Is bad context worse than no context?

In elementary classes we consider it a good thing to be able to move from the abstract to the concrete. We ask students to count and perform arithmetic on objects, even contrived ones. We ask students to group socks, slice pizzas, and describe snowballs. A critical person might suggest these are all examples of pseudo-context, and they’d be right! These are more-or-less contrived scenarios that don’t really require the context to get at the math involved. Why do we provide such seemingly inessential context? I’m venturing a little far away from my area of expertise here, but I’d guess it’s because it helps kids understand the math concept to have a concrete model of that concept in their heads.

My question is this: in secondary classrooms, is there inherent value in linking an abstract concept to an actualized context? Even if the context is contrived?

I mean, yeah that’s bad. Comically and tragically bad. It doesn’t do anything to enhance understanding. I’d say the context actually hinders understanding. The thickness of an ice sculpture dragon’s wing? That’s about three bridges too far.

But what about a slightly less convoluted, but also-contrived, example. Say:


This problem is still most certainly contrived: dimensions of tanks aren’t often given in terms of x. I’m not even suggesting this problem will engender immediate, massive engagement, but it might help students create a mental model of what’s going on with a third degree polynomial. Or at least the context allows students to affix understanding of the x- and y-axes once they create the graph of the volume of this tank.


We provide similarly pseudo-contextual in elementary classrooms in order to enhance understanding of arithmetic and geometry.

From Burns’ “About Teaching Mathematics”

Of course, we also compliment such problems with manipulatives, games, play, and discourse, which secondary math classrooms often lack. In the best elementary classrooms we don’t just provide students with that single task. We provide others, in addition to the pure abstract tasks such as puzzles or number talks.

Perhaps the true sin of pseudo-context is that it can be the prevailing task model, rather than one tool in a teacher’s task toolbox. In secondary math classrooms pseudo-contextual problems are offered as the motivation for the math, instead of exercises to create models and nothing more.

(See also: Michael’s blog post on Context and Modeling)

Critiquing the Common Core on its Merits and Demerits

Screen Shot 2014-11-19 at 10.48.01 PM

Criticism of the Common Core State Standards (CCSS) has sadly devolved into theater, when it and schools would benefit from critical analysis. CCSS criticism is all-too-often hyperbolic while CCSS defense delves in dismissal of concerns or even ridicule. That’s a shame because CCSS could use a critical eye: one that understands the standards as an educator and is able to negotiate the good and the bad. A good-faith critique as it were. That’s what I aim to do here as an educator, a parent, and an instructional coach.

Before we get into it, I suppose I should give a full disclosure of all my work-related comings and goings, because that’s apparently a thing that gets called into question these days: I generally support Common Core. I’m a former math teacher (so I naturally gravitate toward critiquing CCSS-Math) who became employed in my current position starting with a grant awarded by the Bill & Melinda Gates Foundation to professionally develop teachers toward CCSS implementation. I’m still employed at the same non-profit, but no longer under that or any grant.

Here are four things I think about CCSS.

1)   National standards are basically a good thing, but they do pave the way for mass-assessment.

The concern about horizontal and vertical alignment is real. Pro-CCSS folks often point to student mobility from state-to-state as a reason to have nationalized standards, but I’m not even sure you need to go that far. I taught in a district that wasn’t aligned from school-to-school. It would have been nice to have a clear playbook of standards that we were all working from so I knew roughly where kids were (or should be) from day 1.

However, a nationalized set of standards makes it really easy to test and develop tests. While No Child Left Behind (NCLB) was the genesis of national high-stakes testing, a common set of standards may well accelerate it. A nationalized set of standards will make it such that an environment where School X is compared with School Z is inevitable. While I’m a big fan of data generally, that kind of cookie-cutter analysis is troublesome. Even if next generation assessments are “better” (as is alleged, whatever it means), the impetus to benchmark students like crazy will be there.

 It’s also true that killing the CCSS won’t end the over-benchmarking of students via standardized test. Neither will scrapping NCLB.

2)   The standards are generally better than current state standards.

