The Explanation Difference

Brett Gilland coined the term “mathematical zombies” in a comment on this blog:

Students who can reproduce all the steps of a problem while failing to evidence any understanding of why or how their procedures work.

When I think about mathematical zombies, I think about z-scores – how easy it is to calculate them relative to how difficult it is to explain those calculations.

Check it out. Here is the formula for a z-score:


In words:

1. You subtract the mean from your sample.
2. You divide that by the standard deviation.

Subtraction and division. Operations simple enough for a elementary schooler. But the explanation of those operations – why they result in a z-score, what a z-score is, and when you should use a z-score – is so challenging it eludes many graduates of high school statistics. Think about how easily you could solve these exercises without knowing what you’re doing.

That difference brings this chart to mind and helps me understand all of the times I’m tempted to just tell students, here’s how you do it already so now just do it. That’s where the operational shortcuts are most tempting.


All of this is preface to a lesson plan on hypothesis testing by Jeremy Strayer and Amber Matuszewski, which is one of the best I’ve read all year.

Hypothesis testing is, again, one of those skills that’s far easier to do than to understand. As you read the lesson plan, please keep in mind that difference. Also notice how capably the teachers develop the question, disclosing the mathematics progressively, and resisting the temptation to shortcut their way to operational fluency.


It’s spectacular. I’m struck every time by a moment where Strayer and Matuszewski ask students to model an experiment with playing cards, only to model the exact same experiment with a computer later. They didn’t just jump straight to the computer simulation!

Here is a video of an airline pilot landing an Airbus A380 in a crosswind. This is that for teachers.

Featured Comment


I always think of z-scores as a set of transformations from one plain-vanilla normal curve to the hot-fudge-sundae Standard Normal curve. Maybe once you see it this way, you can’t unsee it. To me, that helps make sense of the “why” you would bother standardizing and the “how” it’s done.

David Griswold:

I’m not sure I agree that z-score is so conceptually difficult as to be worth the shortcut. Though I suppose it requires understanding of standard deviation, which is kind of hard. But if you think of standard deviation as “typical weirdness distance” then z-score as the idea of “how many times the typical weirdness is this point” becomes pretty straightforward. A z-score magnitude of 1 becomes average weirdness, less than 1 becomes less weird than average, etc. The bigger the magnitude of the z-score, the weirder the point.

Bob Lochel:

In introductory stats courses, much of what we do simply comes down to separating “Is it possible?” from “Is it plausible?”. We have seen a wonderful growth in the number of free, online applets which allow teachers and students to perform simulations designed to assess these subtly different questions.

NCTM Puts up a Sign

NCTM President Matt Larson wrote an essay last week titled “Curricular Coherence in the Age of Open Educational Resources.” As I read them, the big takeaways are:

  1. Coherence in curriculum is important.
  2. A curriculum is more than just a sequence of activities.
  3. Activities that are downloaded from the Internet vary in quality and often undermine curriculum coherence.
  4. If you’re going to download activities anyway, then download them from NCTM’s lesson site or download them in learning communities where there might be more accountability for coherence.

I’ll co-sign all of the above. I hadn’t thought about collaborative lesson planning as insurance against incoherence. That’s clever. All of that said, I’m disappointed at how much of the essay obliges teachers and how little of it obliges publishers.

It reminds me of desire paths.


Desire paths occurs when people walk off the path you pre-determined and create their own. They create another route because it satisfies their needs better than yours did. In that situation, you have options. You can post signs directing people to stay on the path. You can hire security to make sure people stay on the path. Or you can admit that you messed it up on your first try and pave the desire path.

NCTM is putting up a sign.


Aside from a brief mention of sympathy for teachers who “lack highly engaging, high-cognitive demand tasks or lessons,” the essay doesn’t acknowledge the desire that leads to online activities.

It isn’t as if many teachers are eager to spend nights and weekends cobbling together a curriculum from scratch. What gives? I asked teachers about their desire. Most prominently they want materials a) that are engaging, b) that are scaffolded appropriately, and c) that create high cognitive demand. A large number of them don’t think their core curriculum is particularly coherent.

I appreciate Larson’s leadership and support NCTM’s interest in coherent mathematical experiences for kids. But if teachers – especially at secondary levels – had access to resources that offered those features above, I suspect the desire lines, and consequently NCTM’s sign, would be unnecessary.

So an open question: What would it mean to pave the desire path here? Now that we’re at the point where people are tromping across the lawn, marching towards online activities, what would it look like to say, “Okay,” and then pave that path for teachers.

