foothold situation

I’m reading 5 Practices for Orchestrating Productive Mathematical Discussion and ran across the term foothold. Now, I watch a fair amount of sci-fi so this typically means alien possession within an organization to me so I did have a moment of confusion. That’s definitely not how it’s meant here.

While laying out the case for good task selection as a way to promote equity in the classroom, on p19 Smith & Stein note that

“Once a student has a foothold on solving the task, the teacher is then positioned to ask questions to assess what the student understands about the relationship in the task and to advance students beyond the starting point.”

I really like the term foothold used this way. “Will this task allow all my students to gain a foothold?” Isn’t that a nice question to ask as you plan out tasks for your kids?



Today’s chart that has me thinking

[This whole thing is a ramble, but it’s an oddly accurate representation of what my brain does as I try and assimilate new information and ideas into the current structure. You’ve been warned.]

I mentioned a few posts back (and yes, I’m still thinking about elevator questions–trying to finish a book first that’s helping me thinking about them) that I’m reading The Art of Explanation by Lee Lefever. Throughout the book so far he’s been making use of the following diagram as a way to plan explanations:

p78. I'm reading an ebook so it may not match up.

p78. I’m reading an ebook so it may not match up.

One analogy he uses that I like is that if someone needs to change a tire, they probably already have the big idea down and are over on the right side of the chart needed an explanation of how. On the other end of the spectrum I think about the students I’ve had in my class. How many times did I start with “how” before establishing why? And I’m good at “how”! I can lay out steps like a pro with three examples and then practice time but if that kids is still over at the start of the alphabet, then I suspect that  “how” is just going to cause them to tune out or teach them how to mimic.

The “why” bit means a lot of things to me that I’m still poking at. Why this method? Why this problem? Why should I care? How do I, as a teacher, present problems in such a way that those “whys” are answered? Off the cuff, I think different tasks are going to do this in different ways. Some will, to borrow Dan‘s phrase, perplex the kids. Others will be topics with human interest (“real world” applications). Whatever we choose, we need to get an answer to “why” that a student will follow down the alphabet into the weeds of “how” and that’s where all sorts of mathematical fun can be had.

So I’m still processing, but I’m liking the image as a way to think about where students will start before designing lessons around specific problems. Yes I’m recognizing that it’s like a fancy way to say the kids need to buy into what you are doing, but I like thinking about ‘givens’ from new frame works and the why/how understanding spectrum depicted in the picture is new for me.

Lastly, I’ve worked with a lot of struggling kids who had a hard time getting to “how to be an active, engaged math student” because they were stuck on why they would want to be one and why they would want to worry about doing well in a math class when there were other things going on in their lives. I can’t address every why my kids come in the door with, but hopefully I can not add to their list and give them a place where more understanding can happen and give them something to be proud of.


in which my mind is blown by polynomial long division

My love of polynomial long division is already documented, but now I’ve found more reason to love. To give some context, I am working on plans for my Intro to the Calculus Unit for my Precalc kiddos. I do it right at the end of the 1st semester followed by an Introduction to Statistics Unit because I want my Sophomores and Juniors to make an informed decision about whether to take AP Calculus or AP Statistics next year.

The Calc unit is fairly straightforward: here are some of the big ideas, hey lets learn some notations, oh noes division by zero?!, take some deep breaths it’s just a limit stop freaking out type of stuff. After doing some work with the delightful Chris Sangwin, I have chosen to play around with the following piece of information (and hopefully some geogebra modeling) to take polynomial long division to a new level in precalculus:

Given a polynomial P with degree n ≥ 2, the remainer when P is divided by (x – a)² is the equation of the tangent line to P at x = a.

Boo Yah. Go try a few examples–it’s rather fun. I am going to set up the kids w/ the long division and then have them graph the original equation in geogebra along with the remaider equation and then have them write down what they notice. I have yet to come up with anything practical as it’s not all that useful for finding extrema, but I need to think about it some more.

