The neglected "R" - the resurgence of writing in education

[marite]

<p>

The thing that I find appalling is that on some of the NY assessments a kid who can do the math in his head and puts down the right answer can get a lower score than the kid who explains his work and gets the wrong one.

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<p>This is not restricted to k-12 math.
When S took a college physics class (while in high school), he received back comments on his problem set: “Even in physics, we use English, you know.”
It seemed that “since” and “therefore” were not adequate explanations.</p>

<p>The most valuable part of the AP-Calc test is the Free Response Question where students are asked to explain their solutions (as opposed to multiple choice). They do get partial credit for right explanations even if the computation leads them astray. </p>

<p>I don’t mind a certain amount of writing in math; done well, it can illuminate what the actual problem is that needs solving (it’s amazing how many students charge ahead without asking themselves what the writing prompt–even in the form of a question–really seeks to elicit, even in the social sciences and humanities, let along math or science). However, sometimes, the writing is totally irrelevant to actual math learning.</p>

<p>Sometimes, the question can be fuzzy and trip students. A classic example is that of the 136 soldiers who need to be ferried to some exercise spot by bus. Each bus can carry 42 people. How many buses are needed? This problem was presented to parents when the high stakes tests were first introduced (they were then presented as diagnostic tests, rather than as high stakes ones). The DOE representative said that either 3 buses, or 3 buses with a minivan to ferry the remaining 10 soldiers would be an acceptable answer.
The fact is the question was fuzzy. Was it really to test the students’ knowledge of division? In which case, 3 buses, (leaving 10 soldiers stranded) would be okay. Or was it a question about how to get all the soldiers to the exercise field? In which case, something had to be done to get the remaining 10 soldiers to the field. A parent pointed out that if the question had been about transporting 136 students to a game or a picnic, the students would have unerringly figured out that it was not okay to leave 10 students behind, and would have answered 3 buses and a minivan.</p>

<p>Of course, the question can be simply posed as “Divide 136 by 3.” But then, there are many students who do not know when a particular computation is needed, though they do know their mulitplication facts and their long divisions.</p>

[Tarhunt]

<p>Math is its own language, but the reason so many people do not master it is that they don’t learn it as a language. It is quite possible to graduate high school, having taken even advanced math courses, and know precious little math. Finding the right answer can be as simple as memorizing the steps in the problem sets. You don’t necessarily need to why you are doing the steps or even what you are trying to find. All you need do is memorize the steps and repeat them on the test.</p>

<p>Naturally, one tends to forget the steps as soon as one no longer has to deal with them, because there is no rhyme or reason for those steps in one’s mind. It’s a bit like learning a number of useful phrases in a foreign language without learning grammar, syntax, or vocabulary. </p>

<p>I have a wife who took and passed advanced math courses in high school by getting the right answer, but I have to sit down with her and explain how to convert a fraction into a percentage. She memorized the steps very well, but doesn’t understand the concept. Therefore, she got As having learned exactly nothing except rote mechanics.</p>

<p>Having to explain, in words, why one is doing what one is doing sets elementary mathematics in the context of WHY things work as they do. It also inculcates the habit of thinking of mathematics as a language. Eventually, one gets to the point where words are no longer useful, and mathematical expressions are their own language. But most people never reach that point because they never come to think of math as a language.</p>

<p>In child #8’s elementary school, fifth grade proficiency on state math tests increase from just above 50% to over 80% after revising the math program to require that students understand and explain why the operations they were peforming were being done.</p>

[marite]

<p>I totally agree with Tarhunt. Having to explain why one does a particular operation or need to apply a particular formula is crucial.</p>

[asteriskea]

<p>Too much math?</p>

<p>

Advocates of new math and old math, back-to-basics math and “fuzzy” math might be shelving their differences to collectively focus on what many consider a more pressing problem: too much math.</p>

<p>Maryland math leaders meet today – and D.C. math educators gather tomorrow – to discuss Curriculum Focal Points, a new document from the influential National Council of Teachers of Mathematics that could profoundly influence math instruction in the region and nationwide.</p>

<p>It says the typical state math curriculum runs a mile wide and an inch deep, resulting in students being introduced to too many concepts but mastering too few, and urges educators to slim down those lessons…</p>

<p>The report urges teachers to focus on three broad concepts in each grade and on a few key subjects – including the base-10 number system, fractions, decimals, geometry and algebra – that form the core of math education in higher-achieving nations. Some are calling Focal Points the most significant publication in the field since the 1980s.</p>

<p>R. James Milgram, a Stanford University math professor who is among the harshest critics of U.S. math instruction, said the 41-page report aligns teaching “with what is being done with unbelievable success” in other countries. The curriculum would teach a few topics intensely and have students master them and move on rather than teach many topics briefly and repeatedly over several years…</p>

<p>Refocusing the curriculum will help struggling students, mathematicians say, because they need more time to learn the basics. It will also aid math teachers across the elementary and middle grades, many of whom learn comparatively little math in getting their teaching credentials.</p>

<p>“Wouldn’t it be nice,” Fennell said, “if someone told [them] what’s important here?”</p>

<p>He and other math scholars hope the new publication will, as a bonus, bring together the adversarial camps of math instruction.</p>

<p>One group, led by mathematicians, has argued that children must learn a sequence of basic skills, including times tables and some memorization, if they are to have a fighting chance at college-level math study.</p>

<p>The other camp, led by math educators, has urged children to understand the theory behind math problems and to find solutions in their own way.</p>

<p>W. Stephen Wilson, a math professor at Johns Hopkins University, deems Focal Points “probably the most important contribution to mathematics education in a good long time.” As states begin to review and revise their math standards over the next few years, “this will be the document that states will look at, and it’s a good one,” he said.</p>

<p>Math educators in the region point to local and national tests as evidence that, international comparisons aside, math instruction in America is fundamentally sound. The National Assessment of Educational Progress shows no decline, and some improvement, in math ability nationwide during the past 30 years. The SAT math average in the same period has climbed 21 points, from 497 to 518, after adjusting for a recalibration of scores in the mid-1990s.</p>

<p>Math leaders in some Maryland and Virginia school systems say their math curricula accomplish the key goals of Focal Points: organizing and prioritizing what should be taught. In Montgomery County, for example, the expansive state curriculum is synthesized so that “it doesn’t look like this long list of things that teachers need to do,” said Leah Quinn, mathematics supervisor.

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<p><a href=“http://www.washingtonpost.com/wp-dyn/content/article/2006/12/04/AR2006120401491_2.html[/url]”>http://www.washingtonpost.com/wp-dyn/content/article/2006/12/04/AR2006120401491_2.html&lt;/a&gt;&lt;/p&gt;

[kluge]

<p>Well, here’s an anecdote: I remember reading a police report of an car accident once and seeing where the investigating officer had calculated an estimate of the speed of Car A. The answer was plausible; absent an explanation I doubt I would have questioned it – and there were significant consequences attendant to the answer. But as it happened the officer “showed his work.” There were no calculation errors, but his casual reference to the pythagorean theorum caught my eye. He had correctly added the squares of two numbers together, and calculated the square root of the sum, but… that calculation had absolutely nothing to do with calculating the speed of a vehicle. He had simply applied a formula he knew to a couple of likely looking numbers, got an answer that looked plausible, and voila - there’s your authoritative answer.</p>

<p>So maybe it is important to “show your work”!</p>