Just added: Grade 3 CCSS-PrBL Curriculum Map of Tasks

I don’t often put up a blog post every time I add a new map, but I thought I’d pause a minute and comment on the ol’ CCSS-PrBL Curriculum Maps in general.

I just added a CCSS map of (in my opinion) quality tasks that adhere to tenets of Problem-Based Learning created by teachers, curriculum specialists and math advocacy organizations. However, I want to make sure that I make plain my ignorance when it comes to elementary math education. Having a couple kids get put through the ringer at an elementary school has given me some insight, but nowhere near the elementary teachers that I work with in person and follow online. If the “# of days” is a best guess for Middle and High School tasks, then it’s an even worse best guess for elementary. And that goes triple for my best shot at what makes for quality instruction at the elementary level. I’d guess (but again, I don’t know!) that the task itself is less important at lower grades than at higher. And that the discourse (aided often by the task) is the real thing to get better at as a facilitator.

Also, I’m having more and more difficulty in finding tasks the younger the age groups gets. I’d suggest there are more holes in this Grade 3 map than in any prior. Maybe that’s because I don’t know where to look or maybe there’s just a relative dearth of tasks for certain younger grade standards.

So that’s a way of inviting critique and feedback from elementary teachers that do want to steep their kids in mathematical inquiry, just as I do as a middle/high school centric person.

Anyway, here’s the grade 3 map. Enjoy! And keep a bookmark there for updates to the google docs.

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Better (or at least less destructive) Test Prep

I’m dubious of the effectiveness of last-minute test prep. It might help final scores on the standardized exam, but I’m not 100% certain that it does. That said, I still conducted test prep class days because A) it might help! and B) there was a decree from above to do so. So there you have it: test prep was happening. My question is “how do we make the experience better for students and better for learning?” We have the skeletal structure of the standardized (often multiple choice) items. We want kids to become familiar with the verbiage of the questions. But we also want students to not be bored and unhappy.

So rather than throw up my hands, puff air our of the corner of my mouth, and throw a review packet at them, here are some things I tried to make the weeks we had of test preparation slightly better.

(note: as a teacher from Texas, the original problems are most often taken from released Texas exams; you might want to consider PARCC practice exams if that’s your jam).

  • Remove the question. Provide the problem setup, remove the question and ask “what question do you think they’ll ask?”

Original Problem:

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  • Item Analysis. Ask students to see if they can identify why the test writers chose the three incorrect multiple choice answers that they did.

Original Task:

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  • Turn the words into gobbledygook. Turn the questions into gobbledygook and see if they can still get it (or at least some of them) right.

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One of my favorite quizzes I ever gave was during test prep week. I handed out a quiz that was 10 questions and was barely readable. I turned most of the words into greek (by switching the font to symbol). I usually left the equations and numbers alone. Students were all “you serious Mr. Krall???!!!!”.

Initially they tried to decode the words – you can kind of make out the words from the greek symbols – but eventually they attempted other techniques, like plugging in numbers or sketching things out. Afterwards I gave them the actual items and we compared how right they were, despite it being initially unreadable. Obviously my preference is that they read the items carefully and understand what it’s asking (one little “NOT” or “perpendicular to” could throw everything awry).

This was fun because even if my students were able to get just a few of them it (hopefully) to NOT give up at the first sign of a complicated or unfamiliar word. Also, I’ll fully cop to cherry picking which problems got “symbol’d” (or “wingding’d”). You obviously can’t do this for all problems, but you can do it with some. And all it takes is a couple correct answers on gobblygook questions and you’ll build that persistence, if ever so slightly.

Original Task:

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My version

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Or here’s one from the released Algebra 1 PARCC practice exam:


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Gobbledygook version:

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  • Scavenger Hunt. That thing where you post the answer above another problem, then they have to go find the solution to the bottom-problem and go around the horn until they arrive back at the starting point.

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  • Give the answers, have students to create the question. 

Original problem:

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Certainly follow-up with the big reveal to see what they actually asked. You could even potentially give them hints (i.e. “the original problem includes a data table”).


The following two examples are cribbed from other blogs that I’d like to share here. I personally didn’t conduct either of these, but they seem like they’d go along well with preparing for a end-of-course style exam.

  • Choose your own problem (from David).

I shared this during my NCTM Adaptation talk but it works especially well for long, awful problem packets.

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Clearly some of these techniques work better than others for particular problems. These are a few strategies to potentially prepare kids for the fun of examinations. What are some of the strategies you try?

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[NCTM] Adaptation

Greetings everyone! Below you’ll find resources and some sample tasks to potentially adapt in service of opening the beginning, middle or end. Whether you attended my session or not, let’s have a discussion in the comments about which task you would adapt and how you’d adapt it.

