The [I.N.]Comparable I.N. Herstein - Our Math Icon of the Week Revealed

[Only a few more days to vote in the poll in the sidebar! To see the results thus far, you have to cast your vote.]

Well, the secret has been out of course if you read the comments to Doors and Windows Left Unopened, but it's time to officially recognize one of the great teachers and writers of modern mathematics. Israel (Yitzchak or, affectionately, Yitz) Nathan Herstein wrote, IMO, the clearest exposition of Abstract Algebra I have ever read. His Topics in Algebra was a major influence on my development as an undergraduate math major. Whatever ability I have to write a logical structured proof probably came from reading and re-reading his classic. To this day, I can picture his presentation of theorems, particularly how he developed the set of numbers that can be represented as the sum of two squares. Yitz, you can still run rings around anyone!

Our winners...

kevincp wrote:

Mystery mathematician: I N Herstein. I used his unsurpassed text, "Topics in Algebra" as an undergrad in the 60's. A curiosity I came across when browsing his name today was his acute use of logic to demonstrate the superiority of the latke over the hamantash as quoted in R.F Cernea's "The Great Latke-Hamantash Debate". The final line of the "proof" is: "Has anyone here ever seen anyone eating a hamantash with sour cream? Q.E.D."

and our other winner is...

Hypatia who wrote

...your mathematician is Yitzchak Herstein.

Congratulations to our winners!

I will leave Yitz's image up there a few more days - he deserves that recognition.
I will end my tribute to him by quoting one of his 30 research students:

"He was someone of great warmth who took an intense personal interest in his students and had a knack of getting them to believe in themselves."

[1,2]-3-[4,5]-6-[7,8]...21 Helping Children Devise and Understand Winning Strategies

Do you remember playing those fun counting games in elementary school? No, well, play along as if you do! The teacher or a friend would go first and always seem to win or you would go first and always lose. You knew there was a trick and if you figured it out it was exhilarating - like understanding the key to a magic trick.

Like most parlor games, there's genuine mathematics underlying these counting games. In this post we will describe a few of these and an investigation to help students not only devise a winning strategy (or algorithm) but to come to an understanding how division and remainders play a significant role.

Variation #1: The Game of '21'

Age Group: Certainly appropriate for children even as early as 1st grade (however, devising winning strategies and explaining why they work might be a bit ambitious!)

# of players: 2 is best

Object: To win, make your opponent say some target number like 21

Rules: First player starts counting from 1 and says either '1' or '1-2'; Other player then says the next number or the next two numbers; play continues in this way until someone is forced to say the number '21'. Verbal or written directions here are far more confusing than just demonstrating actual play.

Sample Play: See title of post for a partial play

Winning Strategy (partial): If you go first, say '1,2'. If you don't, your opponent can beat you if she/he knows the strategy.

Further Discussion: For the younger children, let them play against each other in pairs for a few minutes to allow them to feel comfortable with the game. Then you can ask if anyone wants to 'challenge the master' - you, that is! Tell them because you are older, you deserve the courtesy of going first (that will last for about 30 seconds or less!). After playing against students for a while, they will figure out that part of the winning strategy is to go first and say '1,2' but most will not pick up on the rest of the method. To mystify them even more, you can let them go first. You most likely will still win because you know the strategy and they will most likely not catch on for some time! There's always one sharp youngster even in the primary grades whose eyes will start glowing and will say, "Let me go first. I can beat you." At that point, you may want to say, "Game over!"

Winning Strategy: Those of you who are familiar with these kinds of counting games, know that they are all variations on the same basic theme and are simpler versions of the classic game, NIM. In this version of '21', some children will quickly see that, whoever gets to 2o has to win. It will take them a little longer to work backwards from there to see that to get to 20 you have to get to reach 17, which is 3 less than 20. To get to 17, you have to reach 14, which is 3 less than 17. Thus, working backwards, the winning positions, or 'magic numbers' if you will, are 20-17-14-11-8-5-2. Reversing this provides you with a guaranteed win but of course you need to go first and say '1,2'! But learning and using this strategy does not imply that the child understands WHY it works!

