Badly taught math prevents fluency?

Parent Cafe
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<p>At my D’s school, the math teachers all seem keen on “putting in all of the steps.” My suspicion is that this allows peer grading. </p>

<p>This seems to be the opposite of what good mathematicians do - develop a fluency with their subject s.t. they can express ideas or a proof in a few sentences. </p>

<p>Concur?</p>

<p>Is there a cure for a “putting in all the steps” mentality? I’d like my D to develop the fluency. Her HS math grades are fine - she is in the 2nd AP calc this year.</p>

<p>beprepn</p>

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<p>It certainly is possible that math at your daughter’s school is poorly taught. But I would not assume that, and I would not assume peer grading, simply because the teachers like for students to show steps. The teachers probably just want a way to figure out where the student went wrong when they get the wrong answer. It is also possible for a student to be able to leap to the right answer when the problems are just like the samples, but not understand well enough to solve more difficult problems later. It’s easier to fix that early than to wait until it is obvious that the student is floundering but the part they don’t understand is 3 chapters back.</p>

<p>If the teacher only had one or two or three students, they wouldn’t need for the students to show the steps, because they would know from talking to the student what they do and don’t understand. But in a class of 20-30, they don’t know. I’m sympathetic that your daughter considers writing out steps to be a waste of time (so did my son), but it may be a necessary evil from the teacher’s point of view.</p>

<p>A student who is in a learning phase is not necessarily analogous to a practicing mathematician. A better way to look at it is that fluency is assumed for things one has learned in the remote past, but not necessarily what is currently being studied. (ie, your daughter’s calculus teacher probably does not expect her to show the steps that involve 4th grade arithmetic, but a 4th grade math teacher might require her students to do just that). </p>

<p>Here’s an analogy - the foreign language teachers at your school probably do not want the students to rely so much on short-hand idiomatic expressions, even though “fluent” speakers use them, that they fail to master the standard grammatical constructions.</p>

<p>If you would like for your daughter to experience math more like what mathematicians do, she might enjoy the USA Math Talent Search (a proof-writing contest for pre-college kids) or the classes/materials available here:
<a href=“http://www.artofproblemsolving.com%5B/url%5D”>www.artofproblemsolving.com</a></p>

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<p>My D is now a math major (to my surprise). She always had to “show the steps.” It can be aggravating in those cases where you can solve a problem by inspection but “showing the steps” forces you to demonstrate mastery of the concept, not just the ability to come up with an answer.</p>

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<p>The showing the steps thing is interesting. My son once put in all the steps he’d used–and the teacher used it as an illustration of “an alternate method I hadn’t seen before”–and he’d been teaching for 40 years at that point. </p>

<p>As a teacher, I like to see all the steps for several reasons:</p>

<p>*it demonstrates and reinforces understanding
*it reduces cheating
*it helps to understand where a student went wrong, if they did
*and sometimes it teaches me something</p>

5

<p>you always have to show the steps, are you kidding me? </p>

<p>I don’t know, maybe its different everywhere else, but by showing the steps you guarrentee yourself like 80% of the credit for the problem. Alot of the time you’ll doa whole problem, and add a number wrong or something to get the wrong answer, even though it was an arithmatic error. By skipping steps, and just going straight to the answer, the teacher doesn’t know if you understand the material. </p>

<p>Also, while I’m not entirely positive about the AP calc exams, I know that for the AP Statistics exam, the right answer is worth 0 points on the question. As long as you did all the steps right, it doesn’t matter if you made a silly adding or subtracting error, or if you switch up signs by accident. If you just put the answer down, regardless of whether or not its right, you get a 0 on the question.</p>

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<p>Mathematicians learn to show all the steps early. Giving heuristics and short ideas of proof is good only for short papers or when the derivation of an already established result is unnecessary for solving a bigger problem. Most research problems nowadays can’t be proved in a few short sentences, though sometimes the underlying ideas can be efficiently summarized.</p>

<p>In a math class, it’s the process of getting the answer which is most important, not the actual result. To show an understanding of the subject, you need to show all your steps, because many answers in low-level math courses can nowadays be easily obtained with a (f-bomb) programmable graphing calculators Conversely, a wrong answer might be caused by a small mistake in computation or logic along the way, and this will not be heavily penalized when all the work is shown.</p>

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<p>I’m a student, and we all have to show steps, too.</p>