I had a conversation this weekend with a Scientist and kindergarten Teacher. We wound up talking a bit about Common Core. The Scientist was mentioning that he saw one of those Facebook posts where the parent shares a confusing worksheet and then it goes viral and then that’s supposed to be evidence that Common Core is dumb. The Scientist, however, said “I saw the worksheet and was like ‘that’s how I do arithmetic in my head.’ The Teacher was a fan of conceptual understanding, promoted in CCSS in a way that until recently was oft absent in state standards.

Conceptual understanding of numbers and number-sense is crucial for (among other reasons) future Algebraic understanding. CCSS attempt to get at that. However, it leaves many parents – even educated parents – frustrated. Within the past few weeks I’ve had to google “Story Mat”, “Base 10 Drawings”, and “bar model” – which aren’t even in the Common Core Math standards, but rather, idiosyncratic terms developed by curriculum publishers – to help my daughter with her math homework, and I’m allegedly some sort of math expert. I’ll admit it’s frustrating, and there will be a gap between those of use that learned procedurally and those that are learning conceptually. Still, the ability to break apart numbers and recombine them is an essential mathematical skill.

Moreover, state standards are often kind of a mess. They can be a mish-mash of best-intentions, over-prescriptive, lengthy, poorly-aligned, and not terribly well thought-out or research-based standards. Sometimes they look like the worst of things that were invented by committees. I can primarily speak to the context I taught in (Texas), but I’ll say that CCSS-M are fewer, cleaner and simply better standards than the ones I had to wrestle with. There’s an emphasis on reasoning and conceptual understanding that wasn’t there in the previous generations’ standards.

It’s interesting that Indiana, which opted out of CCSS, has adopted standards that look conspicuously like CCSS. It’s one reason that I’m optimistic that even if CCSS becomes so politically toxic that all states abandon it, it will still have been for the greater good. The folks actually in charge of standards and standards-writing generally see the good that CCSS has to offer.

3)   Common Core has had an awful rollout strategy and has been accompanied by virtually non-existent training.

The Teacher in the aforementioned conversation was a fan of Common Core, but did describe that many of her colleagues were struggling to teach math conceptually rather than procedurally. That’s 100% understandable given the means of CCSS rollout, which wasn’t much of a rollout at all.

I can’t say exactly what the “correct” rollout would have looked like, but it wouldn’t have been this. Teachers are often left to interpret and teach the new standards on their own. There’s a gap between how teachers (and you and I) learned (or didn’t learn) math and how teachers are expected to teach. Almost every teacher working today was trained in a decidedly non-CCSS pedagogical environment.

While that’s understandable in any seismic shift in education standards, what’s inexcusable is the lack of time and resources devoted to professionally develop teachers, particularly at the federal level. Race To The Top (RTTT) is kind of ridiculous as an avenue to professionally develop teachers: “show us that you can demonstrate proficiency in Common Core and then we’ll give you money to develop teachers to teach using Common Core State Standards.”

What’s worse is that many states and districts are tying teacher pay and employment to success on standardized assessment. And they’re doing it now, instead of after a few years of trial! I’ll be honest, if I knew my employment was tied to my students being successful on a math assessment, I’d probably “play it safe” and try to push as much algorithmic instruction as possible as a temporary band-aid rather than try a new avenue of fighting for conceptual understanding. So there may even be a misalignment between the current instruction and the current standards.

There have been disparate tools here and there to help teachers out, but no nationalized training or systemic interpretation. It’s been largely grant-based which is, by nature, sporadic and not systemic. Pro-CCSS folks like to chortle at the vitriol directed toward Bill Gates for awarding CCSS-related grants, but grants as a mechanism to drive systemic and ubiquitous change is a sketchy proposition.

But once again, my optimism shines through: now that math education programs and teacher training programs actually have standards (good ones!), they’ll hopefully start being able to prepare teachers properly. There will certainly be a lag time.

