What are "Basic Skills" in Math?

@engineer,

Quote:

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?

Everyone learned them but only she used them. A waste of everyone else's time to provide her with classmates. What about a student who could have become a music or sports teacher but dropped out because they hated quadratic equations (only an example...dont go there...).

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@engineer,

Wrong Engineer. You are confusing two separate issues. Let me explain.

The quadratic equation is ( -B + sqrt(B^2 - 4AC))/2A , ( -B - sqrt(B^2 - 4AC))/2A. I don't even have that memorized any more (although I am quite certain I could derive it if I had to). I just looked it up on google.

I distinctly remember in school getting problem sheets that had me solve problems like

1. 4x2+2x+13 = 6
2. 3x2 + 19 = 5x

To solve these problems I rearranged the formulas to find A, B and C so I could plug the values into the equation above. Doing these problems had nothing to do with my understand of quadratic equations. I had to rearrange the equations a bit to tease out A, B an C. And then I had to make sure that I got the damned parentheses right on my calculator.

The fact is I could have done 100 million of these problems and I wouldn't have gained any real understanding of quadratic functions. They are basically a meaningless waste of time. They have nothing to do with the understanding I use in my profession now. This type of rote problem has nothing to do with anything I have had to do in 20 some years of professional life.

In contrast, the things I do as an engineer involve understanding a problem and trying to whether it is a quadratic function or not. When it is a quadratic function, I need to understand how this affects its behavior in different situations.

The drill and kill exercises that I have done haven't helped me at all in my profession. Even the skill I gained typing little calculator keys doesn't help me much any more.

What has helped is the many different types of thought based problems that I have seen. It is the problems that made me think in a new way, or the problems that showed me flaws in my currently understanding, or the clever new ways to attack a problem that have made me a good engineer. Rote Schmote, mathematicians and engineers are trained to be thinker, creators and innovators, not calculating machines.

1 Reply

@fresco,

fresco wrote:

The "happy clappy/ instant access" philosophy of elementary teaching which has developed on the back of the information revolution seems to have had statistical effect of dumbing down the end product for the majority.

word.

the bulk of students coming out of schools in north america in the last 2 decades definitely seem like a dumbed-down end product. definitely some exceptions to that but I'd say my parents/grandparents/great-grandparents could wipe the floor with any university grad of the last two decades in terms of general and specific knowledge as well as ability to learn

they don't seem to teach students how to learn anymore

1 Reply

@maxdancona,

maxdancona wrote:

Our current education system is outdated. It was designed during the industrial revolution where we didn't need people who could reason and think.

I guess the past generations were just smarter than what is being hatched these days

1 Reply

@ehBeth,

Nonsense ehBeth! When you hear people talking about how good, noble and exceptional the past was, it is almost always mythology. In this case it what you are claiming about the past doesn't make any sense whatsoever.

Assuming that you are about my age, between 90 to 95% of our grandparents never even went to high school (of course for our grandmothers the number was even higher). It is possible that you had exceptional grandparents, but in general our grandparents were not very well educated or very good at math for that matter.

My grandparents were German immigrants from farming community. They worked really hard. My grandfather worked as a prison guard. They knew lots of things about history, but they had some crazy ideas as well. From talking with them, I know it was difficult for them to think critically (i.e. look at things from different perspectives). This is typical of people from that era. Of course the vast majority of the population either worked on farms or on assembly line jobs. The economy, particularly the mix of jobs has dramatically changed since then-- if 90% of people didn't have a high school diploma in this economy, it would be a disaster.

The fact is that far more kids are being taught Algebra then ever before. We have also learned quite a bit about teaching math, and now many methods are research driven with proven results.

We could dig up real numbers if you want to make a claim. I think the high school attendance rates are pretty conclusive as are the crusaders of the 20's and 30's who were complaining about how bad the education was at the time. Many of the points they were making are greatly improved now.

My kids' education is at least as good as the one I received.

1 Reply

@maxdancona,

maxdancona wrote:

Wrong Engineer. You are confusing two separate issues. Let me explain.

The quadratic equation is ( -B + sqrt(B^2 - 4AC))/2A , ( -B - sqrt(B^2 - 4AC))/2A. I don't even have that memorized any more (although I am quite certain I could derive it if I had to). I just looked it up on google.

