Sources to study basic logic for precocious 10-year old?

[nomadreid]

I am advising the mother of a precocious 10-year old who studies on his own and already has some rudimentary algebra skills. I am not tutoring the child, just giving some recommendations. One aspect that seems inadequately represented among resources on the Internet or books that she could order are books on basic logic. That is, if you do an Internet search with the keywords "logic for kids", you merely get lots of puzzle books. (Puzzles are fine, but when they are not explicitly and consistently connected to more significant elements in mathematics or science, then they are nothing more than curiosities.) With keywords "Introduction to Logic", one gets texts which would be too difficult for a 10-year old, despite his (apparent) high intelligence. There are some good books for older students -- The Annotated Alice, the Raymond Smullyan books, etc. -- but one has to already have a grounding in basic logic to understand them.

The logic should include aspects which also allow the student to better understand the way science works beyond the stereotypical simplified description of the scientific method as "hypothesis--experimental test--accept or reject hypothesis -- if reject, new hypothesis".

So, does anyone have any recommendations, either for books that the mother could order or Internet sites that the son could visit? Thanks in advance.

 

[gleem]

This might work. The Fallacy Detective by Bluedorn and Bluedorn.

https://www.amazon.com/Fallacy-Dete...ze-Reasoning/dp/097453157X/?tag=pfamazon01-20

"I really like The Fallacy Detective because it has funny cartoons, silly stories, and teaches you a lot!"--11 Year Old

 

[nomadreid]

Excellent recommendation, gleem! Many thanks! I will immediately recommend that the mother buy this.

(To other readers: I am still open for more recommendations!)

 

[mathwonk]

harold jacobs, geometry, 1st edition, definitely not 3rd edition.

maybe this one:
https://www.betterworldbooks.com/pr...WY_rwkA1Gqd8x27t3SFY2RI5NJU7YPsxoCj6wQAvD_BwE

 

[nomadreid]

Thanks, mathwonk. I shall keep it in mind once the child has some basic algebra skills: as far as I could see, this book requires (what Americans call) Algebra I as a prerequisite. By the way, beyond the expense, what do you find objectionable about the third edition? And what about the second edition?

A side note: the child is British. I wonder how many of the "real-world" examples in the book come from the American "real world"? I thought of this as I saw an example from the book in which they refer to the symbol of one of the American television networks.

 

[berkeman]

I don't know if it's appropriate for your student/friend, but when I learned about the Peano Axioms in undergrad, I was amazed at how powerful the concept was. At the start of that section of the class, the professor asked us basically "How do we know that our development of mathematics is correct? How can we possibly know that?" And he went on to explain the concept of coming up with the simplest set of axioms that we could all reasonably agree on, and then using those axioms to derive much of the rest of math and operations and identities, etc.

And if the topic could be appropriate for your student, I'm not sure what resource is best for learning about them and their applications. The Wikipedia article is a pretty good introduction, but it would be better to find a good online teaching resource that would be understandable and entertaining to your young student.

https://en.wikipedia.org/wiki/Peano_axioms

In mathematical logic, the Peano axioms (/piˈɑːnoʊ/,[1] [peˈaːno]), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.

 

[mathwonk]

I do not have the Jacobs geometry book at hand, but from memory, I recall that the first edition has a chapter early on, covering logical reasoning, using amusing examples from Lewis Carroll, and cartoons from (American) newspapers, like Peanuts, BC,...

The later editions removed much of the useful coverage of logic, apparently dumbing down the book for the modern audience. Since you asked about logic, I suggested the book mainly for that aspect, rather than the geometry it contains, although that is good too.

I do not remember that algebra is actually a prerequisite, almost certainly not for the logic part at least. He also has an excellent algebra text, which did not suffer from dumbing down revisions as I recall.

