9
   

What are "Basic Skills" in Math?

 
 
maxdancona
 
  1  
Reply Fri 12 Nov, 2010 06:51 am
@aidan,
Quote:
You know - WHY does it always have to be one or the other? Why can it never be both? Why do we always have to insist on throwing out the baby with the bath water?

Because I'll tell you something. I've got two kids - one has a naturally inclined math brain and one doesn't. And the one who doesn't has made it almost all the way through school- and I'm talking about good schools in two different developed countries- and she can't add fractions.
Why? I guess because these days it would have been looked at as 'cruel and unusual punishment' to make her learn them- along with memorizing her times tables.


Actually, I am suggesting we throw the soap scum out with the bath water. You are making the assumption, without any proof, that the exercises do any good. There is no reason to believe that these rote exercises do any good for helping people gain the critical thinking or reasoning skills that are at the heart of mathematics.

Our current education system is outdated. It was designed during the industrial revolution where we didn't need people who could reason and think. We needed great numbers of people who could work on assembly lines, meaning they could follow instructions and do rote tasks without asking questions.

Your story about you daughter doesn't support your point. It is clear that she needed attention and special help. The question is what type of help would be helpful. Fractions represent an idea. They are best taught by teaching the idea behind them, we use manipulatives to teach factions as part of a whole. Then we use real, every day problems to show fractions as numbers. Then there are all sorts of ways to address the idea between finding a common denominator.

To master fractions, students need to understand the ideas behind them. If we teach some magic mechanical steps that happen to work most of the time, we may be able to fill out a worksheet, but it makes the use of fractions in the real world impossible and the entire exercise meaningless.

I hope your daughter received help to think about and understand the ideas behind fractions. Rote exercises without understanding would not address the problem, and would be a tedious process with no meaning.


fresco
 
  1  
Reply Fri 12 Nov, 2010 08:08 am
@maxdancona,
The UK data indicates ,for example, that the "better " high schools have been obliged to drop fractions from their entry examinations because many candidates are unable to manipulate numerators and denominators due to poor knowledge of tables. At age 16+, many such high schools aiming at the traditional universities have been obliged to drop the state GCSE syllabus, because falling standards of numeracy have "dumbed down" examinations such that they fail stretch their students adequately. Many have have turned to the more traditional International Baccalaureate.

BTW. Regarding your discussion of the teaching of fractions....I used to teach fractions as a sub-topic of "bases" which gave the students cross linkages using the concept of "packet size". However, whether traditional base work is being practised ( e.g.weeks and days) or discussion of mixed numbers and improper fractions, knowledge of tables was essential.
0 Replies
 
Ionus
 
  1  
Reply Fri 12 Nov, 2010 08:14 am
@maxdancona,
I agree with fresco. In my view of teaching, I found a lot of time was wasted on mathematical exceptions because these were interesting to the teacher. More time should have been spent on the rules rather than the exceptions. If you do this, it is surprising how much ground you can cover. I taught my preschool son binary mathematics and computer flow charting. He has been a talented mathematician and computer whizz ever since.
aidan
 
  2  
Reply Fri 12 Nov, 2010 08:30 am
@maxdancona,
Quote:
You are making the assumption, without any proof, that the exercises do any good.

No, you are making the assumption without any proof, that I have no experience or proof to back up my 'observation'.
Quote:
There is no reason to believe that these rote exercises do any good for helping people gain the critical thinking or reasoning skills that are at the heart of mathematics.

I didn't say they did help people gain critical thinking or reasoning skills - I said that they improved the facility with which calculations could be made and probably (if not definitely) led to increased confidence and enjoyment of mathematical activity in those who experienced this increased facility.

Quote:
Our current education system is outdated. It was designed during the industrial revolution where we didn't need people who could reason and think. We needed great numbers of people who could work on assembly lines, meaning they could follow instructions and do rote tasks without asking questions.

What? This has to be one of the most arrogantly dismissive statements I've read in a while. So you think that people - where - in developed countries, in third world countries, world-wide- were LESS reliant on reasoning and thinking skills in the past than they are now?
My ability to think about it critically leads me to the opposite conclusion. And I think that if you think about the strides in technology that occurred during the industrial revolution, you'd rethink your statement.

Seriously, you find critical thinking and innovation today (in more individuals) more in evidence than during the eighteenth, nineteenth and first half of the twentieth century?
Life is not only about a person's job. Think about getting through the day, eating, and living without reasoning or critical thinking skills then as compared to now.

