The Lawnmower Games


Mow a huge lawn. Earn $75.

Knock down clothes line; shred grandmother’s robe. Pay $30.

For a short while this summer, I’ve been working with a rising ninth-grader, Hat (not his real name), establishing and repairing his foundational math concepts and skills. For one of our episodes, I taught operations with negative numbers using an idea that I’d left undeveloped since I’d thought about it almost a year ago (during the New Blogger Initiation festival!). At the time, I wasn’t sure about Casino Zero’s working mechanism, mostly because I didn’t need one. Now I needed one.

Hat is an entrepreneurial number rocket with access to a riding lawn mower, making semi-mad stacks of cash this season. A simple story and board-game model provided all the control I needed. Here’s the set-up:

  1. The Game is played for pretend money but not with pretend money; it’s played with value chips.
  2. Chips are in two places: the World has chips (in the center of the game board) and the Player has chips.
  3. The Game starts with the World and the Player both having no money, but in the form of zero value, that is, as equal amounts of positive and negative value chips.
  4. Chip denominations are 5 and 25. (I would have also used 10 if I’d had a fourth color.) Those denominations come in both positive and negative, too. So, chips are +5, -5, +25, and -25, with positive and negative denoting increasing or decreasing wealth. That kind of “direction” of monetary value is not so strange.
  5. Positive and negative were used to denote one other direction: the flow of value. I suppressed the urge to talk about magnitude and vectors, but I did define positive flow as meaning “in the direction from World to Player” and negative flow as “in the direction from Player to World.” Hat agrees that this language captures the idea of chips either coming to him (getting paid) or going away from him (having to pay).
  6. The Game is played by rolling a die, moving a marker, and either earning money (a wealth-increasing event) or paying money (a wealth decreasing event), with plenty of ridiculousness along the way. Obviously, this game is easily adaptable to any story you want to make.
  7. Hat keeps a ledger with columns labeled Direction (In/Out), Number of Chips, Chip Value, Change in Wealth, and Total Wealth. He writes an expression to describe what happens for each play, then simplifies it to see the result. For example, you earn $75, so your ledger looks like +, 3, +25, +75, (new sum): +3\bullet +25=+75 . If you had to buy gas, your ledger might look like -, 8, +5, -40, (new sum): -8\bullet +5=-40 . An important supporting piece here is that the result of the operations must match the change in wealth, which is obvious because the game tile says what that change is and also because you can see your rising and falling fortunes for yourself in your pile of chips.  (Hat sees that having equal stacks of opposite chips means having zero value and zero dollars.) With each event, Hat and I look at the outcome of the operations: (+) • (+) = (+), (–) • (+) = (–), and so on, building some sense about the “sign rules” for multiplication.
  8. Finally, the big mathematical moment happens! The World owes you $30 but it has none to give because there are no more +chips in the middle of the board. What can you do? If you go to the cashier right now and cash in your chips (for the pretend money), what will your wealth be? What about the wealth that the World owes you? What can you do with the chips in your hand that will increase the wealth you have in your hand, and so, the money you get from the cashier?
  9. Give away negative chips? Really? What would that look like? So, -, 6, -5, +30, or -6\bullet -5=+30 . Now, when you go to the cashier with fewer negative chips, you’ll receive more money; your wealth has increased: (–) • (–) = (+).

We only had time for one game and never got into some of the more interesting combinations of operations that might be required to create the necessary change in wealth. And for sure, not only would this have been more fun and more kindly chaotic with 3 or 4 players playing at 5 or 6 tables, but Hat and his game-mates would also have had some conversation, argument (I hope), and other social-learning energy to help drive the lesson forward. Having concrete experiences that illustrate abstract ideas helps secure those ideas. Even semi-authentic, not-exactly-real-world experiences can help students create cognitive pegs to hang math concepts on.

To add more support to the idea of negative factors making positive products, I drew Hat into Chris Adams’s great response to Dan Meyer’s call-for-tweets about A Negative Times a Negative. Hat’s gears audibly clicked into place when he played that movie in his head.


If you try a Casino Zero type game with your students, ping back how it went and add your ideas for making it better.



Peace Of Mind To Try Your Hardest

Climbing. Issue 315. May 2013. p. 72.

Climbing. Issue 315. May 2013. p. 72.