<p>If the student gets a question wrong, the teacher likes to know where the error was. Was it a simple addition/copying/computational error? Or a misunderstanding of the concept being tested? There is world of difference between the two.</p>

<p>My math teacher also likes to see how we did the problems. In math, problems can be solved in many different ways, so a method that I use without giving it another thought may be a unique way of doing it. There are only eight kids in my class, so we all know each other super well, so a lot of the time he will show us how one student did a particular problem. I love having comments on my paper like “Elegant solution” or “This is the most difficult way you could have possibly solved this problem. It is also correct” or “6*2=12 not 36.” I am ashamed to admit that I have gotten each of these!</p>

<p>Showing work is also a good habit to get into. For long, complicated problems that you can’t necessarily solve the first time around, you need to be able to look back and re-evaluate what you have done. As Jags said, you also need to show detailed work for AP tests, or you get no credit. </p>

<p>Fluency in math is not skipping steps; it is integrating them into your mind so that they are instinctual.</p>

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<p>D’s math classes have been: "solve all the problems in the book, put in all the steps, next to the step put the justification for the step. "</p>

<p>Maybe the teacher graded these, it is an awful lot of problems.</p>

<p>beprepn</p>

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<p>I teach algebra, and think that writing the steps is important at various times of learning. For example. If I say 2K = 6 most people will instantaneously know that K = 3. If I require students to show the division process they are better prepared to handle 2.45K = 8.085.</p>

<p>Knowing the justifications for steps (math properties) does build fluency.</p>

<p>As my classes progress, for convienence, I will frequently merge steps, accept that some steps are understood ( like the identity properties.) When a student gets a different answer or arrives at the same answer by a different path, it becomes essential to discuss the steps and process followed, along with the justifications for the process. It also can turn copying homework into a lesson.</p>

<p>In many places Geometry is taught by proofs. I do not concur with the OP.</p>

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<p>In the AP Calc test, students are expected to show all the steps in the Free Response Questions. They can get partial credit if the steps are done correctly even if there is an error in computation resulting in a wrong answer. Conversely, even if the answer is correct but the steps are not properly explained, points may be deducted.</p>

<p>My S was always very intuitive and resisted showing steps. It was not until he prepared for the AP-Calc exam that his online tutor, a former AP Calc exam reader, forced him to explain his reasoning. It did him a lot of good. Ever since then, he has taken math classes that are very heavy on proof. Again, points are deducted for not explaining steps correctly. </p>

<p>There are some wonderful Sydney Harris cartoons about math. They are used for T-shirts. Many of them show students having trouble writing proofs.</p>

<p>I don’t know about peer grading. But at Exeter, which heavily recruits top math students, the Harkness method is used for all classes. In math, this involves a student presenting how s/he got the solution to a particular problem, explaining all the reasoning (=steps) that went into the solution.</p>

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<p>I am sorry to say, Mr. B, that very few schools still teach geometry with proofs. Proofs are taught casually and without rigor, I’m afraid; you can see the results everywhere, as formal logic falls by the wayside and is replaced by intuition.</p>

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<p>In the “real world,” people need to “show the steps” behind their calculations.</p>

<p>If you are making a sales projection, a budget request, a tax computation, etc., you need to explain and justify the figures you came up with.</p>

<p>Good mathematicians develop their mathematical expository writing skills, whether their work is theoretical or applied. </p>

<p>Mathematical expository writing that shows “all the steps” is an art that requires careful thought and practice, as well as attention to one’s audience.</p>

<p>What “all the steps” means depends on the mathematical sophistication and background of one’s audience. A math professor helping a student in an introductory class will need to show more basic details in that work than she does when writing an article for a professional journal in her specialty.</p>

<p>So mathematicians need to be comfortable moving back and forth between styles of communication appropriate for the different audiences with which they work.</p>

<p>I agree that it’s entirely appropriate for a math teacher to expect students to show the steps in their work. That is indeed how the teacher can diagnose student misunderstanding and also how the teacher can detect creative and innovative new problem-solving approaches developed by the student. Also, a student who writes up his work clearly and carefully may develop a better understanding of the mathematical solution process. (Often, the student may initially see a messy, inelegant approach but once they start writing it up, they may reformulate a more elegant, streamlined solution.)</p>

<p>The AP calculus exam gives very little credit for the “correct numerical answer” on the Free Response section. Most of the credit is awarded for a clear and logically correct exposition of the method used to arrive at that answer.</p>

