The thinking is what matters approach is invalidated if the students don’t get to the correct answer.

How many students reinvent the Peano Axioms? Or see the necessity of the Dedekind recursion theorem? Or invent the Grassmann induction proof that addition of whole numbers is associative and commutative?

How many of these classes get to correct versions? How many leave out details?

In a subject like math or even more so in physics, there is confusion and also intentional false statements as pitfalls.

Working with the correct versions is hard enough.

What benefit is it to stop at a wrong understanding?

Devlin doesn’t understand that addition and multiplication of natural numbers are just a way to keep one’s place in the count.

In the process of spelling it out, the author, teacher, tutor, etc. learns something and often doesn’t know the correct version unless they spell out the proof or method in every detail.

Computer programmers, users, etc. are always complaining the specs, documentation, etc. has left things out. Do we want specifications for VCRs, computers, etc. that leave things out?

If you look at videos for home repair, they spell it out. Unplug the refrigerator before replacing the automatic ice maker. Do we really want this left out? Which wires to connect? Leave it out?

Should anything involving safety leave out steps and entire concepts? The answer is no. This tells us people can’t invent those steps and safety precautions in all cases.

How many people continue with math courses taught this way? How many people search out Schaum’s outlines to see detailed step by step examples?

How many people learn something in math or applied math from an article that spells out in detail every step and obvious concept and connection?

The Dedekind book on natural numbers spells out each little lemma and its proof. How many have rediscovered it?

Math needs to be spelled out more not less.

Furthermore, important insights into math have been lost in current textbooks. Look at the book Lennes Veblen

http://www.gutenberg.org/ebooks/18741

This book is 100 years old and contains an approach to calculus reflecting the Lagrange and Cauchy way of doing calculus in some points. Most of this is completely lost to teachers of calculus today in high school and college.

In a series called Vanishing Calculus, I will demonstrate a superior way of teaching calculus incorporating these prior approaches. Base Point Test Point Calculus is a series already available in part that emphasizes the logic of limits closer to how they are taught currently.

Base Point Test Point Calculus spells out in complete words the logic of limits in calculus. Most students don’t understand this at all. This disproves the self discovery approach.

Base Point Test Point Calculus at e-vendor.

The don’t explain the details approach lets the textbook writer and thus the teacher off the hook to explain. In practice, such gaps are never made up. Tests are not more detailed and nit picky than are the text materials in practice. I would like to see a case where self discovery was used without a text that spells things out and the test asked for detailed distinctions of the students and marked off for not doing them correctly.

How could a test have the words and terminology to make distinctions that are common to the students taking the test if those words were not taught by the text or class materials?

If each student invents the restrictions and limitations of a method or theorem with their own words, how does the test have common words to use to ask them for those distinctions or mark them off for the wrong words to explain the distinctions?

How about a function is a set of ordered pairs in which no two ordered pairs have a common first element and different second element? Is this self discovered? Including the words?

Or the epsilon delta definition? Including the requirement of a test point in the domain of the function in each interval around the base point? This question illustrates the value of having terms for the parts of the limit definition. Why didn’t these teachers invent these words already? A few have, but the combination of base point and test point in the limit definition is new as far as I know.

“base point” “test point” limit

Hello –

Thanks for responding to my article, though I have to say, I think you’re largely misrepresenting my views. You ask, “what benefit is it to stop at a wrong understanding?” but this is something I never suggested. The very purpose of mathematics is to make sense of confusing circumstances and arrive at greater understanding. To that end, all students must carefully analyze the legitimacy of their reasoning, and construct their arguments with explicit, careful language. If I’m reading you right, you’re suggesting that the only way to reason legitimately or construct a proof is to appeal to the Peano Axioms or first work through the writing of Dedekind. I certainly disagree with this philosophy, especially as it impacts the doing of mathematics for kids, but again, this is not what I had intended for my article, so I’ll omit my critique.

I was simply arguing against highly regimented, overly structured math pedagogy, which aims to perfect the content delivery system. In particular, I had in mind the mistake that many teachers make when they prioritize mnemonics and specific techniques or facts above the broader purposes of a mathematics education: learning to think through confusion, towards clarity.

I’ll end there, but I’d be happy to keep the dialogue going however you’d like. Thanks again.

Thanks for your long comment. I was not intending to represent your views specifically as being criticized, since I don’t know them well enough to do so. Evidently, I meant the link to your page just to be a sample and not explicitly what I was trying to address as a doctrine.

“The very purpose of mathematics is to make sense of confusing circumstances and arrive at greater understanding.”

I think this suggest a different program you have then what I was discussing.

My program is that place value is recursive and depends on addition and multiplication of whole numbers. I see place value and its algorithms as the central core of elementary math. The structure of place value comes from the two operations of addition and multiplication of whole numbers, which are already recursive to start with and then are combined in a recursive way in place value.

I do not believe that many people involved in elementary math see place value that way. I think Devlin’s approach of field axioms for real numbers is very far away from the approach I suggest.

Grassmann was a teacher of high school and lower grades and developed his book in 1861 with the definitions of addition and multiplication of whole numbers using recursion.

