The first part of this school year has been pretty gratifying for me. My students who moved on to new teachers and new classes are all faring well, much better than they were doing at the same time last year. For me it is like seeing the light at the end of the tunnel. Although there are completely new students this year with whom I need to repeat the struggle, I can see that it does take a full year to bring a kid up to speed to become the student that I want them to be (something that my old Math Coach used to always say to me.... words of the wise!). Sometimes, as an impatient person, I want to see immediate results. But, it is just as well to see it a full year later, that the kids are more confident, more independent, more responsible, and that they feel that they can actually tackle mathematics.
More and more I am thinking about the utter importance of confidence in a student's mathematical success. I have always tried to think of the magical formula for teaching, and more and more I believe that confidence is it. There is a high correlation, I find, between a student's general personality and their energy in learning math. If the student is generally confident (and especially in math), they can take risks, challenge solutions, see a problem through different angles, and if they fall behind they can do extra work on their own in hopes of catching up. If they are not confident, then they often try to do the minimum, cannot persist through difficult problems, and cannot begin to enjoy the process of thinking about math. Of course, I am not saying that this erases the need to bring in interesting ways of teaching kids, but confidence makes such a huge difference, and if you can find ways to increase a student's confidence in math, the payoff is in fact doubled in the long run.
Concretely speaking, I increase kids' confidence levels in three ways: I explicitly teach study strategies for mathematics (and we spend class time making a few flash cards, for example, with problems on one side and solutions on the other side, or problem type on one side and strategies on the other); I help kids identify/fix basic skills they are still missing from years prior (such as integers or multiplication) that are causing them frustration in current material; I offer re-quizzes and re-tests as much as the kids want, so that they can feel successful about having gone back and mastered an old topic.
Enough rambling and philosophizing. This year, thus far, I have been fairly successful with teaching linear functions and quadratic factorization to my "low" Grade 9 students. Again, I find that the confidence building is a huge element of my classroom (as many of my students come from backgrounds where for years they have felt largely unsuccessful with math, for one reason or another), but I also tried to layer the concepts this year in a way that eliminates confusion as much as possible. Some specific feedback I have had from parents of students in this class is that 1. Their child is really feeling much better about math this year, 2. They can now actually enjoy math. Some parents are actually less concerned about the learning results, and more grateful that math is not the class keeping their kids from wanting to go to school anymore!! From my side, obviously confidence is necessary, but more gratifying is to see the increasing independence in their work, and in their ability / enthusiasm to discuss and to help each other -- something that they simply were not able to do at the start of the year.
Since I have worked on these units so carefully, I wanted to put them on here and discuss the extra scaffolding that went into them. They are not ideal for your regular lessons, probably, but if you find yourself in the peculiar situation of teaching math to kids who have no prior retention AND who dread mathematics, maybe this would help to see how I scaffold things for my kids.
here), I didn't want to make the assumption that kids knew what linear functions looked like. Actually, I thought that since our kids were coming from all over the place, maybe I shouldn't assume that they knew how to graph points either. So, we started the class with going over some key words (see the box) and copying definitions -- this was a strategy for reaching my EAL learners, as well as the highlighting of key words in each question. Sure enough, on this first day I discovered that we had to review how to graph points (x, y). Problem #8 and #9 introduces the idea of line equations, and ask the kids to look for a visual connection between an equation and an already graphed line, and to begin writing their own equations based on their observation. Problem #10 was a bridge between two different representations (list of points and a table), which again I didn't want to assume was obvious. By the end of the first day, kids were expected to be able to graph lines and to write equations based on a table or a graphical line.
The second day of lines (see above, but actual file is here), we again started the class with introducing some key terms (again for our EAL learners). Then, the worksheet started off with review problems very similar to the problems from the previous day -- this is a confidence builder. I wanted my students to feel successful, like they have by now mastered something from a previous lesson, or at least that they had the resources immediately available to recall those skills. Then, Day 2 is the reversal of Day 1 -- they needed to now go from equations to tables, in order to really solidify their understanding of the different parts of an equation. (By the way, I don't teach the phrase rise over run because I don't think that has any real meaning to a normal child. I always say that the rate is what is happening over how long it takes, kind of like miles per hour, so a slope of 1/3 means "the value goes up 1 every 3 stages"). Anyway, problem #4 and #3 are related, because when they start graphing from line equations, I still recommend that they always make a table first. (And this has shown to work like a charm. All my kids can graph consistently.) #5 introduces the idea of parallel lines (which we come back to at the start of next class).
The third and really fourth day of lines (see above, but actual file is here), we quickly took notes on some new key terms and the worksheet started off again with review problems. At this point, the kids were fairly confident in their acquired skills, so I pushed them a bit further by introducing lines in non-slope intercept form, and also asking them to find equations repeatedly in a diagram. We also went over why vertical lines are x=... (because they "hit the x axis at ...") and why horizontal lines are y=...
Then, finally, to pave way towards more abstract ideas like "finding b", I used this worksheet (see below, and Dropbox link here). This worksheet starts off with a very "simple" idea of points fitting through lines, and then hooks it up to the idea that "b", the y-intercept, must have one specific value in order to allow the point to still fall on that line. Then, we apply the idea of parallel lines and gradients (I took out the perpendicular lines skill for now, but I'll put it back in later this year when we do more integrated practice, once the kids have more confidence in their general abilities).
To help the kids study for the linear functions quiz and test, we did one day of team games (Powerpoint here), one day of mini-whiteboard practice exercises, a couple of days of textbook practice (problems here), and one day of move-around review where I put up problems on index cards around the classroom with answers on the back.
I hope that helps to give you ideas for teaching a low-confidence population! For my quadratics teaching, we focused a lot on identifying which stage a quadratic equation already is in, using an introduction/visual organization chart that looked like this. (I am a big proponent of the box method factorisation, as I think it covers all cases of factorisation and eliminates the need for a student to memorize 10 different methods for essentially the same situation.) Since the kids were motivated after the linear functions unit, they also did an amazing amount of work on their own during the October break, which greatly sharpened their factorisation skills. Right now I am very happy with where my Grade 9 students stand, and I hope that it gives you hope that all students can be reached, if we keep trying different methods and keep trying to build their confidence.