Adaptation NCTM _ title.001

[Slides] Adaptation NCTM

[Handout] NCTM_Adaptation_Krall


Adaptation NCTM _ framework.001

[Tasks to adapt]

physics example

Carnegie Gas Tank Task

which line is parallel example

online video games example

quadrilaterals example

Counterexample Task

Carpentry Problem

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Revisiting the revisitation of the 2000 election

A couple years ago I shared a stats-ish problem idea regarding the oh-so-fun 2000 presidential election. The problem was cribbed from a graduate level stats textbook but the data are straightforward enough for a – let’s say – 9th grader to grasp. What made me want to revisit the task is some fun data tools that Tuvalabs recently released. Tuvalabs, whom I’ve raved about in other forums, now allows teachers to upload their own data so that kids can play with it in a non-Excel format.

So the 2000 election. Yeah that was fun. Anyway, I uploaded the data I had from the original task and went to town. The data show the county-by-county votes for George W. Bush and for Pat Buchannan. You’ll recall that Palm Beach voters complained about the confusing butterfly ballot. Some voters claimed they intended to vote for Al Gore but accidentally punched the chad for far-right candidate Pat Buchanan.

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I reworked the original task in my previous post, and I’ll rework it again so here, in proper entry event form.

So play around with the data yourself. How would you present the task as a teacher? How would you present the analysis as a student? And what workshops and scaffolding would you offer in between?

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Your 2014 Math Blogging Retrospectus

Compiled a tad too late for the Winter break, but just in time for January in-service, here’s your 2014 collection of posts shared by people who came across my post.

[2014 Math Blogging Retrospectus]

Be sure to check out previous years’ retrospecti as well.

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Calling for 2014 Math Blogging Retrospectus Posts!

Ah it’s the time of year again. The time of year when we all start looking forward to fireplaces, family, and chewing up reams of your school’s printer-paper by printing out the Math Blogging Retrospectus.

The impetus of creating the Retrospectus was that it’s so damn hard to keep up with all the great math teaching content being produced. It’s really difficult to make sure you got all the value out of the math blogosphere when new posts and bloggers pop up every day. Thus, the Retrospectus.

All I’m asking you to do, dear reader, is to paste a link to one (or two or three or ten) of your favorite math blog posts from 2014 in the comments. It’s up to you to determine what “favorite” means. Perhaps it was something that you used in your class or want to use in your class. It’s possibly a moving story from a thoughtful facilitator. It could be a post that made you think differently about something, or in a new light.

From last year’s description:

It’s incredibly difficult to keep track of the ever-growing Math Blogosphere. Keeping up with posts is like trying to hold water in your hands. I’m looking for timeless or timely math blog posts that inspired, touched, and/or entertained you. This decidedly NOT a voting thing. It’s NOT a ranking. And for the love of all things holy, it’s not an EDUBLOG award thing. If a blog post touched a single person, I’d like to capture it: chances are it’ll touch another. There are math blogs that I and you do not even know about, but someone reading this does. Let’s all partake in some shared sharing. Share a link to a few (or several!) blog post that you truly enjoyed, I’ll do some of my patented copying and pasting and attempt to assemble it into a tome that can be downloaded or printed out. They could be short posts on instructional practices or problem ideas. They could be longreads of reflections on teaching and systemic issues. Any and all types are welcome.

I’ll start. Throughout the year, I’ve been bookmarking interesting posts. Here are 10 (and only 10) of them.

Now it’s your turn: post a link in the comments below. If you’ve been derelict in your duties of bookmarking your favorite posts, the @GlobalMathDept newsletter archives might be a goof place to get your footing.

Once you’ve done that, feel free to go back and check out previous years’ Retrospecti:

[2012 Math Blogging Retrospectus]

[2013 Math Blogging Retrospectus]

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Critiquing the Common Core on its Merits and Demerits

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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.

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Where inquiry and methods intersect

Had a nice, quick twitter conversation with Anna (@borschtwithanna) yesterday morning. Anna reached out with a question about providing methods in an inquiry-based classroom.

Anna was conflicted due to her students’ unwillingness to deviate from their inefficient problem-solving strategy. Rather than setting up an equation…

Setting aside for the moment that this is actually a pretty good problem to have (students willing to draw diagrams to solve a problem, even at the cost of “efficiency”), it does circle back to the age-old question when it comes to a classroom steeped in problem-solving: “Yeah, but when do I actually teach?”

The answer to that particular question is “um, kinda whenever you feel like you need to or want to?” The answer to Anna’s question is pretty interesting though, and I’d be curious what you think about it. Personally, I never had students that were so tied to drawing diagrams to solve a problem, that they weren’t willing to utilize my admittedly more prescriptive method. I do have a potential ideas though.

Consider Systems of Equations. This is a topic that is particularly subject to the “efficient” method vs. “leave me alone I know how to solve it” method spectrum. Substitution, elimination, and graphing were all methods that students “had” to know (I’ll let you use matrices if you’d like, I’m good with just these three for now).

