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I think most math teacher (and math textbooks) are very bad. Most math teachers say, "This is the formula and this is how to solve it. Memorize it for the test next week."
For people who are really good at it, math often looks like a collection of patterns, i.e., a structure. Math literally seems self-evident to these people. Solving a problem is about seeing the structure and figuring out the pattern. It's easy to memorize because it seems so clear.
On the other hand, people who are more strategic or theoretical, are far more interested in "why" rather than "what." A what without a why is meaningless; it serves no purpose. Unfortunately most math teachers are very tactical and aren't very good at teaching these people. Math is both logical and theoretical. I think if math teachers taught both, more people would understand it.
My son who is a math genius, imo, always had to know why things were done the way they were. Sure he could and did see the patterns, but he still did not just plug things into those patterns and get answers.
My son who is a math genius, imo, always had to know why things were done the way they were. Sure he could and did see the patterns, but he still did not just plug things into those patterns and get answers.
I'm not sure that you could call mine a math genius.... but the pattern recognition thing is critical to understanding math.
But he was never able to deal with math instruction at high school, college, or PhD level graduate school in Quantum Mechanics. He ended of doodling in classes, and then taking the instruction in the book and doing it all himself. He often came up with novel answers to questions. But what set him apart was his very superior pattern recognition skills, and his amazing memory for math that he's seen before.
In his quantum mechanics class(one of the last actual courses he had to take), the prof gave them a last question on an exam. After the exam everyone of the other nuclear physics PhD students agreed that they'd had no idea how to do it and left it blank, except my son. He wrote on the paper: "I don't know how to do this problem but I know one very much like it, so I'm going to solve that for you using your numbers." Turned out that his solution was part one of three parts need to actually solve the problem. The prof never expected anyone to get it right or even start it, but always threw it in on the chance that somebody might at least get it started correctly. Those kinds of things tend to make a name for you in your grad program.
Teachers are afraid of elementary math? :confused I'm thinking basic things like adding, subtracting, multiplying, dividing, some elementary fractions, graphing, decimals, etc. Are people really saying that professionals who have made it through 12 years of public school plus 4+ years of university are inept at teaching first graders to add, or fourth graders to multiply fractions? Seriously?
Yes, that is what I am saying. A lot of people who go into elementary education do not like math, are afraid of it, and do not understand general math concepts beyond the very basics.
This is the El Ed curriculum from a highly regarded teacher program in Colorado. You can see they only need to take 3 math courses.
Teachers are afraid of elementary math? :confused I'm thinking basic things like adding, subtracting, multiplying, dividing, some elementary fractions, graphing, decimals, etc. Are people really saying that professionals who have made it through 12 years of public school plus 4+ years of university are inept at teaching first graders to add, or fourth graders to multiply fractions? Seriously?
Yes. Seriously. Sadly. Hopefully it's getting better as states have raised standards and required elementary teachers to pass exams like the Praxis.
Adding and subtracting, okay. Multiplying and dividing, maybe they can do it but not really understand it. For most people, that's enough. To be a good teacher, it isn't.
Plenty of first and second grade teachers are afraid of decimals and fractions. They remember some rules, but don't have a real understanding.
By fourth grade, the thinkers are capable of asking some good questions: "Would it work to . . .? " or "Is this like . . .?" or the big one, "Why . . .?" The teacher who is just getting by cannot answer and may even live in fear of these questions. This attitude, whether stated or not, teaches kids not to ask, not to try to understand. The teacher who only knows one way to do a problem will discourage students from discovering other ways, and thereby learning mathematical truths beyond the basic lesson presented.
Some teachers aren't afraid to tell the kids they don't know but will find out. One of the fifth grade teachers at my school came to me to find out why a number to the 0 power equals 1 - because her students wanted to know (needed to know) and their textbook only stated it as a fact. I explained it to her a couple ways and gave her a way that would make sense to fifth graders. Unfortunately, not all teachers will do that - and some have much larger holes in their understanding.
An algebra example: in eighth grade we learned the quadratic formula. I refused to use it and continued solving quadratic equations by completing the square. Why? I did not know where the formula came from. It wasn't until I figured out that it was the same thing that I accepted the formula. What if the teacher can't explain that?