Quote:
Your story about you daughter doesn't support your point. It is clear that she needed attention and special help. The question is what type of help would be helpful. Fractions represent an idea. They are best taught by teaching the idea behind them, we use manipulatives to teach factions as part of a whole. Then we use real, every day problems to show fractions as numbers. Then there are all sorts of ways to address the idea between finding a common denominator.

Laughing Laughing She got a lot of special attention and help with her math. I did have her tested for a math disability - she didn't qualify. I helped her as much as I could in all the creative ways I could think to help her and then, much to my surprise, when I got her classroom teacher changed, thankfully she ended up with Mr. P. (a wonderful teacher) who subjected (!) her to timed multiplication tests- yep - and that's how she finally learned her times tables.

If you think she's special in terms of not knowing fractions or being conversant or adept at the sort of math that people her age were adept at twenty years ago - you should check again. How long have you been out of education?
If you take a look these days - she's the rule - not the exception. That's why I don't get that upset about it - although I do think it's a shame.

maxdancona
 
  1  
Reply Fri 12 Nov, 2010 08:33 am
@Ionus,
Quote:
I found a lot of time was wasted on mathematical exceptions because these were interesting to the teacher.


Could you give an example of a mathematical exception?

My position is in favor of ideas and understanding over rules. I don't know what an exception would be.

maxdancona
 
  1  
Reply Fri 12 Nov, 2010 08:34 am
@Ionus,
Quote:
I found a lot of time was wasted on mathematical exceptions because these were interesting to the teacher.


Could you give an example of a mathematical exception?

My position is in favor of ideas and understanding over rules. I don't know what an exception would be, but it sounds like something I wouldn't be in favor of unless, of course, it illuminated something about the ideas behind mathematics.


0 Replies
 
engineer
 
  3  
Reply Fri 12 Nov, 2010 08:44 am
We hashed some of this out on another thread several months ago. Here were my comments from there.
engineer wrote:

ebrown p wrote:

I have never needed to know my multiplication tables. When you understand what multiplication means... you don't need to memorize tables.

Count me in those who think you need to memorize those tables. Math education is moving the way you suggest with much less emphasis on repetitive problem solving and much more on understanding "what multiplication means". The downside of that is when you are working on large, real life problems those who know the multiplication tables don't waste time on the mechanics and focus on the theory while those who don't get bogged down in pulling out their calculators. When I'm working with engineers of my generation or older, we routinely do basic math to a couple of significant digits in our heads while working out problems. I just watch the younger engineers struggle with the basic math. It's not that they don't know how to do it, just that they can't do it as second nature and it becomes a stumbling block. I remember one extrememly bright engineer who had to pause the discussion to open a spreadsheet to do a very basic math problem. I was stunned.

If you are very competent in multiplication, it makes it easier to focus on the fundamentals of algebra. If you are very competent in algebra, it makes it easier to focus on learning trig and calculus. It's hard to build up a solid math learning structure without a strong foundation. While not everyone needs this approach, I disagree that the exceptions disprove the rule.


engineer wrote:

ebrown p wrote:

Kids grow up to be Engineers in spite of grades, not because of them. I know this from personal experience... I grew up to be an Engineer. Most of us have a natural love of and aptitude in mathematics and a surprising number of us got poor grades in mathematics. Yet, we solve problems very well... and we even, at times, sit around and invent problems for ourselves to solve.

That may occasionally be true, but the majority of engineers I know got very good grades in math and science primarily because of that natural love and aptitude in mathematics and science. My experience is that those with that aptitude blow through those rote math problems even though they don't find them particularly challenging. I would hope that there is value to those without particularly strong math skills in solving a tough problem about two trains leaving cities at different times, even if it is only the reward of a tough challenge overcome. Will you ever have to do that in real life? Of course not, no more often than you will be called upon to solve a real life jigsaw puzzle. I do believe that those who can see the logic in that train system would be more likely to see the logic in more complex systems and be better candidates for harder classes. You'd be hard pressed to show that those with C's in math in high school will perform equally to those with A's in a science or engineering curriculum although I'm sure individual examples abound. Grades aren't a perfect predictor of future performance, but I think the correlation is pretty decent, whether the student got the grade due to hard work or talent.
Ionus
 
