Facilitating the smart aleck

I5 Lite Bright

We talk about wanting classes full of critical thinkers. I think this is a great goal and I am generally very high on empowering students to authentically think through complex situation.

But it’s not all pros. Cons do exist. (Nothing… nothing… is all rainbows and unicorns.) Especially when you consider that critical thinking is a skill we are becoming more equipped to foster and practice, while discernment is a skill that is best taught by life experience and generally comes along much more slowly.

So, we need to make sure that we are embarking up the critical thinking mountain soberly. The fact is that “critical thinking” is an easily-transferable skill set — this is why it is so attractive to us. But what happens when the students decide that they want to turn that critical thinking on you as the teacher? When our undiscerning young learners want to practice critical thinking in an authentic setting?

Are you being fair? Are your instructional decisions reasonable? Did your grading of that test make sense? Your work becomes much more scrutinized when you have 25 sharp-minded critical thinkers on your journey with you. And with their lack of discernment that almost goes without saying (students don’t behave professionally), you are almost empowering smart alecks – on purpose.

Let me give you an example of what I mean. This came out of a third grade class.

The teacher is trying to talk about division. The book uses the example of someone who knits glove who makes the fingers of the gloves separately. The question was if the person created 25 fingers, how many gloves could they make. Seems like a pretty straight forward intro-to-division word problem. Except…

One of those kids was a critical thinker.

And she said, “Why would that person make 25 fingers? That’s five gloves. Wouldn’t she make 5 more fingers? Or quit at 20? Why would someone make 5 gloves?”

Interesting questions. Critical thinking questions. And for some teachers that would be a great question. For others, it would sound like the wonderings of an tangentially-on-task smart aleck.

In reality, it might be nothing more than the mental overflow of a student who is really exploring a context they way we taught her to. And the context didn’t immediately make sense, so she asked a clarifying question. That sounds like critical thinking. But, in the ears of some teachers, the word “critical” is in bold. And while, I suppose, it is possible that the student was intending to be disrespectful, I think it is exceedingly more likely that she wasn’t.

We wanted her to engage context. She did. We wanted her to think critically. She did. We wanted her to apply her answer to check for sense making. She did.

But it’s not going to stay in math or science or reading. This skill set transfers, remember. So, you may want to consider each of the following:

  • Making sure your grading policies are properly aligned to the messaging about teaching and learning in your classroom. (Critical thinkers can poke holes in inconsistencies.)
  • Making sure your student discipline policies are consistent with your messaging about teaching and learning and applied equitably. (Critical thinkers see patterns and draw conclusions from them.)
  • Making sure that each of your activities is meaningful and has value toward the learning goals you have for your students. (Critical thinkers tend to be more comfortable making their own decisions about what is and is not worth their time.)
  • Make sure you develop a habit of adjusting your planning based on their feedback (or at the very least, have a darn good reason why you won’t and be willing to be transparent. Critical thinkers ask questions and know the difference between useful information and useless information.)
  • Be prepared to sell your coursework and learning targets and spend some significant energy inspiring and compelling students to engage it. (“Because it’s going to be on the SAT” isn’t a natively meaningful sound byte for many of them. So, if this is the best you got, you are going to have to at least take this argument to the next level. They will if you don’t.)

In general, these are things you should be doing anyway. But, if you are properly fostering critical thinking in your students, you may find that some of the elements of you coursework that you felt were “good enough”, may not stand up to the scrutiny of 25 critical thinkers with still-under-development discernment and very little professionalism (as we understand it) looking to poke holes in it.

Embrace that. The smart aleck in your room might well be practicing the skills that we want him or her to have. The trick is to recognize what we are looking at. They might be trying to be critical thinkers and apply their new skills. Take their effort. Analyze it with them. Teach them how. Model respect.

Throughout geometry, we’d explore proof-writing (which is basically a formalized, mathematics version of persuasive writing.) I used to sell it to my students by saying “Stick with me and I will teach you how to win an argument with your parents.” And I’d refer to that all year. “Let’s look at the argument you are trying to make.”

And rule #0, is if the argument is going to work, the LISTENER needs to change his or her mind when you are done. That means keeping them with you the whole time. That means not doing or saying anything that will shut them off. So, if you are trying to convince your parents (or other adults) to change their minds, you have to present your case in way that won’t shut them down. Now we are talking respect, evidence, cool heads, eye-contact, word choice… That’s proof writing. That’s argumentation.