I distinctly remember in school getting problem sheets that had me solve problems like

1. 4x2+2x+13 = 6
2. 3x2 + 19 = 5x

To solve these problems I rearranged the formulas to find A, B and C so I could plug the values into the equation above. Doing these problems had nothing to do with my understand of quadratic equations. I had to rearrange the equations a bit to tease out A, B an C. And then I had to make sure that I got the damned parentheses right on my calculator.

You have the order backwards. The first thing you learned was how to do FOIL. Then you learned how to reverse that to basic factoring. From there, you learned how to compete the square. Once you learned how to complete the square, you were showned the derivation for the quadratic formula which you can derive to this day because you were shown the principles behind it. After all of that, you were never asked to factor a quadratic again because you had learned all the skills necessary to solve any type of quadratic equation. Not only did you learn how to solve quadratic equations, you learned a process for solving not just quadratic equations, but all types of problems. That you had to do a few examples to see the beauty of it doesn't seem like a high cost. Of course, someone could have just given you the quadratic equation and told you to crunch it. I don't think that would have inspired the love of mathematics that you suggested is a basic skill we should be promoting though.

maxdancona wrote:

The fact is I could have done 100 million of these problems and I wouldn't have gained any real understanding of quadratic functions. They are basically a meaningless waste of time. They have nothing to do with the understanding I use in my profession now. This type of rote problem has nothing to do with anything I have had to do in 20 some years of professional life.

So I can't argue your personal situation. I find that what you deride as a meaningless waste of time was the equivalent of learning while working on the programming challenges you mentioned earlier. Each problem you solved caused you to reach out and use a tool, just like that cheat sheet posted on your board. After a while you didn't need the sheet anymore. We all learn at different speeds, so maybe you mastered it all after 20 problems and your classmate needed 30, so the extra 10 were a waste for you. The problem is that for the next topic, there was another cheat sheet required and then another after that. If you still need that first cheat sheet to get by ten levels later, you are in trouble.

maxdancona wrote:

In contrast, the things I do as an engineer involve understanding a problem and trying to whether it is a quadratic function or not. When it is a quadratic function, I need to understand how this affects its behavior in different situations.

Of course! Everyone in your profession is assumed to be completely competent in those basics. You say that once someone explained the concept, no drill was required, but I argue that if you did not achieve the fluency that you did when studying basic Algebra II, that you would not be nearly as successful today.

maxdancona wrote:

The drill and kill exercises that I have done haven't helped me at all in my profession. Even the skill I gained typing little calculator keys doesn't help me much any more.

Probably not, but at one time, it did help you - a lot. The first time you solved a quadratic equation, you learned a lot. The next time, less so. Soon, you got very little and it was time for the next type of problem. Just like your daughter doesn't get much anymore from looking at those infant books doesn't mean that when she was younger, she wasn't pulling real information out of them. Now that she has outgrown them, you've moved on to more skill appropriate material, but that doesn't mean that those books were useless. She absorbed the information through repetition. She learned those letter sounds through the repetition of hearing you say them and seeing them on the page. That you made it fun and intuitive is to your credit, but it doesn't mean it wasn't repetition.

maxdancona wrote:

What has helped is the many different types of thought based problems that I have seen. It is the problems that made me think in a new way, or the problems that showed me flaws in my currently understanding, or the clever new ways to attack a problem that have made me a good engineer. Rote Schmote, mathematicians and engineers are trained to be thinker, creators and innovators, not calculating machines.

But don't schools do that as well? Aren't school books looking at more ways to make math more relevant in their word problems? I certainly see that in what my children bring home. Several months ago, we had a roaring debate on A2K over a word problem and whether it was too complex for the age level. I think that is more common than sheets and sheets of crunch the numbers workpages. Maybe not in K, 1 and 2, but certainly in grades higher than that.

1 Reply

@maxdancona,

Quote:

It is possible that you had exceptional grandparents, but in general our grandparents were not very well educated or very good at math for that matter.

Ignorance will get you nowhere fast.

Your grandfather isn't an example of learning during his generation. He is an example of how people often lose that ability when they get older and set in their ways.

Quote:

The fact is that far more kids are being taught Algebra then ever before.