The third edition of the geometry, "...doing, understanding", has a rave review on Amazon from a mathematician who seems to have been involved in the production of the book. My critical review used to be there, but seems to have disappeared. As I recall, I believe I sadly observed that while it was still a good geometry book, it was not as good as it had been. But for logic instruction, it was in fact not nearly as good, as that had largely been removed. I am not sure about the second edition, but as with almost all math books, I highly recommend the first edition. Changes to later editions are almost always made with pressure from the publisher, to increase sales to less gifted readers, or just to make inconsequential changes designed to give the false impression that one should buy the latest version at a higher price.

If you are open to a geometry book with logical reasoning, indeed the source of all such books in western culture, you might try Euclid, in the green lion edition. I taught from it to a group of very bright 10 year olds at epsilon camp in 2011, and it went very well.

Here are my free notes from that course:
https://www.math.uga.edu/sites/default/files/inline-files/10.pdf

the books for the course were Euclid, and a guide to it by Hartshorne:
https://www.greenlion.com/books/EuclidsElements.html

https://www.amazon.com/Geometry-Euc...product-reviews/0387986502/?tag=pfamazon01-20


my own introduction to logic was in high school, from a chapter of a 1955 book called Principles of Mathematics, by Allendoerfer and Oakley. Used copies seem pricey.
https://www.abebooks.com/servlet/SearchResults?an=allendoerfer, Oakley&cm_sp=SearchF-_-home-_-Results&ref_=search_f_hp&sts=t&tn=principles


As to prerequisites for Jacobs geometry, I began teaching my own 8 year old son, who knew no algebra, from Jacobs geometry many years ago, and he took to it easily. But when my other son complained of my somewhat pushy home schooling of him in algebra, I decided to back off with both kids, and hence did not continue the lessons in geometry. The moral is that Jacobs works fine for young bright kids, but such instruction is only recommended if they really want it.

 

[nomadreid]

I don't know if it's appropriate for your student/friend, but when I learned about the Peano Axioms in undergrad, I was amazed at how powerful the concept was. At the start of that section of the class, the professor asked us basically "How do we know that our development of mathematics is correct? How can we possibly know that?" And he went on to explain the concept of coming up with the simplest set of axioms that we could all reasonably agree on, and then using those axioms to derive much of the rest of math and operations and identities, etc.

And if the topic could be appropriate for your student, I'm not sure what resource is best for learning about them and their applications. The Wikipedia article is a pretty good introduction, but it would be better to find a good online teaching resource that would be understandable and entertaining to your young student.

https://en.wikipedia.org/wiki/Peano_axioms



View attachment 347848

Excellent suggestion, but as I am only briefly advising this student (or more exactly, his mother) rather than tutoring him, I would need a source with a softer introduction than the Wikipedia article, as the student is only 10 years old . However, you are right that the axioms are a great introduction to a lot of important logical concepts -- I introduced them to a student (whom I had the advantage of tutoring, so I was less dependent on sources) last year, when she was (a precocious)14, --- not only that went over well, but also I intend to use them this year with her to highlight the difference between 1st and 2nd order sentences.

Also thanks for the domino example. I shall use that with my older student.

 

[nomadreid]

I do not have the Jacobs geometry book at hand, but from memory, I recall that the first edition has a chapter early on, covering logical reasoning, using amusing examples from Lewis Carroll, and cartoons from (American) newspapers, like Peanuts, BC,...

The later editions removed much of the useful coverage of logic, apparently dumbing down the book for the modern audience. Since you asked about logic, I suggested the book mainly for that aspect, rather than the geometry it contains, although that is good too.

I do not remember that algebra is actually a prerequisite, almost certainly not for the logic part at least. He also has an excellent algebra text, which did not suffer from dumbing down revisions as I recall.