Oh, and please feel free to shoot me down if this is wrong. My working knowledge of Taylor Series is rough at best and that’s where this idea comes about.

in which i present about why i only twitter with math teachers

Today I did my first ‘real’ presentation of sorts to a professional math meeting, NWMI. I was asked to talk about twitter and blogging and how I use it to stay connected professionally. The following is an abreviated version of my talk along with promised links for those looking to get into the edutwitterblogosphere. Gesundheit.

I chose prezi as a presentation format. You can find my prezi here. I tried to keep the text minimal and just use the presentation to make points and show a few graphics, so I’m not sure how easy the prezi will be to follow without having seen the actual talk.

the highlights

I stumbled onto blogs in a move of desperation my 3rd year of teaching. I’ve talked about this before, though, so I’m not going to belabor the whole story here. Suffice to stay, I found online professional development that wasn’t about clock hours that I could partake of on my terms. A few summers later I got to go to PCMI and Sam Shah got me into the world of twitter and blogging myself, instead of just lurking. I hadn’t really looking into twitter outside as something I would see occasionally on blogs or mentioned in the news media in a joking context. Sam also gave a presentation on twitter and blogging at PCMI that you should really go read/watch.

Twitter has been an amazing experience me both professionally and on a more personal level. I have people I consider friends (as in, I would let them sleep in my guest room if they needed a place to stay) all around the country and several outside of the country due to twitter. I find this very cool.

the links

During the presentation I showed a variety of links of places to start for new twitter/blog math people.

First off, if you want to really leverage twitter, you will need to go and make an account. Once you have a twitter account, check out this great list of tweeps compiled by @Fouss.  You can follow as many people as you like and you can make lists if you only want to read certain things at certain times.

I also recommend checking out the hashtag feature. Hashtags are a way to tag your tweets for a specific channel of conversation. For example, #mathchat is something you can search for and see just tweets about math education. #anyqs is another great channel of people sharing videos/pictures of mathematics in the world. Check out Dan Meyer’s post for a great explanation of this channel.

For blogs, I use WordPress, but there are other sites out there. If you want to find blogs to follow, you can start with the blogroll I have at the right of my page. From there, follow their blogrolls, and so on and so on. There’s really all I did in the beginning and it’s lead to a pretty filled reader that I turn to for inspiration weekly. I’m not as good about blogging as I want to be, but I’ve always felt when I do get a post out there it’s really helped me think about the issues and then any comments I get are just icing on the reflective process I’ve already taken part of.

One other thing I shared out was the Virtual Filing Cabinets people have created. Sam Shah has one of the larger ones I’ve come across. Bowman in Arabia even shares out some sweet Geogebra resources in his. If you just want a place to start reading some quality posts, go check out Riley Lark’s Conference on Core Values from this past summer. The topic was about what is at the center of ones classroom and posts came in from all over.

And if you’re wondering how to join in the conversation, don’t feel like you have to do something like write some amazing blogpost and submit it to a virtual conferences. Just get out there and post on peoples’ blogs. Give feedback. Ask questions. Be respectful. In short, come enjoy being a professional with people who are as dedicated to this profession as you get. It’ll change how you teach in the best ways because there’s nothing like knowing your tweeps have your back, good days, bad days, and all the grey in between.

In which my folded plans are turned into bad kirigami

What do you do when a kid cheats? Or worse, steals?

I am away at a conference right now (it’s super cool, but that post will have to wait for the week to be complete), and I made sub plans for 4 days. I had a folder for each class for each day. Seating charts. Annotated keys. I left cocoa for my sub as a thank you.

Tomorrow the kids are taking a midterm. I know it’s more than awkward to have them take a midterm when I’m not around, but they know exactly what the problems are and have spent the past two months working with them. Each day was planned out with review problems like ones they had seen before to practice and a full key to check their work with at the end.

But that’s not the distressing part of this tale.
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