4)   There are legit concerns about the appropriateness of grade-specific domains

I’m uncomfortable suggesting that “Every student should know how to do X by the end of first grade.” Kids do come in at very different levels. What’s confusing about CCSS (Math) is that after Grade 8, they do away with grade-specific standards and give general domains such as “High School: Interpreting Functions” and “High School: Number and Quantity”. It’s as if after 8th grade, suddenly students and schools have the agency to figure it out on their own.

I appreciate having those benchmarks of what students “should” know by the end of each grade. However, the consequences of students not being able to demonstrate proficiency on those standards – particularly in the early grades – can be disruptive. And while Pro-CCSS folks would argue that we need to separate the standards from the assessments of those standards, the assessments and consequences are a natural outcrop of nationalized standards. One naturally follows the other. And I’ve no idea how to alleviate those consequences. Districts, States, and the DOE will not simply afford more resources to schools with students that fail to meet those standards. They’ll shut them down. Common Core State Standards is part of a system that potentially greases the skids for school closures and community disruption. These disruptions are essentially mandated in NCLB as federal law, before the CCSS existed. My fear is that CCSS will be used as the tool that NCLB uses to disrupt communities.

It’s also not fair to pin blame on the standards themselves. The goal was to develop a set of national, easy-to-follow, research-based, appropriate standards that would ensure students would build toward conceptual understanding of mathematics and problem solving, and I believe they achieved that goal.


I’m not terribly optimistic about the long-term sustainability of CCSS as a national set of standards. Steve Leinwand once said that if Common Core becomes political, it’s dead in the water. It’s certainly political now (even if it doesn’t really move the needle electorally). I am optimistic that the folks in charge of evaluating and writing standards, such as those in State Departments of Education, have tended to see the importance of conceptual understanding, among other things.

I’m hopeful that 10 years from now either A) my concerns and the concerns of others will have been addressed or B) the residual of the failed-implementation of CCSS remains embedded in state-level standards. Either way, it’s about time we have a conversation about Common Core that is based in actual teacher input and student outcomes.

(I’m happy with comments on this post with the intention of continuing conversation. But c’mon, hysterical comments have no chance of getting published.)


Thanks to Christopher, Tracy, and Mike for their feedback on this post.

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.

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.

Inheriting the wind; these are two of my favorite books about math


Baseball Prospectus

I never liked baseball as a kid. Maybe it’s because I wasn’t any good at it. Maybe it’s cause I never went to a professional game. Maybe it’s because it is quite boring when you watch it on TV. Then in the late 90’s the sabermetric revolution upended the stuck-in-the-50’s baseball establishment by using data to prove and disprove various myths that were pervasive in the game. From roster construction to in-game tactics, the sabermetric community was one or two steps ahead of the rest of the game. It was this data-driven analysis that served as my entry point into the game. Eventually the data-movement coalesced under the Baseball Prospectus name. Housed at, the writers produce an annual that is my notice that Spring has arrived.


The annual contains copious amounts of raw numbers, advanced metrics, data tables, projections, as well as an approachable and easy-to-digest writing style that I blast through every March. This year was no different. This is one of my favorite books to read every year.


Here we have another book grounded in mathematics: Paul Lockhart’s Measurement. Here we have a rich text of mathematical creativity and imagination. In fact, pretty much everything in the book is developed in the author and reader’s imagination.


The problems posed (some of which even have Lockhart’s proof to accompany them), are decidedly abstract in nature. The problems rely on ingenuity for a solution. Lockhart is more likely to use mental shape-folding than a two-column proof to describe a mathematical concept, let alone a spreadsheet of data. This is one of my favorite books.

Both of these books (in Baseball Prospectus’ case, the annual publication) are quite dear to me. They also represent two entry paths through mathematics. One uses messy numbers and data to explain why things are the way they are. The other uses clean, imaginary shapes to explain why things are the way they are in our imagination. They both feature clever, humorous, conversational writing including analogies and storytelling.

Inherit the Wind

Then there is the final scene of Inherit the Wind. After psuedo-successfully defending a high school teacher who dared to teach evolution in the classroom, the protagonist of the true-story, defense attorney Henry Drummond picks up the Bible in one hand, Darwin’s The Descent of Man in the other, and exits the courtroom. Both of these books reveal things about the nature of man, and you’d be a fool to entirely discard either.

