Thought experiment: combine Algebra 1 and Physical Education

(Part of the reason I started this blog is so I’d have a place to play around with ideas, no matter how non-field-tested they may be. Consider this one of my many half-baked ideas that I haven’t fully thought through.) 

One of the hallmarks of a New Tech Network school – the network of schools to which I am happily attached contractually and emotionally, and spent part of my teaching career teaching at – are teaching using a Project Based Learning approach within combined courses: World Studies and English, Biology and Literature, and so forth. The first math class I ever really enjoyed taking was my combined Physics and Calculus class my HS senior year.

While I’m not suggesting that mathematics is impossible to combine with other courses, it is often fraught with peril. When we were starting out our journey as a New Tech school, the Science teacher and I splayed out our content standards on the table to see around which we could build projects around. We had a couple ideas for projects, but that would have left over half of our content standards either not combined in a project, or combined in contrived and unnatural ways. Often many of the math standards don’t play well with others.

Moreover, in a PBL classroom, it’s easy for math standards and skills to get dwarfed by the project’s product itself. That was part of my discomfort with PBL and began experimenting with what we now call Problem-Based Learning. It’s doubly easy for math standards to get dwarfed by the lab report, the prettiness of the art exhibit.

That said, I do think students learn the content better when it’s connected to other content. I got more out of my Calculus class by chucking things off the roof and bouncing tennis balls and seeing that the acceleration and the derivative of the speed magically matched. How do we reconcile the value in connecting math content to other physical, tangible subjects while maintaining fidelity to mathematical standards and quality pedagogy?

Here’s a class I’ve never seen implemented (at least, not implemented the way it exists in my head): combined Algebra 1 and Physical Education. That’s right the nerds and the jocks, hanging out together! The more I think about it, the more I like it – again, with the full disclosure that I’ve never seen it taught, never taught it myself, and haven’t even totally thought it through. I’m not sure I’d even consider this half-baked. This is a more 1/8th baked idea.

Still, here’s what I like about it:

The tasks themselves. The content can play pretty well together. I’ve created a couple of tasks just my little old self around physical fitness, and I’m not terribly fintessy. The tasks could either be directly about a student’s physical fitness or about sports and fitness at large. This allows for long term data tracking and regression. Even standards that don’t seem to play well with physical fitness still have physical fitness-like applications (like, say, quadratics … or… quadratics).

maiwwage has started making their SportsVU data public and it’s changing the way the game is played. Slow and fast people are running at the same time and it’s on video. Teams aren’t punting anymore. There’s fitness equipment to be constructed. There are NFL plays to be scripted.

For the PBL-practicing Physical Education teacher, this may hopefully push you beyond the “make a new sport” or “teach other kids sports” projects.

Seriously, why are we letting all this precious data from PE go to waste?

The way you could structure your weeks.

Another nice side-benefit of a combined course is that they are largely double blocked, giving you a full hour and a half or so a day. Seems to me a weekly schedule could look something like this.

Monday: do something physical that gives you data (and some math practice after cool-down, now that the brain has oxygen and blood and stuff)

Tuesday: do something mathematical with that data

Wednesday: do something physical that gives you more data (and some math practice after cool-down)

Thursday: do more mathy things with that data

Friday: spend 45 minute “maxing out” (or whatever) on that data-producing physical activity. Spend 45 minutes analyzing performance

Or just go halvsies the entire week and plot the progress of the students in whatever physical activity they’re doing.

Reduction of status issues in the math classroom

This might also be fraught. I mean, the only place that creates and supports status issues than a math classroom is a physical education classroom, right? On the flip side, it might allow students who are perceived to be low-level achievers in math to finally take the lead. You might get the athletes wanting the “smart kid” on their team, in their group.

Preparing the Brain for Cognitively Demanding Tasks. Physical activity makes for great pre-work for creative mathematical problem solving or a nice interruption from cognitively demanding tasks. Physical activity releases all sorts of good chemical stuff that make you more productive, more creative, more engaged, less stressed and presumable, more capable of taking on cognitively demanding work. It’s also a really nice way to break up an otherwise plodding work day. Here’s an example of a school that is trying to keep kids’ heart rates up for the sole purpose of preparing kids’ brains for learning (hat tip: @JimPa23). Even if the math task has no relation to the PE task, I’d rather have a bunch of kids who have just been exercising than kids who have just come back from the Taco Bell Express in the lunchroom.


There are also lots of problems to address with such a mash-up. How much time would a class spend changing in and out of their gear? Would combining math and PE compound, rather than equalize status issues for some students? Do the facilitators have a similar vision for the class and what the kids should be getting out of it?

I also know that there are a ton of data in exercise to make good math tasks and physical might help prepare students’ brains for the cognitive ask that complex problems require.

What do you guys think? Is this feasible, or just one of those ideas that should stay in the “fun-to-think-about” realm?

Equalizing Practice and Assessment (Part 2): What You Value Should Be What You Assess

Have you told your students how much you value honest attempts at solutions to a problem? Even incorrect solutions? Then you have to assess this way.