Using questions to help children begin to grasp the underlying idea: Children will immediately see why '20' is a winning position but ask them to explain why 17 also is (Possible student response: "Because if you say '17, then the other person can only get to 19 and you will be able to get to 20"). Continue to subtract 3 to obtain other 'magic numbers.' Ask the children why subtracting 3 is critical. Why 3? Children, even older ones, will soon see what is going on. Some may ask if one has to memorize all of these numbers. Don't answer that! Just smile and let them figure it out for themselves. Allow the children to practice the winning strategy on each other until they feel comfortable. They will surely want to try this out on other friends, teachers or family members!

Underlying Concept: At what grade level are children expected to grasp the essential idea that repeated subtraction is equivalent to division? Thus, in our problem, working backwards, starting from the winning position of 20 and continually subtracting 3, is equivalent to dividing 20 by 3:
20 ÷ 3 = 6 with a remainder of 2.
This can be interpreted to mean that after performing six subtractions by 3, the number 2 will remain! Of course, the repeated subtractions reveal all of the winning positions so children may not be appreciate the benefit of division. Help them to see that the remainder does reveal that one needs to go first and say '1,2' to guarantee a win.

A Million Variations
Well, maybe not that many in this post, but I'm sure you can see the possibilities are endless. You may want to ask children to devise their own version and a winning strategy as an outside project or assignment. They may invent something really cool no one has thought of! You might want to first ask the group how they could modify the game: "If you were going to invent your own game, what might you change about the game of 21?"
Some suggested variations:
(1) Whoever says '21' wins
(2) '21' loses but this time students can say the next number or the next two numbers or the next three numbers. Thus, if you go first and say '1,2,3', I would say '4'; then you might say '5,6' and I would say '7,8'. Am I guaranteed to win if I play correctly?
(3) Start from some number like 50 and allow children to subtract any number from the set {1,2,3,4,5,6}. Then you subtract one of these 6 numbers from the result and repeat play until one player reaches the number '1'. That player wins. This 'Game of 50' is also famous and will mystify adolescents as much as younger children! I'll let our readers explain the strategy and why it works! By the way, don't underestimate how much reinforcement of basic subtraction skill this game provides!

Find all combinations of 3 distinct primes whose average is 13

[Have you voted yet in the survey in the sidebar? Time is running out...]


Just an isolated middle school mini-challenge to get the day started? Perhaps...

Those of you who are familiar with this blog know that MathNotations is dedicated to providing activities/investigations for middle and high school teachers to use or modify (provided proper attribution is given of course). In this post, I will demonstrate how one can build an extended or richer activity from a math contest or standardized test problem.

It is important to remind our readers here that these kinds of activities and problems do not constitute a curriculum. Students need to first develop proficiency with skills and procedures. These explorations are only intended to extend and enrich student learning. They can be used in part or in whole, as a long-term project outside of class, a team activity in the classroom or a myriad of different ways. All of this is at the discretion of the educator.

First of all, the problem in the title, in its present format, would not be an SAT or a standardized test question, unless the standardized test included free-response or open-ended questions.

In SAT format, the question might be changed to:

Which of the following can be expressed as the sum of three distinct primes?
(A) 6 (B) 9 (C) 12 (D) 15 (E) 17

Not a particularly challenging problem, but some students would struggle with comprehending the wording or paying attention to details ('distinct') or because of lack of knowledge about primes. This type of question is fairly common.

Let's return to the original question:

Find all combinations of 3 distinct primes whose average is 13.

I've administered this type of question to students and observed their methods. Sadly, some do not immediately recognize that the problem is equivalent to:

Find all combinations of 3 distinct primes whose sum is 39.

Most students do see this at once, but there are a few in middle and high school who have not developed sufficient conceptual understanding of averages or have simply not been exposed to enough problems.

As far as methods and approaches go, I'm always surprised that many middle and high school students use fairly random listing methods rather than a systematic approach. After all the years now of instruction in problem-solving techniques, one should expect that students would make an organized list as follows:

2,2,35 Discard this for two reasons! Would most students recognize the logic behind concluding that 2 cannot be one of the three primes?

3,5,31
3,7,29
3,11,25 (discard)
3,13,23
(I'll let the reader finish the list!)

If I were to assess the value of this single question, I might give it a 7 on a scale of 10. I'm sure some would rate it as 1 or 2 since some perceive these kinds of questions as useless. However, my feeling is that the question does develop mathematical thinking and there's something to be said for attention to detail and a systematic approach.

