### Speed versus accuracy in math

Just go read this very interesting story at Ragamuffin Studies blog: Taming the speed demon.

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Just go read this very interesting story at Ragamuffin Studies blog: Taming the speed demon.

Someone asked me recently to make a post about logarithms. So here goes. I already answered the person in an email but I thought I could include some interesting history tidbits here, too.

Logarithms are simply the opposite operation of exponentation.

For example, from 23 = 8 we get log28 = 3, and we read it "base 2 logarithm of 8 equals 3".

So it's not difficult: if you understand how exponents work, logarithms have the same numbers, just in a little different places.

Just as in exponentiation, a logarithm has a base (2 in the above example). Remember that in 53, 5 is called the base and 3 is the exponent.

Other examples:

53 = 125 and log5125 = 3.

104 = 10,000 and log1010,000 = 4.

2x = 345 and log2345 = x.

As you can see from the last example above, you can use logarithms to solve equations where the*x* is the exponent:

4x = 1001

x = log41001.

Then you'd get the value of x from a calculator.

However, the base of the logarithm can be anything and most calculators only have two bu…

Logarithms are simply the opposite operation of exponentation.

For example, from 23 = 8 we get log28 = 3, and we read it "base 2 logarithm of 8 equals 3".

So it's not difficult: if you understand how exponents work, logarithms have the same numbers, just in a little different places.

Just as in exponentiation, a logarithm has a base (2 in the above example). Remember that in 53, 5 is called the base and 3 is the exponent.

Other examples:

53 = 125 and log5125 = 3.

104 = 10,000 and log1010,000 = 4.

2x = 345 and log2345 = x.

As you can see from the last example above, you can use logarithms to solve equations where the

4x = 1001

x = log41001.

Then you'd get the value of x from a calculator.

However, the base of the logarithm can be anything and most calculators only have two bu…

I was just sent a link to this site; all it is, is a handy one-page printable conversion chart for various US measures, metric measures, and US vs. metric measures. Includes even a comparative Fahrenheit vs. Celsius thermometer.

http://metricconversioncharts.org/

http://metricconversioncharts.org/

(Updated in 2018)

People sometimes ask me of my opinion or review of Saxon math. What I've written here applies in particular to Saxon Math's high school courses and middle grade levels. (The grades K-3 are by a different author and are quite different; more on that below.)

Saxon Math uses an "incremental approach" where math concepts are studied in little pieces over several lessons, and those lessons are strawed over a long period of time, intermixed with lessons about other topics.

In other words, if one lesson is on some particular topic (say, percentages or inequalities), it's almost guaranteed that**the NEXT lesson is NOT on that topic**. It jumps around from topic to topic constantly, and this is by design.

Saxon's method also includes a feature where after a lesson is taught, there are very few practice problems about the topic of the lesson. Most of the problems are mixed review problems, and they practice concepts from earlier lessons, not the concept …

People sometimes ask me of my opinion or review of Saxon math. What I've written here applies in particular to Saxon Math's high school courses and middle grade levels. (The grades K-3 are by a different author and are quite different; more on that below.)

Saxon Math uses an "incremental approach" where math concepts are studied in little pieces over several lessons, and those lessons are strawed over a long period of time, intermixed with lessons about other topics.

In other words, if one lesson is on some particular topic (say, percentages or inequalities), it's almost guaranteed that

Saxon's method also includes a feature where after a lesson is taught, there are very few practice problems about the topic of the lesson. Most of the problems are mixed review problems, and they practice concepts from earlier lessons, not the concept …

## Comments

Thanks for the reference. Nice read.

I appreciate and enjoy your dedication to teaching an sharing Math.

I commented the following on Ragamuffin's blog. Your readers might be interested in as well:

-----------------------------------

Hi,

Your post was very touching.

I've always been very good at Math but certainly not the quickest at doing arithmetic.

A couple of comments that may be helpful to you and others:

1. Mathematical reasoning and doing arithmetic quickly are not the same skill. Arithmetic is largely an algorithmic mechanical process. Some people can mechanically process data more quickly than others. Others can do abstract reasoning better but not do so well at arithmetic. Einstein was a prime example of this. He could come up with brilliant formulas but then needed help to do the calculations.

2. Making arithmetic errors can have different causes and different solutions. I realize that making errors in calculation is different than being able to do arithmetic quickly.

I have a couple of articles I've written on my blog, WildAboutMath.com that may be of interest:

Good storytelling ability related to good mathematical skills.This article touches on how Math is different from arithmetic although it's not the main focus of the article.How to get past "stupid" Math mistakes.This one discusses some possibilities for where errors in calculation come from and ways to address them.Sol