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Continued from the previous post:

The same person wrote a follow up comment:

“you are not pretending i’m stupid!!!!! Okay is a base the number you can multiply by?????
“example: base two is 2,4,6,8,10,12,14,16,18 ?????????i don’t know what you mean!”

Yeah, keep trying to convince me that you’re stupid. From your grammar and your tone, you’re starting to make some headway. 

But I generally don’t believe a child can be stupid. Misguided, full of anxiety about themselves and the world, OK, but stupid is reserved for adults (where a lot of people make up for lost time). 

Maybe I wasn’t clear enough, so let me try again.  A base is a way to write a number using place value (columns). The amount of digits you decide to use in the columns determines the number of the base. If you use ten digits per column, the number will be in base 10. If you use three digits per column (the digits 0, 1 and 2), the number will be in base 3. You will understand this better as you read on. (more…)convert base 10 to base 2 what do we use bases for

photo by lsiegert

A curious reader asked this question:

What is a base?? I’m sorry but I’m in the sixth grade and never heard of a base and then all of the sudden it’s in my homework. Will you please explain to me in easy fifth or fourth grade words what a base is? Pretend I’m stupid or something!

 Professor Homunculus replies:

Well, that’s going to be hard to pretend, because you are obviously smart enough to ask for help. You also did a good job expressing your question, so here goes:

Bases are different ways to express numbers. Like languages are different ways to express thoughts. You could say, “butterfly” in English, or “mariposa” in Spanish, “papillion,” in  French, or “schmetterling,” in German, but they would all mean the same thing, just different names for it.

You can write the number 11 in base ten, or as 21 in base five, or as A in base eleven, and they all stand for the same amount.

Just as in different languages, there are specific times you need to use different bases. That is a little hard to understand, right now, I know, but first you must learn how to translate into different bases, before you can understand anything about them.

Fortunately, it is much, much easier to learn how to translate from base to base than from language to language.

A base is the amount of digits we use to represent our numbers with.

We normally use what it called the base ten system. As you know, we normally use only ten digits - 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 to make up all of our numbers. After 9, we have to start a new column (called the “tens” column, because it tell us how many tens we have). 

(more…)base 10 base 2 bases what are bases what is a base

Hey, you droogs,

There was an interesting post on the Whallah! blog about an article in the Associated Press, concerning the education of math teachers in public schools.

Apparently the National Council on Teacher Quality has done a comprehensive study to come to the conclusion that everyone who is not an “expert” has known for years: Teachers are not being taught math adequately, and generally fail to teach it well to their students. (Do tell…)

Isn’t it funny that the “establishment” will never admit that? It takes an expensive academic “study” to show what is already known, yet Universities (in general) will not do anything about the way they teach teacher how to teach math. They will try some new, expensive methods that some textbook company has lobbied for, of course. But they won’t try anything that might actually work.

(more…)math ed math education math pedagogy multiplication public schools

A mom recently wrote in to ask this question about standard and expanded notation.

“How do you know when you are writing in standard form, expanded form? For example, is the expanded for of 30,048
30000 + 40 + 8 ?
Or for 29,486, the expanded form = 20000 + 9000 + 400 + 80 + 6 ?”

Professor Homunculus replies:

Precisely! Oddly enough, there is no “standard” for “standard.” What I mean is, for 30,048 the standard form could also be considered:

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photo by jonathan229

I haven’t been posting much. Sorry. Been in a kind of existential funk.

But today I had to include this. It is a link to an article that was in our local paper today, about a heinous phenomenon in a local school. Apparently, some second-graders have been actively plotting to kill a little girl in the school. You can read about it in this digg.com post.

OK, no big deal right? After all, this is America, where everyone is “entitled” to their lunacy, no matter how depraved.

But this is in a rural, upstate New York school. No inner-city, no whacked out Waco, no out-in-the-hills survivalist community.

I have done afterschool Math Mojo programs in this school. It is a nice place with (generally) nice kids. I’m not amazed, though, because our society has become all about abuse of power, from the highest, to the lowest, levels.

I know that you can’t make a sweeping judgement about public schools in general from a local anecdote. But the anectodes are getting to be pretty thick in our public schools.