  1  
Reply Fri 12 Nov, 2010 08:45 am
@maxdancona,
I dont have a book handy...may I struggle forward using descriptions ? Lets say you are teaching "solving quadratic equations" at secondary level. The main rules are the quadratic formula, solving by visualisation, plotting the graph, (phew! this was decades ago) and any others I left out. The exceptions are all the tricky little problems that are done slightly differently...example, rearranging the quadratic formula when you have already covered how to rearrange formulas. Too much time is wasted on that when some students need to be going over the rules again. This generalisation of the difference between learning the rule and doing countless variations and exceptions is a major reason for the eyes glazing over like a feeding shark. More broader education of important rules (insert concepts), less detail on exceptions and fine tuning.
maxdancona
 
  0  
Reply Fri 12 Nov, 2010 08:47 am
@aidan,
Quote:

Seriously, you find critical thinking and innovation today (in more individuals) more in evidence than during the eighteenth, nineteenth and first half of the twentieth century?


I would suggest the percentage of workers working on assembly lines has dropped precipitously in the past 100 years. In the same time the percentage of workers working on information jobs has gone up significantly.

maxdancona
 
  1  
Reply Fri 12 Nov, 2010 08:55 am
@Ionus,
Quote:
The exceptions are all the tricky little problems that are done slightly differently...example, rearranging the quadratic formula when you have already covered how to rearrange formulas.


That's interesting. I think I disagree with you.

There is a difference between mad libs, and original writing. In a mad lib, you pick nouns and adjectives etc. as requested to fill in blanks left in a template. You can pick different instances of a word, but the basic template is always the same.

Mad libs may be fun, and they do produce a story. But, doing a mad lib isn't writing and it certainly doesn't teach you much about good writing.

A writer can apply their skills in any topic to express many different ideas, even when the particular application isn't one they have seen before.

There is a common view of mathematic "rules" that is very much like mad libs where you give kids a bunch of preset templates with blank lines where numbers go. Kids can create and solve an expressing by picking appropriate values, but the basic template is always the same.

But filling in the blanks in a preset template isn't math. Someone with math skills can apply their skills in any area to solve many different problems or express many different ideas. If you understand quadratic equations, you should be able to exactly what you describe, being about to use the base concept to solve problems in a form that you haven't seen before. These aren't exceptions, these are the whole point. This is what people who do math do.

Students should be able to work without a template, either in writing or in mathematics.


aidan
 
  2  
Reply Fri 12 Nov, 2010 09:04 am
@maxdancona,
As I said, I'm not talking about jobs exclusively - I'm talking about life in general.
And I'll ask again - do you really believe that it took less critical thinking for the average person in any sort of habitat or environment to sustain him or herself and a family, and make it through an average day of life 150 years ago (apart and aside from his or her job) than it takes the average person to feed, care and sustain him or herself and a family today?
0 Replies
 
engineer
 
  2  
Reply Fri 12 Nov, 2010 09:08 am
@maxdancona,
maxdancona wrote:

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.

Tests don't teach, they measure. No one ever learns grammar from a grammar test because that's not the purpose of it. (I know this is a carryover from another thread and I haven't forgot about that one, I just want to have the time to give it the attention it deserves.) Your daughter's accomplishements are certainly impressive, but she is memorizing and using skills, she is just not doing it from flashcards. If she is sounding out elephant, then she knows the sounds that the letters make. Maybe she didn't learn that F sound from a flashcard, but knowing it is a basic skill that she has picked up. I don't think you are arguing that it is not necessary to know the sounds of the letters, only that the structured technique of using flash cards is not a great way of doing it. (Please correct my assumption if it is not correct.) Likewise, if you can test basic multiplication without using flashcards or multiplication tables, then that's great. Personally, I found the table part of multiplication tables to be very intuitive, much better than flashcards.

My opinion on basic skills in general can be reflected by continuing my basketball analogy from the other thread. I enjoy playing basketball. I do not particularly enjoy dribbling, passing or rebounding, but if I can't do those things I can't enjoy the game of basketball. If you want to cultivate a love of mathematics, then you have to be competent enough with the basics to enjoy the entire process. You have to know how to multiply and divide. You don't have to learn that through flash cards if another way works better, but you have to know it. If you have to pull out a calculator every time you see 4x6, you aren't going to enjoy ripping through a good problem because you can never really get going.

Quote:
- 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 like these, but I'd argue that these are advanced skills that are developed over time with layers of subtlety added as the student develops more basic skills. The love and excitement part is not so much a basic skill to me as a personality trait, certainly something to be developed and nutured, but not a skill per se.
sozobe
 
  1  
Reply Fri 12 Nov, 2010 09:14 am
@engineer,
I've been reading about this, can try to find the articles I have in mind if anyone is interested.