Are you telling me there isn't any algebra in this 8th grade test from 100 years ago?

http://jonb.blogspot.com/2008/07/8th-grade-test-from-100-years-ago.html

Quote:

I think the high school attendance rates are pretty conclusive as are the crusaders of the 20's and 30's who were complaining about how bad the education was at the time. Many of the points they were making are greatly improved now.

Since the time of cavemen, parents have complained about how the next generation isn't as smart. Their complaints don't prove anything about the actual education system anymore than the complaints today prove how bad it is.

1 Reply

@parados,

Quote:

Are you telling me there isn't any algebra in this 8th grade test from 100 years ago?

http://jonb.blogspot.com/2008/07/8th-grade-test-from-100-years-ago.html

Absolutely. I am telling you that there isn't a single question on that test that has anything to do with Algebra. These are all mad lib type questions. The kids memorize a fill-in-the-blank template and then plug in the right numbers. My kids, in eighth grade were doing much more complex and interesting problems involving setting up equations and solving for missing variables.

Snopes has a similar reaction to this alleged test.

Quote:

Consider: To pass this test, no knowledge of the arts is necessary (not even a nodding familiarity with a few of the greatest works of English literature), no demonstration of mathematical learning other than plain arithmetic is required (forget algebra, geometry, or trigonometry), nothing beyond a familiarity with the highlights of American history is needed (never mind the fundamentals of world history, as this exam scarcely acknowledges that any country other than the USA even exists), no questions about the history, structure, or function of the United States government are asked (not even the standard "Name the three branches of our federal government")

....

An exam for today's high school graduates that omitted even <I>one</I> of these subjects would be loudly condemned by parents and educators alike, subjects about which the Salina, Kansas, students of 1895 needed know nothing at all.

Would it be fair to say that the average Salina student was woefully undereducated because he failed to learn many of the things that we consider important today, but which were of little importance in his time and place? If not, then why do people keep asserting that the reverse is true? Why do journalists continue to base their gleeful articles about how much more was expected of the students of yesteryear on flawed assumptions? Perhaps some people are too intent upon making a point to bother considering the proper questions.

http://www.snopes.com/language/document/1895exam.asp

2 Replies

@maxdancona,

That was an eighth grade test right?
Unless you're in accelerated math, most highschoolers take Algebra 1 in 9th grade, Geometry in 10th grade, and Algebra 2 in eleventh grade...if you're accelerated you start everything a year early unless you're REALLY accelerated and then you work with an area university - at least that's the way it worked in the public school systems I worked in (in the US).

So, it's not really fair to compare like to unlike - in other words a person whose education consisted of eight years of instruction as compared to a person whose education consists of twelve years.
All of the stuff this guy said these people didn't learn would have happened in high school - which apparently most people didn't attend 100 years ago.

It'd be more enlightening to compare an eighth grader back then to an eighth grader today.

If eighth graders back then could pass that test, I know where I'd put my money.

1 Reply

@maxdancona,

Quote:

Would it be fair to say that the average Salina student was woefully undereducated because he failed to learn many of the things that we consider important today, but which were of little importance in his time and place? If not, then why do people keep asserting that the reverse is true?

Right back at you, Mr they didn't learn anything back then.

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@maxdancona,

Many people seem to think that memorizing equations is the same as learning concepts. Richard Feynman (Manhattan Project) called this the “Euclidean Viewpoint” meaning the love of mathematics, as opposed to the “Babylonian Viewpoint” where concepts are more important than equations.

Math alone is not a genuine explanation. Math is just a tool, it can be a crutch for those that only want the final numerical result. Math itself does not confer expert knowledge.
----------------------------------------------------------------------------------------------------
Richard Feynman
If we will only allow that, as we progress, we remain unsure, we will leave opportunities for alternatives. We will not become enthusiastic for the fact, the knowledge, the absolute truth of the day, but remain always uncertain. In order to make progress, one must leave the door to the unknown ajar.

Stephen Hawking
If what we regard as real depends on our theory, how can we make reality the basis of our philosophy? But we cannot distinguish what is real about the universe without a theory, it makes no sense to ask if it corresponds to reality, because we do not know what reality is independent of a theory.