It is depressing as hell. I don’t mean to depress you. There has got to be a solution, and I believe that readers of this blog are generally part of it. Homeschooling, unschooling and afterschooling are good, positive movements.

The big difference is the amount of parental involvement. If your child knows that you truly take an interest in them by spending time with them, your conscience becomes part of their conscience, without having to lecture them or make them feel “watched.”

But you know that. I just want to say that from the comments this blog gets, and the e-mails I receive, it’s people like you that give me hope. I hope I give you some to, at least as far as encouraging you to play around with math.

To that end, I went out and bought a cool little piece of software at the Apple Store yesterday. It’s a Wacom writing tablet, and I hope to make some really easy-to-follow math tutorials for you with it in the next few days. I’ll have one up here by tomorrow probably.

See you then (if I didn’t bum you out too much.)

The previous post was about the value of learning conceptually before you start practicing for skill.

There is an alternative argument that argues for the opposite. Many pedagogues try to plead the case that first you must teach the “basics” (meaning the basic skills, like the “multiplication facts”) before you can expect a child to acquire any meaning about it.

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There seems to be a big “fight” about “which should you teach first, math skills or math concepts.” A popular example is the “multiplication tables” versus the concept of multiplication (as repeated addition, for example).

It’s a pretty good bet to say that when memorizing things it’s easier if you can relate the objects. Like if you went shopping and had to get toothpaste, a toothbrush and dental floss, that would be easier to remember than if you had to get shoe polish, armadillo meat and an f-string for a lute (do lutes even have f-strings?)

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“The essence of mathematics lies in its freedom.

- Georg Cantor

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Not much math this time, but there is a math joke at the bottom.

Today is our little dog’s birthday. Our “little” dog is an eighty-pound Golden Retriever named Maia . (Our hundred-ten pounder is a Golden Retriever named Galileo.)

I went down to the local slaughterhouse (yes, we have one in our little rural village, and it’s right along the railroad tracks) and got a bag of bones for Maia. I came home with them and she was ecstatic. She even shared one with Galileo.

I couldn’t resist putting a video of her up today. I hope you like it.

Happy Birthday, Maia!

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I found this joke at http://www.danielsen.com/jokes/Mathematicians.txt

• A graduate with a Science degree asks, “Why does it work?”
• A graduate with an Engineering degree asks, “How does it work?”
• A graduate with an Accounting degree asks, “How much will it cost?”
• A graduate with a Liberal Arts degree asks, “Do you want fries with that?”

Nice title, eh? Let me preface this with the admission that I know just about nothing about dyslexia. Clinically, I mean.

The reason for this post is that Angela (Mother Crone) left a very interesting comment on yesterday’s post concerning how mental math has helped her daughter, who is dyslexic.

How many screwbulbs does it take to light in a dyslexic?

(Yes, that was unbelievably cheap.) Although I have no insights into clinical dyslexia, I have fought my whole life against certain dyslexic-like symptoms. I also suspect that any person who is at least mildly aware of his or her thought-processes struggles with similar symptoms.

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“In mathematics the art of proposing a question must be held of higher value than solving it.”

Georg Cantor

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About a month ago, Penny commented on this post.

Here is an except from that comment:

“Frankly, I don’t care if an elementary school child can add long columns of numbers in their head - it is an almost worthless skill. I do care if they can think about mathematical concepts.

Better to teach them to come up with simple proofs (not memorized proofs) of basic facts in math.

Better that they should understand what a prime number is, and why we care about prime numbers.

Better that they should learn to enjoy slow, deep thought about puzzles and concepts.

That is where the gold standard in math education is.”

I wanted to revisit this thought, because Penny brought up some great points. I don’t disagree with any of them. But I must say that I, as well as a lot of the readers are coming from a different place. Penny is a brilliant research mathematician. A lot of us, on the other hand, basically have a history of thinking that we sucked at math (at least until we came upon Math Mojo, and learned that almost no one sucks at math, but some sometimes the way math is taught sucks.)

I wanted to address some of the points Penny made, because those points made me think a lot this month. Here’s

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