Over a late night dinner at B-Dubs last night, my friend Dave, and I were talking a little about Algebra 2 and how much it just made us want to kill ourselves this year. We really tried. There were 4 of us that met a lot and discussed targets, essential questions, the ACT, and tried to create a fluid coherent Algebra 2 curriculum.
My biggest problem with the whole endeavor was that in the end the whole thing just turned out to be a list of skills much like this one. We expected C students to be able to do the straight skill work. We expected B students to be able to things like resolve an argument. Something like “Dave claims that bla bla bla and Aaron claims bla bla bla: who do you agree with and why? Then we expected A students to be able to use the skills to solve some sort of problem that we had not considered at any time during class. It might be a unique application or maybe we flipped a problem around so they were looking at the back end and had to figure out the front end.
I liked the way we assessed but getting from assessment to assessment was a huge chore. It just seemed like there were too many topics to make decent connections with. But my biggest issue was that it was hard for me to justify a lot of the content. Who needs to know all of that crap and why? Really? Who cares about complex solutions to quadratic equations? The ACT? Then who? I just don’t have the acting skills to pull some of those topics off. BTW, see Shawn Cornally’s latest post for more.
What Dave and I have spent some time talking about is why we don’t let the technology we have do the grunt work for us. If there is a need for complex solutions to a polynomial function maybe my average 11th graders don’t need to know how to find them by hand if they can do it with the technology. Maybe they don’t need to factor anymore because the technology can do it. Solve a quadratic by hand? What the hell for?
The interesting thing to me is thinking about what we do with our time if we aren’t learning how to solve systems of equations by elimination. What if we were solving complex and interesting problems? Problems that you would never get to in the 11th grade curriculum because you are hamstrung with skills and procedures. We should be using technology where it exists to do the grunt work for us so we can get on to the more interesting and important things.
Two final thoughts:
I’m not sure yet what the complex and interesting problems look like because we have never had to think this way.
I’m convinced that teaching the students to use the technology has a whole lot more benefit (mathematically speaking) than we think.
In her last session, Gail mentioned using technology as a tool. We need to help students:
Confront their misconceptions.
Structure their knowledge.
Identify and clarify key concepts.
Play with ideas.
Students learn if:
They are actively involved in choosing and evaluating strategies.
They explore contrasting cases and notice differences.
To get students to do these things, Interactive Math Boxes on the TI-Nspire are a great way to start. The first thing we do is create a set and then evaluate a function using those set values:
The discussion then will center around how we know that the two sets that came up equal are actually algebraically equivalent. We would chalk-talk a little about what we could do to prove equivalence. Going back to the Nspire then we can have the students find another expression that is equivalent and the Nspire will give instant feedback. So pick a misconception (in the above example we were getting at the idea that some kids might add 3+4 first before distributing) and build a math box to confront it.
For the next example we made a slider and entered an expression:
Can I make it zero? Can I put in a y to make 22? Can I make an equivalent expression? These are all things you might do with the slider model.
Next we used an interactive math box to roll some dice:
In the last example we looked at, we considered how weird it would be to walk into a Starbucks and find that 31 out of the 52 people there were female:
One final video showing the same example above but using the randbin command:
Don’t trample any flowers or rip apart any trees or bushes. All containers are easily accessible.
Replace the container exactly where you found it.
Caches are located between Kern and Bauer Lodges.
So, this was fun. We were given a set of problems to work which gave us coordinates for 8 different caches. All of the solutions led to coordinates which could be input into a GPS device. The units we used did not have an electronic compass built in so we had to be moving in order to be pointed in the right direction by the GPS. Our biggest problem was organizational – we just ran out there like a bunch of giddy little girls and never really checked off the ones we found and we didn’t keep all the coordinates on the same sheet of paper.
My colleague and I talked afterward about the fact that you could really tailor this kind of thing for anything you are studying. Give the kids some drill and kill problems and just make sure that you somehow make the solutions into the coordinates you have hid the caches at. Jim used old plastic pill containers or Altoids boxes that were covered in duct tape. Some he ran a wire through so he could hang them from trees or bushes and some he put magnets on so he could stick them to something metal. Inside the cache he had candy and a little sign in sheet. I would imagine that you could even include a clue to the next cache.
Jim offered up these web-sites as useful to check out:
Day 3 of Green Lake is a completely perfect day. The Lake is beautiful.