There are context-rich ways of posing mathematical tasks. There are entirely abstract ways of posing mathematical tasks. There are interesting and engaging ways of posing problems. There are dry and uninteresting ways of posing math problems.

I hated math in high school. All the way up until my integrated Physics/Calculus class threw an old computer off the football stadium and recorded it on video and we were able to successfully approximate the gravitational constant. I’m not sure if I had been presented abstract math in more interesting ways I’d have latched on to it. I was a pretty detached kid. It’s possible, but I was also more prone to think about mid-90’s Indians baseball and Carlos Baerga’s VORP. That would have been a way for me to engage with mathematics in High School. In fact, I did engage in it when Rob Neyer would publish a new column. But that was me.

I’m not saying that every task needs to be grounded in the real world. But the number of contextual mathematical tasks that should be provided any given year is certainly greater than zero. It’s probably greater than 10. Maybe the tasks need not even be that authentic. I mean, is throwing a computer off a building a “real-world” situation? Or was it just a fun thing to do that we then did a bunch of math on? It was “real-world” in the sense that we saw it happen. We interacted with it physically, visually, inter-personally  and mentally. I have not, to this day, had a professional reason to toss a computer off a stadium.

Mathematics is so wonderful and ubiquitous that anyone can have some sort of entry point into the subject. It’s too vast to be constrained to a single context or a single person’s imagination. Our access points to the subject change from person to person, from age to age. You’d be foolish to eschew context-dependent scenarios to explore mathematics. You’d be foolish to toss aside all imagination from the content. You’d be foolish not to explore any and all avenues of all ways to provide access to this remarkable subject to your diverse students. You’d be inheriting the wind.


The Struggle for Productive Struggle


This NPR radio spot confirms much of what we already know about struggle. There’s so much good stuff in this report, I’d encourage you to go listen to it or read it. Here are a couple nuggets I found particularly illuminating (emphasis mine).


For example, Stigler says, in the Japanese classrooms that he’s studied, teachers consciously design tasks that are slightly beyond the capabilities of the students they teach, so the students can actually experience struggling with something just outside their reach. Then, once the task is mastered, the teachers actively point out that the student was able to accomplish it through hard work and struggle.

“And I just think that especially in schools, we don’t create enough of those experiences, and then we don’t point them out clearly enough.”

“We did a study many years ago with first-grade students,” he tells me. “We decided to go out and give the students an impossible math problem to work on, and then we would measure how long they worked on it before they gave up.”

The American students “worked on it less than 30 seconds on average and then they basically looked at us and said, ‘We haven’t had this,’ ” he says.

But the Japanese students worked for the entire hour on the impossible problem. “And finally we had to stop the session because the hour was up. And then we had to debrief them and say, ‘Oh, that was not a possible problem; that was an impossible problem!’ and they looked at us like, ‘What kind of animals are we?’ ” Stigler recalls.

“Think about that [kind of behavior] spread over a lifetime,” he says. “That’s a big difference.”

Great, teachers say, I know students learn from struggling through a complex problem, but my kids won’t do that. They won’t participate in productive struggle. They don’t come pre-loaded with productive struggle software. Therefore, the thinking goes, I need to address the kids where they’re at in terms of willing to struggle through a task.

Talk to any math teacher, and one of their frustrations will no doubt be that students give up too quickly on a complex problem. Mathematics is unique among all subject areas in its potential for cultivating a positive attitude toward struggle. It’s also the subject that tends to shut kids down for that very reason.

In other words, it’s a struggle to get kids willing to struggle.

And it’s one we need to struggle through.

Teachers, particularly math teachers, know that most learning occurs in the midst of struggle and overcoming struggle. But all students’ life they’ve been assessed and encouraged to solve rote problems quickly. How can we undo years of low-complexity problems? How might we develop students’ desire, ability, and disposition to struggle? What should students be doing? What should teachers be doing? What kind of tasks should kids be engaged with to encourage positive struggle behavior?

