For parents concerned about experimental progressive math ruining your child's education

The Coffee Shop
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The most annoying thing was not being able to help my 5th grader with her math homework. It wasn’ the right answer they were looking for, it was some whacky way of estimating that I didn’t understand.

My nephew was given a question as follows on a test.

“Why is it helpful to make an estimate before finding the size of a piece of circle graph”.

His answer

“I don’t think it is”

He got it wrong.

As I told my brother, I think he should have gotten it right. Estimations are only helpful if you either don’t have time at the present or will never need to know the exact number. Find an estimation and then finding the exact number isn’t very helpful :slight_smile:

It took me a while to figure out what they were trying to accomplish with all this “Estimation” stuff but when I did I realized two things.

They were trying to allow the kids to come up with an “Estimation” so that they would not have to actually do the math to get the exact answer…because that requires more work and an understanding of the basic math. Also that once they understood the basic math they no longer need all the “Estimation” tricks because they could figure out the exact answer just as quickly.

~Matt

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“Estimations are only helpful if you either don’t have time at the present or will never need to know the exact number.”

They are also helpful in evaluating whether you made a mistake in the calculation that you used to get what you thought was the exact number. For example, they’ll help you discover a misplaced decimal point.

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They are also helpful in evaluating whether you made a mistake in the calculation that you used to get what you thought was the exact number. For example, they’ll help you discover a misplaced decimal point.

That is also true. But I always preferred working the problem another way and seeing if it came out to the same answer.

It’s also very helpful to keep your impatient and or irrational boss or customer happy when they just HAVE to have numbers NOW…cause the extra couple of minutes to get the non estimate may cause the world to collapse :slight_smile:

In either case however “Estimation” does very little, if nothing, to teach the actual math, at least in my experience.

~Matt

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“In either case however ‘Estimation’ does very little, if nothing, to teach the actual math, at least in my experience.”

Agreed. But are you sure the teachers ever intended it as a substitute for the main math calculations? Or did they just happen to be in the middle of a unit on the topic of estimation?

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His answer

“I don’t think it is”

He got it wrong.

As I told my brother, I think he should have gotten it right. Estimations are only helpful if you either don’t have time at the present or will never need to know the exact number. Find an estimation and then finding the exact number isn’t very helpful

I think if he followed up his answer with your explanation, he should have gottne full (or almost) credit. But just answering “Because” is insufficient. THAT is probably the lesson that he needs to learn. A question that asks “why” implies that a cogent argument is expected, whether or not the conclusion is agreed upon.

I was taught to use estimations on a test “to see if the decimal pt was right (for example)” because working the problem a completely different way was something one didn’t have time for.

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In either case however “Estimation” does very little, if nothing, to teach the actual math, at least in my experience.

I agree with you that estimation doesn’t teach the actual math, but estimation is an important math topic that should be learned in addition to everything else, IMO. I use estimation in real life very often. If I want the exact number, then I use my calculator in most cases. But, I quite often want a quick estimate which I’ll do in my head.

I also agree that it helps you figure out if you made a mistake in your calculation. If you know your number should be in the hundreds, but your answer is 54.3, then you know you have a problem and can fix it.

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I also agree that it helps you figure out if you made a mistake in your calculation. If you know your number should be in the hundreds, but your answer is 54.3, then you know you have a problem and can fix it.

But what do you fix, your estimation or the figgers used in the calculation?

Too deep for my little head.

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Agreed. But are you sure the teachers ever intended it as a substitute for the main math calculations? Or did they just happen to be in the middle of a unit on the topic of estimation?

Clearly I can’t speak to what was supposed to be happening in the class, but in my daughters case it was quite obvious she had not been taught, or at least didn’t understand, the basics underlying these “Estimation” exercises.

After working with her for a couple nights she picked up these basics rather easily which leads me to believe they had never been gone over in class.

That left me with the impression that this method was a substitution for teaching the basics or best case scenario that they were doing estimation BEFORE teaching the basics which, IMO, is bassackwards.

To the contrary in the school my other kid is in now it was clear they covered the basics before the “Estimation” although he is not at the shown level of math yet. The “Estimation” was simply guessing, but none the less he already knew numbers and what they represent etc.

~Matt

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I use estimation in real life very often. If I want the exact number, then I use my calculator in most cases. But, I quite often want a quick estimate which I’ll do in my head.

I pretty much stated this already. If you don’t need the exact number, or don’t have time to find the exact number estimation is a perfect substitute. Estimating to “Double check” is also a great way to see if you have major errors or not.

I don’t think the point that is being argued is on the value of estimation as much as the value of estimation as teaching tool to teach basic math or the value of it when you know you need the exact number.

In my daughters case I believe, although can’t 100% say, they were attempting to use estimation as a teaching tool to teach basic math skills. In my nephews problem case it asked if it was useful to estimate if he had to figure out the exact number…which it’s not really unless he wants to check for gross errors.

~Matt

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My only concern here is that in the past I’ve encountered people who dismissed such matters as learning alternative number bases (binary, hexadecimal, etc.) as impractical “new math.” As most of us here are aware, such number bases are a valid and important part of the modern world. Let’s just be sure we’re not throwing out the baby with the bathwater, if you’ll pardon the cliché.

In high school I was taught under a program known as SMSG, and I notice that the Wikipedia article puts it under the umbrella of “new math.” The emphasis was on proving theorems and understanding the reasoning rather than merely learning by rote. That approach certainly didn’t hinder my understanding of math, to put it mildly.

