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What are "Basic Skills" in Math?

 
 
Reply Thu 11 Nov, 2010 01:56 pm
The term "Basic skills" in mathematics was used in another thread. It got me to thinking about a new way to express my philosophy in education.

I think that the term "basic skills" is generally used to refer to things that I find quite useless. They include memorizing multiplication tables and learning the mechanical process long division.

As an engineer and former educator, I find a focus on these skills as "basic" to be worse than useless. First, they are a waste of time since they aren't necessary. Second, they bore the hell out of students at a time when we should be getting them excited about math. Third, they fail to teach students what real math is about. After all, what mathematicians and engineers really do has very little to do with rote memorization.

I never learned my multiplication tables (although I have a college degree and work as an engineer). I was lucky enough to be in a math program in elementary school that focused on problem solving. We were given puzzles to solve. Some of them were logic puzzles. Some of them were number problems and I am pretty sure that some of them were Algebra in the though processes (without the X's and Y's).

The great thing about focusing on problem solving (i.e. real math) rather then rote memorization was that not only laid a foundation for the real things I do now, I developed a real love and excitement for math.

So what are "basic skills in math"?

My list is

- Love and excitement for the ideas and the ability to solve problems.

- Critical thinking skills. Being able to ask questions and to think about problems from a new perspective.

- Logical thinking. Being about to test your own solutions and to discuss it intelligently with your peers and teachers.

- Abstraction. Being able to model your ideas and to express ideas in different ways. And, being able to understand and discuss other people's models.

I propose that if you compared to these basic skills, rote memorization is a meaningless distraction.

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Type: Discussion • Score: 9 • Views: 4,039 • Replies: 87

 
PUNKEY
 
  2  
Reply Thu 11 Nov, 2010 02:05 pm
I once saw a group of adults stymied by the task of how to create a bowling chart on a piece of paper. Seems they could not figure out how they were going to make a chart to log scores for 18 people playing 6 games on a piece of paper, 8 /12/ x 11 inches.

Lack of basic skills, I'd say.
0 Replies
 
aidan
 
  1  
Reply Thu 11 Nov, 2010 02:08 pm
This is directed to the opening poster - not Punkey - I accidentally clicked on reply to all:
Have fun trying to do algebra or any other math without being able to compute - add, subtract, multiply and divide - yeah - basic skills in math.
I guess I would also add the ability to count and understand place value.

A skill is different than a method- which is what your rote memorization is- yeah YOUR offering of a basic skill - not mine. I said nothing of the sort.
Ceili
 
  2  
Reply Thu 11 Nov, 2010 02:12 pm
BEDMAS
maxdancona
 
  1  
Reply Thu 11 Nov, 2010 02:21 pm
@aidan,
Quote:
Have fun trying to do algebra or any other math without being able to compute - add, subtract, multiply and divide - yeah - basic skills in math.


Agreed.

The question is how do you teach someone to add. I would never have my kids memorize an addition table. No, I teach them what addition means, starting with problems that they want to solve.

My five year old is doing this now and there are all sorts of problems we work out. We play games with two dice, we figure out how many cookies we need for a tea party with 3 dolls who each want 2 cookies.

The focus is not on the mechanical act of addition. The focus is on the meaning of addition. She has the ability to figure out how she wants to attack the problem (which for her is counting).

I would never think of making my daughter memorize these numbers. She has too much fun, and does too well, just figuring out these problems for herself.

maxdancona
 
  1  
Reply Thu 11 Nov, 2010 02:25 pm
@Ceili,
Ceili,

Your BEDMAS example is instructive (for those who don't know it is a mnemonic for the order of operations).

Can you explain why BEDMAS is not EBDMAS or SABMDE? There is a interesting mathematical idea behind this that memorizing the mnemonic doesn't give you.

The point is that understanding the mathematical idea would eliminate the need for the mnemonic.
aidan
 
  1  
Reply Thu 11 Nov, 2010 02:43 pm
@maxdancona,
I don't know anyone ever who has or was taught addition by memorization!
Are you saying you do?
maxdancona
 
  1  
Reply Thu 11 Nov, 2010 02:47 pm
@aidan,
That's my point Aidan.

If you focus on the real "basic skills", meaning understanding, reasoning skills and problem solving, the other "basic skills" are a needless and mindless distraction.

