring what we explore… Well, have at it. 🙂


These will all be dead links until the actual sessions have occurred (and the post-session writeup post has been written and posted), but eventually easy access to each session post will be found below.

Session 1 (August 28, 2013)

Session 2 (September 4, 2013)

Session 3 (September 11, 2013)

Session 4 (September 18, 2013)

Session 5 (September 25, 2013)

Session 6 (October 2, 2013)

Session 7 (October 9, 2013)

Session 8 (October 16, 2013)

Session 9 (October 23, 2013)

Session 10 (October 30, 2013)

Session 11 (November 6, 2013)

Session 12 (November 13, 2013)

Session 13 (November 20, 2013)

Session 14 (November 27, 2013)

Session 15 (December 4, 2013)

Session 16 (December 11, 2013)


I’ll close the comments for this post, but leave them open for the individual sessions (partially because I will share the course description/goals in the Session 1 post). If you have thoughts on how to make this experiment more useful to anyone involved, please share them. If you have recommendations for our reading assignments (articles, blog posts, books, Twitter chats, etc.), let me know. In the session posts, of course. 🙂

Transition to Integrated Course Sequence with CCSSM

This is the sort of post that in years past I would have scribbled in some word processor or private blog. I often write to clarify my thinking and set personal and professional goals, and I’ll do so again here. My reasoning for making this round of reflecting and planning public is twofold:

  1. It will force me to consider my assumptions, goals, and specific game plan even more carefully knowing that others might read what I write.
  2. It may help others process through their own transition, whether to an integrated course sequence or a more traditional slicing-and-dicing of the CCSSM content standards.

Whether the second of these two reasons will actually play out, I don’t know. But the value I’ll derive from the first (reflecting and planning publicly) is reason enough to proceed, so here goes.

My Schedule

Our school has seven academic periods during the normal 8 am to 3 pm day. A full teaching load is six classes, plus one period to prepare. In recent years I’ve elected to work during my prep for a slight pay increase (diapers are expensive!).

This year my schedule is filled with five classes and two periods we’re calling “Program Development.” During these two periods (2nd and 7th) my task is to redesign our 7-12 mathematics program to align with the CCSSM content and practice standards. That’s a lot of planning time each day, but it’s a fairly monumental task, especially considering that we’re transitioning to an integrated course sequence for grades 9-12.

My Assumptions

This could get out of hand (lengthwise) rather quickly, so I’ll jump right in with the bullets to share some of my assumptions:

  • An integrated course sequence in grades 9-12 will be more difficult to design and more difficult to teach, but (if done well) will provide students with a richer, more connected mathematical experience (provided I don’t settle for what @NatBanting describes here).
  • Due to the small size of our school and the constraints on budget and staffing (there are two faculty members—myself included—in the entire 7-12 math department), we need to make the transition to CCSSM content in grades 7-12 all at once. (In other words, we don’t have the staffing necessary to transition one course/grade level at a time, or to make the transition gradually over a number of years, essentially running two programs side by side in the interim.)

I’m calling these assumptions because that’s what they are, at least in part. It might be better to call them semi-researched opinions/positions. In any case, I hope some of you will push back and play devil’s advocate, especially on the second point above. If you think it would make more sense (in my small school environment) to roll out the transition one, two, or three courses/grade levels at a time, please share!

My Goals

By June 2014 I want our course offerings to include:

  • Integrated Math A
    (CCSSM Grade 7 content standards)
  • Integrated Math B
    (CCSSM Grade 8 content standards)
  • Integrated Math 1
    (CCSSM high school content standards)
  • Integrated Math 2
    (CCSSM high school content standards)
  • Integrated Math 2 Honors
    (CCSSM high school content standards, including STEM (+) standards)
  • Integrated Math 3
    (CCSSM high school content standards)
  • Integrated Math 3 Honors
    (CCSSM high school content standards, including STEM (+) standards)
  • AP Calculus AB
    (aligned to the College Board’s AP Calculus Course Description)
  • AP Statistics
    (aligned to the College Board’s AP Statistics Course Description)

Two notes:

  • Students who intend to take AP Calculus AB will be required to complete Math 2H and Math 3H (where, theoretically, they’ll learn the STEM (+) standards and other topics necessary for success in Calculus)
  • I’m not sure if we’ll offer a Math 1H course. If we do, it probably won’t include any of the STEM (+) standards, and I’m currently running short on ideas for how to differentiate it from the non-honors section of Math 1.

While aligning our courses to the CCSSM content standards will be an important task, I consider it even more crucial that we infuse all of our 7-12 courses with the eight Standards for Mathematical Practice. I want our courses to help students grow in their ability to make sense and persevere, reason, argue and critique, model with mathematics, etc. The content itself is important, but it’s the habits of mind that will last.