Even though dubbing this the “East” vs. “West” mindset is lazy terminology, it does conjure up a sort of spectrum of student and teacher attitude toward struggle. How can we nudge students (and teachers) a bit eastward on the spectrum? What kinds of tasks can we provide for kids so they may learn persistence.

struggle specturm

* – “My Favorite No” refers to this video from

** – “The Mistake Game” refers to this post from Kelly O’Shea.

While, as the article cited suggests, I hate to place a value judgment on the East-West mindsets presented. That said, I don’t think we’re particularly lacking of Western-style classrooms. Do you have particular tasks that help kids get away from a Western attitude toward struggle and smarts? Even if it nudges kids just a bit further “eastward”, please share. Because we all know that struggling and persisting are good things, but we’re not always sure on how to engage students in this manner.

The Problems have become self-aware: Introducing the Skynet line.

I had a great twitter conversation tonight with a bunch of people about the topic of “authenticity.” That is, what’s the relationship between pure mathematical investigations (like, say, this one – a problem I absolutely love), versus a more concrete, applicable problem (like, say, this one – a problem that I also love). It’s a piece of curriculum design that’s never settled quite for me. Am I developing problems that are not authentic enough? Am I artificially making problems authentic? And does doing that make them inauthentic by definition? Is it a balance? A 2-to-1 ratio? Which way does that ratio point? I was going to blog about it, but A) I should probably have something mildly-intelligent to say about a topic before I write about it, and B) David (@delta_dcalready did a year ago in a much more coherent manner than I could have attempted.

I began to think about “good authenticity” vs. “bad authenticity” which I made up on the fly. Bad authenticity is when a mathematical concept gets something real world-ish grafted on to it. You know, psuedo-context. It got me thinking about something Dennis Littky says about “fake real world problems” vs. “real real world problems.” Contrary to Littky though, I think “fake real world problems” are fine, so long as they understand that they’re “fake real world problems.” The famous cabbage-chicken-fox-boat problem is “fake real world” but that’s ok since we all understand that it’s a fun mental exercise. Ditto for the farmer/fencing problems present in many Geometry classrooms. You don’t need to take a field trip to a farm so kids can have a real world experience to figure out the optimum shape of fencing for a pig pen.

It is here that I propose the following axiom:

It’s fine if a problem is inauthentic or contrived, so long as it knows it’s inauthentic or contrived.

Obviously we want kids to have tons of authentic, real-world experiences in math. Practically though, we are limited by budgets, time, logistics, standards, and creativity. So we ought to pick and choose what we want to be supremely authentic and what is contrived. And whatever is contrived ought to be up front about itself.

To that extent, I’d like to introduce the Line of Problem Self-Awareness, or Skynet Line.

Basically the way it goes is this: problems vary in “real-worldliness”, which is fine. But problems also vary in how real world the problems think they are. Ideally a problem ought to be self-aware of its own “real-worldliness” whether it’s super authentic or a contrived problem scenario intended at getting into math content. Ideally, a problem lands directly on the Skynet Line (y=x). Problems above the Skynet Line think they are being authentic when they’re really not (sorry peeps, architects don’t use Geometer’s Sketchpad to design a house, nor do they require that there be a hexagonal room). Problem below the Skynet Line are shortchanging their own authenticity and could probably stand expert input or more physical investigation.

Obviously similar problem scenarios could move all over the graph depending on how they’re posed, facilitated, and concluded, so don’t get too bent out of shape about any of the particular items on the graph above.

I always want to push teachers to make their problems more authentic. It’s invaluable to have a student see a straight line connection between a mathematical concept and something they can see, touch, and behold. I certainly don’t want to give teachers an easy out by invoking Paul Lockhart’s name and washing their hands of all authenticity.

However, believe it or not, sometimes students enjoy contrived problems. Ever played Clue? It’s basically one big contrived logic problem and they made a movie out of it. So while I want to push for authenticity, I do want teachers to at least know whether or not their problems are self-aware.