Ceili
 
  1  
Reply Thu 11 Nov, 2010 03:14 pm
Basic skills are basic skills... that is why they are called basic skills.
Has your child memorized numbers? letters? Will you expect them to learn the basics of grammar? spelling?
I don't think everything should be memorized, like Einstein said, why remember stuff when you can right it down. Some concepts though should be memorized or at least be second nature. A child might get the concept of multiplication, but if he/she can only count so high, only so many fingers and toes... When you get to 6x7 (for example) you start to run out of body parts.
BEDMAS incorporates the most basic concepts of all math. I have no problem at all focusing on problem solving but I think it becomes difficult without the basics.
plainoldme
 
  2  
Reply Thu 11 Nov, 2010 03:22 pm
There was a movement away from memorizing the multiplication tables in the early 80s. It was found to be a mistake. Unless the tables are memorized, it takes a would be problem solver much longer to complete the task at hand.

Furthermore, memorization in and of itself is mental exercise.
maxdancona
 
  1  
Reply Thu 11 Nov, 2010 03:30 pm
@plainoldme,
Quote:
There was a movement away from memorizing the multiplication tables in the early 80s. It was found to be a mistake.


By whom? I got away with not memorizing the multiplication tables. I have spoken to several engineers who had the same experience.
0 Replies
 
maxdancona
 
  1  
Reply Thu 11 Nov, 2010 03:31 pm
@Ceili,
Quote:
BEDMAS incorporates the most basic concepts of all math.


Please explain Ceili. Why do you think it BEDMAS instead of MESDAB?

If you can't answer this basic question, I would say "BEDMAS" didn't do anything to help you understand the "most basic concepts" of mathematics.

And that is exactly my point.
maxdancona
 
  1  
Reply Thu 11 Nov, 2010 04:02 pm
@Ceili,
Quote:
Has your child memorized numbers? letters? Will you expect them to learn the basics of grammar? spelling?


My five year old reads. She has never had a test on the letters, what she has learned, she learned because she was focusing on her desire to read.

Strangely enough given my philosophy of education, I made flash cards for her when she started reading so-called "level 1" books (really simple three letter words). She hated them and wouldn't use them.

She learned to read by reading. Seeing the words in the story was just fine for her.

My daughter's grammar is remarkably good given that she has never had a grammar test. Although she doesn't know what a prepositional phrase is, she uses them flawlessly in normal speech as do most 5 year olds. You don't need to be quizzed on adjectives to use an adjective.

My daughter is starting to write. I don't care about her spelling. When she writes a story about a elefant who bilds a house, I am damn proud. Funny enough she is a perfectionist and is always asking how to spell each word correctly, but this is her motivation and she is motivated just fine.

She is writing and I am impressed. The fact that she has never had a spelling test is not a problem.





parados
 
  2  
Reply Thu 11 Nov, 2010 04:29 pm
@maxdancona,
Different people learn different ways. Memorization isn't right for everyone but it is right for some people.

If nothing else basic skills is a way to check what the machine is doing. Writing a program isn't much good if you can't check the math yourself. An error in programming may not crash the program but it could give you garbage. Unless you have basic skills you won't recognize that it is garbage.

A couple of weeks ago I was buying a sandwich. It cost $8.52. I gave the person behind the counter a $20. They entered it into the machine as I dug into my pocket and came up with 2 cents. The machine said I got $11.48 in change. I gave them $.02 and they gave me back $11.46. Basic skills. I found humor in it at the time.
0 Replies
 
parados
 
  4  
Reply Thu 11 Nov, 2010 04:33 pm
@maxdancona,
Knowing you multiply before adding is memorization. It can't be figured out from looking at an equation that needs an answer.

6 + 2 * 12 = ?
aidan
 
  2  
Reply Thu 11 Nov, 2010 04:49 pm
@maxdancona,
Quote:
That's my point Aidan.

If you focus on the real "basic skills", meaning understanding, reasoning skills and problem solving, the other "basic skills" are a needless and mindless distraction.


If you could see me right now, you'd see the confusion on my face...

A person can be a very logical and reasonable, solution oriented person but if they don't know that 1 represents one which represents a single item as opposed to 2 which represents two and is equal to twice as much as one - what are they supposed to be reasoning and understanding?

In order to understand the simplest algebra - as in 3n+8 =29- Do they not have to know that + means add and = means equivalent to?

How are they going to reason mathematically if they don't have the basic concepts and skills and language of mathematics down first?

And I don't think I ever had a particularly creative math teacher and yes I was abusively forced (I'm being facetious) to actually memorize my times tables - and I still loved math and found it extremely satisfying and fun.

