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?

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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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Designing Problems: Linking a standard to a context

Context isn’t everything, but it’s often a good thing. Or at least, it’s a thing, sometimes only a starting point. While contextualized scenarios aren’t necessarily the key that unlocks engagement they may allow students to model, activate students’ interests, activate your own interest, or simply serve as a starting point from which to develop a non-routine problem or project.

But where to start?

I’ve tried the stare-at-the-standard-until-inspiration-hits-you-like-a-bolt-of-lightning technique, but it’s not terribly effective. Not often, anyway.

I find there’s often freedom in constraints. A while back, Chris Jackson from College, Work Readiness Assessment (CWRA) had us design a performance assessment task based on some psuedo-random nouns and verbs (“fox”, “politics”, “measure”, “travel”, and the like). As difficult as it is to think of a contextualized problem in absentia of any guidance, as soon as the “shackles” of these nouns and verbs were placed on us, our group got straight to work. In a 15 minute time frame, we developed the idea of a performance assessment task in which the student is to analyze data on potentially contaminated milk (or something) and write a letter to a politician advising him/her of a possible political advertisement. Did we fully develop the task? No, we were at a conference and had, like, 15 minutes. Did we design the data? Nope. But those quick constraints allowed us the freedom to think deeply about content.

That felt like an interesting way to begin to design a task: by placing artificial constraints.

I had the pleasure of spending a day at a middle school where the teachers were dipping their toes into the wide waters of PBL. One of the thing I love about my current job is that it forcibly removes me from math-world relatively often. We were brainstorming PBL ideas as an entire staff. We discussed environmental impacts of war (all subjects), skeletal remains identification (systems of equations, biology, social studies), health fairs (all subjects) and more. It’s fun being with teachers that are excited about finding connections across content area, so I was pretty jazzed.

Upon looking at all the brainstorm ideas, I began to think about a schema in which we can place these ideas, potentially to aid future brainstorm sessions. Here’s what I came up with.


What do you think? Consider a content standard (or a cluster of standards) and try to develop a context allowing you to place it in one or several of the grids. Shoot, make it a game: try to get a tic-tac-toe, or BINGO. Maybe even consider your entire curriculum: are you spending an overabundance of time in one particular row, column or grid space? I’d prefer to have this thing blacked out, but that may just be personal preference.

But back to brainstorming, this seems like a potentially useful process check, ensuring you cover all your bases before you throw up your hands and declare a standard entirely devoid of contextualized meaning.

So I may play around with this framework a little bit. It’s probably not terribly revolutionary: all it does is place things on a grid according to temporal and geographic location. Feel free to give it a test run along with me.

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Reduced Fat ‘Nilla Wafers are an Empty Canvas for Problem-Based Learning Models

It’s probably not exactly the Great Double Stuf Oreo Controversy of 2013. We’ll be bouncing our grandchildren on our knees talking about that one. But here I am with some Reduced Fat ‘Nilla Wafers, thinking about multiplying percents and fractions.



Rest assured, I purchased these on accident. I meant to get the Original Wafers in all their full-fatted glory, like so:


Poor stage lighting aside, I want to do something with this, but I’m sort of facing decision paralysis. There are so many great models of Problem-Based Learning or Problem Solving Tasks, it’s difficult for me to settle on one, so I’ll just go ahead and create them all and see what sticks.

Would You Rather

Would you rather eat 5 Reduced Fat Wafers or 2 original Wafers?









Multiple Choice

Here is a plate of original ‘Nilla Wafers.


Which of these Reduced Fat ‘Nilla Wafers plates has less fat in total?

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Multiple Choice: You Make up the Question

Here are a bunch of pictures of ‘Nilla Wafers (original and Reduced Fat). Make up a multiple choice question & answer key.