Isaac Asimov
Man's greatest asset is the unsettled mind.

Bertrand Russell
The fact that an opinion has been widely held is no evidence whatever that it is not utterly absurd; indeed in view of the silliness of the majority of mankind, a widespread belief is more likely to be foolish than sensible.

Bertrand Russell
What we need is not the will to believe but the will to find out.

Edward de Bono (May 19, 1933)
“The need to be right all the time is the biggest bar to new ideas. It is better to have enough ideas for some of them to be wrong, than to be always right by having no ideas at all.”

Le Chatelier's Principle (1850 - 1936)
"Any change in status quo prompts an opposing reaction in the responding system."

Lenz's law
Is simply a physical interpretation of the choice of sign in Faraday's Law of Induction, indicating that the induced emf & the change in flux have opposite signs. Heinrich Lenz formulated the law in 1834. It simply just a special case of Le Chatelier's principle

Karl Popper (1902-1994)
“The better theory is the one that explains more, that explains with greater precision, & that allows us to make better predictions.” The central feature of science is that science aims at falsifiable claims (claims that can be proven false, at least in principle). However no single unified account of the difference between science & non-science is widely accepted by philosophers.

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@ehBeth,

Quote:

I guess the past generations were just smarter than what is being hatched these days

A friend of mine here reiterated what I saw someone mention on the British thread - and that was there is a theory or thought going around that since benefits have been introduced to young single women having children on their own, it has encouraged a rise in the birth rate to less motivated and educated people at the same time the birth rate has fallen among those who have more education and the population is showing signs of that 'dilution', in terms of intelligence.

Don't know if it's true, but that's the impression apparently that some people here have.

I myself think it has more to do with differing expectations surrounding work - and by work, I don't mean job - I mean expending energy to achieve or attain something- even knowledge.
I don't think our society instills or requires that as much from people today as it used to.
It doesn't seem to me that effort and the achievement that can result is seen as universally as enough of a reward anymore.
And of course, there are exceptions to that, but I do see even between the generations of my parents, as compared to mine, as compared to my childrens' - a definite and measurable change in expectations of work and effort from self.

1 Reply

@engineer,

We have a basic disagreement Engineer, I think we have hashed it out as much as we ever will. I will tell you my kids know that they will get hit if they ever say the word "FOIL" in my house. The idea behind foil is distribution of terms, and that is the way to teach it. If you understand distribution, you don't need the FOIL trick. Conversely, knowing the FOIL trick without understanding distribution of terms is meaningless. I suspect you disagree with me.

Quote:

Probably not, but at one time, it did help you - a lot. The first time you solved a quadratic equation, you learned a lot.

This really sums of the heart of our discussion. I don't believe that plugging values into a memorized equation helped me at all.

To me there is a big difference between the mechanical tricks of arithmetic, and understanding of ideas behind mathematics. I don't think they have anything to do with each others.

You seem to be saying that by doing rote problems using mechanical tricks you will gain insight into the mathematics. I don't think think this is true because if you are using the mechanical tricks, you don't need to reach for any understanding.

This is why I feel strongly that the ideas at the core of mathematics must come first and must be focus.

My daughter is just starting to develop an understanding of multiplication through tea parties (it is amazing how much math there is to be done in a tea party). For example, we want to figure out how many biscuits are needed for 3 dolls. I gently push her along while she works it out, it is a very good word problem on a real world situation that she is interested in.

She is learning about multiplication because the idea interests her and she wants to solve a problem that requires it. All this, and she has never seen a multiplication table.

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@aidan,

These are the Algebra requirements in the Massachusetts Curriculum Standards for 8th grade math. I am omitting the Advanced section for space.

Quote:

Proficient
Students recognize and differentiate between variables, expressions, inequalities, and equations. Students evaluate expressions and solve formulas and proportions. Students plot and give the coordinates of points on a number line and a coordinate plane. Students translate word problems into equations and solve single variable equations and inequalities. Students use a variety of tools and techniques to communicate the reasoning used in solving problems.

Basic
Students identify variables, expressions, inequalities, and equations. Students sometimes evaluate expressions and solve formulas and proportions. Students plot and give the coordinates of points on a number line and a coordinate plane with limited success. Students sometimes translate word problems into equations or translate equations into word problems. Students sometimes solve one-step equations and inequalities.