…is a safe way to start a route, especially if the first moves are difficult…Plus, it will give you peace of mind to try your hardest.

Rock climbers are taught safe ways to climb because, once reassured of their safety, they can more easily look for, and find, the focus, strength, and perseverance they need to move ahead and exceed the limits of their skill and knowledge. Hmm. This goal has a familiar ring to it. What will I do to make sure my students feel safe enough to risk blazing trails into places where they thought they couldn’t go? Watch them. Talk to them. Know them. Govern them well. It all starts there.

Bad Governor

M is an eighth grade, special ed student whom I support (as an ed tech) in two general ed classrooms. He’s easily distracted and does not readily accomplish his goals. But M loves his computer and he loves his Minecraft. Seen that?
The students are watching a movie that parallels the book they’re reading, They’re on the look-out for the usual suspects: comparisons, contrasts, where the movie and text meet, where they don’t. M is on his machine, angled away from sight, almost certainly not taking notes. I confirm this: his not taking notes is certain. Minecraft is in effect.
This is not a freakishly rare event. I speak to M often about this sort of machine mismanagement. But this time I said nothing, just closed the lid, took it off his desk, and put in in the resource room where he could have it next period.
Losing machine privileges for a while was a legit consequence (M is cut a lot of slack but he is not supposed to be blatantly Minecrafting during a lesson). My taking the machine away was a legit action. Nothing here is technically wrong. But everything here is wrong. I should not have wordlessly taken his machine, or anything else. That was appallingly bad governance.
Good governance should inspire self-governance. Good governors should lead instead of drive. The difference is a matter of respect for those being governed: my students. Good governance guides students to their best behavior instead of reducing their chances to behave better.

Dalai Lama for Governor

I could have spoken instead of mutely exercising my power: I have so much more power than you, I don’t even have to say a word. Groan. I could have first spoken with M: What’s going on? Who knows what I might have found out or where I would have seen to go next? Speaking and seeing; those were two senses that I shut down.
I could have explained to M what he needed to accomplish and how, if his machine was getting in the way, if it was too big of a rock to get around that day, that I could have held it for him until he needed it. I could have said something that would have squared M up with his responsibilities instead of stripping away all hopes for him having his own own control of the situation. As a last resort, if necessary, I could have respectfully asked M to give me his machine. My silence was not golden, nowhere near it, more like iron.
I checked in with my supervisor, who understood my unhappiness and said, maybe as balm, that it might have been good for M to have had that experience, to have been reminded that he does not have the freedom to act in ways he knows are wrong.
Meh. I should not have done it. Now I know, and the next time with anyone, anywhere, will be better. Good governors figure things out. They make new mistakes, not old ones.

More on SBGs Four

My district’s committee is working hard to corral the wild mustangs of Performance Based Education and Standards Based Grading, or Reporting. I have a lot of questions, but this new perspective on the 4 helps a lot. This example comes straight from a workshop led by Tom Lafavore that makes extensive use of Marzano’s work as well as that of the Maine PBE cohort schools, I believe.

For me, this example of an assessment task helps to clarify a lot about what SBG is and isn’t, what those numbers (2, 3, 4) actually mean in terms of instructional goals, and what we are gauging when we assess. Here’s the example (I’m not sure whose work this is):


Grade 8


Measurement Topic: Literary Comparisons and Source Material

Learning Goal: Understands how to use details from grade level text to compare and contrast.

2 Task 

  • Describe the following terms: Use a Venn diagram to compare the similarities and differences between fictional and informational text.
    • Compare
    • Contrast
    • Differences
    • Categories
    • Characteristics
    • Synopsis

3 Task

  • Find two opposing views in the newspaper or through other media sources. Using the detailed graphic organizer, identify the similarities and differences between the two points of view. Write a brief synopsis of what you discovered.

4 Task

  • View selected scenes from The Wizard of Oz and The Wiz and use details from each to compare and contrast each director’s purpose.