Recommended. Tyler Auer’s analysis.

Also recommended. The comments of the essay, where Larson is taking questions and adding commentary, including what appears to be an answer to my open question above.

Featured Comments

Please read Matt Larson’s comment.

Michael Pershan:

While Matt’s piece didn’t rub me the wrong way, it does seem to me that treating this as a challenge for people who want to influence what teachers do is going to be a better framing than trying to convince teachers to change their online patterns.

Steve Weimar:

One of the projects that the Math Forum at NCTM is beginning to plan within NCTM concerns the idea of an online collaboration space and supporting repository through which the community can move along the continuum from sharing good tasks to identifying and playing with sequences and instructional practices that lead to the engagement, depth of understanding, rigor, and competence we all seek for our students.

Patty Miloro:

Curricula coherence is my biggest struggle. I work in a small charter high school and we have not purchased a commercial math curriculum. Instead of only 20% of my time spent planning, I am lucky to have the freedom to spend lots of time attempting to develop a cohesive unit by vetting online materials. This autonomy is both wonderful and terrifying.

Henri Picciotto:

One way NCTM might help with both coherence and quality is to offer a really-core curricular framework (as opposed to the much-too-massive CCSSM), and within that offer curated links to high-quality online materials. A positive contribution along those lines would be a lot more useful than anxiety-provoking warnings about coherence.

Elizabeth Statmore:

One last deep thought — I sure am getting tired of being blamed for the incoherence of standards and curricula that are way above my pay grade. Unfortunately, the way all of this has been set up (or not set up), everything rolls downhill into my yard.

Jason Slowbe:

Perhaps the “next big opportunity” for NCTM is in connecting members online around its quality content. NCTM’s average member age is 55 years and fewer teachers are buying memberships, opting instead for free online connectivity with other teachers that is still quite good overall.

Christopher Danielson’s “Which One Doesn’t Belong?” Now in Print


Stenhouse just released Christopher Danielson’s book, Which One Doesn’t Belong?

It’s a must-have if you’re a parent or a teacher with any interest in helping your children or students learn to speak mathematically.

There are few tasks that offer so much mathematical value yet require so few instructions as Which One Doesn’t Belong?

You see four mathematical objects. You ask kids, “Which one doesn’t belong?” You help them negotiate their overlapping and conflicting answers, developing vocabulary and the capacity for argument and abstraction along the way. That’s it.


You can find loads of great WODB prompts online but you can’t find Christopher’s unique presentation, narrative, and teacher’s guide, which is its own kind of graduate-level course in pedagogy.

Highly recommended.

Three Types of Questions

Peter Liljedahl:

Students only ask three types of questions: (1) proximity questions – asked when the teacher is close; (2) stop thinking questions – most often of the form “is this right”; and (3) keep thinking questions – questions that students ask so they can get back to work. Only the third of these types should be answered. The first two need to be acknowledged, but not answered.

[via Dylan Kane]

Featured Comment


When my students ask “Is this right?” or express uncertainty about their answer I like to ask them what their confidence level is. 25%? 50%? 75%? etc… If it’s less than 50% then I ask them why they are unsure. If it’s over 50% then I ask them what gives them confidence.

Great Classroom Action

Jennifer Abel creates a promising variety of card sort activity:

Basically, after dealing the cards, the basic idea is for kids to pass one card to the left while at the same time receiving one card from the player to their right. The object of the game is to collect all cards with the same suit/type/category. Here are two examples that I recently created for next year.

Julie Morgan offers three sharp lesson-closing activities. My favorite is “Guess My Number”:

I choose a number between 1 and 1000 and write it on a piece of paper. Each group takes it in turns to ask me a questions about my number. The questions can vary from “is it even?” to “what do the digits add up to?” to “is it a palindrome?” (my classes know I like palindromes!) When a group thinks they have figured it out they write it down and bring it up to me. Each group is only allowed three attempts so cannot keep guessing randomly. I like this for emphasising mathematical knowledge such as multiples, primes, squares, etc.

Pam Rawson contributes to #LessonClose with both a flowchart that illustrates her thought process at the end of classes and then some example exit polls for both “content” and “process” objectives:

As a member of the Better Math Teaching Network, I had to come up with a plan – something in my practice that I can tweak, test, and adjust with ease. So, I decided to focus on class closure. Since I don’t have an actual process for this, I had to think intentionally about what I might be able to do. I created this process map.

Robert Kaplinsky offers the #ObserveMe challenge:

We can make the idea of peer observations commonplace. It’s time to take the first step.