That’s taking the smart aleck’s skill set and turning into functional critical thinking.

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Proofs (and writing) are difficult

The moment I started to have success helping student really learn how to write proof in geometry was the moment I realized that”The Proof” is nothing more than a persuasive essay converted to math class. It’s disciplinary literacy. And thinking of them as mathy-style essays can help us isolate some of the reasons the students struggle with proof in general. My experience leads me to think that many of the struggles are the same the ones the students experience with writing outside of math class. They don’t understand the structure, they don’t appreciate the value of honoring their target audience, and they don’t understand the content well enough.

Luckily for us, the ELA department has often hit those three points really hard. As math teachers, we just need to help the students bridge the gap.

What’s the structure to an essay? Thesis, supporting paragraphs, conclusion. Or, in math, “Given angle a is congruent to b, I’ll prove that segment a is congruent to segment b. Here’s the evidence I’m using to support my claim. And here’s what I just proved.”

Who’s the target audience? In math, it’s often either someone who doesn’t understand or someone who disagrees with you. That explains why you need to back up each statement with theorem, definition, or previously proven statement. Take nothing for granted or you’ll lose your reader.

And as for content? Well, have you ever read an essay from someone who plain ol’ doesn’t know what they are talking about? The best structure in the world isn’t going to save them if they can’t define the words they are trying to use.

So, when it comes to proof-writing, I think we math teachers need to appreciate that “writing” really is at the core of it, and the better we make that connection explicit to our proof-learning students, the more likely they are to be successful. And perhaps there’s a role for some meaning collaboration between high school math and ELA departments.

And with that, enjoy the latest “Instructional Tech in Under 3 Minutes” discussing, of all things, writing.

Don’t forget Geometry when teaching Algebra

Right now, Michigan educators are trying to sort out the implications of the state switching to endorsing (and paying to provide) the SAT to all high school juniors statewide instead of the ACT as it had previously done. The SAT relies very heavily on measuring algebra and data analysis. This leaves plane and 3D geometry, Trig and transformational geometry under the label “additional topics in mathematics.” The new SAT includes 6 questions from this category. Compared to closed to 10 times that from more algebraic categories.

This comes up in every SAT Info session I lead. Should we just stop teaching geometry? All Algebra? Could geometry become a senior-level elective? If it’s not going to be on the SAT, then what?

These questions reflect a variety of misconceptions about the role of testing in curriculum decisions. In fact, these are the same misconceptions that are driving an awful lot of the decisions that are being made. And in this case, I think there’s more at risk than simply over-testing our students.

It would be a real shame to see Geometry become seen as an unnecessary math class. Because it’s not.

And to illustrate that, I’m going to tell you a story about a conversation I had with my daughter. She’s 6.

She wanted to try multiplication. She’d heard the older kids at school talking about it. So, I taught her about it and let her try some simple problems to see if she understood. And she did, for the most part.

So, I started to not only make the numbers bigger, but also reverse the numbers for some problems that she’d already written down. She had already computed 2 x 3, and 4 x 2, and 5 x 3, but what about 3 x 2, and 2 x 4, and 3 x 5. Were those going to get the same answers as their reversed counterparts?

She predicted no. So, I told her to figure them out to show me if her prediction was right or wrong. To her surprise, she found that switching the factors doesn’t change the product.

“Daddy… why does that happen? It should change, shouldn’t it?”

Translation: Daddy, how do you prove the Commutative Property of Multiplication?

How would you prove it?

The geometry teacher in me thinks about area when I see two numbers multiplied. We can model (and often do) multiplication as an array. It’s just a rectangle, right? A rectangle with an area that is calculated by it’s length and width being multiplied.

What happens if we rotate the rectangle 90 degrees about it’s center point? Now it’s length and width are switched, but it’s area isn’t. Because rotations are rigid motions. The preimage and image are congruent. Congruent figures have the same area.

If l x w = A, then w x l = A. 

