Based on this plot, can you tell me when it started snowing? Or, can we fit a sine curve to it?

Now, I’m biased of course since I got a MS degree in Atmospheric Science, but I feel like weather and climate is one of the biggest untapped wells of potential math problems. I think there are two reasons for that (aside from the “lack of expertise” / “it’s not in the textbook” angles):

1) The math involved is often considered “too advanced” and/or involves copious amounts of data.

2) There is significant amounts of uncertainty involved.

If I’m right, then this is troubling. It’s troubling that either of these aspects of weather and climate would make people shy away from using it as a tool to promote mathematical thinking in the classroom. Students should absolutely be exposed to and explore the concept of uncertainty. Ditto for data processing. I’d argue that the ability to assess uncertainty and/or large quantities of data is more relevant than the Pythagorean Theorem. Also, possibly more interesting. Consider this: would your students be more engaged by the good ol’ ladder problem, or by analyzing climate data or incoming cold-fronts?

Now, there are probably a myriad of other reasons that weather and climate, which is such an important part of all peoples’ lives that we constantly engage in it conversationally, isn’t studied in the classrooms (it’s difficult to visualize or represent fluid dynamics on an atmospheric scale; the field is still relatively new and there are many open questions still out there). But don’t let either of the above “problems” with weather and climate science make you shy away from utilizing it, if at all possible. I’d love to see a math curriculum incorporate scores of atmospheric science … or an atmospheric science curriculum incorporating scores of math concepts.

/steps off soap box

OK, sorry about that tangent. Here’s your artifact for today.


This plot of temperature and dewpoint.

And, I dunno, maybe this picture too.

Guiding Questions

  • When did it start snowing?
  • What is dewpoint? And what does it have to do with snow?
  • That temperature plot looks pretty pattern-like until 10/25, what kind of curve could be fitted to that plot?
  • Can we use some sort of pattern-deviation to determine when “weather will happen?”
  • So how much snow is that? Wait, how do they measure snowfall totals?

Suggested activities

  • Based on the data from 10/21 go 10/24, develop an equation that represents the “normal” or “expected” temperature.
  • Use Weather Underground to find your local temperature history for a given time to do the same.
  • By the way, that can also be done to determine monthly average temperatures and dewpoints. Create a sine wave for monthly averages?
  • Convert the dewpoint plot to relative humidity – the preferred measure of atmospheric moisture content by the public.
  • After deciding when the snow began falling, go back and check your solution with actual weather reporting. (group closest to actual time gets free sno-cones?)
  • Based on that photo, estimate how much snowfall was measured (hint: snowfall totals are measured according to the amount that’s left after it’s melted).

Honestly, when it comes to weather and climate in general, I’d be interested to hear what students are interested in. Is it climate change? Tornadoes? Blizzards? All of these are going to incorporate aspects of data collection and uncertainty. That’s probably a good thing.

An oversimplified model of an inquiry-based lesson, with visual aids

Last week, I mentioned that, having begun to attempt to slay one of the two giants of inquiry-based math instruction, I’d be steering into a potentially trickier aspect of inquiry based instruction: namely that of instruction and facilitation.

Most of us learned math like this.

We have decades of evidence suggesting that this method of instruction is not only ineffective, but damaging – both to students’ confidence and love of the subject (see Jo Boaler’s awesome “What’s Math Got To Do With It” for more). But honestly, I think that battle has essentially been won. Most of us (I think) are in agreement that this isn’t the ideal way to teach math, or any other subject. But the question is “how?”

Even in more-or-less traditional high schools nowadays, you’ll see something more like this.

Students need to be actively and collaboratively involved in the problem solving process, but what does that really look like? How do we give students both the freedom of solving a problem collaboratively and in novel ways, while still providing enough support to help students along the way?

That’s where the real art of teaching lies. One has to be nimble to adjust instruction based on student need, but prepared enough to be able to anticipate and address the need.

So let’s start with the “ideal” inquiry based lesson, start-to-finish, then in future posts we’ll go back and analyze the process further.

A model of inquiry-based instruction.

Step 1: The problem is posed.

Usually the problem is introduced along with some sort of class or group discussion facilitated by the teacher where students identify key components of the problem and begin strategizing.

Step 2: Students begin work on the problem.

As the students work together toward a solution, the teacher checks in with each group and each student, probing for understanding and answering any clarifying questions.

Step 3: Questions begin popping up from the students.

As the students are working through the problem, questions related to the intended content begin to crop up. As students begin to struggle with the problem a “critical mass” (or “tipping point) of perplexity occurs.

Step 4: Appropriate scaffolding and/or instruction is provided by the teacher.

Based on the students questions, the teacher provides instruction in some format. Maybe it’s a lecture, maybe it’s groups sharing out, maybe it’s analyzing samples of student work, maybe it’s a research resource, maybe it’s an investigation, activity or lab.

Step 5: Students work on the problem some more, after being provided instruction.

Having acquired the requisite content knowledge from the instruction, students proceed to work on the problem.

