I've been teaching math to at-risk high school students for the last 10 years. I have spent more time helping students understand that they are not stupid, that something just got in the way of their learning at one point, and they never understood anything after that. I'm going to use your quote in some of these conversations now.
What most of my students think: "I could never do math, I fucking hate it, and I might drop out because I will never finish my math credits. I can't do math because it's stupid and meaningless and I will never get it."
What really happened to get people off track?
- Some just didn't follow one topic in some early grade, nothing else made sense after that, and no teacher was prepared to get them back on track.
- Parents split up, student couldn't focus in school for 6 months, they got off track.
- Parent/ sibling/ significant person passed away when student was young, couldn't focus for 6 months-2 years, no way to get back on track.
Any number of other external events happen, and it is perfectly reasonable for students to get off track in math.
a systematic way to deal with the discontinuities when they strike, especially that first time
Exactly. I would like to see every elementary school have a math specialist, who knows advanced math, to help students with their overall understanding when they get off track. Helping a kid master some mechanics does a little to get them back on track, but diagnosing misunderstandings takes more math expertise than most elementary teachers have.
I could go on forever; thank you for putting some of these issues so clearly in focus.
The classic example of this is "Benny's Rules", eg. http://math-frolic.blogspot.co.uk/2012/11/bennys-rules.html
Note: This response is US-centric.
These individuals are exceedingly rare (if they exist at all). In fact, I would be absolutely shocked if 100 such people existed. College students who go into Elementary Education are stereotypically terrified of mathematics, and they have (at most) one required math course. This course is a "general methods" course that essentially acts as a survey of the elementary school mathematics that they'll be teaching.
Until these positions exist, are respected, and are not first on the chopping block the next time budget cuts roll around, these people will continue to exist under the radar. (I'd gladly transfer into a Elementary Math Specialist position, if I was sure it wouldn't threaten my family's livelihood.)
[1]: http://www.nctm.org/resources/elementary.aspx
[2]: https://www.terc.edu/display/About/Mission+and+Vision
But maybe I'm misinterpreting your point. Do you have links to what you've written?
Advanced math, on the other hand, is a different matter. And as far as HS education is concerned I believe the focus should be on building mathematical thinking skills and not worrying about preparing students for a particular subject they're unlikely to ever use.
For example, here is a lecture that I give to HS math students on graph theory [1]. You'll notice there's no algebra, no geometry, no calculus, almost nothing is required except the idea of a function (and even that is technically not required, and I tell them not to worry if it's confusing). What is in this talk is a whole lot of mathematical thinking, and I do believe (though this sounds like bravado) that if I were to put my mind to it I could model a year's worth of HS education around developing this kind of mathematical thinking. It would also have some highly nonlinear components to it, organized instead primarily around proof techniques.
[1]: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t...
So you're saying there's nothing fundamental about the typical HS math sequence. I agree. But I also don't think there's that much of a compelling reason to change it, because there are going to be difficult portions no matter how you arrange it.
But I think it's not exactly true that ellipses and congruent triangles have nothing to do with calculus. Graphing ellipses is meant to help understand functional thinking, which is crucial to calculus. Those miscellaneous facts about triangles are as examples to motivate understanding of mathematical logic -- also crucial.
In other words, most of what we learn in math are really intended to illustrate underlying mathematical concepts with some level of concreteness. Otherwise, we'd just start with category theory in kindergarten and derive all other math from that :)
I can certainly understand the perception that these things are often taught solely as ends in and of themselves. I think that part of the challenge is that there is a tradeoff between taking the time to provide a concrete motivation for every math concept upfront versus saving time by dealing with math concepts in their own world to cover more ground. For instance, the seven bridges problem serves as a great motivator for graph theory (and is used very often for this purpose), but can we really afford to find a similar motivating problem for every single graph theoretical concept?
I think if everyone agreed that the goal is to teach critical thinking skills, and have the factual knowledge be a byproduct (and elementary facts are very easy to pick up if you have critical thinking skills), then it would make a world of difference.
As to the motivations, after the students get going they don't need more real world motivation. They seem to be interested enough to ask their own questions about graphs, try to answer them, or come up with their own relations to the real world. This is where I think a lot of the critical thinking happens, not in learning facts about graphs. The facts (what degree means, what planar graph means, etc) come as a byproduct of following these paths of thought.
It didn't go well. Somehow this thing that had seemed so natural and intuitive now just seemed totally incomprehensible, mainly because I really just didn't understand the level the abstraction was at or something. I did poorly and it really shook my confidence. I changed majors and wound up a designer instead.
It's not a surprise to me that I've worked my way back into a field with deep math roots, though I think I would have found it much sooner if it hadn't been for that initial roadblock.
It gave me a new appreciation for what many of my classmates were struggling with in the early math that I breezed through in grade school. It's very difficult to see what the concepts you're learning are building towards if you don't have a sense of the bigger picture.
I thought I was good at it until I hit 17-18, doing advanced mathematics. I really didn't understand Taylor series, and I just froze up on the calculations. You fall behind, and then the class just moves forward and it stops being fun anymore
I don't know the solution, but as you said - I went back into it later and it was much easier. Catching up those years inbetween was hard though!
Some people find eventually find their way around this first road block, and future discontinuities in understanding become less stressful, and eventually understood to be a completely normal part the process.
But the usual experience is that a person's math confidence is blown and as the math truck barrels on ahead, they never catch up. They understandably accept the identity of not being "good at math".
What's missing in math pedagogy at most schools is a systematic way to deal with the discontinuities when they strike, especially that first time. We can prepare students to deal with that panic. The tough part is that the math teacher probably has 90 students on roster, but the discontinuity could hit pretty much any given lesson, for some given student.
I know so many people who have come back to intermediate math later in life and breezed through it, armed with intellectual confidence gained from other fields. They look back and wonder how they came to be so intimidated by math in their younger days. We've got to give younger people the tools and knowledge for overcoming this intimidation at a younger age. We've got to kill "I'm just not good at math".