The third edition of the geometry, "...doing, understanding", has a rave review on Amazon from a mathematician who seems to have been involved in the production of the book. My critical review used to be there, but seems to have disappeared. As I recall, I believe I sadly observed that while it was still a good geometry book, it was not as good as it had been. But for logic instruction, it was in fact not nearly as good, as that had largely been removed. I am not sure about the second edition, but as with almost all math books, I highly recommend the first edition. Changes to later editions are almost always made with pressure from the publisher, to increase sales to less gifted readers, or just to make inconsequential changes designed to give the false impression that one should buy the latest version at a higher price.

If you are open to a geometry book with logical reasoning, indeed the source of all such books in western culture, you might try Euclid, in the green lion edition. I taught from it to a group of very bright 10 year olds at epsilon camp in 2011, and it went very well.

Here are my free notes from that course:
https://www.math.uga.edu/sites/default/files/inline-files/10.pdf

the books for the course were Euclid, and a guide to it by Hartshorne:
https://www.greenlion.com/books/EuclidsElements.html

https://www.amazon.com/Geometry-Euc...product-reviews/0387986502/?tag=pfamazon01-20


my own introduction to logic was in high school, from a chapter of a 1955 book called Principles of Mathematics, by Allendoerfer and Oakley. Used copies seem pricey.
https://www.abebooks.com/servlet/SearchResults?an=allendoerfer, Oakley&cm_sp=SearchF-_-home-_-Results&ref_=search_f_hp&sts=t&tn=principles


As to prerequisites for Jacobs geometry, I began teaching my own 8 year old son, who knew no algebra, from Jacobs geometry many years ago, and he took to it easily. But when my other son complained of my somewhat pushy home schooling of him in algebra, I decided to back off with both kids, and hence did not continue the lessons in geometry. The moral is that Jacobs works fine for young bright kids, but such instruction is only recommended if they really want it.

Wow, super answer, mathwonk. Many thanks. I shall therefore pass on your recommendation about the Jacobs book to the mother, emphasizing this difference in the editions. (This kills two birds with one stone -- the price difference being the other bird.)

I have also just downloaded the Green Line Edition of Euclid; it looks very useful for this student, and so shall send this to the mother as well. (I will also use it for an older student that I am tutoring.) Your notes will also be handy bit by bit.

I noticed that you included a remark about calculus in your notes. Although you do not actually do the calculus, such a remark might scare a student who has been taught that this is something for much older students. This is a shame, given that many calculus concepts are accessible to students at younger ages than they are presently introduced to them. However, the Hartshorne book definitely looks university level.

 

[mathwonk]

thank you. In fact, if you look into my day six notes I feel that I do actually use calculus methods to find volumes there, see e.g. the paragraph titled Newton's approach. As you say, calculus ideas can be explained to anyone at any age, if the person explaining them understands both the ideas and the knowledge level of the listener.

the way I introduced it to these kids was to re-examine Euclid's results on areas of triangles. I.e. given two parallel lines, Euclid showed that if we draw the base of the triangle on one line, and the vertex on the other line, then the area of the triangle is the same no matter where on the second line we put the vertex.

Euclid proved this by decomposing the two triangles and re -assembling them to give the same rectangle. This method did not work for 3 dimensional solids, where he had to make an infinite approximation by decomposing into smaller and smaller pieces.

Then Archimedes discovered the right way to do it. Namely given two triangles with the same base, and with vertices on the same line parallel to the base, then any intermediate line parallel to the base will cut both triangles in a line segment of the same length. That guarantees they have the same area.

This phenomenon does generalize to 3-dimensional solids. I.e. if two solids in space are such that every plane parallel to a given plane, cuts both solids in slices of the same area, then the solids have the same volume. This principle, understood by Archimedes, which is now called "Cavalieri's principle", after an Italian who lived centuries after Archimedes, is the basic idea of integral calculus.
It is illustrated by looking at a deck of cards, and observing that the volume of the deck is the sum of the volume of the cards, hence does not change if we shove the cards over to form a slanted deck. This is true no matter how thin the cards are.