Step 6: Students solve the problem.

Students finalize their solutions. Usually some sort of informal sharing out or presentation is accompanying. The teacher asks probing questions to get students to make generalizations about their work and promote sense-making.


Now, this is clearly an over-simplified model of what a classroom actually looks like. Every step along the way is fraught with different challenges and obstacles to understanding which need to be addressed. I hope that the simplicity of this model does not imply that inquiry-based instruction is simple: far from it! For example, in our fantastical little classroom above students appeared to be all having the exact same question at the exact same time. Obviously that doesn’t ever happen in classrooms. (also, it looks like we lost a couple students from the first image of this post to the next)

I would also like to formally declare that I don’t have all the answers. Frankly, I’m not sure I have very many answers at all. I do have a lot of questions though.

In order to address the monumental challenges, we’ll be looking at each step in depth over the next few weeks, discuss particular challenges, differentiation strategies, etc. My preference would be to get your input and suggestions, since I’m far from an expert.

But before I do, what do we think of this little utopian situation? Did I miss anything? Would you swap out one of the steps for something else? In my desire for simplicity I may have glossed over something or left something out entirely. Please chime in in the comments.

Inquiry-Based instruction, in a PNG-nutshell

In talking to math teachers about an Inquiry- , Project- , or Problem-Based approach, these are the following questions that come up most often.

1) How am I supposed to cover all the standards using this approach?


2) “So, when do I actually teach?”

An attempt at the first question is reflected in the Great Inquiry-Based Curriculum Mapping Project, from a couple weeks ago.

The second question can be a bit loaded, especially when you move the emphasis from word to word (as I did, by emphasizing “actually“).

We can discuss the second question a bit more in-depth going forward, but I’d like to attempt to simplify the nebulousness of “inquiry-based instruction.” When teachers ask “when do I actually teach?” I think they’re asking when do they stand up in front of the class and demonstrate examples and processes? And is such a time ever appropriate for an inquiry-based classroom environment?

But before we get into the weeds of such a rich topic of discussion, let me posit this to you: I would suggest that the change from a “traditional” approach to an “inquiry-based” approach may be as simple as moving from this

to this:

(apologies for the computer science jargon)

Now, obviously there’s a lot more to it than just a couple diagrams, but the point is this: instruction still happens, but it simply happens after students have attempted a problem and within the context of a problem. Instead of saying “Today class, we’re learning about slope, here’s a lecture,” followed by a lecture, followed by a problem set, the practice is in some sense, simply reversed: “Today class, here’s a problem,” followed by instruction about,say, slope.

So yes: instruction is still useful and necessary for an inquiry-based environment. And I would also say yes: lecture or direct instruction is often a appropriate tool to transmit mathematical knowledge in an inquiry-based environment. (Although, I would warn against it’s overuse, lest it become the default mode of instruction.)

The deeper questions of when and how do I instruct is a bit more of a dance that I hope to at least partially address in the coming posts. But in the meantime, let me hazard a broad-brush answer at these.

When: after students have had a goodly amount of time to discuss the problem with each other, and at least begin to attempt a solution. Maybe at least 30 minutes?

How: it depends in part on the number of students struggling with the content. If every group is having difficulty even starting the problem, then a whole-class lecture may be appropriate. If half the class is struggling, maybe some share-out, gallery-walk, and/or group-student-exchanges may be appropriate (or better yet: Kate’s “Speed Dating” activity). If only a few students are unable to jump into the problem, a small workshop may be necessary, while groups discuss and assess their solutions.

But these are broad-brush, haphazard solutions to potentially a much bigger question. I’d love to begin aggregating and categorizing math scaffolding activities and to have a discussion about when they may be appropriate.

7th and 8th Grade additions to the Great Inquiry-Based Curriculum Mapping Project

If you haven’t read last week’s introductory post or description of the Great Inquiry-Based Curriculum Mapping Project, please do so. Here’s the short description: there are some open google docs that tie math standards to activities, problems, projects, blog-posts and/or ideas. There have already been some great aggregation of ideas and the documents continue to grow.

Moreover, the number of documents continues to grow. Huge props to Julie (a.k.a. @shyj) for picking up the ball and running with it. She created open google doc curriculum maps for 7th and 8th grade standards. So once again, we’re looking for ideas from the blogosphere, twittersphere, and general internet-sphere for potential ways and resources to address math standards. Anything and everything is welcome. Let’s A) stop working independently and b) start collecting some practical activities we can use in our classrooms.

Thanks to all the “anonymous users” that have thus far contributed. This project really excites me as someone who often struggles with brainstorming alone for math activities and interesting math problems. I’m so glad I have the ability to brainstorm with like-minded people, many of whom have never sat in the same room as me.

Here are the 7th and 8th grade documents:

7th Grade Curriculum Map

8th Grade Curriculum Map

Here are the documents from last week:

Algebra 1 Curriculum Map

Geometry Curriculum Map

Algebra 2 Curriculum Map

Pre-Calculus Curriculum Map

Calculus Curriculum Map