The problem with that system, s well as most progressive systems, is that it is often taught at the expense of the fundamentals. Hate to say it, Rob, but number theory and proofs are largely useless for all but the smallest percentage of the population. The theory isn’t too different from the modern math programs: get them doing complicated and confusing stuff with numbers and magically understanding of the entire subject will occur.

What most kids simply need is to be taught how to add, subtract, multiply, and divide. From there you move on to areas, decimals, and fractions. From there you move on to basic algebra, geometry, and trig. If they make it beyond that, you introduce calc and/or stats.

Proving theorems is fine to throw in there for faster paced kids.

As for rote - it is the tried and true best method. We can sit here and talk about running technique all we want, but the fastest runners will be the ones who get the most miles under their belts. I’m not saying to learn via repetition with an absence of explanation. ie…learning to memorize rules for adding negative numbers by doing it over and over again is a poor substitute for teaching the concept of a negative. But at the end of the day, as an old phd physicist colleague of mine once said, “You don’t learn with your eyes and ears. You learn with your arm.” ie by doing lots and lots of problems.

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“The problem with that system, s well as most progressive systems, is that it is often taught at the expense of the fundamentals.”

We were put in the SMSG program in ninth grade, by which point we had long since learned the “fundamentals” as you seem to be defining them (“how to add, subtract, multiply, and divide…areas, decimals, and fractions”) and a lot more. So it certainly wouldn’t be fair to say that the program was “taught at the expense of the fundamentals.”

Not long ago, BTW, we were talking with my nephew about the college algebra courses he had recently taken. Wondering whether he had actually learned anything, I asked him if he knew about the quadratic formula. “Oh yes,” he replied. But when I asked him what it was, he couldn’t tell me. I then prompted: “x = minus b plus or minus the square root of…(etc.)” and he was able to parrot back to me what I had just said. So I asked him what you would use the formula for, and he hadn’t the slightest idea. Further questions made it clear that he wouldn’t have any idea how to solve a second-degree polynomial equation, even after we had explained to him what that was. He’d never heard of the concept of “completing the square,” or any other method of solving the problem. This kid’s dad, BTW, is highly skilled in math and several fields that apply it, so one would suspect the kid would have plenty of native ability in the subject.

In any case, in the program I was following, by the end of ninth grade we were very well drilled indeed in several alternative approaches to problems of that sort.

“but number theory and proofs are largely useless for all but the smallest percentage of the population… Proving theorems is fine to throw in there for faster paced kids.”

If, as you imply (and I don’t necessarily agree), it’s only the “faster paced” kids who are capable of deriving logically correct conclusions from premises, that would seem to be a strong argument against universal suffrage. The methodology of proof isn’t just for mathematicians.

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The problem with your nephew are exactly the very fundamentals I’m talking about. There are three likely scenarios regarding his understanding of math:

  1. There is very little teaching happening and lots of free time in class.

  2. Two much time is spent on crap like in the link I provided in the OP that not enough time is spent on what you mentioned above.

  3. Your nephew is in a slower paced class because he isn’t good enough at math.

  4. is crapy teaching and 3) is a crappy student (no offense intended), neither of which is the product of an educational philosophy.

#2 is what we are talking about in this thread. “This is the quadratic formula, this is how it is derived, this is how to understand it, and this is the practical application of it,” + lots of practice is what I consider good, sound, fundamental teaching for an algebra II class.

That’s also called “direct instruction” which is often frowned upon in educational academia. If this was where I taught, I would have spent half the class on an activity involving algebra tiles working in groups with the goal of them figuring out the quadratic formula on their own. Then they’d have to write a paragraph explaining what they think it is and what it is for. After all of that time is wasted, they would then be shown how to do it on a calculator.

At least that was what I was supposed to do. Following that system, those students wouldn’t be able to answer your questions either.

As for “new math” in the 60s, a lot of time was spent on theorems and number theory, as well as other math tricks…again, wasting a lot of time when they could be learning the quadratic formula and it’s applications.

If, as you imply (and I don’t necessarily agree), it’s only the “faster paced” kids who are capable of deriving logically correct conclusions from premises, that would seem to be a strong argument against universal suffrage. The methodology of proof isn’t just for mathematicians.

You grossly overestimate the abilities of the common person. I will guaranty you that 90% of the LR can’t write a geometric proof (and contend that the LR represents a well above average strata of society).

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“As for ‘new math’ in the 60s, a lot of time was spent on theorems and number theory, as well as other math tricks…again, wasting a lot of time when they could be learning the quadratic formula and it’s applications.”

Actually, the whole point of my anecdote regarding my nephew was that those of us who were schooled in the SMSG program, at least, learned very well what the quadratic formula was and how to use it. I’ll grant you that not all “new math” may be that effective, but it’s foolish to assume that nothing new can be good in the area of math pedagogy.

“You grossly overestimate the abilities of the common person.”

Au contraire. The common person grossly overestimates his/her ability to make wise decisions about everyone else in society via the political process. That’s a principal reason why our political system is in such a mess.

I do believe, however, that the abilities of the common person could potentially be much higher if the learning environment were improved.

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I wanted to add that your example above was a good example of where “rote only” education goes wrong. Memorizing steps isn’t learning. Unfortunately, because of this, rote has gotten a bad wrap among the educational hirarchy…so, as always, they throw the baby out with the bath water. “Rote only” is bad, but “rote” is necessary to reinforce good practices and help set the lessons into long term memory.

I do believe, however, that the abilities of the common person could potentially be much higher if the learning environment were improved.

I agree…emphasis on “improved” rather than simply “changed.”

On a side note, some schools are experiementing with a 4 day a week schedule. I told my wife if they really want to fix education, make our kids go to school as much as the other westernized countries. Our kids attend 10-20% less school than most of these countries.