My Game Plan

It’s rather easy for my to become overwhelmed by the magnitude of this whole undertaking. But I also get incredibly excited when I think about chipping away at specific tasks in transforming our program, my courses, my teaching, etc.

With those two ideas in mind, I believe it will be helpful to break down the entire project into a sequence of smaller, more manageable tasks. In theory, this will keep me sane, on track, and encouraged. (We’ll see whether that’s the case.)

I also hope that by planning in this way it will be easier to share resources with others (in both a “give” and “take” sense), and that I’ll have more opportunities to collaborate. For example, if I ask on Twitter, “Who wants to help me develop a CCSSM-aligned course sequence for grades 7-12 with integrated courses for high school,” I’ll probably hear nothing but crickets. However, if instead I ask, “Who wants to help me create a concepts and skills list for, say, an integrated course for Grade 9,” I might have a few more takers.

So with that background, here is my plan of action, laid out more or less in the order I’ll proceed:

Curriculum (Draw the Big Picture)

  • Arrange the high school standards into courses (whether that involves adopting something like this as is, using or modifying California’s integrated pathway (see pages 95-123 of this document), or starting from scratch, I don’t yet know)
  • Identify the three or four “big ideas” in each course (and later, develop six- to 12-week units around them)

Note: I see myself reading more of this blog and these books in the near future.

Assessments (Set the Targets)

  • Develop performance task assessments for each of these units (emphasizing “synthesis skills”)
  • Write a “concepts and skills list” for each course (possibly by using these as a starting point)
  • Develop assessments for each of the items on the “concepts and skills list” (ideally, assessments worth posting here)

Lessons (Work Out the Details)

  • Create a list of individual topics (based on the “concepts and skills” list) for each “big idea” unit
  • Select, adapt, or create a rich task to launch each “big idea” unit (one that we can refer back to throughout the unit)
  • Sketch a rough outline of individual lessons for each topic
  • Write individual lessons for each topic (this should only take, roughly, forever)
  • Select, adapt, or create appropriate homework assignments for each lesson (though I probably should read this—currently sitting at my bedside table—before forging ahead)

That’s All for Now

If you need to tackle any of those smaller projects and you’d like to join forces for a bit (whether we collaborate through Dropbox, Google Drive, Hangouts, or some other tool), I’d love to have some help and/or lend a hand with your transition.

Drop me a line in the comments, or send me a note on Twitter (@mjfenton) if you’re interested.

Rich Math Tasks

Thursday evening I asked Frank Noschese (@fnoschese) and Elizabeth (@cheesemonkeysf) a question about whether they thought there was any value in students using a reductive, drill-and-kill math practice exercise platform, provided that it was accompanied by rich tasks and assessments in a classroom that demands synthesis and critical thinking, and provides students with opportunities to develop mathematical habits of mind.

I received a reply, but I’m more interested in a tangential question @NatBanting asked in the middle of the exchange:

I’ve been throwing that phrase (“rich tasks”) around a lot more lately, in tweets, blog posts, workshops, and conversations at school and at home (my wife is either amazingly longsuffering or genuinely interested—maybe both?—when it comes to talking about math education). So what exactly is a “rich task”? What do other people have in mind when they use the phrase? What do I mean when I use it?

I told Nat that I’m still working that out in my own mind, but I think it’s time to clarify what I mean, if for no other reason than to have a better sense of what I’m looking for when I try to find, adapt, and/or create rich tasks for my own students.

Here’s What I Do Mean

A rich mathematical task is one that…

  1. Has a low floor and a high ceiling. The first of many ideas I’ve stolen from others, this one from so many sources I don’t even know who to credit anymore. Most recently, Dan Meyer has me thinking about this in his Makeover Monday series. The bottom line: everyone can start, no one can truly say they’ve exhausted the problem’s potential (at least not in a 50-minute period).
  2. Has multiple entry points, invites use of multiple representations. Student A starts by exploring numerically, Student B begins by investigating graphically, Student C jumps in by reasoning algebraically, and I don’t have to tell two of them that their approach is a dead end because—even if they don’t always make it—there is fruitful territory a little further down the path in any of their approaches.
  3. Has multiple solution paths, provides opportunity for rich discussion. If there’s only one way to solve the task students lose out on the rich discussion of making connections between various approaches and teachers lose the opportunity to build a mathematically coherent, concrete-to-abstract storyline as they orchestrate these discussions.
  4. Integrates multiple topics. I owe a lot of what I’m thinking here to a single word Daniel Schneider used in a post about assessment. After my initial foray into standards based grading left me dissatisfied with an overly fractured curriculum, I’m now placing a high priority (philosophically, at least) on tasks and assessments that bring multiple topics together. A rich task, in my estimation, should demand that students wrestle with multiple topics from multiple domains (if I can use the term in the CCSSM sense).
  5. Engages student interest, is mathematically/cognitively challenging. I’m a little mixed up here, because I believe engaging students’ interest is massively important, but I’m not ready to throw away tasks that fail to generate buzz among students if I know they nevertheless provide great opportunities for exploration and discussion.