<p>I concur with Texas137’s recommendation of the Art of Problem Solving (AOPS) website and the USAMTS problem-solving contest. </p>

<p>I also highly recommend their article on mathematical expository writing here:</p>

<p><a href=“http://www.artofproblemsolving.com/Resources/AoPS_R_A_HowWrite.php[/url]”>http://www.artofproblemsolving.com/Resources/AoPS_R_A_HowWrite.php&lt;/a&gt;&lt;/p&gt;

<p>Lots of good practical advice here!</p>

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<p>Thanks for the link, Wisteria. I recommend AOPS all the time, but had not before stumbled on that particular article. (another classic AOPS article, this one decrying the rush to calculus is here: <a href=“http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calculus.php[/url]”>http://www.artofproblemsolving.com/Resources/AoPS_R_A_Calculus.php&lt;/a&gt; )</p>

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<p>beprepn,</p>

<p>Doing all the problems, showing all the steps and justifying them may not stop anytime soon for your daughter. I have taken several upper level grad school courses in math and engineering that operated that way. It all depends on the teacher.</p>

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<p>Well, it is also possible that they just put too many problems. It’s good to have a lot of practice but eventually it can get redundant and really annoying. When the problem sets are too long, it can really kill even a good student’s motivation.</p>

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<p>marite-
DS also had the frustration in early years of having to “show his work/steps”, made him nuts. Calc teacher explained the free response question and how points are calculated so son finally understood why he had to show his steps. He is in calc 4 and other stuff and he and the same teacher are still arguing over his need to show his steps. He knows he has to do it for the tests but for homework and in class discussion he says his hand can’t write fast enough. For now, the teacher allows him to do it without the steps, but for tests he has to show his work. His teacher says it isn’t for now but for later when he is college/grad school he will have to show is work.</p>

<p>Maybe MIT won’t make him always show his work!!! Now Harvard they may make him show his work… I don’t know…</p>

<p>Kat</p>

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<p>Kat:</p>

<p>I don’t believe it’s a Harvard vs. MIT thing but it depends what kind of math is being taught. For engineers, it is more important to know the correct algorithms than proving why they work. When S took MV Calc and Linear Algebra, he had to take the basic course which, in college, is intended for physicists, engineers and other scientists rather than math majors. He thought the course was very “plug and chug” and did not require rigorous proof. He then audited a course that was designed specifically to teach how to write proofs. Similarly, in his math camps, the problem sets were all about how to write proofs: the solutions were given. The students had to prove them. S is currently in a study group in which the students help one another with their problem sets. Explaining their reasoning to the rest of the group is a big part of it.</p>

<p>Part of S problem when in elementary school was that he was so much more advanced than his classmates. He is now well used to showing his work.</p>

<p>MIT has a very strong pure math department, so not all math courses are geared to engineers and scientists. In fact, the instructor who taught the math proof course had just gotten her Ph.D. from MIT.</p>

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<p>I’m surprised that geometry isn’t focused on proofs. When I took geometry freshman year in highschool it was like uber proof. Almost the entire first semester was doing proof work. I hated it lol, but it is a good logic builder.</p>

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<p>I took pre-ib geometry and it was very proof heavy…maybe this is something happening in the last 2 years or so?</p>

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<p>I agree. Both MIT and Harvard have some math courses which are more proof-oriented than others, depending on the motivation and interest of the students.</p>

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<p>Even in an engineering course, students are still expected to “show the steps in their work.” </p>

<p>Although they may not have to prove the validity of the fundamental algorithms they use, they will typically apply those algorithms to complex problems and they will need to show the details of their work to demonstrate that they have applied the algorithms correctly, that they have set up their problem in a valid way, that they have made reasonable simplifying assumptions to make the problem tractable, etc.</p>

<p>EDIT: To confirm my belief that MIT does indeed expect its engineering students to “show their work,” I visited their Open CourseWare website, which shows the problem sets and model solutions for the problem sets. You can check out those for yourself. Here’s a link to the assignments for a pretty typical class in mechanical engineering:</p>

<p><a href=“http://ocw.mit.edu/OcwWeb/Civil-and-Environmental-Engineering/1-050Fall-2004/Assignments/index.htm[/url]”>http://ocw.mit.edu/OcwWeb/Civil-and-Environmental-Engineering/1-050Fall-2004/Assignments/index.htm&lt;/a&gt;&lt;/p&gt;