<p>I just had a look at Scott Foresman Exploring Mathematics for grades 6, 7 and 8. Editions were from the early 1990s. In the Grade 6 book, square roots were an enrichment exercise. In the Grade 7 book, they had two pages on it with a little on estimation. In the Grade 8 book, they showed an algorithm for estimating irrational square roots - there was considerable coverage and plenty of exercises. I would consider these textbooks average and ordinary. Some of the estimation, critical thinking and number sense was new with the previous revision of the NCTM standards.</p>
<p>S1 was not allowed to use a calculator for his calculus placement exam at UChicago, nor for his in-class exams during the year, if I recall correctly. Relying on calculators will likely not pay-off in college.</p>
<p>^ Oh dear…</p>
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<p>Yes it is, as it should be. </p>
<p>Calculators can be very useful in many situations and I by no means would suggest banning them completely from the classroom, but if one can’t perform basic maths without a calculator then yes a remedial maths course seems only appropriate.</p>
<p>The GRE is indeed allowing a calculator on the revamped exam, but it is their calculator. It’s a simple on-screen 4 function one.</p>
<p>For my Calc I and II classes I could not use a calculator on exams, and students were discouraged to use them on homework. I understand that this is standard practice in the math department at my college. While I had an adjustment from “calculator” high school math to “non calculator” college level math, I am so glad that I know how to use my brain for mathematical thinking now. As an added bonus, I know that my work is wrong on my math exams if I’m getting a funky number.</p>
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BC, when I was in high school algebra our teacher had a thing for the method of finding square roots that was some kind of algorithm similar to long division. I can’t remember now how it was done, but I know she was big on making us find answers to 3 decimal places, and that I have never seen the procedure since then. Do you remember learning something like that? It wasn’t one where you guesstimate high and low and then keep reducing the interval.</p>
<p>Newton’s Method.</p>
<p><a href=“Newton's method - Wikipedia”>Newton's method - Wikipedia;
<p>No. It was something weird. Ok, Wiki details it under the digit-by-digit calculation section: <a href=“http://en.wikipedia.org/wiki/Methods_of_computing_square_roots[/url]”>http://en.wikipedia.org/wiki/Methods_of_computing_square_roots</a></p>
<p>I have Cohen’s Numerical Analysis on my desk (bought this in my mid-teens) and it lists bisection, Newton’s method, Bairstow’s method, Deflation (synthetic division), and Lehmer’s method (which I spent hundreds of hours trying to implement but was unsuccessful). Besides these methods which are described in the book, there are listed:</p>
<p>The Secant Method (I think that I can imagine how that works)
Lin’s method
Graeffe’s method
Bernoulli’s method
Laguerre’s method
Muller’s method</p>
<p>There are references to four additional books for further methods. There are also some methods described in the exercises. Most of these methods are generally aimed at solving polynomials, not finding square roots.</p>
<p>There is one notable approach with an iteration formula that converges cubically but the arithmetic is more complicated than Newton’s method.</p>
<p>I think that Newton’s method is best for doing hand square roots. It’s easy to remember, converges quickly and doesn’t entail complicated arithmetic.</p>
<p>I have a feeling the digit-by-digit calculation would not lend itself well to programming, which is the intent of the typical numerical methods. I have always found it interesting that this one teacher was so fixated on finding square roots of numbers with this method, and no one really does these calculations by hand anymore. Any type of computer based approach uses Newton or one of these other nice ones you have mentioned, more amenable to computer capabilities.</p>
<p>Well, it’s pretty easy to just use the IEEE floating capabilities in modern processors to handle arithmetic but it’s not terribly easy to use discrete approaches to math too.</p>
<p>Consider Javascript which has to handle numbers with arbitrary bases. So it has to handle hex numbers where digits go from 0-9 and A-F and either switch back and forth between integers or floating point numbers or just manage the math in the human representation. It’s more coding work but I wouldn’t call it difficult.</p>
<p>Here are some methods. The best are in the comments below the original post.</p>
<p>[Calculate</a> square root without a calculator](<a href=“http://www.homeschoolmath.net/teaching/square-root-algorithm.php]Calculate”>Calculate square root without a calculator)</p>
<p>In my math class, we pray that we aren’t allowed to use calculators. Exams w/ calcs are always much harder because you’re expected to be able to do all the mundane stuff instantaneously!</p>
<p>I know in my BC Calc class we’ve only has 1-2 exams where we could use calculators, and for one of them it was only part of the exam. </p>
<p>Calcs are great for chem… I try and avoid them for math. I’ve been really lucky in having great teachers who test on theory and not number crunching, although pre-highschool was terrible. Middle school math teachers are the worst tbh…</p>