I think that there has to be a combination -- having some mathematical facts (like multiplication) right at hand, and then also a deep understanding of mathematical concepts (not just rote). My kid is very good at math in general and great at the why but was having problems with the rote stuff in multiplication -- she got every question right, but took too long to figure it out. And that's the thing, she'd figure it out for all but the simplest ones. 6 x 8 wasn't an automatic "48," it was breaking down the component parts and putting it back together.

The goal this year (4th grade) is for them to know multiplication tables (up to 12) "as well as their own name." We recently had a conference where my kid had to identify something she needed to work on, and she chose improving her times on the multiplication tests. We've been practicing more at home (I'll spring one on her randomly) and she's been doing these games at www.multiplication.com (yes it's a site!). She just HALVED her time after like a week of practice. (She used to have a hard time doing 50 questions in under 3 minutes, she just did it yesterday in a minute and a half.) (And was very happy about it.)

Note, she got all the answers right in both instances, it was just a matter of memorizing what she already understood.
engineer
 
  2  
Reply Fri 12 Nov, 2010 09:17 am
@maxdancona,
maxdancona wrote:

My position is in favor of ideas and understanding over rules. I don't know what an exception would be.

I think most people would agree that understanding is the goal, but the question is how to get to understanding and what is the most effective technique to use in a large class. If you have twenty students and each one has a different optimum technique for learning, you could spend time with all twenty to understand what technique is absolutely optimum for them or y0u could use a technique for all twenty that demonstratably works for most children even if it isn't optimum. For my children (and it sounds like in your family as well), I have the time, energy and desire to help them find that optimum technique. The results in both our households reflect that. I think the studies that show the impact of single parent homes on education reflect that as well in the opposite direction. The school system doesn't have that luxury so that use techniques that might not be best for all students, but that are effective for most students.
0 Replies
 
maxdancona
 
  1  
Reply Fri 12 Nov, 2010 09:20 am
@engineer,
Well Engineer,

I think you and I pretty much agree on the outcome. The question is the focus.

You are absolutely correct that in learning to read and write, my daughter (or any other kid) memorizes quite a bit. The question is about teaching and learning, do you focus on the memorization and then let the understanding stem from that? Or, do you focus on the understanding and let the memorization stem from that.

I think this is our basic disagreement.

Quote:

I like these, but I'd argue that these are advanced skills that are developed over time with layers of subtlety added as the student develops more basic skills. The love and excitement part is not so much a basic skill to me as a personality trait, certainly something to be developed and nutured, but not a skill per se.


This sums up the discussion pretty well, and I would say the opposite. Critical thinking and understanding are, to me, the basic skills. I personally start there, the core ideas, the reasons why are the important part. The specific knowledge I need will come as I need it.

I recently learned Spring (a new techie way make web applications). I didn't spend even one second memorizing the multiple new keywords I need to make the thing work. I have a reference card taped to my wall for the exact words I need that I look up when I need it.

When I jumped in to learn this brand new technology, I wanted to know the key concepts. I looked at diagrams on what is happening, and right away I imagined new ways that I could use it. My learning was to study each idea, and then to play with the thing to make it work in ways that I thought up. I learn the most through exploration, getting things I want to do to work and by poking around to see if my assumptions are correct.

Now sure, now I happen to know the basic keywords because I am using them a lot. After looking them up on my handy chart two or three times, they kind of stick by accident. And sure, this make things a bit quicker when I don't need to look them up.

But the key is the focus is on getting a deep understanding of the important ideas behind the technology. The details are just details. I never even thought about sitting down to memorize keywords. This would have been a complete waste of my time.
0 Replies
 
engineer
 
  3  
Reply Fri 12 Nov, 2010 09:30 am
@sozobe,
Congrats to her! I can completely understand because I was there. I remember in elementary school trying to compute 6x8 (very specific memory there) by building it up. Unfortunately, I took forever to do my work even though I had a great understanding of what multiplication was and how it was used. I finally took a multiplication table on a green card on a family trip. When I came back I could nail those problems and it's been an asset ever since. I firmly believe that without doing that, I would have been continuously frustrated, slowed down by the mechanics instead of enjoying the challenge.
0 Replies
 