Geometry helps prove this. Geometry also helps support a variety of other algebraic ideas like transformations of functions within the different function families. Connections between slope and parallel and perpendicular lines. There’s also the outstanding applications of algebraic concepts that geometric situations can provide. Right triangle trig, for example, is often a wonderful review of writing and solving three variable formulas involving division and multiplication. (A consistent sticking point for lots of math learners.)

As a teacher who spent years watching how Geometry presents such an environment for real, effective and powerful mathematical growth, eliminating it will leave a lot of holes that math departments are used to geometry content filling.

Coke vs. Sprite – One Class’ Response to Dan Meyer’s #wcydwt Video

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Almost three years ago, I highlighted the Coke vs. Sprite video that @ddmeyer made. It is a very intriguing question (which glass contains a greater amount of its original pop) without a clear answer.

Today, I let a class give it a go and here’s what they came up with.

First, they chose to model it with integers. Said one student: “I pretended it was a jar with 10,000 marbles.” 2014-04-03 11.07.36

Then they assigned “a dropper-full” to be 100 marbles. So, the first dropper took 100 red marbles and placed them into the jar with the 10,000 green marbles.

Now we assumed the stirring made the mixture homogeneous which meant that there was a consistent 100:1 green-to-red marble ratio in the right jar. So, when we pull another 100 marbles to put back, (with a little bit of rounding) we pull 99 green marbles and a single red marble back into the left jar.

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A little number crunching reveals (at least in this model) that there would be 9901 original-colored marbles and 99 other-colored marbles in each jar.

My original assumption matches their investigation. It seems (at least by this model) that at the end of the video, each glass is containing equal amounts of their original soda.

Another teacher Chris Hunter (@ChrisHunter36) also battled with this video and created an excellent read about his experiences. Check them out.

 

The Power of Network: Triangle Similarity

I want to share a story that shows the power of an effective PLN.

In a previous post, “proof and consequences: circular reasoning“, I begged for help solving a problem with students struggling to see their own logical crisis that was leading to predictable and consistent problems.

Several people reached out to me with suggestions. Thanks for that. I would like to highlight one specific suggestion that I tried to today and it worked just exactly as the designer predicted.

The suggestion was made by @nerdypoo.

From the comment:

“(i drew an example of this on the train home from work where i drew two isosceles triangles — the first had congruent legs 2 & 2 and the second had congruent legs 3 & 3, so scale factor of 1.5, but the first triangle was an isosceles right triangle and the second had an angle of a bit more than 90. i can send a jpeg if you want!)”

I loved this idea. And yes… I did. I did want that .jpg.

So, here’s a portion of what she sent back.

Dutch Triangle Idea Original

Today, I tried it in class. I began by putting up this image…

Dutch Triangle Idea Starter

… and asking the students to vote on whether the triangles were similar, not similar, or we don’t know. Overwhelmingly most students voted that they were similar. The thoughts they articulated were mainly that they could find the legnth of the missing side (which they claimed to be 3 cm long) and then could use SSS to show a consistent scale factor.

Then I showed them this image and asked them to vote again.
Dutch Triangle Idea

A lot of votes changed. Many changed from “similar” to “not similar”. A few others changed from “similar” to “don’t know”. An additional piece of information revealed an assumption. The assumption was that finding a consistent scale factor in two pairs implied the third. Perhaps an assumption that the angles were congruent.

It was essential that I made sure the students knew that I wasn’t changing the situation from the first question to the second. I was simply revealing information that was hidden. Those angles were never congruent. They simply didn’t know that, but most them assumed they were. But every person who voted that the first two sets of triangles were similar were making an assumption, an assumption that they didn’t recognize before. An assumption that shouldn’t be made because sometimes it’s incorrect.

Proof and Consequences: Circular Reasoning

I’m frustrating my students in ways that I don’t want to. I’m not sure exactly what to do about it. In geometry there’s proof. With proof comes a certain logical structure. Once you know this structure, it is terribly difficult to unknow.

Currently we are dealing with similarity, which involves using SSS, SAS, and AA postulates to prove whether or not two triangles are similar.

Suppose I gave this image to a student and asked them to find whether or not triangle FES and triangle GHS were similar.

Similarity4

Let’s suppose the student divides 54 by 24, and also 58.5 by 26. Both times the student gets 2.25 as a solution. The student assumes this is a scale factor and applies it to GH, finding that FE = 45. The student then divides 45 by 20 and gets 2.25 for a third time. That’s three pairs of proportional side lengths and BAM! Similarity proven by SSS.