If anything, the ease and speed with which I could compute added to the pleasure of being able to quickly and competently problem solve.
And I gained that ease and speed through rote memorization and practice-that's what they did when I was in school.

But as Parados said - my memory is a strength- so it worked for me. My son's memory is a learning strength for him - even more than it was/is for me.
Just because someone can remember well and uses it to his or her advantage doesn't automatically mean they don't also understand and apply the underlying logic and concepts.

maxdancona
 
  1  
Reply Thu 11 Nov, 2010 04:53 pm
@parados,
Parados,

Let's start this discussion by explaining there is a dig difference between arithmetic and mathematics. Arithmetic is a way of using a known technique to solve some problem that fits a known template. Mathematics is an understanding of how values and objects work and how they can be used to solve problems.

In arithmetic, you just follow the template. There is no need to understand why-- you just do what you are told. Of course, the problem is that if I give you a problem in a form that you haven't seen before, you are pretty much screwed.

My belief is that we should focus on teaching mathematics. If you start with mathematics, the arithmetic comes for free.

So let's ask why? What is the reason that you multiply before you add? I think this is an important mathematical idea, the idea that you do it because you were told is wrong and limiting (it will make it much difficult to understand systems that are different, for example postfix notation).

Kids should be learning to think about mathematics. They should understand the concept of a term, and understand objects and operations. If you focus on these things, then not only is BEDMAS obvious, it is completely unnecessary since the order of operations will stem from the ideas, rather from a silly rule.




maxdancona
 
  1  
Reply Thu 11 Nov, 2010 05:08 pm
@aidan,
Quote:

A person can be a very logical and reasonable, solution oriented person but if they don't know that 1 represents one which represents a single item as opposed to 2 which represents two and is equal to twice as much as one - what are they supposed to be reasoning and understanding?


The focus should be on understanding and problem solving, not on memorizing mathematical "facts" or methods. Obviously students are going to learn the facts they need, but the focus should be on the problem solving and grasping mathematics as a way to express ideas.

Quote:

In order to understand the simplest algebra - as in 3n+8 =29- Do they not have to know that + means add and = means equivalent to?


What does equals mean? You are bringing up an interesting problem for teaching algebra (and one that current math teachers do a horrible job with). Most students are taught to deal with an expression like the one you give with a preset set of operations without thinking about what is going on.

Let's start with 5th or 6th grade. Kids are taught that the equals sign is an operation, they are given problems like 3 + 2 = ______ where they are supposed to fill in the blank.

We gave kids a true/false test such as the following

1 + 2 = 3
2 + 4 = 7
5 = 5

Kids get the first two correct (saying the first is true and the second is false). Most kids will say (incorrectly) that the third expression is false. This shows that they are unable to understand the equals sign "equivalency". They only think it makes sense as the answer for some operation they have seen.

Later on we really confuse kids. We show kids equals as "assignment", with x=3. This is an assignment, we are actively defining what x is (this statement is neither true nor false).

Finally we add functions as in f(x) = x + 3. Of course, functions are the real key to Algebra, when mathematicians or engineers use Algebra, they almost always are using functions. Of course, by the time we introduce functions we have changed what the equals sign means so many times that most kids have trouble ever grasping it.

But the real frustrating part was, as a teacher, seeing a kid mechanically start subtracting terms from both sides of an equation without even thinking about whether this made sense or not.

Math is about understanding and expressing ideas. It is not about mechanically following memorized instructions.



parados
 
  3  
Reply Thu 11 Nov, 2010 05:30 pm
@maxdancona,
The how and why of memorizing something doesn't change the fact that it is memorized. If I know something and don't have to calculate it, then it is memorized.

I prefer to know WHY something is. That doesn't mean I didn't memorize certain things because simply knowing is a lot faster than thinking through why before doing it.
maxdancona
 
  1  
Reply Thu 11 Nov, 2010 05:38 pm
@parados,
Sure, I agree with this. Of course if you know the how and why you don't need to memorize. But you are correct that often memorizing makes things more convenient. I won't argue that memorizing isn't often very convenient, although I feel strongly that it shouldn't be a focus of education.

Here is the question. Which is the "basic skill", memorization or understanding?

My point is that understanding concepts and using ideas to express or develop insights on mathematics is a key part of mathematics. Being able to quickly fill out a fill in the blank sheet of "multiplication facts" from memorization is not. In my experience, cheap shortcuts like BEMDAS, take the focus away from the understanding.


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