Ionus
 
  1  
Reply Fri 12 Nov, 2010 09:45 am
@maxdancona,
Quote:
But filling in the blanks in a preset template isn't math.
I chose a very simple example. You do understand what I meant though ?
Quote:
being about to use the base concept to solve problems in a form that you haven't seen before.
This is where I disagree. I think it is more important to have a little amount very well known than how to apply it to all the exceptions. I think back to my maths classes and they just did problem after problem.....I wanted to learn major concepts, not do every possible example. If I had a choice I would take kids as far as maths can go before I went back and showed them every possible problem that may arise in every area.
I would like to see a class where broad areas are covered quickly, then the slow ones cover it again whilst the fast ones explore greater depth. The worst person to select a maths program is a maths enthusiast. They want all the thrills and excitement of really numbing detail and this is at the cost of large numbers of students shutting down and turning the lights of.....keep the pace up and do it in big brush strokes.
maxdancona
 
  1  
Reply Fri 12 Nov, 2010 10:12 am
@Ionus,
I think the example of solving quadratic equations is a very instructive example.

Let me ask this question. You obviously learned to solve quadratic equations in high school, but what is the last time you have had to solve a quadratic equation? Actually in the past 40 years or so, human beings have very rare solved quadratic equations outside of school.

What is the purpose? What is the benefit of learning to solve quadratic equations (something that even people in math and engineering jobs never do any more).

Well, let me tell you. As an engineer I never solve quadratic equations (I have two devices on my person, plus 3 or 4 programs on my computer that can do that in seconds). But, I use my understanding of quadratic equations all the time, for example to discuss how fast one algorithm will be compared to another. But every use is different, I am not given template-form problems. Rather, typically I am proposing a design for a computer program that I am responsible for, and I am answer questions. Each situation is different.

So I rely on my ability to use the core concepts of quadratic equations every day. I never need to solve them. Each time I encounter a quadratic equation it is in a different situation and a different form to solve a different problem.

Quote:
They want all the thrills and excitement of really numbing detail and this is at the cost of large numbers of students shutting down and turning the lights of.....keep the pace up and do it in big brush strokes.


You are confusing pace with content. My point is that the concepts are basic and that on each subject, you start with the core ideas, not mechanic methods or memorized details.

When I taught quadratic functions, the basics are what the roots and vertex mean particularly in real world applications. Getting students to understand when functions describing real world situations are quadratic is key to any meaningful understanding.

Understanding the key concepts means synthesis, that I can give a completely new situation that we haven't discussed and the student can give me a thoughtful answer about whether it is modeled by a quadratic function or not. This is a key skill in math and engineering. Then they should be able to explain what the roots and vertices signify in that specific situation.

Teaching them to solve the quadratic equation is important only if it provides insights into understanding some key concept. Factoring polynomials is very interesting and informative although the ability to factor polynomials is worthless. Memorizing the quadratic equation is a complete waste of time.




Ionus
 
  1  
Reply Fri 12 Nov, 2010 10:22 am
@maxdancona,
Quote:
what is the last time you have had to solve a quadratic equation?
When helping my kids learn quadratic equations and when making a graph of non experimental data.
Quote:
As an engineer I use quadratic equations all the time
I am almost surprised to meet an engineer who is not in a managerial position.
Quote:
You are confusing pace with content.
No. I am saying the pace should dictate the content and not the content slow down the pace.
0 Replies
 
engineer
 
  3  
Reply Fri 12 Nov, 2010 10:23 am
@maxdancona,
maxdancona wrote:

I think the example of solving quadratic equations is a very instructive example.

Let me ask this question. You obviously learned to solve quadratic equations in high school, but what is the last time you have had to solve a quadratic equation? Actually in the past 40 years or so, human beings have very rare solved quadratic equations outside of school.

What is the purpose? What is the benefit of learning to solve quadratic equations (something that even people in math and engineering jobs never do any more).

Well, let me tell. As an engineer I use quadratic equations all the time, for example to discuss how fast one algorithm will be compared to another. But every use is different, I am not given template-form problems. No, I am proposing a design for a computer program that I am responsible for, and I am answer questions. Each situation is different.

So I rely on my ability to use the core concepts of quadratic equations every day. I never need to solve them. Each time I encounter a quadratic equation it is in a different situation and a different form to solve a different problem.

That is why you had to solve all those quadratic equations in school. You learned how to do it upside down, in zero G, in the dark and now when you see all those different types of problems, you can adapt your skill set to solve them. If they had only showed you the concept and never had you apply it, do you really think you could apply those concepts with the dexterity you do today? Your skill set has evolved so far from where you were that you can't see building blocks anymore. What is utter simplicity today is only that way because you have collected a mountain of skills, each build upon the other, to apply to the problem.
 

 
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