Why I like this example:

  1. It connects Thinking Skills (Marzano and Kendall’s New Taxonomy) directly to learning targets and assessment tasks.
  2. It shows how a 4 Task can be specifically designed to address higher order cognitive functions, or “utilization of knowledge.” That means that teachers could create opportunities for students to earn a bona fide 4 rather than waiting for them to come up with an inspired act of inference all by themselves, as I somewhat oddly suggested earlier. (I have even heard the suggestion that successfully demonstrating at least one 4 Task performance could be a requirement for graduation. Interesting.)
  3. It reminds me that the way I tried to use SBG as an intern, which was how I saw my mentors using it, which was: carefully writing a very good rubric and then figuring out some distribution system, e.g. all correct or just 1 wrong = 4; most correct = 3; some correct = 2; few or none correct = 1, that this method, although it seemed like a good idea, isn’t really SBG at all. Using that method, I ignored the hierarchy of thinking skills and just smeared all of them, along with the standards, over one assessment “surface” and then resorted to the same, old, percentage-based scoring system to determine a grade, except that I had restricted the results to only 4, 3, 2, or 1.

Playing Pong ON Downtown Philly

Pong in the City

Pong in the City.

  1. Pong has been enlarged to be played on the side of the Cira Centre in Philadelphia. If the controller was enlarged as much as the video screen, how big would the controller be?
  2. Find a picture of something in the world that is as big as the controller would be and create a Photoshop file that shows it along with the Cira Centre building as if it was really there and you had taken that picture.

SBG: Less Than or Equal To Four

I’m trying to see what I know about SBG, a 3, and a 4.

Suppose I’m teaching the Pythagorean Theorem to eighth graders. [8.G.B.6.]

After unpacking the standard and identifying the learning targets and thinking skills, I design an assessment that can demonstrate to me, and to my students themselves, that this standard has been met. Let’s say I have decided to use a performance assessment.

Student P does very well. P has produced a play with hand puppets. Awesome job. Excellent understanding. Nailed the performance assessment rubric. Standard met. P scores a 3.

Student Q also does a great job, also has employed hand puppets. Q has used a 3D printer to produce the puppet costumes. Look closely and you can see that the costumes’ surface designs incorporate Fibonacci numbers. Q asks why his score is a 3 and not a 4. I explain that Q’s efforts have done nothing to explore in some deep, creative way, the concept of the Pythagorean Theorem. Q has spent a lot of time making wonderful icing. Q has met the standard, which is excellent and sufficient, but that’s all. Furthermore, I say that I have heard that Q’s Arab Spring project has been in need of substantial revision, and I ask Q if the decision to spend time making clever costumes that in no way addresses the learning target was, in hindsight, a good one (now working those metacognitive skills).

But wait. Students R and S (who both have scored a 3 for this standard) come to me and tell me that they want to write, or are writing, an entertaining stage play that will explain, or at  least show, the Pythagorean Theorem to the 6th graders in the building. By the time the curtain goes up during a special assembly, R and S have been advised by both the drama coach and me. Also, students T, U, and V have joined the project. They have written original music (flute, bass, and bongos might be nice) for the performance, which lasts 20 minutes. The sixth graders howl with delight, some of them are drafted into the play as performers, and they all leave the show with vivid notions about a powerful mathematical idea. Students R and S have each earned a 4 for this standard. The musicians might also have earned 3’s or 4’s for a music composition standard.

This is way more than icing. This is reaching deep into a concept, figuring out how it works, taking it apart, putting it back together for a specific audience, and then communicating with that audience. [I don’t see this as exceeding the standard so much as exploring the standard in greater depth.] Schools should acknowledge that this kind of creative energy is valued in the world, that societies depend for their vitality on there being people who want to make fires like this, that creativity brightens the world. A 4 is just one kind of acknowledgment, but it’s also the kind that colleges can see easily during a quick scan of a student’s record. When schools openly and physically celebrate this kind of creativity, that is, not merely as a number in a data field, other students may be inspired to reach high and go deep towards their own creative adventures. How great would it be if every graduating HS student had a 4 somewhere in their data field?

Mistakes? Yes!

[I’m writing reflections on my student teaching, still.]

…But I should put more faith in my judgment of the kinds of lessons that can be created by following these sorts of provocative leads. If I had judged poorly, then I would have learned something about teaching mathematics. If I had judged well, then my students would come to expect that unusual but productive forays into thinking were the usual practice in our community. In either case, I would have demonstrated that mathematics is alive and that we, as students of mathematics, are willing to take risks, to challenge our assumptions, and to live a little in our math classroom. That means: we can and do make mistakes. This is not a new idea. It should be our practice to make mistakes as often as necessary, but no more, and not the same ones as before.