Gamifying: Really strange playing cards

The point is, I’ve been trying to be less myopic when it comes to PrBL. There are so many great, differentiated models of Problem-Based Learning, I think it would be silly to get sucked into one and one alone. While I do think there is power in iterative routines, such as using a relatively consistent problem solving framework, it would be silly to neglect the power of “Any Questions” or the “Know/Need-to-Know” processes.

Going a bit further, the lines between the task vs. the scaffolding vs. the assessment are probably best when blurred. Considering just these wafers, we could probably place them in any of these three slots.

nilla spots


But alas, here I am, back to my decision paralysis. I suppose it’s a good problem to have.


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Larry Ellison, billionaire CEO, makes unsound business decisions with regards to his basketball playing on his yacht.

Larry Ellison, co-founder and CEO of Oracle, has gobs and gobs of money. How much money? Well enough that he can do this.


Boy that seems wasteful, doesn’t it. I mean, when I’m playing basketball on my yacht and I lose a ball into the ocean I just purchase an extra basketball. Wouldn’t it make more sense for Ellison to just buy a bunch of basketballs and grab a new one every time he loses one overboard? So my question is this: How many basketballs would Ellison have to lose in order to make the expense of basketball retrieval worthwhile?

Here is some of the board work we generated during the initial Problem Defining and Know/Need-to-Know process.


It’s critical that we understand our ultimate goal here: we want practice developing a mathematical model based on a given scenario. A model should, among other things, simplify a complex situation. We wound up focusing on only two variables: the cost of the annual salary and the cost of a basketball. A couple variables that folks tossed out ended up not being explored mathematically. As you may have experienced, when given a modeling scenario, students might throw out potential variables to tack on in perpetuity. There comes a tipping point where the mathematical model ceases to simplify a complex scenario and only confuses it further. I find this pretty typical of “make a budget” tasks or other accounting-type tasks (“what about sunscreen costs? what about health insurance? what about the yacht food? etc etc etc.”). When you’re facilitating the brainstorming process, I’d suggest you restrict the number of variables you’re including to two or three. This way, the entire class is focusing on the same few variables, keeping the focus on the model development, not the number of ingredients you can toss into the stew.


Here’s the initial PDF file if that embedded version looks goofy.

Basketball Overboard – Problem Solving Framework (pdf)

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[NCTMNOLA Processing Session 5] Networks and Silos

This will be the fifth and final NCTMNOLA Processing Session. It’ll be short too, just a quick debrief.

I vacillate between the poles of “math is different” and “math is just like other subjects.” Sometimes I wonder if math teachers use its alleged differentness as an excuse to teach it in an overly linear way. On the other hand, it sure seems different, doesn’t it? I’m not convinced either pole is correct, at least not for more than 72 hours at a time. I will say that math does feel especially silo’ed. I mean, here we were at a conference full of math educators and pretty much only math educators. We have our own vocabulary, our own best practices, our own standards of practice, our own conference, our own software. Yet still, we struggle as a profession to do the basics: get students to talk mathematically. Are we too buffered from other disciplines? I have the incredible opportunity to spend significant time with non-math teachers and much of what I do is taken from them: the way I conduct my debriefs, a See/Think/Wonder routine for interpreting works of art, fishbowls, Critical Friends for peer editing and solution review. These are protocols and facilitation moves usually reserved for non-math disciplines, to math’s detriment. 

In general, I wonder about the long-term sustainability of effective math teaching if single teachers are the unit of change, instead of systems. Here is what I mean: Seattle and University of Washington have an amazing system in place to keep their practice of Complex Instruction rolling, even as teachers move on. Most communities don’t have that systemic approach. When they lose a teacher, they have to start from scratch, hiring, professionally developing, and inducting (or shielding) that teacher into (from) the school culture.

However, what we do have is a network of educators online, on blogs and twitter, all the time at our beck and call. Maybe this is our permanent system that will outlast those of us who gracefully exit the classroom. 

Thank you to all my online collaborators that I got to meet in person and all the online collaborators that I have yet to meet.


