The Monsters I Have Created

Our Geometry course includes lots of discovery. LOTS. As a result I have 55 kids who like to have a ruler and a compass at hand when they're asked to investigate anything. Which is totally fine and awesome. When they want to check if things are equidistant or sketch a circle and they're like "can you get out the compasses" my heart sings. I can't even tell you.

So today in an uncharacteristic head-fake toward efficiency over laborious, painstaking discovery, I threw them this:

If you didn't notice, take a look at those measurements and that diagram and you will see the problem. I did not think much about it, because diagrams are not to scale and perfect reasoning on imperfect pictures and blah blah blah. A few years ago my students wouldn't have batted an eye at this, they would have just plugged and chugged, but these guys were NOT. HAVING. IT.

"What is going on with your diagram?"

"That is pitiful, Nowak."

"This is not physically possible."

This made me ridiculously happy. Just that they view the quantities we work with as physical things they can measure and verify and use their spatial intuition on. And they don't see them as arbitrary rules devoid of meaning that they have to memorize because the grumpy lady makes them. Maybe this is obvious and it just means that I'm now a mediocre Geometry teacher instead of the worst Geometry teacher ever. But, you know, baby steps. I may or may not have also shown them Rico Suave on youtube. It is the end of May. We are punchy.

Adventures in Using Math Nspired Materials

TI has some really decent investigations available on their Math Nspired website. The quality is a bit uneven, but there is some good stuff in there. However I keep finding things that look good on first read-through, and seem straightforward when I try them, but don't work so well with kids.

For example, their activity that explores arcs intercepted by central and inscribed angles. I am very grateful that they make the student investigation handout available as a Word document, so that I can edit the bejeezus out of it. Thank you for that, TI.

The first thing I normally do is insert a problem-to-find after every big conclusion.
They usually ask the student to summarize the finding in words, but "summarizing in words" often turns into "write some BS," at least with my tenth graders. They're just not sure what's expected, so they put some ink on paper and try to make it look good. Asking them to apply the concept to a problem right away really helps.

For example, their culminating question for relating central angles to an intercepted arc looks like this:



and then they move on to inscribed angles. I added this:

The second thing I do usually happens after I've run the activity in class. I notice something that could just be arranged much better to promote student understanding. For example, they put this sequence of questions on two different pages:

And you can forgive a student for being like, what? So I took out question 4 and just added a row to the table:

Another thing that was unclear was how they lead them to notice the relationship between a central angle and an inscribed angle, or the equivalent idea, an inscribed angle and its intercepted arc. Why didn't they put in another nice table, so the numbers are written out and staring them in the face? Were they trying to conserve paper? So I fixed that by adding in another table.

So, conclusion... caveat emptor? There is some fine stuff in there. It just doesn't feel like it was tried out on kids before they published it. Or maybe it's just a case of how you can't really ever just adopt someone else's stuff wholesale without modification, but you have to work through it on your own, modify it, try it out with your kids, and modify it some more.



And Furthermore

Dan said:

I’m surprised by how many people have found me and I’m glad people find my ideas meaningful. I guess I’m just amazed at how easy it’s been to have a voice in the semi-anonymity of the internet.

Other Dan said:

If you're just getting into teaching, there are plenty of worse ways to invest your time than in blogging, tweeting, and building your own faculty lounge.

Kate says:

If you are in a room full of math ed bloggers and you don't remember someone's name, try "Dan." 

And also, I want to share this: I received an unsolicited email from a department head this week that read in part

"...if/when you get the urge to move, the math department in _____ would love to talk to you. We always need great people with a passion for improving their craft (and especially doing so in such a public fashion.)" 

So, you know. For the past long while it seemed like admins and hiring-decision types paid no mind to my blogging. But that's changing. People are paying attention, and more importantly, it's people who value the same things we do: continuous learning, reflective practice, learning out loud. I was asked about specific posts on f(t) in the interview for my new job, which not only helped them get to know me, but heightened my opinion of them and their school as a promising place to work.