Here’s What I Don’t Mean

To clarify what I think a rich task is, I’ll share a few thoughts on what I think a rich task is not:

  1. A well-crafted, but constrained guided-discovery activity. I value this kind of activity, but when students are guided along a specific path to a specific goal, it pushes the lesson into a different category for me.
  2. A thoughtfully constructed lecture or an engaging presentation. If the instructor is doing the heavy lifting during class, I would say the students are not engaged in a rich mathematical task. I’m not opposed to heavy lifting, especially in preparation outside of class, but students need to play an active, central role in exploring/solving/reporting if I’m going to use the “rich task” label for an activity.
  3. A challenging problem for which students already have a tried-and-true method. I have a large stack of started-but-not-finished books, some related to math and education, others not. George Polya’s How To Solve It is on the list, though I’ve read enough of it to be provoked and inspired, particularly by the distinction Polya provides between a problem (solution/method not known) and an exercise (solution/method already known).

I’ll Close with a Link…

Incidentally, after typing out the post, I Googled “what is a rich mathematical task” and found this.

…And an Invitation

I would love to hear what you think are the key characteristics of a rich mathematical task. Drop a line in the comments or send me a line on Twitter.

A New Kind of First Day

Today marked the first day of the 2013-2014 school year for me. It was, all in all, a wonderful day. Teaching at a relatively small K-12 school, I have an opportunity to work with some students for two, three, or even four years in their 7-12 grade experience. Seeing students who I’ve come to know so well over the past few years walk through my classroom door on the first day of the new year is a tremendous blessing, and it’s always fun to see how they have matured over the summer months in their transition from 7th to 8th, middle school to high school, or maybe even junior year to senior year.

The Highlight

I won’t bore you with the details, but the highlight of my day came in watching my AP Calculus AB roster (which appeared to have only 6 students when I checked online late last week) grow to 11 students by the end of the school day on Monday. This included three unexpected, last-minute additions, and I’m thrilled to have these students join our small but amazing group. (For reference, I typically have 10 to 15 students in Calculus, and our graduating class has been around 50 students in recent years.) I’m excited and honored that these students (all 11, really) have decided to challenge themselves with another year of math, and have even sacrificed other classes they were interested in to make it possible. Plus I think we’ll have a blast this year.

The Struggle Continues, Intensifies

Despite all the good vibes, I struggled through some parts of today. I’ve worked relentlessly over the past nine years to create as strong a math program as I could imagine at the school. If my today-self could time travel (with a USB flash drive or a link to a Dropbox folder) to meet, say, my 2006-self, that older version might think something along the lines of, “Wow! Most of what I’ve imagined for the math department is a reality. That’s super swell.”

Here’s the problem, though, and I have many of you to blame for it: The ceiling on what I can imagine has been blown off, and I’m now confronted (in the middle of nearly every class) by a dozen or so notions of how the lesson I’m smack-dab in the middle of could be improved, become less terrible, etc. These thoughts don’t even allow me the grace of finishing an example or an activity; they jump right to the front of my brain even as I’m presenting/discussing/guiding.

So after a few months of relatively stress-free hanging out in the MTBoS, I see the tension between what I believe should happen in the classroom and what I’m currently doing (the struggle between my philosophy and my practice) not only resuming, but (rapidly) intensifying.

Determined to Grow

While the MTBoS is largely to blame for this increased tension, it’s also likely to provide the inspiration for much of the personal growth I’ll experience in the coming months. It may be exceptionally frustrating at times, but I’m determined to make this struggle (my first full year of MTBoS-fueled, classroom-based struggling) exceedingly productive.

Goals for 2013-2014

I spent the summer alternating between tuning in to and checking out from Twitter and blogs, including my own. Inservice at my school begins in two days, and students will arrive August 12. As the summer winds down I’ve started thinking about my blog-related goals for the school year. This will be my first full year hanging out in the #MTBoS, so I want to be intentional about the ways I engage, particularly in how I use this blog to grow as a teacher.

These are subject to change throughout the year (and possibly even during the course of this post), but right now my framework for getting better has three five categories:

  • Implementing
  • Creating and Sharing
  • Reflecting
  • Collaborating
  • Transitioning

I’ll share a few goals in each category, not only to let others know what’s bouncing around my head as the school year begins, but also to force myself to organize my own thoughts and build in some personal accountability by leaving a paper (er, page) trail.


