I asked today in class. Probably less than half of them have been doing their homework. I think I'll ask to look over some of the homework binders, so I can get a sense of how well they're choosing problems.
I'm teaching 3 sections of Beginning Algebra; they need review on fractions, integers, distributive property, and order of operations, which I'm now calling FIDO (instead of 'chapter one'). Last week on day two I gave them a problem I've given for years now. You've bought 3 lots at a campground with your partner. You've broken up. You now own 1 1/2 lots. Each lot is 2 1/3 acres. How much acreage do you own? I require them to start out by drawing a picture, and they work in groups of 4. We get to think carefully about fractions.
We did some more fraction stuff last week, and today's warmup was a nested fraction challenge problem:
Can you all guess the most popular error? I hadn't seen it coming, but I should have... Lots of students wanted to cancel the 1's. When that was the first suggestion of how to start, I started to say something, and caught myself. I think not many noticed.
We talked about what belonged on top if the 1 was canceled. A zero? No... And it looked pretty strange with nothing on top... The first class had someone tell me that the problem was with the addition. I have in the past explained about not canceling when there's addition and subtraction in the fraction, but students all the way up through calculus keep doing it. Way too tempting... Perhaps the fact that they were trying to figure this out together will help them resist temptation? In the second class, when no one could tell me what was going on, I left it on the board for us to ponder. I did not give them the 'answer' (that the addtion is why you can't cancel). We finally got it.
I was excited about how it went today. We spent so much time on this weird problem that we didn't get much time for integers. There will be time tomorrow.
Posts
Do most people/teachers attribute the cancellation error to a fundamental gap between mathematical technique (simplifying fractions) and mathematical reasoning (multiplication as the inverse of division and associativity)? Or is it a problem with not understanding notation and order of operations?
ReplyDeleteRe: Roundabout way of getting math formulas into images
ReplyDeleteIf you have the symbols in word, you can also make a jpeg using the Print Screen (Prnt Scrn) button and then pasting into your favourite image editor and cropping it to size. There's an even simpler method in MacOS for screen grabbing but I don't remember it.
If you get Math Type (I finally broke down and did) there's an export as option that takes your equation and makes it automatically into a .gif for you. I just discovered this over the summer--I think it's likely I'll be using it this year.
ReplyDelete(Yep, print screen would be even easier. Next time.)
ReplyDeleteHao, that's a good question. I sure don't know the answer. What I do know is that students don't think much about why 'canceling' works; they just do it whenever they think they can.
I think they've focused too much on procedure, and if it looks like canceling would simplify their expression, they do it, whether it makes real sense or not.
Sue, I know what you mean about the doubts! I'm doing the same thing with 1st semester calc. Not collecting homework, not requiring a particular book. I listed text homework problems that are good examples, but I feel a little blind without their homework to look at. I am doing sbg with them and I am having doubts about that, too. After two assessments of their precalc skills there are still many with scores of 1 or 2 out of 5. I'm committed to keeping this up, but my excitement about it is wavering. Maybe I needed to lurk and learn longer before jumping in? Someone please tell me this is normal and I will get more confident about it if I just keep going and pretend I am positive it will work!
ReplyDeleteHi Kelly,
ReplyDeleteI'm asking Shawn Cornally to come over here. If you need someone to support your efforts, I think he can do that.
I had a great day yesterday, and I've reminded them all they need to get a text. I'm feeling better about it.
If you'd like to chat by phone or email, I would enjoy working together on this.
Tomorrow I test on the pre-algebra skills for Fractions, Integers, Distributive property and Order of Operations.
It's not very exciting stuff for them. (Ditto the pre-calc skills you've tested on.) If they don't test well on it, I won't be too surprised. I'm teaching college, and have said they have to come to my office to re-test. I won't worry if this one takes them a while to fix.
Often lower level skills can be learned in service to a harder, more exciting problem where they're needed.
Thanks Sue, I teach college too and will be having them reassess in office hour. I think the sbg philosophy applies, I am just a little worried about the actual application of it. I really appreciate being able to use you and all the other sbg'ers whose blogs I have been reading as my support system! The theory of it was very exciting as I read all summer. Reality is messier. ;-)
ReplyDeleteAgreed. I am very lucky to be teaching three sections of one course (and nothing else) this semester. So I have more prep time than usual. I'll need every minute of it.
ReplyDeleteSue:
ReplyDeleteThis all sounds very exciting. My only advice is to not worry as students may seem to flounder. You're trying to break a decade or more of counter productive habits where homework and practice have been unfriendly bedfellows. Stick with it and make sure your philosophical basis is strong for what you're implementing.
The other thing I've learned after a few semesters is to make it clear what homework they should be doing, if they are choosing to do so. I then spend a lot of time working practice problems that students ask me to do during class. This validates their work without making it about points and bringing all of those negative aspects into play.
Good luck! Let me know how things play out. I've never seen SBG done in a college algebra setting.
=shawn
It's college, and it's algebra, but it's not college algebra. ;^) (Beginning Algebra is the college title for a course that's pretty similar to Algebra I.)
ReplyDeleteThanks for your thought, Shawn. I am tempted to give them 'points' for doing homework, but I think I can resist that.
Kelly, I gave that test today, and my students may have done worse than yours. I'm scoring on a percent scale, and if there's hope for an 85%, I'm willing to give partial credit. Most students did badly enough that I didn't have to think about what partial credit to give.
The chance for a retake means there's not the heavy, oppressive atmosphere there usually is after a bombed test.
Sue,
ReplyDeleteNew here to your blog (came from Sam Shah's site) and I have an opinion on the canceling business. I don't know how much it helps, but I always try to be really strict with myself and my kids about vocab. I try to say things like 'Oh, we've got x divided by x here' rather than saying 'Oh we can cancel these x's here' It is way too easy to confuse the identity elements of addition and multiplication in these situations.
On a related note, I am new this year at my current high school and I am interested in starting an SBG conversation in my department. Any advice on a concise primer of these ideas for me to share with my colleagues?
mrdardy
Check out Think Thank Thunk. Here's his SBG section.
ReplyDeleteRegarding canceling:
I hear a lot of students refer to 'cross canceling'. I don't even know what they mean. That term seems like a recipe for disaster.
But 'canceling' seems like an accepted term for a clear process, so I use it. However, I often talk about why it works, and show the numbers that become a 1, that we don't need to write any more.