Lately

Geometry: Wrapping up similar triangles including proofs, finding missing proportional sides, and the mean proportional. I finally gave up on an investigation I wanted to like so much I tried it for three years that I called Hands on the Mean Proportional. They would cut out two congruent right triangles, and then slice one at the altitude to make three similar right triangles, and then use the cut out shapes to investigate which segments were geometric means. The whole thing was just too confusing. This year I modified a more straightforward investigation originally created by a colleague here at school. It worked nicely. On the second day, we solved some by setting up a table of the three triangles and three sides to see which would make a proportion we could solve.

Alg2/Trig: Finishing up statistics, which is basically standard deviation and the normal distribution. A new thing this year was I had students submit three survey questions they wanted all their classmates to answer. These had to go through a few revisions to remove ambiguity (for example, "How long does it take you to get ready in the morning?" turned into "How many minutes from when you wake up until you leave the house?"), and so that we would get numerical answers (for example "Do you use drugs?" turned into "What drugs have you tried?" turned into "How much money do you spend per month on drugs that are illegal for you to possess?") I whittled the questions down to the dozen or so most common ones, and all the students responded. I put each survey into its own Spreadsheet page on the Nspire, and sent the document out to all the kids' handhelds. Boom! I still did the remaining lessons with the required coverage as planned, but instead of made-up who-cares examples, we had all this interesting data to explore. We were able to grasp the meaning of sources of bias, standard deviation, histograms and box plots, and distribution functions in context. I think it really helped the cherubs assign meaning to the concepts.


I know you are all dying to know how my work visa process is going. I mean, it's half of what I think about, so it must be important to everyone else too, right? Well, update: I no longer feel like a rat in an unsolvable maze. Now, I feel like Zeno. Every time I think I am one step away from having all the paperwork in order, I find out there are two intermediate steps that have to happen first. There's not a government official with a stamp at the town, county, or state level that I have not met in person. They are nice people, if a bit harried. Tip: the Onondaga County Clerks like donuts, but the Manlius Police prefer peanut butter cups. The New York State Department of State (yes, that's a real thing) are all on low-carb diets. The Syracuse Police do not like anything.

In other moving news, my new school sent a list of things I should think about bringing or else live without being able to buy for the next year. (I'll save you lots of reading: clothes, shoes, OTC medicine,  skin and hair care products, and peanut butter.) That threw me into a stress tailspin from which my pancreas has not yet recovered. I never thought I would be ordering luggage in bulk, but that seems to be the next step.

Go to the Math Circle Summer Institute

If you are looking for something fun and educational to do with yourself this summer, you should consider the Math Circle Teacher Training Summer Institute. It is run by Bob and Ellen Kaplan (founders of the Boston area math circles, and authors of Out of the Labyrinth: Setting Mathematics Free, along with Amanda Serenevy (founder of Riverbend Community Math Center.)

You will learn about how children can learn math through exploring open-ended problems, playing, and asking questions. There are basically two components. You get to participate in a math circle, working on some challenging, fun, unfamiliar math, and you also get to formulate a math circle and try it out with real-live kids. During down time you can tour the impressive campus, walk/swim, or if you are not mathed out, there are books and puzzles and origami stuff galore JUST LYING AROUND.  People also tend to get together in the evenings (and long into the night) to keep working on the unresolved math from the day's activities. Notre Dame's cafeteria feels like Hogwarts and the food is actually quite good. And VERY cool and smart people attend this thing. This is where I met some of my favorite math-enthusiast buddies: Ben, Alex, Jesse, and Sue. This is also where I got the idea for the Delving Deeper article I wrote for Mathematics Teacher that was published in December.

It takes place at the University of Notre Dame and is only one week long : arrive Sunday, July 8 and depart Saturday, July 14. It is a good deal at $850, which covers both tuition and room and board. Learning about the Math Circle approach will make you a better teacher, mathematician, parent, and human being. But if you don't believe me and want to find out if it aligns with your goals, your best bet is to read Out of the Labyrinth and see what it is all about. I hope you decide to go, and you love it as much as I did.

Squareness, continued

Yesterday there was lots of grappling with what we might mean by the "squareness" of a rectangle, and how one might come up with a way to express squareness with a single number.

Today, I was a little bit grumpy with my first class right off the bat, because first thing this morning a Resource teacher came to tell me a girl complained about her group yesterday not working and talking about weed and making her uncomfortable. Does anyone know how to do group activities and guarantee everyone is on task at all times? Because I don't. (And people who claim they do are lying. Let's be real.) I didn't know how to respond to her. "Oh ok I'll seat them in rows and demand silence every day" is the normal level of sarcasm of my internal monologue, but I'm a professional, so I just said "Okay, I'm sorry, thanks for letting me know."