Archimedes in fact famously used this principle to derive the formula for the volume of a sphere. I.e. since the slice areas of a hemisphere and a cone add up to the slice area of a cylinder circumscribed about them (by Pythagoras), the volume of a sphere is the difference between that of a circumscribing cylinder, and of a (double) cone also inscribed in the cylinder

To avoid the fear factor inspired by scary words, one gives the simple explanation first, and then remarks that what they have just seen and understood, is called the "calculus" approach. The fundamental theorem of calculus as applied in college calculus classes is merely the observation that once formulas are introduced, both for the volume and for the various slice areas of a solid, it is possible to look at them, and to see how to pass from the slice area formula to the volume formula. they call this "integrating".

I.e. Archimedes knew that the volume of a solid is determined by the family of its slice areas. But he didn't use algebraic formulas. Newton added a method of passing from a formula for the general slice area to a formula for the volume, i.e. "anti-differentiating" to give the value of an integral.

Notice by the way, that although my last exercise, just before the beginning of the epilogue, is to "use calculus" to compute the volume of a bicylinder, in fact I have taught them all the calculus they need in the previous two paragraphs. I.e. one can easily do this exercise without ever having seen calculus, but having read these notes.

In fact, Newton's definition of the "derivative", which connects these two formulas (the slice area formula is the derivative of the volume formula), is merely a precise version of Euclid's definition of the tangent line to a circle. Note that Newton apparently read Euclid shortly before creating his own definition. I.e. if you parse it carefully you can check that Newton's limit definition of a derivative is merely Euclid's Prop.16, Book 3. *(see below))

It is very helpful to encourage people to just plunge into books without worrying whether they are ready to understand them or not. If not, that will quickly become clear.

Sorry if some words make my notes scary, but they were an attempted record of an actual explanatory lecture. I would also strongly recommend Hartshorne's university level book as a companion to Euclid, from a course which he taught to sophomores who were not very high level in math. At a certain point Hartshorne will begin to use sophisticated abstract algebra, but not at first. Only in chapter 3 does he begin even to assume a knowledge of real numbers. These are represented by infinite decimals. The main idea is that, given an origin point and a unit length, real numbers correspond to points on an infinitely long line with no breaks or gaps in it.

enjoy. good luck with your tutoring.
come back anytime for more comments if wanted.

* I want to explain how Newton's definition of the tangent line by limits is exactly Euclid's, made precise. I will progress gradually from Euclid's characterization to Newton's.

Euclid shows in Prop. 16, Book 3, that the straight line T perpendicular to the circle's diameter ending at p, lies outside the circle and has the property that, "into the space between that line T and the circumference of the circle, another straight line cannot be interposed." He concludes the line T is the tangent to the circle at p.

Now what does that property say?
Euclid's "into the space between the line T and the circumference of the circle, no other line through p can be interposed", means that:
given any other line L through p, hence making a positive angle with T, the circle eventually comes in between L and T.

I.e. given any line L making a positive angle e with T, if we look close enough to p, say within a distance less than d, the points of the circle that near to p will lie entirely between the lines T and L.

I.e. given any e>0, there is a d>0 such that every point of the circle closer to p than d, defines a secant making angle less than e with T.

This rephrasing of Euclid is exactly (the modern precise "Weierstrass" version of) Newton's definition of the tangent line T at p, as the limit of secants through p, as the second endpoint of the secant approaches p.

Thus properly understood, Euclid already contains ideas of limits exactly as they are used in differential calculus today, and Archimedes uses the main idea of integral calculus. Hence Euclid plus Archimedes is the best preparation for calc. Moreover, although there are no algebraic formulas in Euclid, his geometric decomposition results give geometric versions of basic algebra formulas such as (a+b)^2 = a^2 + b^2 + 2ab. In fact he also solves essentially arbitrary quadratic formulas. As a geometry text, Euclid is vastly superior to all subsequent ones, including Jacobs, but Jacobs is easier and more fun. The right book is the one the student will read and enjoy.