Anyway, so I’m supposed to teach these three different methods for solving the same genus of problems. I want kids to know all three methods (generally), but also want to give them the agency to solve a problem according to their preferred method. Here are a few possibilities to tackle this after all three methods are demonstrated:

1) Matching: Which method is most efficient?

OK so matching is kind of my go-to for any and all things scaffolding. It’s my default mode of building conceptual understanding and sneaking in old material (and sometimes new material!).

In this activity students cut out and post which method they think would be the most “efficient.”

Students could probably define “efficient” in several ways, which is ok in my book. Also, it’ll necessitate they know the ins and outs of all three methods.

2) Error finding and samples of work

This is another go-to of mine. Either find or fabricate a sample of work and simply have students interpret. If you’re looking to pump up particular methods, consider a gallery walk of sorts featuring multiple different methods to solve a particular problem. The good folks at MARS utilize this in several of their formative assessment lessons. These are from their lesson on systems.

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Students are asked to discuss samples of student work and synthesize the thinking demonstrated, potentially even to the point of criticism.

That’s a couple different ways to address methodology and processes that may turn out to be more efficient, while still allowing for some agency and inquiry on the part of the student.

What do you have?

Posted in algebra, scaffolding, tasks | 3 Comments

Quick Hits: Razor Blades and Fractions

A potential fractions task because because middle schoolers probably really struggle with the high cost of shaving blades. Not a super complex task, but maybe good for a warm-up?


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Suggested Knows/Need-to-Knows:

We know…

  • Dollar shave club sells razors at a price of four for $6.
  • “Their” razors cost 1 1/2 for $6.

We need-to-know (or, we’d like to know)…

  • How much do both companies’ blades cost per blade?
  • How much do “theirs” cost for a pack of four?
  • Where did Dollar Shave Co. get these prices for “theirs”?

Quick commentary:

One of the things I like about this is that you potentially have a fraction within a fraction. That is, one can calculate the cost of “their” razors by dividing $6 by 1 1/2 razor blades.

Also – and I don’t say this often – please don’t make this a hands-on activity.

(9/20) Update / Conclusion?

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Assessment via audibles: OMAHA! OMAHA!

It’s both the first question and the last one when developing an inquiry-driven classroom, ostensibly featuring significant groupwork:

How do you keep individual students accountable while working in groups?

While that’s a huge bear of a question that is better addressed via a book, I want to take a stab at a small slice of it. I’m going to ask myself this question instead:

How do you assess individual students in group settings?

While also a big question better served by a myriad of strategies, interventions and norm-setting, I’d like to share a brainstorming “aha” moment that I had a couple years back with Jessica (@bloveteach).

We were discussing her thematic Problem-based unit on solving systems of equations featuring diagramming and developing football plays. (Note: you can read about the unit and the awesome task author here from the local paper.)

Jessica and I were trying to come up with a way to adhere to the norms and boons of groupwork and collaboration while developing the unit with an individualized literacy prompt. In groups, students would analyze an assigned football play (specifically the wide receiver routes). The groups would develop a linear equation to model the play. They would prepare a presentation discussing their assigned football plays and whatever additional attributes of the play they’d add on: receivers running parallel routes, crossing patterns and so forth.

denver play orig

Then what? As a network of schools, we’ve pretty much decided that every effective PBL/PrBL unit requires an individualized disciplinary performance task, preferably one as engaging as the tasks themselves. 

After discussing and racking our brains, Jessica and I came up with the idea of audibles. That’s a footbally thing, right?

Students presented their plays as a group, but then Jessica called an audible, in which students were assigned a different, but not totally dissimilar play in which they were to write an analytical report assessed via a rubric.

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(Note: I recently went back and prettied the plays up. You see, these tasks were originally developed back when desmos was only a gleam in Eli’s eyes.

They used the same concepts from their groupwork in a similar scenario to ensure they had gotten the mathematics concepts down pat.


I had a similar conversation about this task from the Shell Centre. 

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This is one of my favorite tasks, so much so I threw it in my PrBL Starter Kit. I was sharing this task and then the question came up: what next? How will I know if individual students have learned anything? Again, setting things like equitable groupwork and norm-setting aside, the easiest thing a teacher could do is go back to the prompt and just do this with it.

security camera task add-on

OK, now you’ve done it in a group, let’s put your understanding to the test.

Shoot, let’s make this the test. It took five minutes to rework the diagram (badly!) with the help of inkscape (free!) and now I’ve got a similar problem for individual students to undertake. Perhaps the embodiment of “we do, you do.” Instead of Peyton Manning calling an audible, Mr. Shopkeeper blew out his east and west walls in order to expand the store. Feel free to use your notes from your groupwork.


The audible method probably isn’t earth-shattering, but it is quick, easy and implementable. And congrats! You just saved yourself the writing of a test.

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