Previous Processing Sessions:

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[NCTMNOLA Processing Session 4] I may have missed my calling

Those most mathematical fun I had at NCTMNOLA was in sessions focused on Elementary School level math. Either by accident or by impulse I found myself drifting into sessions that one would associate with K-5. Well, if that’s the case, why did I have so much fun?

David and Kathryn led us through a gallery walk of sorts of mathematical games for K-2 Common Core standards. Immediately, selfishly, I and Alyssa began trying to adapt them for our contexts. Maybe we should have been more in the moment, and maybe that’s our High School education brains kicking in, where everything is a nail to our math hammer, but manipulatives and games by and large don’t happen at the high school level. A few months ago I had some non-math teachers do a little card matching game with functions and she remarked that she had just assumed that these manipulatives activities were for the kids in Talented and Gifted. It’s understandable where one would get that impression. Anyway, David and Kathryn were great hosts and provided several activities in a relatively short amount of time.


Marilyn Burns is as exceptional in person as you’ve come to know from her books and websites. Her presentation focused largely on her wealth of experience with math talks, presenting video of various methods kids use to mentally solve 99+17. Every time a kid solved 99+17 in a different way my heart fluttered a little bit. This looked like so. much. fun. After watching a few of these videos I wish my entire job was to ask kids how to solve 99+17 all day. As soon as I arrived on my home doorstep I asked Mrs. Emergentmath and emergent kids #1 and #2 to solve 99+17. Why did I feel like this was revolutionary? Why was this so much fun? Was it because these kids were practicing invaluable Algebra skills they will need later in school? That’s probably part of it. It also hearkened back to the idea, first presented to me by Bryan Meyer, of Mathematical Play. Dang, there’s that word again: play.

And speaking of Mathematical Play, perhaps the most fun I had in any single session was Christopher’s Hierarchy of Hexagons. Double points for an 8am start time. Christopher began by having us mentally sorting four different visual polygons into two groups (we hadn’t gotten to the hexagons just yet). Some people chose to sort the regular polygons from the irregular, others chose to sort by reflective properties. Either way, it gave us a good bit of practice until the real fun began. ENTER THE HEXAGONS.



We chose a hexagon that, in Christopher’s words, “spoke to [us].” I chose the one that kinda looked like a fox, or the FiveThirtyEight logo. Brandon had no clue whatsoever.


Anyway, once we chose our hexagon, Christopher asked some participants to describe the hexagon they chose, why, and then led us through some authentic definition-making, that makes the hexagon belong (or not belong) to a category. Participants’ definitive categories included “waffle cones”, “reflectors”, “utah’s”, and more. From there, we created a flow chart/Venn diagrammy thing that showed which hexagons belonged to which categories and which were mutually exclusive. We’ve all done this with quadrilaterals (“a square is a rhombus, but a rhombus isn’t necessarily a square”). But hexagons are (as it turns out) an Undiscovered Country of polygons. It was fun and rewarding to explore those uncharted waters for a while. Here are half of those hexagons.


Explore. Play. I remember playing. I remember learning new things by conjecture, trial and error, sketches, etc. Why was I having so much fun in these sessions that weren’t in my self-defined “wheelhouse?” Did I miss my calling? Should I have been an elementary school teacher? Probably not, but it was fun to occupy that space for a while.

After volunteering in my daughter’s 3rd grade class for a year, and after these sessions, I’m more convinced than ever that we need to blur the lines between elementary and secondary math education. There’s no reason I couldn’t have done the Hierarchy of Hexagons in my Geometry class. There’s no reason I couldn’t have started off with a number talk in Algebra 2 once a week. If only for the fun of it.



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[NCTMNOLA Processing Session 3] Summer School is Dead, Long Live Summer School

In between the instant they opened the door to Jo Boaler’s talk on “Promoting Equity Through Teaching for a Growth Mindset” and when she began speaking, I, Eleanor, and a few others in my row banded together as Jo Boaler groupies. We were those people at rock festivals who get to the stage several acts prior to your favorite band to ensure that we are front row, center. And to be honest, it kind of felt like that: like we were waiting for a performance for a favorite musician.