But this is not the end. Suppose the educator finds this problem in a book or math contest or online. How can one extend it to a richer experience for all students, not just the accelerated, honors or gifted child? Although it may appear at first that the primary intent of the question is to encourage a systematic approach (making an organized list) or reviewing ideas about averages or primes, the content of the question is essentially about writing a number as a sum of 3 primes, distinct, in fact. Is this an important question that has occupied the minds of our greatest mathematicians for years? Uh, actually, yes! Look here!

Students need to be encouraged to ask more questions after the problem is solved. The instructor guides this exploration by modeling some of the questions students need to ask: Is there anything special about 13? Can every prime be written as a sum of three distinct primes? Every odd? Three primes, not necessarily distinct? Does the original number 13 have to be prime or even odd for that matter? Why are we using three primes in the sum? Why not two? Your turn, boys and girls!

You get the idea. This isolated problem becomes a springboard for deeper mathematical research. Here is one possible assignment:

Write your own challenge problem of this type? Make sure you can solve it and be prepared to present it to the class!

What would you expect your students to come up with? You can't be sure until you try it of course, but can you anticipate some of the responses?

By the way, I have already heard most of the arguments for why this type of research is impractical in a math classroom:

"My students don't even know their basic facts and you want them to become mathematicians!" "This is for the math team geniuses."
"I don't have time for this - I have a real curriculum to cover and if this not going to be tested..."
"Teach children the basics, not this 'fuzzy' math!"

Oh well, enjoy it anyway!

Fascination with Pyramids again...

Do you recall the post about pyramids from last April? This is a continuation of that investigation and is a problem that has appeared on standardized tests. There are several approaches and the learning objectives for geometry students are many:


(1) Develop spatial reasoning
(2) Review terminology of space figures, pyramids in particular
(3) Make connections to the real-world problem of finding the height of an Egyptian pyramid
(4) Apply the Pythagorean Theorem or special right triangles
(5) Justify (prove) one's methods

STUDENT/READER PROBLEM
The figure attempts to depict a special regular square pyramid.
The 4 lateral edges and the 4 base edges all have the same length x.
Show that the height PT of the pyramid has length x(√2/2).

Notes:
(a) It may be instructive to encourage students to approach this by more than one method. One could ask students to find 30-60-90 as well as 45-45-90 triangles in the pyramid (lines may need to be constructed of course).
(b) Many students may assume ΔPTS is 45-45-90. Challenging them to prove it is an important objective here (there's more than one way).
(c) One could begin with a specific value of x, such as x = 10 (see the original pyramid post).
(d) I strongly urge you to have students research the Great Pyramid of Giza. Is it approximately a regular square pyramid? Are its faces equilateral triangles as in this post? There are many classic math problems associated with this Wonder of the World and it's not all about geometry!

Doors and windows left unopened...

Many many loose ends...

1. The poll in the sidebar is still ongoing. Have you voted yet? It is Super Tuesday after Super Sunday after all! Jonathan suggested I submit the link to this poll to the Carnival of Education. I will consider that as well as the upcoming Carnival of Mathematics on 2-8-08 over at 360.

2. The poll has generated some interesting comments on this blog and elsewhere. Some readers and highly knowledgeable individuals are offended by the implication that the poll is somehow suggesting that teachers must teach a particular way and students must learn a particular way. I do feel there is a fairly wide range of options from extreme traditional to extreme reform, but others may see it as more biased. Similarly, some are bristling at the implication of Standards. I won't get into an exposition of my views at this point but I will share my wife's insights. She cuts through all the b******* -- that's her style. When I told her that some are having a problem with the poll, because it seems to suggest restricting what or how children should learn, she looked at me incredulously and stated:
"First of all, you're surprised someone is offended by a position you're taking! I'm not an educator, but, why do all of you make things so complicated. Who decided that the way we learned was broken and needs to be fixed? I see it like this -- Children should learn the traditional method, then when they can do that, they can be shown other ways. If they like another way better, they can decide for themselves but only after they know one method well."

She then added the following: "Perhaps those who are upset about children and educators being 'boxed in', fail to recognize that leaving everything open-ended is just another box." My wife has always seen things differently - I guess she thinks outside the box! She also added that if her position offends anyone and she gets attacked, she can handle that. After all, it's not her blog!

3. Anyone notice there is a new Mystery Mathematician in the sidebar. Daniel Gorenstein has now been replaced by another contemporary mathematician. Sorry, no hints at this point, other than to say that he has had a profound influence on myself and many students of undergraduate mathematics.