You can’t tell students that you value their incorrect attempts at solutions when you take off points when they get an answer wrong. Worse, you can’t say you value the process over the solution when you assess using a multiple choice or short answer examination. Students are too smart and they will see right through that facade, as well they should.

“Mr. Krall, you say you want us to be persistent problem solvers and you value our mathematical thinking, but you still took off half-credit for my solution attempt.”

“And you said the highest I can make on my re-test is a 70.”

“One of those new age feel gooderies.”

I’m not suggesting an “everyone gets an A” or a “crocodile in Spelling” method of assessment, but just that one needs to put a grade where their mouth is. 

Similar to Dan’s “What Do You Worship?” question, I’d ask what do you, the facilitator, value? Value, both in an ethereal “boy I sure would like this!” way and a “yes, this is what you will be assessed on” way.


In the Why/How/What framework, “why” has been addressed all over the blogosphere (but here’s a thing), “how” was partially addressed in my previous assessment post. As for “what”:

You will be assessed on your growth. You will be assessed on your persistence. You will be assessed on your various methods of solutions. You will be assessed on your communication.

You will not be assessed on the correctness of your answer. You will not be assessed based on the boxed number on the right side of the page. You will not be assessed using rote tasks that are easily solvable using a formula chart. 

Also, this goes beyond “I allow retakes.” Retakes is a way of saying that students have one more chance to get it right (usually accompanied by a significant numerical penalty). It’s not penalizing a student for a wrong answer whatsoever. Or at least honoring the solution attempt that isn’t actually a penalty in disguise (i.e. “partial credit”). This is a huge assessment shift, and requires a more sophisticated assessment tool than an answer key can provide (such as these).

It’s really difficult to switch gears like this in the middle of the year though. There’s a certain foundational work that needs to happen first. And frankly, it’ll probably take a few rounds of assessment before students even believe you. You’re probably not the first person to say they value honest – if incorrect – solution attempts, only to turn around and dock students in the name of “well, the SAT doesn’t allow redo’s”.

Getting Better: I can improve anything for students, but I can’t improve that

I can get better at almost everything. You can get better at your practice, regardless of your teaching style. I know I often come across as dogmatic with regards to

Figure 1

Problem-Based Learning (see Fig. 1), but really, it’s all about steady improvement, irregardless of your teaching style. My personal preference is inquiry and complex task oriented groupwork 100% of the time (even if I fall short), but yours might be different. You can get better at it. You can improve it. 

Like to do inquiry learning? You can improve that.

Like to utilize real-world tasks? You can get better at it.

Like to do Project-Based Learning? 3 Acts? AnyQs? You can improve at that. And you can improve the stuff provided to students: better projects, more compelling videos and pictures,

Like to use a textbook? There are ways of improving its use.

Shoot, like to do worksheets? I know that’s allegedly a bad word but man, some of the worksheets – YES, WORKSHEETS – that Sam (@samjshah) and Jeff (@devaron3) have put together put most PrBL lessons to shame (or are included in PrBL-ish lessons!).

Whatever you find compelling, you can get better at. And, you can mix-and-match, depending on the day/week/content area.


Except math instructional software.

I’ve always had a problem with instructional software and I think I’ve found the root cause: you can’t make it better or adjust it to your students’ interests or curiosities. Sure, you can adjust it according to their needs, most, like ALEKS, Cognitive Tutor, and Khan Academy can be adaptive to do that for you, but not according to students’ interest or curiosities. You can’t change on the fly. And districts spend so much on this software, or invest in so much PD in this software, that you, the teacher, kind of have to use it. Maybe this isn’t news to you but to me – who has had the pleasure of working with teachers who use such software expertly – it was an “aha” moment.

I’m not suggesting these tools have no use. But that their use is quite limited by nature. ALEKS can determine and teach a lot of things about and to a student, but it can’t determine what the student finds compelling about math. And that’s kind of the whole ballgame.

Equalizing practice and assessment

I’ve made it a habit to retweet this once a month or so from Jenn (@DataDiva) who I look up to as a leader in the field of teacher- and student-friendly assessment.

Citation: Martin-Kniep, G. & Picone-Zocchia, J. (2009) Changing the Way You Teach: Improving the Way Students Learn.

I retweet it because it’s a good reminder and, hey, it’s easy to miss in the never-ending scrawl of twitter. It’s so crucially important that it’s one of those things that should be shouted from the rooftops (on a regular basis, apparently). (PS: anyone know how to get rid of my dumb tweet? I already tried unchecking “remove parent tweet” but to no success.)

One of the side-benefits of transitioning to an inquiry-based, problem-based classroom is that you can slowly start to scrap those old entire-class-day-killing tests. Ideally, once you’re humming along, the Assessment Problems and Problems for Learning will be largely indistinguishable.


It took me a while to realize the power of this. It wasn’t until my final year of teaching that I have a single task to students for their final exam. Students worked on groups and developed a presentation on how to solve a particular complex task; it was assessed with a rubric, which was exactly how the class was structured throughout the year.