What’s Math Got To Do With It is the first book I recommend to math teachers and parents vaguely interested in math education. Its combination of research-based practices, accessibility, and price make it – in my opinion – the premier text on math instruction. And that was before the Boaler-Dweck tag-team was unleashed in full.

I’ve been thinking a lot about “death knells” recently, things that basically signal there’s virtually no going back. Remediation is a “death knell” for many students. Very few students who are labeled as needing remediation ever get caught up with their peers, fewer still ever exceed their peers. Some of that may be the remedial label itself, much of that is the methods in which these remediation classes are taught.

“These kids are so far behind, we need to do more traditional math to get them caught up!”

That is a sentence uttered by someone who doesn’t understand irony, and yet is the pervasive “methodology” (if one can call it that) for reaching students who are “behind.” The same math that got these kids behind in the first place … but more of it. More packets, more computerized instruction, more “I do/you do.”

I’ve heard enough “but these kids lack the basic skills to do complex math” to last a lifetime. First off, if you ever said “these kids” around Kelly Camak, you would probably never be heard from again. Second, the experience of doing challenging, fun, creative math is exactly what “these kids” need.

Jo Boaler shared a video of “these kids” in one of her Summer sessions. The students in the video persisted on a pattern problem for, according to her, 70 minutes. We saw about 5-7 minutes of three students working on a pattern, doing complex algebra, sharing ideas, and being 100% fully engaged in math. An individual problem packet would have not fostered that level of mathematical engagement.

More striking than the video and the numerical total was this:

Kids that have been told they are remedial know that they’re probably getting rudimentary math. Even if you call the course something cheery like “Gateway to Algebra!”, they know. They’ve been told, possibly by their math teachers, possibly by their peers, that they don’t have what it takes to be an exceptional math student.

Here’s a quote from Ilana Horn’s phenomenal Strength in Numbers I shared in my presentation.


So what of “these kids”? What about the kids that fail (and presumably have or will fail) math? They’ve lost credit in math for the year and must spend their entire summer sequestered in Summer School.

Let’s try revamping the Summer School experience. Boaler shared a small clip of a summer math experience that allowed kids to experience math in a rich way, possibly for the first time (though, to be fair, some of the kids in the class hadn’t failed and were there for some other reason, I’m not quite clear). My recommendation would be not to attempt to re-cover 8th grade math in a more rote way, but rather consider this an intervention, a lifeline, for math. Throw out the packets and books and spend a few weeks combing through and giving kids the following:

Summer School is one of those “death knells” for students. Students that get sacked into Summer School often are doomed to repeat it. At that point, cramming a year’s worth of mathematical content isn’t going to bring them back from the brink. It requires an entire rebuilding. I’m so thankful Boaler has offered teachers and students so many lifelines, via a low-cost book, a no-cost MOOC, and of course, a beautifully designed website for introductory resources and videos.

Instead of saying “these kids can’t do X,Y,Z”, let’s try this instead.

Thanks to Fawn and Jo and Ilana and Kelly, and my fellow #joboliebers.


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[NCTMNOLA Processing Session 2] What math teachers applaud

I realize now I’m recapping NCTM in order of sessions that force me to process things. Dan’s presentation definitely forces me to do just that.

Dan’s talk focused on the lessons he learned while playing countless hours of Angry Birds, Portal, Flight Control, and Stickman Golf. I loved how he pointed out that the lessons of Angry Birds’ go beyond parabolic motion. Quadratics are a fun application of Angry Birds, and has resulted in great work, but there are other things at play here. With Angry Birds and these other games, Dan posits these six lessons learned.


Through the eyes of his in-laws, who are now among my favorite people in the world, he demonstrates these lessons adeptly.It was incredible how quickly the hour flew by. I hardly had time to tweet or think before he was on to the next great narrative.