Anyway. I wanted students to evaluate the validity of some conjectures, so that they will have a vocabulary to use when we start proving conjectures about similar figures. And hopefully learn and/or recall some geometry along the way.

I also didn't want to just drop the idea and move on to something else, even something else related, because I want them to know that what I ask them to do in class has value and meaning, so that they trust that what they are being asked to do is for a reason, even if the reason isn't immediately clear. (I have paid closer attention to this principle this year, and it seems to have paid off...some days more than others. Keep reading.)

I gave them the following on a 1/2 sheet of paper, a piece of graph paper, and access to any geometry tools they wanted to use.

1. Consider the following rectangles with given side lengths:



(a) 5 by 2 
(b) 12 by 9
(c) 15 by 6
(d) 8 by 8
(e) 16 by 4
 
2. Draw them on graph paper. Come up with what you think the ranking should be, from most square to least square, just by visually looking at them.

3. Here are some methods for ranking squareness that your classmates conjectured yesterday. Some of them are valid, and some of them are not. Use each of them on all the rectangles above. List the ranking from most to least square produced by each method. Which methods are valid and why? Please respond to this in writing, right on the graph paper. Any tools you need are available on the table.

(f) The absolute value in the difference in length and width. closer to zero, the more square it is.

(g) Longer side divided by shorter sides; closest to one is square

(h) Find the areas of the shapes and compare how close it is to a square number.

(i) Create a diagonal and measure the angles on either side of the diagonal. A square's diagonal bisects the 90 degree angle in half, so the measure that are closest to equal would be the square/rectangle that is closest to an actual square

(j) Draw both diagonals. Measure the smaller angle created at the intersection of the diagonals. The closer it is to 90, the more square the shape is.

(k) Attempt to circumscribe a circle around the rectangle. The closer you can get all four corners to touch the same circle, the closer to a square it is.

I found today more frustrating than yesterday. I had to be a bit of an ogre about "I'm going to collect this and grade it" because otherwise, I was sensing kids giving up in response to a little frustration, instead of working through the frustration to understanding. I hate being like that because it sucks all the fun out of the room. But honestly I don't know what else to do. I get that there are more fun things to do with similar figures (go outside and measure shadows and find the height of the flagpole! etc), but honestly I find those kinds of activities a bit juvenile for tenth grade - middle school-ish, if you will. Not that they are bad, but it depends on your goals. We are going for more formality in their Geometric arguments at this level, and I don't know of any of those more-fun activities that stand up to the rigor required to work through what we did today. And honestly, once they engaged and understood the assignment, they were good to go. It just took some prodding to get them to that point.

Squareness

I was looking for a way for Geometry students to wrestle with similarity ratios. Daily, I try to escape the "this is how you do this kind of problem" mindset that plagues them, that they prefer, that is a trap we keep falling into. I suspected that Avery's "rank these rectangles according to squareness" problem would be worthwhile. I was excited by what the kids did with it.

Here's the handout I gave out, that I basically copied from Avery:

We had to really unpack 3. "devise a measure of squareness" to get them beyond "it just looks closer to a square." They were able to understand what I meant, but only after I stated it several different ways.

 - Come up with a single number for rectangle E, and a single number for rectangle D, that shows that E is more square than D.
 - Find a way to do a calculation for each rectangle, so that you can compare all the results, to see which is the most square.
 - etc, rephrased a dozen different ways.

Ideally I would like to find a concise way to phrase this for students, that would get them closer to understanding what we are looking for by themselves.

Eventually, lots of groups came up with either or both:
 - shorter/longer, and closer to 1 is more square.
 - longer - shorter, and closer to 0 is more square.

No groups on their own noticed that you could break the subtraction method by choosing uncooperative rectangles (and I don't know how to get them to do this organically.) But after we reported out our methods, and I said "I'll tell you that the subtraction method doesn't always work. See if you can find some examples of rectangles where the subtraction method gives you misleading results." they were interested in looking for ways to break it, and lots of groups found counterexamples.