The prime notation 2′ = 3 is not more complicated to understand then the addition and multiplication signs.

Once you start using the prime notation to do elementary math of whole numbers, you find it is extremely useful to convey the concepts.

I think if you look at elementary math materials they are mostly drill sheets. There is very little explanation. Can you point to existing books on elementary math that explain place value notation and its algorithms in a logical manner?

If k-6 is mostly drill sheets, then it is hard to see it as directed to concepts. I think the claim it is conceptual is invalidated by it falling back to drill sheets as the bulk of materials available for k-6.

There is also an inability to write K-6 math in words. This arises from the lack of the conceptual understanding of the algorithms in terms of addition and multiplication as recursive.

I think it is hard to write descriptions of elementary math concepts all in words with no symbols if you do not have a good grounding in Peano Axioms and the elementary operations as recursive operations.

Place value is about keeping the place in the count in one way of looking at it. This arises from addition and multiplication of whole numbers being about keeping place in the count. This is most directly because addition and multiplication are defined in terms of counting one more in one of the input numbers. But it is also because natural numbers are all about keeping the place in the count that goes one by one ie the successor function. Since the successor function is in the Peano Axioms as the fundamental object of them, everything that arises from it is about counting by one in some way. Without this idea, one can’t understand elementary math of whole numbers including place value.

Maybe that should have been not so evidently.

Thanks for your clarification. I think I understand what you’re saying now. You’re actually making a really valid point about how different notations can help illuminate certain aspects of mathematics (or at least to a particular student). It sounds like this is exactly what happened for you when you rethought arithmetic in terms of the successor function and prime notation. This point actually pushes me to loosen the constraints I place on notation for my students. There are many paths to understanding and many ways to communicate and represent our ideas. Formal axioms represent one aspect of mathematics, and while I definitely see its value, it’s not everything.

After all, what are we to make of all the incredible math that came in the thousands of years before Peano?

Well thanks for you kind words on prime notation, which presumably is a sidestep from your still disagreeing with my main points of contention.

On the pre-Peano past.

If you compare Euler’s algebra from 1765, he tries to start with whole numbers and fractions but actually fails to explain them. On whole numbers he wants to do something like Peano Axioms but doesn’t know how.

If Euler can’t do it, neither can others, except by going down the Peano path.

By chance, I have a post on his book and his attempt to explain whole numbers.

http://newmathdoneright.com/2012/08/01/eulers-elements-of-algebra-gaps-in-logic/

He doesn’t know how to do it. Euler doesn’t know how to define addition in terms of counting or at all.

With fractions, Euler just says that we know they exist from the line. This is no better than grunting, in fact worse. It makes people think the rules for fractions come from pointing at a line, when they don’t.

Students often have trouble with fractions because the teacher in fact doesn’t know how to explain them. Better students learn to imitate the rules, but poorer students may need the idea. But the idea of where fractions come from is difficult.

http://newmathdoneright.com/category/fractions/fractions-as-functions/

I also discuss Euclid’s approach to fractions, although my understanding of his ideas was vague to begin with and possibly still poor at the end.

http://newmathdoneright.com/category/euclids-elements/eucliss-concepts-of-number/

Euclid is somewhat flawed though and his flaws are still taught in many cases I think. The way to avoid those flaws is to have a better starting point in natural numbers. As we saw, Euler doesn’t have that. But Peano does. So we use Peano.

Wrong explanations are not the answer. Grassmann, Dedekind, Peano figured out what was missing in Euler’s book which was well known. So there is no point teaching Euler’s answers today. But in fact, that is what is done. That turns into pointing and imitating algorithms because the conceptual meaning of natural numbers is not in Euler.

Anyone wishing to leave snarky comments at my expense, please do so. Although I do insist on tasteful language that would be appropriate if a minor chanced on this page sometime before the latter recesses of the century.

Do you mean me? I don’t think I was snarky or distasteful. I was hoping to be explicit and clear.

When I mean you, I will try to remember to say so. Although if I seem senile from my writing, you may be right. However, I mean anyone else who finds my style or manner irksome and has an overwhelming urge to express that. I don’t mean you. Or rather, I did not mean your comment was snarky or distasteful. However, I did include you if you want to leave a snarky comment.

Another inquiry I will post here for those interested is how to shorten “less than or equal”? An example is bounds. So 3 bounds 2. 3 bounds 3. 3 bounds the set of 0,1,2 and 3.

“Less than or equal” needs a pause which means you can’t convey a thought compactly. I have spent a number of years on questions like this. Writing elementary math in words or calculus in words requires this type of thinking.

I have written over 500 pages of calculus in words and in the process have dealt with such questions.

Most explanations in math are take the number add it to the number and get the number. When you write a solid block of math with precision in words you have to go up a learning curve. I have done that. This turns explanations from this that this that this into meaningful sentences. My blog and book are full of examples of this. Look at Peano Axioms for an example. I believe I have gone up a ladder and am willing to let others climb it through this blog and the books. But I think people are so satisfied with explanations of this that this that this that they don’t realize the need of improving their explanations or understanding.