We have two responses for you
Those are excellent questions that have been the subject of a large body of research. I don't think anyone knows the definitive answer. On the other hand, lots of groups have tried to find different ways to help. This website is one of those. For another example, with some different resources, look at http://www.maa.org/pmet/resources.html. A third place to look is http://www.cms.math.ca/Education/teachers. Have you seen the movie "Stand and Deliver"?
I think you have touched on some important issues. If a student is allowed to move on without a satisfactory understanding of fundamental material then her/his long term interests are not served. Undoubtedly, the deficiency will bite them later on.
I think one underlying reason can be found in societal attitudes towards mathematics. No one would think it is acceptable for an adult to be below the grade nine level in reading, but somehow this is deemed acceptable in math. Children's early exposures to math are far too frequently guided by well-meaning individuals who either don't understand it themselves, or who subtly communicate the message that it is ok not to understand, or who create a punitive, negative atmosphere in some way (for example "you can't
Changing the culture will be a very slow process. Your message suggests that you're up to do your part. Good for you. My suggestion is to come at the challenge with knowledge, empathy, and whatever approaches best suit your personality. Participate in professional development. Read what many authors have to say and decide which make sense in your situation. As your situation changes, reassess what you are doing and alter your methods accordingly.
My opinion is that one should try to motivate the material by using a wide variety of real examples, and by communicating to the students where the curriculum is headed. I think it is much better to put the problem solving and mathematical modelling front and centre (we're going to figure out the amount of laminate flooring needed for a room that is 10.75 feet by 12.14 feet ...) than it is to just present the material straight up (today we will do all of the problems on page 53 because that's what my lesson plan says to do). I think that, far too often, students are reduced to trying to survive unmotivated material they don't understand by memorizing algorithms (often wrong), or by trying to guess "rules" or anything
Others might disagree, but I think teachers should simultaneously be kind, fun-loving, sympathetic, highly knowledgeable, and demanding of actual understanding. And lots of other things. It is a very important job.
Don't take what I say as the last word. Listen to others. Do what's best for you.
Hi Cindy. There are many studies and methods that try to address this problem.
For what it is worth, I think one of the main problems is parents and other adults telling kids "it's okay: I was never good at math either." I feel that that statement tells the student not to expect to do well and so gives them permission not to try.
There are a couple of really good approaches I am looking into myself: Spirit of Math: http://www.spiritofmath.com which some middle school teachers who are my friends report works really well. It basically uses "drills" to get kids comfortable with numbers but does it in a really non-intimidating way. As well, the JUMP math program has many advocates for encouraging kids during middle school. See http://jumpmath.org/ for more information.
Kids often ask "what possible use is this to me?" There are several resources on the web that help to encourage kids to see math as relevant. One is our own "Math Beyond School" section. It lets students
You are right that most math is cumulative, so falling behind can have significant impact as time goes on. However, there are areas of mathematics that aren't quite so dependent. Take a look at things like fractals, mobius strips, polyhedra, tesselations, etc. Conceptual mathematics (rather than computational) are sometimes easier for students to grasp, particularly if they have a visual rather than abstract component to them.
Math camps, which in Canada are generally very affordable and sponsored by universities and the Canadian Mathematical Society, try to emphasize the fun and tactile side of mathematics whenever possible. The
I hope these ideas, whether you take them up directly or use them as inspiration for your own efforts, are fertile ones for you.
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.