<p>
katwkittens - don’t get your hopes up. Starting with linear algebra, most college math courses for math majors everywhere are “proof-based”, which is just a more sophisticated version of “show your steps”.</p>
<p>My homeschooled son is taking AP calculus at a public high school this year. He always complains about losing points when he gets the right answer–because he didn’t show all the steps. He says, “this is stupid, I did it in my head. . .” I told him to do what the teacher says, she must be teaching for the AP test.</p>
<p>atomom:</p>
<p>Show him the AP Central website for AP-Calculus. If you log in as an educator, you will be able to access information about how the AP-
Calculus Free Response Questions are scored and why. </p>
<p>Your S’s teacher is not just teaching to the AP test. She is teaching to every college math class. The idea is not just to find the answer; it is to prove it. Consider the story of Fermat’s Last Theorem. </p>
<p>Fermat’s last theorem
From Wikipedia, the free encyclopedia.</p>
<p>Fermat’s last theorem (sometimes abbreviated as FLT and also called Fermat’s great theorem) is one of the most famous theorems in the history of mathematics. It states that:</p>
<pre><code>There are no non-zero integers x, y, and z such that xn + yn = zn in which n is an integer greater than 2.[1]
</code></pre>
<p>The 17th-century mathematician Pierre de Fermat wrote about this in 1637 in his copy of Claude-Gaspar Bachet’s translation of the famous Arithmetica of Diophantus: “I have a truly marvellous proof of this proposition which this margin is too narrow to contain.” (Original Latin: “Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.”) However, no correct proof was found for 357 years.</p>
<p>This statement is significant because all the other theorems proposed by Fermat were settled, either by proofs he supplied, or by rigorous proofs found afterwards. Mathematicians were long baffled, for they were unable either to prove or to disprove it. The theorem was not the last that Fermat conjectured, but the last to be proved. The theorem is generally thought to be the mathematical result that has provoked the largest number of incorrect proofs, perhaps because it is easy to understand.The theorem was written down by the French mathematician. It’s taken several centuries to prove it.</p>
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<p>These do not pass my D’s teacher’s rules. There are multiple statements on a line. Each line does not give the reason, the theorem, that got you to it from the line before.</p>
<p>beprepn</p>
<p>
</p>
<p>The teacher’s rules are typical of those used in high school curricula.</p>
<p>The conventions used in college are different, because professors want students to develop an expository style appropriate for use in their professional life.</p>
<p>You can see a very nice illustration of the beginnings of that evolution in MIT’s Open Courseware for Calculus I with Theory. The model solutions for the first assignment conform to the high school teacher’s convention. As the semester progresses, the style becomes more typical of that used by professional mathematicians.</p>
<p><a href=“http://ocw.mit.edu/OcwWeb/Mathematics/18-014Calculus-with-Theory-IFall2002/Assignments/index.htm[/url]”>http://ocw.mit.edu/OcwWeb/Mathematics/18-014Calculus-with-Theory-IFall2002/Assignments/index.htm</a></p>
<p>MIT also has a whole course on “Mathematical Writing.” There are some excellent articles on that subject here:</p>
<p><a href=“http://ocw.mit.edu/OcwWeb/Mathematics/18-091Spring-2005/RelatedResources/index.htm[/url]”>http://ocw.mit.edu/OcwWeb/Mathematics/18-091Spring-2005/RelatedResources/index.htm</a></p>
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</a></p>
<p>Actually, I misspoke above. The course I linked is “Mathematical Exposition,” not “Mathematical Writing.” It covers oral as well as written communication in math and looks very useful to me. (I wish some of my professors had taken a class like this before they began teaching! I also wish I’d had the opportunity to take a class like this one.)</p>
<p>This post started off mentioning fluency in math. Fluency comes from understanding the rules and some repetition that brings those rules easily into your thought process whether it is ordering a meal in French or solving a quadratic equation. The teacher might be teaching to a test (there is so much pressure to do that these days) but in this case it also seems the best way to teach the material.</p>
<p>I am sorry to hear that proofs are not part of geometry in some schools. They really help to train a mind to approach problems orderly and methodical. I still ponder tri-secting an angle.</p>
<p>D’s Geometry in high school was very proof intensive. As I recall, it wasn’t only doing the proofs, it was “discovering” them.</p>
<p>beprepn, has any of this changed your opinion of your daughter’s math teacher’s approach?</p>
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</p>
<p>I’m not sure. I still need to look at the link that shows the progression from “all the steps” to “professional mathematician.”</p>
<p>It seems to me that the entire class is being treated as if they need remedial work. If I have a kid that is struggling, I might have them include every step no matter how small and put the justification for it next to it. This seems awfully tedious, however, for the kids that are doing well.</p>
<p>I guess I think that combining steps is a skill that is part of math fluency.</p>
<p>I’m a little worried that my D is being trained to be a plodder, only thinking one tiny step ahead at a time.</p>
<p>beprepn</p>
<p>PS. The question of the moment is, can I be doing something with her between now and college to work on creativity/fluency. </p>
<p>The evidence that D needs “help” is meagre. SAT I math test score below her verbal scores, AP calc score inconsistent with her class performance.</p>
<p>
Yes! Take the earlier suggestion by both myself and Wisteria and check out the links, materials, and courses here: <a href=“http://www.artofproblemsolving.com%5B/url%5D”>www.artofproblemsolving.com</a> </p>
<p>Competition math is much more interesting, creative, challenging, and requires a much higher level of critical thinking that most high school math courses. Even if your daughter does not want to compete, she can get a lot out of the resources available for those who do. While you are on the site above, be sure and read “The Calculus Trap”, which it sounds like might apply to your daughter. <a href=“http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calculus.php[/url]”>http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calculus.php</a></p>
<p>My D, who is a second-year Math major in college, says that “showing steps” is required. She’s been marked off when something seemed so obvious and she didn’t show the work. Fwiw, she’s not a plodder.</p>
<p>Fwiw, some things that are tedious are still necessary.</p>