Eighth graders today are doing far more advanced math then what is in that silly test.

1 Reply

@aidan,

Quote:

A friend of mine here reiterated what I saw someone mention on the British thread - and that was there is a theory or thought going around that since benefits have been introduced to young single women having children on their own, it has encouraged a rise in the birth rate to less motivated and educated people at the same time the birth rate has fallen among those who have more education and the population is showing signs of that 'dilution', in terms of intelligence.

1 Reply

@maxdancona,

I believe the Brits are in the process of cutting off benefits for the unemployed.

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@maxdancona,

Quote:

Eighth graders today are doing far more advanced math then what is in that silly test.

Yeah, but do they know the volume of a bushel?
Laughing

I'm just kidding - I'm not trying to downgrade kids today - I love kids.
But I think we could incorporate some of our old practices and values along with the new and hit paydirt...
Just my own opinion.

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@maxdancona,

maxdancona wrote:

She is learning about multiplication because the idea interests her and she wants to solve a problem that requires it. All this, and she has never seen a multiplication table.

That again seems to be presenting a false choice, though -- that because this is a good thing, memorization is a bad thing. I think that this is a good thing AND memorization has its uses.

I think that what you describe is great way to learn multiplication, and that's how my kid learned it too (at least part of the way, and how she was introduced to it at the earliest stages).

And it worked, in the sense that she is good at getting the right answer.

That said, I also don't see anything wrong with her now, as a fourth grader, memorizing the multiplication tables. She gets the process. It's just much more efficient for the number "48" to come to her immediately when presented with "6 x 8," than for her to laboriously figure it out each time. And multiplication is needed for so much other math.

1 Reply

@maxdancona,

maxdancona wrote:

I don't believe that plugging values into a memorized equation helped me at all.

but education, in a global sense, is not about your abilities or your daughter's abilities

people learn in many different ways

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@maxdancona,

maxdancona wrote:

We have a basic disagreement Engineer, I think we have hashed it out as much as we ever will. I will tell you my kids know that they will get hit if they ever say the word "FOIL" in my house. The idea behind foil is distribution of terms, and that is the way to teach it. If you understand distribution, you don't need the FOIL trick. Conversely, knowing the FOIL trick without understanding distribution of terms is meaningless. I suspect you disagree with me.

Not so much disagree as seeing that there is room for both. You can both understand the basic principle and have a mnemonic to remember the process.

maxdancona wrote:

Quote:
Probably not, but at one time, it did help you - a lot. The first time you solved a quadratic equation, you learned a lot.

This really sums of the heart of our discussion. I don't believe that plugging values into a memorized equation helped me at all.

To me there is a big difference between the mechanical tricks of arithmetic, and understanding of ideas behind mathematics. I don't think they have anything to do with each others.

But I didn't say the first time you plugged numbers in a formula, I said the first time you solved a quadratic. Repetition doesn't have to be plugging in numbers repeatedly. You didn't do that as a teacher and my children don't bring home that kind of work either. The real work is setting up the problem and you solve the quadratic to get the final answer. Repetition has value, boring repetition less so.

maxdancona wrote:

You seem to be saying that by doing rote problems using mechanical tricks you will gain insight into the mathematics. I don't think think this is true because if you are using the mechanical tricks, you don't need to reach for any understanding.

It really depends on whether you are given the tool or shown how it is developed and why is works. I was not given the quadratic formula and told to plug in A, B and C. My class followed a specific path of learning concepts until we could derive it. Only then could we use it and it wasn't just plug in A, B and C, but as a way to get the answer more quickly after setting up the problem. Still, we solved a great number of problems of all different types where we used the quadratic formula at the end to get the final answer. There was value in the process of developing the formula and there was value in repetition to master its use.

maxdancona wrote:

She is learning about multiplication because the idea interests her and she wants to solve a problem that requires it. All this, and she has never seen a multiplication table.

That is great, she is lucky to have someone who can help her along like that. To Soz's point though, at some point she will either have those tables memorized because you have put her through hundreds of problems like that (massive repetition even if it enjoyable) or she will want to memorize them because it makes doing those Algebra problems more fun.

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What are "Basic Skills" in Math?

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