 

[nomadreid]

Thank you, mathwonk. After writing my previous comment I did look some more at your notes and did indeed notice that you use calculus methods without naming them explicitly as such, which is a very good approach. Also interesting was the historical link between Euclid and Newton, which I had never delved into.

 

[WWGD]

Nomad Reid: Are you considering too, big picture type books? Ian Stewart is good at motivation of concepts , treads well, imo, the line between rigor and informality. Maybe the student can read these together with other, more formal books

https://www.goodreads.com/author/list/10336.Ian_Stewart
?

 

[nomadreid]

Thank you, WWGD. I remember reading Ian Stewart's column in Scientific American many years ago, and enjoying his writing; he certainly deserves to be introduced to students at some point in their studies. As the link you gave does not offer any samples from the respective books (and as I do not have access to a library containing them), it is difficult for me to judge whether his writing is suitable for a 10- year old. You say that the books have pictures, which is always good. Indeed, I am in fact looking for more informal introduction to concepts [ in a way that a stricter formalization will not prove the informal introduction false -- alas, simplification down to falsehood is too often the case]. Therefore, his approach sounds good, but diagrams do not necessarily mean simplicity; I suppose my best option is to try to find some samples of his books somewhere on the Internet in order to judge whether to recommend them to the student's mother.

 

[Muu9]

I am advising the mother of a precocious 10-year old who studies on his own and already has some rudimentary algebra skills. I am not tutoring the child, just giving some recommendations. One aspect that seems inadequately represented among resources on the Internet or books that she could order are books on basic logic. That is, if you do an Internet search with the keywords "logic for kids", you merely get lots of puzzle books. (Puzzles are fine, but when they are not explicitly and consistently connected to more significant elements in mathematics or science, then they are nothing more than curiosities.) With keywords "Introduction to Logic", one gets texts which would be too difficult for a 10-year old, despite his (apparent) high intelligence. There are some good books for older students -- The Annotated Alice, the Raymond Smullyan books, etc. -- but one has to already have a grounding in basic logic to understand them.

The logic should include aspects which also allow the student to better understand the way science works beyond the stereotypical simplified description of the scientific method as "hypothesis--experimental test--accept or reject hypothesis -- if reject, new hypothesis".

So, does anyone have any recommendations, either for books that the mother could order or Internet sites that the son could visit? Thanks in advance.

EMF math presents elementary math via logic; they can try the first module free: https://www.elementsofmathematics.com/

If you're looking for accessible introductions to logic in a novel medium, you might like Lewis Carroll's game of logic:

 

[nomadreid]

Thank you, muu9. I will pass on the two recommendations to the family involved.

I do not know whether they wish to register in order to get the free lesson of EMF math, as I know that, despite the assurances of many websites, they use the information to pester the family forevermore to get them to buy their product. That will be for them to decide.

As far as the Logic Games, the video would be a bit difficult to follow if one did not know the game they were talking about. Thankfully, I see that Project Gutenberg offers a legal copy of the original book, and Wikipedia has a brief explanation of it. I think it would be something for the parent to digest first, as the rules are presented by Carroll in a rather long-winded fashion.

 

[Muu9]

despite the assurances of many websites, they use the information to pester the family forevermore to get them to buy their product

This can be solved with a disposable email or the "unsubscribe" link all automatic emails are required to have

 

[nomadreid]

This can be solved with a disposable email or the "unsubscribe" link all automatic emails are required to have

Yes, good ideas, although they both involve some additional unwanted hassle. I will mention the options to them. I suppose I should first view this free lesson to see if it is worth it, to see if it navigates between the Scylla of too much formality and the Charybdis of too little formality.

 

[WWGD]

This can be solved with a disposable email or the "unsubscribe" link all automatic emails are required to have

Good luck with that. Several require you enter the email you want unsubscribed. Suspicious.