Except…

Me, the teacher, is there is tell the student that he or she isn’t quite right. (You see the mistake, right?)

The student assumed similarity before it was proven. Then proceeded to use the assumed scale factor to find the missing side length, which ensured that the third quotient was going to be the same as the first two. This is circular reasoning. They are similar because FE = 45. FE = 45 because they are similar. I have seen this play out countless times.

I have addressed it with little success. I can’t seem to make sense to the students why that argument is weak. It sounds like “geometry teacher says we can’t, so do what geometry teacher says.”  (especially when the very next question asks for FE, which is 45… because the triangles ARE similar…). I can’t stand using my authority as a teacher to enforce a math idea that the students are perfectly capable of actually learning.

I’m trying to decide how picky to be with this. I have a hard time allowing that circular reasoning argument to be called correct, although it is clear that the student has learned a lot about similarity, proportionality, and the structure of a proof.

But the more I push the point, the more frustrated I get and the students don’t seem to be getting any significant gains. I just continue to enforce that “math teachers are just picky like that.”

I am hoping for some help on this one. I’ve tried a lot of things, but that thing you like that works really well for you… I haven’t tried that one. Toss it my way. I want to see how well it works.

For your students’ sake: Don’t stop being a learner

Yesterday, we designed an Algebra II lesson using 3D modeling to derive the factored formula for difference of cubes. As we began to finish up, Sheila (@mrssheilaorr), the math teacher sitting beside me made a passing reference to being frustrated trying to prove the sum of cubes formula. Me, being a geometry teacher by trade decided to give it a try perhaps hoping to offer a fresh perspective. I mean, I was curious. It looked like this:

2014-02-20 13.26.00

On the surface, it didn’t seem unapproachable. I quickly became frustrated as well. Most frustrating was the mutual feeling that we were so stinkin’ close to cracking the missing piece. Finally, Luann, a math teaching veteran sat down beside us, commented on her consistently getting stuck in the same spot we were stuck and then, as the three of us talked about it, the final piece fell in and it all made sense (it’s always how you group the terms, isn’t it?)

Then this morning, it happened again. Writing a quiz for Calculus, I needed a related rates problem. Getting irritated with the lousy selection of choices online, I decided that I needed to try to create my own. And I wanted to go #3Act and after some preliminary brain storming with John Golden (@mathhombre) (Dan Meyer’s Taco Cart? Nah… rates of walkers not really related…) we found some potential in Ferris Wheel (also by Dan Meyer)! Between my curiosity and my morning got mathy in a hurry.

First I tried to design and solve the problem relating the rotational speed in Act 1 to the height of the red car. That process looked like this:

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Meanwhile Dr. Golden found a video of a double Ferris wheel, which was pretty awesome. Seemed a little it out of my league, so kept plucking away at my original goal.

It clearly wasn’t out of Dr. Golden’s league, as he took to Geogebra and did things I didn’t know Geogebra was capable of. (You’re going to want to check that out.)

So, what is the product of all of this curiosity and random math problem-solving? As I see it, these past 48 hours have done two things: Reminded me of what makes me curious and reminded me what it’s like to be a learner.

I have a feeling my students will be the beneficiaries of both of those products. There’s a certain amount of refreshment that comes from never being too far removed from the stuff that drew us to math in the first place. The problem you want to solve just because you want to see what the answer looks like.

And this curiosity, the pursuit, it feeds itself. In the process of exploring that which you set out to explore, you get a taste of something else that you didn’t know you would be curious about until it fell down in front of you. (For example, Geogebra… have no idea what that program is capable of, which is a shame because it is loaded on all my students’ school-issued laptops…)

And this process breeds enthusiasm. Enthusiasm that comes with us into our classrooms and it spreads. I’m not trying to be cheesy, but much has been said about math functionality in the modern economy, how essential it is in college-readiness and the like with few tangible results. Let’s remember that there are kids who are moved by enthusiasm, who will respond to joy, who will pay attention better simply because the teacher is excited about what they are teaching. It won’t get them all, but neither will trying to convince them of any of the stuff on this poster.

Now, who’s going to teach me how to use Geogebra?