Glen teaches at BSU in Bemidji, Minnesota. I just noticed his shirt. It says WE ARE # .9! (with a bar above the nine). If you Google Glen’s name his site will be the first one that comes up. Glen had us look at some distance versus time graphs and had us try to walk them while the CBR kept track of our movements. The before and after shots look like this:
As I look at these I’m thinking about linking the discussion to transformations for the times when someone walks the graph in the correct shape but they started too close or too far away.
In Stage 2, I was facing away from the graph so I couldn’t see it and my 2 teammates described what I had to do. We started the graph and they walked me through it.
Here students start to use the language of slope. “Walk backward toward the ranger about 2 meters in 1 and a half seconds.” The beauty here is that the math serves the conversation. It is compelling and necessary to start using mathematical language. As a side note Glen mentioned that for the students he teaches (in a higher education setting) it makes the most sense to use Excel. He sees the kids getting more algebra benefit (than from, say, a graphing calculator) because they are labeling the columns in Excel and also seeing that a given input produces an output.
That was about it for the “slope” portion of the session and we spent some time talking about some recent research by University of Minnesota about NSF vs. traditional students that tracked them through high school and college. I’ll try to find this and link it up here. We also talked about the fact that Calculus should not be the holy grail anymore, but rather statistics. Glen mentioned that 80% of college students will need to take at least 1 to 3 stats classes in college for almost all professions. He also related a story about talking to the Engineer of the year in MN who hasn’t done any calculus since her sophomore year in college but does statistics every day.
If you send Glen an email he will send you an Intro to Mathematical Sciences Class Notebook which he and Dr. Derek Webb worked on. Ask about the islands activity also.
See also the Discrimination or Not? activity from the Navigations in statistics series.
We’re looking at tools that Google offers and how they can integrate into the math classroom. Google Docs is up first: word processing, spreadsheet, forms, and presentation tools. Useful for graphical organizers like for the properties of quadrilaterals. Students can share and edit in real time with each other. When it is done it is clean and easy to save and/or print. Sharing features allow you to make it completely public or share it only with people who have the link to it.
Using the spreadsheet for collecting data is nice also. There are a lot of ways to display the data and of course it is still dynamic and collaborative. Google forms are a way to collect data by creating questions with possible answers. The answers are then compiled and displayed for you in a spreadsheet. Flubaroo is a tool that works in conjunction with google forms that will check student responses against predefined acceptable responses. This is a video demo of how it works.
Google Earth and Sketchup are also on the list as well. Here in Wisconsin we have access to the Pro version of Sketchup. Using the ruler tool in Google Earth allows students to measure buildings or whatnot and find their actual measurements. Real World Math is a site that is exclusively for use with Google Earth. Sort of a project based learning site with some practical applications.
Google Squared is a tool that helps you quickly build a collection of facts from the Web, for any topic you specify. It can be used to compare similar objects by creating a grid or square which can be customized and shared. Try going to the link and typing in “quadrilaterals”. Here is a square I created by just typing in the word “mathematics”. Seems like there is some potential here but I’ve got to play around with it some more.
Well, here we go. The Lake was beautiful on the way in and I am very excited to start another year at Green Lake. Apparently we will be the first group in WI or IL to use the new CX (color) version of the Nspire.
Ray’s main points:
The TI-Nspire is a computer, not a calculator and you have to adjust your thinking.
The relationship between Documents – Problems – Pages is key.
Anytime something is bold, it is defined.
Escape and Tab are key. If one of them doesn’t do the trick, the other one probably will.
Look at the corners of the screen to give you an indication of what is happening and what mode you are in.
In list and spreadsheet pages, a list has a name and spreadsheets are blank.
Adding a “problem” is only done one way (Doc –> Insert –> Problem) because it is a special thing to do that requires you understand item 2 above.
Adding a “problem” clears out all variables and you can start fresh.
delete, delete, delete will clear your history or graphs or whatever you’re looking at
Geez, let’s do some math already!
Ok, not quite to the math yet but here are some quotable quotes:
“The commutative property of exponentiation. The one property that every kid knows”
“The kids won’t always believe me, but they will always believe their calculator.”
Here’s a little math for you:
I’m doing this on my computer but it is exactly what the kids would do and see on their TI-Nspires.
Finally! The color CX’s are here.
They are quite a bit thinner and completely rechargeable (No AAA batteries!).
How about importing a picture and fitting a curve to it:
Lastly then, I’ll show you the last activity that we did. I’ve left out any discussion of the pedagogy of the activity and focused on what the Nspire can do. You can determine the value and implications for yourself. Cheers!
