## “The best part about math is that, if you have the right answer and someone disagrees with you, it really is because they’re stupid.”

I found the above in an amusing list of quotes from an honors linear algebra class that was linked from Epsilonica, a blog I recently discovered, via his great post In Praise of Proving that Zero equals One.

Danielle posted a question on my About page that I thought I’d answer in a separate post.  She wrote:

Hi MathMom!
I have a question… my daughter gets number triangles for math warm-up. E.g. Use each number only one time (1-9) and the sum of the numbers along each side of the triangle must equal 20.

She’s been doing them by trial and error, but that takes quite a long time the more complicated they get. Is there a formula or method that she’s missing?

Danielle

Danielle,

There isn’t really a formula that can be used to solve generic puzzles like this, but the key to solving them is thinking about which numbers get used more than once.  In a triangle puzzle, this is the numbers in the corners.  Each of these numbers gets included in 2 of the sums, while the rest of the numbers get included only once.

In the example you gave above, your daughter should first think about what the sum of all the available numbers is if they were each used only once: 1+2+…+8+9 = 45.  Since we need the total of the 3 sides to be 60 (3×20) we know that the numbers that are used twice must add up to 15 (the difference between 60 and the 45 we’d have if each number were used just once).  So, the corners of the triangle should add up to 15.  Unfortunately, there are a lot of ways of making 15 out of 3 of the numbers from 1 to 9.  But it does cut down the trial and error a fair bit to start.  I tried a few combinations and was able to find answers pretty quickly using that constraint.  I hope this insight makes these puzzles a little more fun and less frustrating for your daughter, Danielle!

Here’s another type of fill-in puzzle.  You don’t use sums to solve it, but the key to it rests in realizing which spots are “special”.  I’ll leave it at that for now so as not to give it away for those who’d like to try to solve it.

In case anyone was wondering, I’m still here.  Real life intervened over the summer and distracted me from blog reading and writing.  But now I’m preparing to go back to my volunteer role in teaching math problem solving at my kids’ school, and I’m excited about that.  Hopefully I’ll find time to write about it as I go.

I found this forum due to the fact that someone there linked to my last blog post. (The link is on page 10 of the discussion if you want to check out the context.) There’s a great discussion there about some non-orthodox math education options (mainly, I believe, intended for homeschoolers or parents of gifted kids wanting to supplement at home).

The thread was started by a parent who had just heard a presentation by a local mathematician who advocated that kids shouldn’t learn math using a textbook at all — it should all emerge naturally from various scientific/engineering explorations. Although I don’t particularly agree with that idea (not least because “natural” situations tend to involve ugly numbers, which make hand computation impractical, and argue for the use of calculators — I’d be in favor of calculator use in such cases, but not if it’s the only exposure to computation the student is going to get!), the thread contains a lot of good ideas about math education, many relevant to both gifted and non-gifted students, and many as relevant to classroom instruction as to homeschoolers.

On page 11 of this discussion, there is a link to a great-looking free online math course intended for “adult learners and high school teachers”. NOT a course in computation or algebra or trig, but a course in cool math topics, such as prime numbers, combinatorics and game theory. Although they are aiming it at HS teachers, I think it could be a great resource for math-phobic elementary teachers, not because they would necessarily pass on that particular content to the kids they teach, but hopefully to help improve their attitude toward math, so that they could pass on some excitement about it (and also give age-appropriate introductions to many of the topics, even at the elementary level).

I haven’t had a chance to actually go beyond the overview of the course — perhaps I will have a chance to review it more completely at a later date. But it definitely seems worth a look.

Over at MathNotations, there is yet another raging debate about math education. I wasn’t going to get involved, but I finally broke down. 😉 I started this as a comment on that post, but it is so long that I decided to post it as a blog post here.