My dad was quick to learn math in school because he had a good memory. He majored in it in college, got all A's (in math), and never asked questions. It was easy. My mom, on the other hand, learned in elementary school that she would never get math - that math made no sense. She dutifully memorized the algorithms, but never was satisfied. She asked the teachers, "Why . . .?" and the teachers said, "Because that's how you do it." With better teaching, my mother could have excelled.
Most teachers can do all right teaching kids like my dad, but they don't know what to do with those like my mom.
By fourth grade, the thinkers are capable of asking some good questions: "Would it work to . . .? " or "Is this like . . .?" or the big one, "Why . . .?" The teacher who is just getting by cannot answer and may even live in fear of these questions. This attitude, whether stated or not, teaches kids not to ask, not to try to understand. The teacher who only knows one way to do a problem will discourage students from discovering other ways, and thereby learning mathematical truths beyond the basic lesson presented.
My point exactly. I think the problem is that most teachers in any subject don't understand "why;" they only know "this is how it's done." To use a different example, most grammar textbooks say that there is no such thing as "case" or declension in English. It became obsolete hundreds of years age. This is not true. We use different declensions in every other sentence, for example "he" and "him" or "who" and "whom" or "good" and "well." If you don't understand the theoretical concept of "case" you will never truly understand how to use pronouns correctly. This is a very, very common problem, even among well-educated people.
No English teacher ever taught me about case and I never really understood English grammar until I studied Latin and Greek. Of course, most people don't study Latin anymore, but many students would benefit from a general discussion of comparative grammar (say Latin, French and German). It would answer "why" English has so many anomalies and exceptions that are so hard to remember.
They're hard to remember because, as they are currently taught, they require random acts of memorization, rather than a theoretical understanding of the evolution of the language.
Similarly, from pedagogic perspective, I think it's more important to understand what the quadratic formula is, what it does and how it was developed rather than simply how to solve it. It would teach students so much more, i.e., how to think.
Yes, that is what I am saying. A lot of people who go into elementary education do not like math, are afraid of it, and do not understand general math concepts beyond the very basics.
This is the El Ed curriculum from a highly regarded teacher program in Colorado. You can see they only need to take 3 math courses.
Again... this is just fearmongering untruths made up. I live with a kindergarten teacher who teaches advanced math to her kids. She never loved math, but she loves teaching.
Besides that, the textbooks are so specific that any college graduate could teach them, probably many highschool grads. Even I could do it, and I've not had a math course for 44 years.
And besides that, any decent district has a k-12 curriculum that we are required to teach, and monitored by the state tests for AYP.
This is just made up by people who have no idea what education is actually like in 2010. Maybe in 1978, not in 2010.
Again... this is just fearmongering untruths made up. I live with a kindergarten teacher who teaches advanced math to her kids. She never loved math, but she loves teaching.
Besides that, the textbooks are so specific that any college graduate could teach them, probably many highschool grads. Even I could do it, and I've not had a math course for 44 years.
And besides that, any decent district has a k-12 curriculum that we are required to teach, and monitored by the state tests for AYP.
This is just made up by people who have no idea what education is actually like in 2010. Maybe in 1978, not in 2010.
Zarathu
I'm going to have to go ahead and agree here. As a homeschooler, I can attest that the teachers' guides for teaching math are very, very self-explanatory. If you can read, you can teach elementary math. I find it disconcerting to think that there are people who believe that teaching first through fifth grade math is too hard for the average teacher. Do you really think that the average person, college-educated or not, is so woefully unprepared that they don't understand how multiplication works? This is not rocket science, LOL.
My son who is a math genius, imo, always had to know why things were done the way they were. Sure he could and did see the patterns, but he still did not just plug things into those patterns and get answers.
I am very good at math. While I see patterns easily, that's not how I, usually, do math. I've done enough math that sometimes I just recognize the answer and that is pattern recognition but I learned math more as puzzles to solve than patterns to recognize.
I'm going to have to go ahead and agree here. As a homeschooler, I can attest that the teachers' guides for teaching math are very, very self-explanatory. If you can read, you can teach elementary math. I find it disconcerting to think that there are people who believe that teaching first through fifth grade math is too hard for the average teacher. Do you really think that the average person, college-educated or not, is so woefully unprepared that they don't understand how multiplication works? This is not rocket science, LOL.
Again, yes, I do. Talk to a few random people. A LOT of college educated people don't understand math. Teaching math is more than "just the facts, ma'm".
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