4. Ok, so anyone can prove that they made some prediction about the Super Bowl that sorta kinda turned out to be true. I suggested a couple of days ago that perhaps the number 8 would play a role in the game. Since I'm a 'digit-man', I'll let you scoff at the fact that the number 17 played a key role in the final score and Plaxico's uniform. Well, what do the digits of 17 add up to? Yes, we all know how fortune tellers make their predictions! Here's a little more coincidence:

Last night, as the Giants were disembarking from their plane at Newark's Liberty International Airport and boarding their team bus, my son was disembarking from his flight at the same terminal at approximately the same time and getting on his team bus going back to his high school after a competition. So what, you ask? Uh, well, my son's name is the same as some MVP quarterback. Alright, we all know how common that name is...

5. Many other loose ends from 'boring a hole in a sphere' to clock problems, but I'll stop here. Good morning and g'day!

My dad was born 2-4-08. He would have been a centenarian today...

Happy 100th dad. I will never forget you. (My dad passed away in '88). You will always be the greatest teacher I ever knew...

GIANTS BEAT ALL THE ODDS - NUMBER 17 IS TRULY PRIME!

No post - the title says it all ! I'm still in shock...

'Left-Overs' before the Super Bowl: Crazy Eights, Squares, Remainders and Algebra

Ok, so most normal people are not thinking about the significance of the digit '8' in 2008 the day before the Super Bowl. Sorry, but in this post there will be no predictions about the score, no 'over-unders', no boxes, no betting at all. You do have to admit that this is a great time for lovers of mathematics. People are actually interested in mathematical odds and chances of all kinds of weird number combinations occurring in the score on Sunday night. However, this post will focus instead on the number 8, the units' digit in 2008. The Super Bowl comments above will no doubt soon become outdated but the mathematics below will live on! Who knows, maybe the number 8 will turn out to have special significance on Feb 3, 2008? Remember, I said that here before the game!!

2008 is a special number for so many reasons, being divisible by 4 of course: Leap Year, Prez Election year, Summer Olympics and much more. In fact, 2008 is divisible not only by 4 but also by 8 itself. In the good ol' days, some students were even taught the divisibility rules for 2, 4 and 8:
Divisible by 2: If the 'last' digit is divisible by 2 (of course!)
Divisible by 4: If the number formed by the last TWO digits is divisible by 4
Divisible by 8: If the number formed by the last three digits is divisible by 8.

Let's demonstrate this for 2008:
2008 us divisible by 2 because 8 is divisible by 2
2008 is divisible by 4 because '08' is divisible by 4
2008 is divisible by 8 because '008' is divisible by 8

A little weird with those zeros and not particularly interesting, right? Anyone care to guess a rule for divisibility by 16? Interesting, but none of this is the issue for today....

BACKGROUND FOR PROBLEM/INVESTIGATION/ACTIVITY
Today, we are are interested in the squares of numbers and their remainders when divided by 8. Notice that 4² is divisible by 8 but 6² is not. So we cannot say that the square of any even number is divisible by 8. What about the squares of odd numbers when divided by 8?
1² leaves a remainder of 1 when divided by 8
3² leaves a remainder of 1 when divided by 8
5² leaves a remainder of 1 when divided by 8
7² leaves a remainder of 1 when divided by 8

What is going on here? That's for your crack investigative team to decipher.

TARGET AUDIENCE: Our readers of course; Middle schoolers through algebra

PROBLEM/INVESTIGATION FOR READERS/STUDENTS
1. Discover, state and prove a general rule for the remainder when the square of an even number is divided by 8.
2. Discover, state and prove a general rule for the remainder when the square of an odd number is divided by 8.

Comments:
(1) These are well-known relationships and not very difficult questions. Just something to extend thinking about divisibility, remainders and the use of algebra to deduce and prove generalizations. Prealgebra students may be able to explain their findings without algebra!
(2) 'Discovering' or stating the rule for question (2) is transparent from the examples above. Instructors may prefer 'data-gathering' and making a table first. That is, have students develop a table for the squares of the first 10 positive integers and their remainders when divided by 8. Proving the result for the squares of odd integers is more challenging, even algebraically. Most will see the remainder when dividing by 4, but 8 is slightly trickier.
(3) Those who are more comfortable with congruences and modular arithmetic can approach these questions another way.