I asked each group to summarize their method in a google doc...here is the raw output.

We had some really interesting methods, too!
 - Draw a diagonal and measure the angles made by the diagonal and the sides. The closer the two angles are, the more square. (Way to draw a connection to the properties of special quadrilaterals! I pounced all over that like...something that pounces enthusiastically on something. So cool.)
 - Attempt to circumscribe a circle around the rectangle. The closer you can get the vertices to all touching the circle, the more square it is. (We did not have time to delve into this today but MAN, THAT IS GOOD. I would really like to find a way to get them to all think deeply about that.)
 - A somewhat crazy method involving comparing the average of the four side lengths to the length and width separately. I encouraged this group to write their method algebraically, but they ended up with an expression that always reduces to zero. However, it doesn't always reduce to zero when they do it with numbers. Would like to spend more time on this, too.

In a nutshell, I had a great time this morning. Philosophical side note: I have been engaging in these thought experiments recently, often, for some reason, about the idea "Could I be happy...?" (Could I be happy if I won the lottery and didn't have to work? Could I be happy working at Walmart? Could I be happy being a housewife? Could I be happy as a math teacher forever?) and what I have come up with so far is that I can be happy as long as I am spending a non-trivial amount of my time learning something fundamentally new - about myself, about other people, about the way things work... ("fundamentally" implying, well, fundamental...specifically not something stupid like how to navigate the menus in a new app or what Loft has on sale this week.) In order to be happy being a math teacher forever, I need more days like today, where I am getting kids excited about learning. As ultimately cheeseball as that sounds.

Here Are Some Videos

So...I realize the date for this post is inauspicious, but this is really a thing. About a year ago, I was contracted to make an Algebra 1 course in the form of <10 minute videos by a company that didn't ultimately get off the ground. I completed part of a course - up through factoring trinomials. I didn't get to solving/graphing quadratics, rational expressions, radicals, or right triangle trig. I made a halfhearted attempt to finish it up on principle but MAN, there are so many more interesting things to do with my time. (And also necessary, boring things taking up my time, like getting a work visa for Argentina, a process not unlike one of those unsolvable mazes designed to induce frustration paralysis in laboratory animals.) I retained the rights to the videos so I put them up online with a Creative Commons license, because I don't have the time or inclination to do anything else with them for now.

One of this company's innovations was for the videos to ask questions and pause and wait for a response. So, within each there are questions with a short pause before the answer is revealed. Each one took me about a week of time outside of school to plan and record and cut. The quality gets better within the first few, due to me learning some things, and also acquiring a decent microphone. Also... I don't think anything like they are the answer to any real or perceived problem in math ed. I don't think they are the best way for anyone to learn (the first 5/8 or so of) Algebra 1. They are the best I could do with the format and something I did for some extra money because, hi! teacher. I'm not un-proud of them, but I know they are imperfect. I'm not inclined to go back and re-record or fix anything, so feel free to criticize but I am not likely to react.

But, they're a resource and maybe they can do some more good than they were sitting on my hard drive. I hope someone finds them useful.

What to Do with All the Technologies

Among people who know me professionally I have a bit of a reputation as a technology person. It might be because when people walk by my room, there's usually laptops and calculators all over the place. Or it might be because I put it all over my CV because people like that sort of thing. Or it might be my third robotic arm. Who knows.

But when people talk to me about the technology I have to constantly Reframe the Issue and explain how I'm not all pro any technology for its own sake. You don't go, "Oh here's this cool technology let me shoehorn it into my classroom." Instead you go, "I think I have thought of the best way to teach this, and it would be impossible in an analog world, but I know enough about the technologies to realize this idea." You don't go to a twenty-minute inservice about xyz.com and go "I'm going to make an xyz.com lesson." You use xyz.com for your own purposes, or you suspect its utility and put it in your back pocket, until your awesome instruction idea needs xyz.com in order to exist. Your lesson is the fuel and xyz.com is the oxygen.

So here is a lesson that would not exist without dynamic geometry software and classroom polling. It does not matter what sort of dynamic geometry software. I've done it with Sketchpad and I've done it with Nspire and next year I'll probably do it with Geogebra (damn, that sentence makes me sound disreputablllllle.) But I don't think you could get the same effect without the technology. Maybe you could give them diagrams on paper and rulers and protractors, but there's no way to make those not static, even if there's a lot of them.