In the few months I’ve spent engaging with other math teachers through reading blogs and following conversations on Twitter, I’ve been exposed to a wealth of rich mathematical tasks for the classroom. If the members of the #MTBoS are a chorus of angels on one shoulder, urging me to break out of my direct instruction-heavy approach to incorporate more rich problems and tasks, then my own experiences as a student, my initial teaching style, my tendencies toward control and perfectionism, and my at-times overwhelmingly-varied course load (typically four to six different preps) are a drove of naysaying demons on the other shoulder.

However, thanks to what I’m learning in Smith and Stein’s Five Practices (and reading on blogs), I’m gradually working up the nerve to take what I hope will be major strides this year. I’m finding the thoughtful intentionality of the five practices reassuring, as I’ve always feared relinquishing control of the mathematical flow of my classroom and assumed (incorrectly, I now believe) that this was a necessary part of implementing tasks and fostering student solution-centered mathematical discussion. As I try and fail and tweak and try again and retweak and find some measure of success along the way, I’ll reflect on these experiences here on the blog.

If you’ve had rich, engaging mathematical tasks on your “maybe later” list for a while, join me this year in making concrete plans to include them in your classroom on a regular basis. I hope you’ll reflect on your own experience, preferably by blogging about it, or at the very least by leaving the occasional comment on this blog. Speaking of concrete plans (and practicing what one preaches), here’s my goal:

One rich task each month in each course

Given my teaching schedule next year, that means 40 tasks. Excuse my while I go hyperventilate.

Okay, I’m back. And while I’m a bit freaked out by the prospect of shifting a core part of my teaching approach, I’m also excited about these 40 opportunities for growing in my craft next year. I think I’ll keep some paper bags in my desk at school, just so I’m prepared.

Creating and Sharing

The #MTBoS is full of amazing people. I regularly feel out of my league in this diversely awesome group, particularly in two categories: (1) thoughtfulness, completeness, and coherence in educational philosophy and (2) relentlessness in creating (and sharing) amazing resources. This year I want to shift my interaction with this community from primarily receiving to a combination of receiving and sharing. Not only out of a sense of gratitude for all of the excellent things others have made that I’ve enjoyed, but also because I think my quality as a teacher will grow through the practice of creating, sharing, receiving feedback, revising, etc.

As the school year begins, I’ll turn the lights back on at the Better Assessments blog. I have big plans for September (more on that later). I will also begin organizing a series of proportional reasoning challenges (tentatively titled The Running Game) so they’re available to other teachers as I create them throughout the school year for my own students. My 101qs radar will remain up, and I’ll try my hand at a few more Three Act tasks. This is distinct from my first goal since many of the bumps on the “creating and sharing” road will stem from my imperfect tasks (rather than imperfect implementation). At any rate, I’ll blog about both experiences here, and hopefully grow as a teacher through the process.


I suppose most of my posts in the coming year could be filed under this category. But it’s worth mentioning separately. When I started the blog earlier this year and saw a spot for a subtitle, I picked better through reflection. I want to get better at this teaching gig, and I know that reflection is a key means to that end. Over the years I’ve tinkered with different approaches to reflection, but nothing I’ve tried has been as helpful as working through my thoughts in a public forum. Knowing that someone else may read a post in which I reflect on the effectiveness of, say, my approach to homework leads me to be that much more thorough in my self-examination. I look forward to continuing more of the same this school year.


My first real #MTBoS buzz came from collaborating with Justin Lanier and Dan Anderson on Daily Desmos. Inspired by a tweet from Dan, I suggested a daily match-my-graph project. Several days later we had a head of steam, an unofficial endorsement from Desmos, and a growing team of collaborators. The entire Daily Desmos experience has been one of my favorite thus far in my #MTBoS tenure, probably because my involvement on this project fits more on the “give” side of the give-and-take scale.

While this next project has stalled (I’ll blame myself and summertime), I’m excited to kick start Better Assessments back into action. I hope others are interested in joining the conversation, but even if I have to fly solo for a while, I plan to forge ahead with some ambitious Algebra 1 assessment makeovers in September.


I’m looking for other ways to collaborate next year, particularly as I transition our department from the old Mathematics Content Standards for California to CCSSM. I’ll have some prep time set aside specifically for working through this transition (designing our courses, creating and curating tasks, developing SBG and performance assessments, etc.), and I’m hopeful that by collaborating with others on various projects (asynchronously, I assume) I’ll be able to multiply my own productivity and serve other teachers, school, etc., with the materials I/we create.

Join In, or Hold Me To It

So there you have it. My goals for the upcoming year. I invite you to join me in thinking about how to make the most of next year by writing your own Goals for 2013-2014 post, or by dropping a line (or link) in the comments. And I certainly hope people will hold me accountable now and again by asking how things (e.g., The Running Game, the Better Assessments blog, etc.) are coming along.