However, I’ll describe one thing I didn’t do that is crucial, but I need to get in to some rubric weeds.

There ought to be two sections for most assessment tools:


One thing I did not do throughout my classes that represents a huge gap in my practice was assessing against common standards of quality (“super-standards” is a term that I just made up that I need to sit with before I start using). I strictly assessed students against the particular content that was being taught at the time. “Demonstrated how this diagram proves Pythagorean’s Theorem? Great! PROFICIENT.” “Failed to simplify the quadratic into its simplest form? DEVELOPING.” What was missing was tracking growth in particular mathematical proficiencies over time. More generalized mathematical proficiencies such as “Developing a model”, “Using mathematical literary conventions”, “Representing scenarios in multple ways” that are ubiquitous across most worthwhile problems. Think Bryan’s Habits of a Mathematician. Shoot, think Common Core Standards of Mathematical Practice. By using indicators that lie outside the realm of the particular content addressed in a problem, students can demonstrate growth over time, and learn what it is to be a mathematician (and probably better articulate it).

Here’s an example of what I’m talking about: the top row is specific to this particular problem, the succeeding rows are to be assessed periodically throughout a course.

But this brings us back to equalizing the assessment and instruction. If these are the things you assess, then these are the things you have to teach. And it has to be ongoing.

Also be sure to check out Raymond’s analysis of Shepherd’s The Role of Assessment in a Learning Culture (2000). From which, I’m going to straight up crib his block quote:

“Good assessment tasks are interchangeable
with good instructional tasks.”

When to scaffold, if at all

It’s been a while since I’ve revisited the Taxonomy of Problems I threw together a while back, but I think it’ll be helpful to spend some time there when considering the following Most-Wanted question around Problem-Based Learning:

At what point after allowing the students to work on a problem do I scaffold the content knowledge?

It’s probably important to identify exactly what type of problem you’re implementing before deciding this.


One of the reasons I wanted to think about this as a potential framework is to address scaffolding (I’ve already addressed assessment). It might not be perfect or precise, but here’s what I basically envisioned.

taxonomy w scaffolding

Unintentionally, this kind of mirrors the ideal progression of both a PrBL Unit as well a classroom and high school experience.

So once you’ve figured out where you are on the taxonomy, where you are in the unit, you can think about your scaffolding.

What & When

I’ll toss out a couple broad-brush rules that oughtn’t be universally applied.

When you’re at the left end of the spectrum – the Content Learning Problems, I’d suggest the following.

If the need for the content is germane to the problem, intervene relatively quickly and with the entire class.

If the need us for an ancillary concept or “side-topic”, consider holding back and/or offering small, differentiated workshops. 

For example, I threw Dan’s Taco Cart task into my unit on Linear Equations.

However, use of the Pythagorean Theroem is required to develop your linear equations to model. There will no doubt be a need for some – probably not all – students to revisit or relearn the Pythagorean Theorem. That is ancillary content knowledge: essential, but not the targeted content knowledge skill. Consider holding off on scaffolding that – another groupmate might be the better vessel to explain the concept. Or, if you deem yourself the ideal vessel, consider jigsawing that concept or holding a small pullout workshop with one groupmember per group (the groups’ “student-teacher liaison” as it were).

If the knowledge is germane and is the targeted content knowledge of the task, the scaffolding might need to be more prescriptive, more whole-group. You certainly could lecture (Grant Wiggins has an exceptional post on that), but you could also offer one of these scaffolding tasks. I’m a huge fan of manipulatives and students evaluating student work samples.

Ah, but when do you offer that scaffolding? How much productive struggle should we allow students before intervening? This is where teaching is more of an art than a science. Although if it is truly germane to the problem and it’s a Content Learning problem, I’d err on the side of quick-intervention. Twenty minutes after a problem is launched, perhaps? Thirty?

More important than a time demarcation for instruction is probably some classroom behavioral evidence. Here’s a short list of things to look for to initiate INSTRUCTION MODE:

  • Over half the groups or students asking the same or similar thing
  • Loss of cognitive demand in the attempted solutions
  • Attempted solutions going totally off the rails

What have I missed? What are some indicators that it’s time for you to intervene with scaffolding? Or do you have a particular system or time-frame when considering when to cease the productive struggle time?


If your problem is more to the right on that arrow above – Exploratory or Conceptual Understanding problems – the question might not be “what and when” to scaffold but “if”. There is inherent value in an unscaffolded, nonroutine, “ill-structured” problem with a lugubrious associated standard. For these problems consider restricting yourself solely to small workshops devoted to ancillary content knowledge. Or perhaps follow up the problem with a standalone scaffolding task – perhaps, again, a manipulative or evaluation of work samples. Scaffolding for Assessment problems should focus on revision and peer-editing.

The tension between inquiry and instruction shifts from day-to-day, problem-to-problem, so I wouldn’t hold anyone to a hard-and-fast rule. I hope you’ve appreciated my self-indulgence as I continue to try to figure this out and establish a few basic tenets of solid PrBL practice. As always, feedback and commentary is appreciated.