<p>As an elementary teacher, I try very hard to get my kids to THINK about what they are doing with math and not just “get the answer”. </p>
<p>Here’s a question that used to be on our standardized test but was “released” and is able to be used for examples.<br>
“Mrs. Smith put her students into groups. One sixth of the students were in each group. How many groups were there?”</p>
<p>What grade do you think I teach? (okay I’ll just say it) I teach second graders, 7 year olds. This is a very simple question if the students understand the concept of fractions. If not, there will be problem. </p>
<p>When my kids try to cut corners and not learn the process because the smaller number is easy to do by counting up or something instead of working the process on paper, I show them a much larger problem and dare them to try it their way. Once they understand the numbers and how the process works they can do anything. They love harder and harder problems!</p>
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<p>Here’s the thing, though. Assuming that the 22 on the ACT (roughly comparable to a 520 on the Math SAT) is a correct assessment of your daughter’s math ability at this time, she’s not yet ready to take calculus. She needs the remedial class. People with 22 ACTs don’t know enough algebra to be able to do calculus. This has nothing to do with calculators. If she took the calculus class right now, she probably wouldn’t pass it.</p>
<p>Discrete Probability would be a challenge too, I imagine. Those probability and counting problems require some mathematical imagination.</p>
<p>There are many math courses between remedial math and calculus. My guess is most college students never take calculus even if they have never had a remedial course.</p>
<p>Most college students don’t take calculus, but for her major the OP’s daughter needs to take Calculus, or she needs to take both Statistics and Discrete Probability. So she needs to suck it up and take the preparatory course.</p>
<p>And if she can’t come up with a pretty good estimate of the square root of 95 without a calculator in her hands, that’s good evidence that the placement test was perfectly correct.</p>
<p>My friend said she was very GLAD her kids had to take remedial courses in college to help them brush up on English and math. I’m glad mine have not had to, as both had thorough preparation in HS. </p>
<p>I recall my sophomore year in college, when I was a teaching assistant in sociology. I was very concerned that many of the short answers for class assignments were very poorly written (hard to even understand what they students were trying to convey) and shyly suggested to some of the students that they consider taking English courses. A few of the students were quite offended and proudly said that their English teachers considered them great writers (made me shudder).</p>
<p>I also remember a college statistics course for non-science majors I took where some people asked if we could form a study group and I could lead it. Neither of the other students knew the relationship between a fraction and a decimal. By the end of the term, they both had a good understanding of statistics, acquired SOME number sense and all of us got As in the course.</p>
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<p>Fractions are a big problem area in elementary-level arithmetic. I have seen students that
have trouble with problems involving fractions because their calculators don’t natively
represent them in any form other than as decimals.</p>
<p>Fractions are quite complicated as presented. Adults usually don’t see it that way as they
are familiar with the implicit framework around fractional operations.</p>
<ul>
<li>We assume the decimal numbering system. I think that most would have difficulty doing fractional operations in base 16 or base 11. Most would probably convert to base 10, do the arithmetic and then convert back to the original base instead of working natively in the other base.</li>
<li>We assume a certain amount of the rules of number theory. When we see 15/40, we mentally simplify because we know that there is a common factor to the numerator and denominator.</li>
<li>Many people, children and adults, have trouble adding fractions with dissimilar denominators. The algebra is fairly simple for those that remember it but sometimes you don’t want to work with large numbers (the product of denominators) so you try some factorizations. There are processes, procedures and algorithms that adults take for granted that kids don’t know.</li>
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<p>In elementary school, we teach kids facts and procedures and sometimes we toss in properties. In general, I don’t believe that the terms associative, commutative, distributive, existence, identity and inverse are used but the ideas are. They are presented as laws or facts implicitly in the process of learning to do operations. Most adults do operations knowing these things implicitly. Many are aware of them as they are covered in middle-school and high-school but they are not covered in a formal way. The first chapter of Spivak’s Calculus does a pretty good job at introducing the student to a more formal understanding of arithmetic.</p>
<p>It isn’t surprising that a lot of students get confused in fractions as they don’t see the whole picture and framework. They can certainly learn fractions well enough so that they become proficient in operations but I sometimes wonder if the process would be better with exposing young children to some of the formal stuff.</p>