Fidelity in Math

In Algebra 2, the topic today was an overview of functions. Some students were having difficulty understanding the "each input has exactly one output" condition, and the previous example (percent score --> letter grade) just wasn't cutting it.

The follow-up example was much better. The domain was a set of boys' names, the range was a set of girls names, and the mapping was "dating". And, of course, one of the boys had an arrow pointing to three different girls. We discussed why this was not a function, and one student said, "So to be a function, they have to be faithful!". Exactly! I took her up on this, and had them add to their notes: Functions are Faithful! This instantly made sense to them, and this language carried forth through the rest of the lesson. I then added another boy pointing to one of the girls that was already in the list, and asked if everyone was still faithful. They said no, and we clarified things; our new "taken-as-shared" idea was that only the boys (the inputs) have to be faithful for it to be a function. (I mentioned that if all the girls were faithful too, then it is called a one-to-one function, and we'd look at that later.)

It was really amazing - even when we did examples involving decontextualized numbers, they were still very comfortable using the analogy: i.e., that set of ordered pairs is not a function because the 4 is being unfaithful! It even made the vertical line test a breeze to teach.

It's always nice to find something new to add to the bag of tricks.

Happy Tofurkey Day

If you've read my blog in the past, you may have noticed I haven't been writing much lately. I've had some bad health problems over the past few months, which have made it difficult for me to get anything but the essentials done. It's finally under control, and I am feeling relatively human again. I am thankful for strong drugs and medical advances!

So, my lack of energy along with the typical November doldrums have made my freshman Numeracy classes less effective and positive than I'd like. I hope that this mini-break will give us all a chance to recover a bit, and come back to end the semester strong. Here are some of the major things I need to work on:

1) Multiplication Tables.
I decided not to focus on teaching the tables this year to the whole class, as it is a waste of time to the 1/2 to 2/3 of the kids that know them. And, I'm not sure how to do it really effectively for those that are in high school and still don't know them. ALEKS doesn't deal with multiplication tables.

I've given students who need them 12 x 12 tables to keep in their binders and look at as they do things like reducing fractions. I need to find a way that they can actually work on improving, and I think this will be different for each student. Some combination of flash cards, games, incentives, and quizzes will be needed. But how to work this seamlessly into the class? Hmm... Needs more thought..

2) Classroom Culture / Readiness
I've posted about this in the past, and my readiness checker was working very well. But, recently, class hasn't been getting started very efficiently. The students recognize this, and freely admit that, now that they are well into school and have made friends, there is a lot more temptation to hang out between classes and avoid getting into their classroom or seat in the room until the last possible second.

I don't sweat it if we miss out on the first minute or two of class, but more seriously and annoyingly, school supplies seem to be growing ever more sparse (kids don't have pencils or binder paper, and act shocked when I ask them to take these things out!). It's like they buy some new stuff in September, and when they're out, that's it for the year. Also, binders have gotten to be a mess (i.e. holy terror) again - and that's for the kids who still have binders.

I need to come up with an incentive system to get this all back on track - and a way that forces me to stay on top of it.

3) Problem Sets - Singapore Bar Models
I overshot greatly with my initial stabs at assigning students problem sets. My goal was to get them started on longer-term planning, while assessing their ability to use the bar model method to solve word problems. My initial idea was to assign ten problems on Monday, due the following Monday. I would grade and return them, and they would have till the following Monday to do revisions to increase their scores. Sounds reasonable? Well, it was still way too much for them to handle. The second time around, I made mandatory progress checks during the week, to help keep them on track. I got more turned in the second time around, but they were still not very good. And, there is a big problem with cheating. Unfortunately, students don't really understand all the time what cheating is, and it's hard to get them to see it. I want them to work together to help each other; many of them think that copying someone's answers who is "helping you" isn't cheating. This is going to require a lot more coaching. Here are my initial ideas for how to run problem set #3:
- Reduce the number of problems from 10 to 6.
- Continue with the progress checks, for regular homework credit.
- The day the assignment is due, give a one- or two-question quiz with selected problems from the assignment, but with numbers changed. This will help me see which kids actually understand the work, and which kids copied.