But like I said, I need to process a bit. Please indulge.

Beginning from the end, Lesson 6 makes a beautiful argument for standards-based grading, or lessons on assessment in general. I love the idea of an open middle.

“Lesson #2: The Real World Is Overrated” drew – by far – the loudest applause. Audible hoots and hollers arose from audience the likes of which were not heard at any other point in the conference, let alone the talk itself.

I don’t have a problem with the point itself. I fully concur that the “real-world”, however one defines it, can be overrated. Before you pillory me in the comments, the following is not a criticism of the lesson: it’s an admittance of fear of what teachers take away.

I’ve blogged before about the value of the real-world, be it significant or insignificant, as a way to – among other things – begin thinking about redesigning your curriculum. A non-sugar coated version might read: a way to start giving a crap about your students and their interests. Considering your students’ interests may be the first of many necessary reparations in their introduction to mathematics. And students’ interests are generally not in abstract math, for good or ill.

To hear so many math educators cheering this lesson above all others was dismaying. I suppose it’s not surprising: teachers have witnessed or viewed enough artificial applications of math in their day; they may have even been forced to design some. But I suddenly felt like I was in a room full of teachers who were a bit too excited to have an excuse to stop making their math curriculum relevant and important to kids. I don’t know if there are analogous conferences in other disciplines, but I can’t imagine Social Studies or Science teachers cheering the news that they should’t worry so much about making their curriculum tangible to students the way that the Math teachers did. It was uncomfortable and convicting as a fellow math teacher. What does this say about math educators?

I’ve no problem with the lesson. I have no problem with abstract tasks. Shoot, I awarded Mr. Honner’s equilateral-er triangle problem as my highly coveted Problem of the Year (first and only ever winner!). Dan has provided great abstract tasks that are engaging, interesting, complex: all things I certainly want in my PrBL curriculum.

However, teachers – by and large – don’t and aren’t able to create nifty Adobe animations to stoke student curiosity. We’re not choosing between this ….


[Real World] Super-boring.

… and this:


[Decidedly not Real World] Kinda cool.

We’re choosing between this …


[Real World] A Tuvalabs investigation on amusement park attendance.

… and this:


[Decidedly not Real World] An easily find-able worksheet on box plots.

If this juxtoposition feels a bit straw-mannish, I’d implore you to google “box and whisker plot exercises.”

131112_1Using an example of a snowboarder artificially grafted on to a problem regarding growth in popularity of the sport sort of destroyed my opening slide (but I still stand by it). No, artificially plastering a picture of a snowboarder won’t suddenly get students interested in growth models, but it might give them a contextualized understanding of parabolic motion, the way a basketball shot, and yes, angry birds might.

Again, this is not a criticism of the Lesson #2 (although, one wonders what lesson can be drawn with the successful Grand Theft Auto, Call of Duty, Madden, and other franchises steeped in realism. Or the highly acclaimed and hyper-realistic Gone Home, Polygon’s 2013 Game of the Year winner.) Dan’s not arguing against engaging real-world tasks, I’m not arguing for boring real-world tasks. No one is arguing for crappy tasks, even as our schools are currently festering in them. This is a fear about what teachers, specifically math teachers, tend to internalize and take home to their students. If I were to judge purely by applause, many internalized Lesson #2 so much so that I’m curious if the other five lessons had any room to find purchase.

In Gone Home, the protagonist explores her own house in search of her/your past.

So here I am, criticizing folks who I feel were over-focused on Lesson #2 by over-focusing on people who were over-focused on Lesson #2. Maybe I need to take my own advice. This is why I need to write to process. More of that coming, rest assured.

So thank you, Dan, for (yet another) engaging and challenging talk. I prefer presentations that force me to think, argue, and justify rather than just tickle my ear. I can think of few other educators who has advanced by thinking on math instruction. This presentation only further solidified that.

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