We are discovering additional properties of special kinds of parallelograms. So everyone starts with a sketch of a parallelogram (that you give them, or in our case, that we constructed the day before.) And the children are in groups. Each group gets a different set of questions to explore. Writing these exploration-y questions is a bit of a dark art. You don't want to send them on a chase of the wild-goose variety but you don't want to set them down too much of a pre-defined path either.

Example: Group 1. Start with your sketch of a parallelogram. Construct both diagonals. Drag points around until the diagonals are perpendicular to each other. You will have to decide what to measure so you can be sure. What is the name of the special kind of parallelogram you get as a result of perpendicular diagonals? Now you need to find at least two NEW properties of this shape. They must be NEW properties that are NOT properties of any old parallelogram. You will have to measure some stuff. If you can't find two new properties, keep staring at it until you get a good idea. Write them down. Verify them with measurements. See if your group members agree. Drag vertices to make a different parallelogram where the diagonals are still perpendicular. Are your new properties still true? Challenge: It is possible to construct a quadrilateral with perpendicular diagonals that is NOT a parallelogram. Open a new page and construct such a shape. What other properties does it have? 

They get ten or so minutes to play around. It is helpful to give them some verbal marginally-hysterical (at least in my case, they always feel slightly deranged) instructions like "It is not a square! Nobody has squares! The correct answer to any question is not "square!"" and you also need to run around like a crazy person and make sure everyone knows how to grab points and drag them around (you might as well admit that screaming about rhombuses to a roomful of 15 year olds makes you a little bit of a crazy person.) Because you KNOW there are at least three maybe four kids who try one thing for half a second and it doesn't work and then they will sit there and stare at their desk for twenty minutes unless you interrupt that little party.

Once they have had ample time to explore, and the faster workers are getting bogged down in the challenge questions you put at the end, you ask them to respond. I have queried each group verbally in front of the class, one by one, and that doesn't work so hot. Usually in a class of 30+ hardly anybody likes to talk in front of everybody. Better... send them a link to a Google Form where they can type an answer to each question. Or come up with your own response system that your existing tech will support. This year I used TI Navigator polls. They are annoying because the TI's don't have a qwerty keyboard. (Lesson number 9,125,698 learned the hard way.) 

Once all the groups have had a chance to report by whatever method, then you write down your notes of properties of rhombuses and rectangles. And then you give them a bunch of problems to find missing measurements in rhombuses and rectangles. They can reason it out now. You don't have to show them example problems first. It feels like kind of a magic trick.

THAT's what the technology is for. 

Nerding Out with the Dictionary

In my Spanish studies, I recently came across radical-changing verbs. When Mark, the teacher (I'm using, among other sources, the excellent Coffee Break Spanish) first said "radical-changing" I first thought WHOA, RADICAL, they must be really extremely different. But here's how it works. In conjugating a standard verb, the stem stays the same, and the ending changes. For example, bailar - to dance: bailo, bailas, baila, bailamos, bailáis, bailan mean I dance, you dance, he dances, we dance, you-all dance, they dance. The stem stays the same and the ending changes. But in a radical-changing verb, there are spelling changes to the stem as well. For example pensar - to think, the "e" changes to "ie" sometimes: pienso, piensas, piensa, pensamos, pensáis, piensan mean I think, you think, he thinks, we think, you-all think, they think.

Why this is interesting to me, is it's another clear example of "radical" referring to a root. "Radical-changing verb" is referring to changes in the root of the word, as well as the ending. With numbers, an example is the square root of a number, like how "radical 9" means "the square root of 9." The symbol for which is , and that symbol is derived from a stretched out "r." The rationale I've heard for this word choice is, picture an upright square resting on the ground:

If the square has an area of 9, the root, the part resting on the ground, has a length of 3. The square root of 9 is 3.

So I went and looked up "radical," and behold the first definition:

rad·i·cal  adjective /ˈradikəl/ 

(esp. of change or action) Relating to or affecting the fundamental nature of something; far-reaching or thorough

not, as I might have guessed, extreme or far-fetched, which are included in the definition, but farther down.

Anyway, you all probably knew that already. But I thought it was a neat connection.