So… here are some of my thoughts on the issue of math education, particularly for elementary and middle school kids. Let me say right up front that I am not arguing for or against any particular math curriculum here. My kids and I do not have extensive experience with any particular math curriculum. My own kids are not taught math from a single curriculum, but are rather taught using a wide variety of materials, exercises, explorations, etc. I have friends who are very pleased with Singapore Math. (I have used some of the workbooks for my younger sons as well, just for fun.) I have other friends whose school does an awesome job with Everyday Math, though I realize that that appears to be the exception and not the rule with that curriculum. What is key is great teachers, in either case. But that is all I am saying about particular curricula.

There is a problem that great teachers like Dave and Jonathan seem reluctant to admit, and that is that many otherwise great elementary teachers are poor math teachers. Elementary teachers are generalists, and for many of them, math is something to “get through” and to get kids through. It is not something they ever enjoyed, not something they are comfortable with, not something they are good at teaching. These teachers destroy curricula like Everyday Math that really require a great teacher. They would probably also be poor at teaching Singapore Math. So… as much as great teachers should not be restricted, or heaven forbid scripted, in the way they teach math, we have to do something about the fact that many elementary kids are getting their fundamentals from instructors who hate math and who are no good at teaching it. Some people think the answer is a highly scripted curriculum. I think that sounds horrible! I think the solution probabaly requires thinking outside of the box a bit more. Perhaps even elementary students need to have specialist math teachers, just as they often have specialist art and music teachers. (I’m focusing on elementary here because this is less of a problem in middle and high school where math teachers are generally specialists. There are better teachers and there are worse teachers, but few middle school, high school or college math teachers hate math.)

Steve H is concerned about kids who don’t have mastery of basic number facts (addition, multiplication, etc.) at an age where he thinks they should. He blames Everyday Math for the fact that kids in his child’s school don’t know their facts, but I’ve never seen any curriculum that builds this drill and practice in. In my experience, this has always been done in addition to following whatever “curriculum” the school is following, and I would think it could be done just as easily in conjunction with Everyday Math as it could with Singapore Math or any other curriculum. It also generally happens mostly at home. Teachers must expect students to study their facts at home, and must assess their progress, but parents must get out the flashcards, or get their kids onto a practice game or website, and get their kids to practice at home.

Class time should not be taken for memorizing number facts, no matter what curriculum you’re using. A few “mad minutes” a week are enough to assess how that is going and keep the kids motivated, and only takes a few minutes out of the math instructional time.
There is an argument that states that mastery of number facts and procedures (such as long division), performed accurately but without necessarily any understanding of why they work, is the most critical job of any math curriculum, and should be addressed before taking on anything else. These are certainly basic skills that all kids should learn. But to be honest, as much as I hate calculator use in school, in this age of calculators and computers, efficiency at hand computation is not, IMO, the most critical math skill for kids to learn. I am NOT saying that it should be ignored, or that kids should be allowed to skip it, and just use calculators in class (see rant linked above). But it is not, IMO, the be all and end all of math education, nor is it a prerequisite, IMO, for studying anything else.

What I consider even more important is a strong sense of number. I want kids who know immediately when the answer they got (either by hand computation or with a calculator) is way off. I want kids who have an instinctive understanding of the distributive law before it is ever formally taught or named (12 sevens is obviously the same as 10 sevens and 2 more sevens). I want kids who know when the amount of change handed to them makes no sense. I would rather have a kid who can multiply 64 x 25 mentally (by halving 64 twice and doubling 25 twice, to see that it’s equal to 16 x 100 = 1600) than a kid who can sit down and carry out the long multiplication with pencil and paper, by rote.

You can’t just sit down and teach kids “number sense”. Certain mental math tricks can be taught and practiced, but the way to achieve real numeracy involves lots of experience playing with numbers, with manipulatives, with measurements, etc. (Ok, I said I wasn’t going to talk about curricula, but I will say that from what I’ve seen of Singapore Math, it seems to do very well at guiding kids toward the development of good number sense.)

“Mastery” is also a slippery concept. I’ve seen kids “master” skills and then promptly forget them. I’ve seen kids who can easily do a page of long division quickly and without errors, who 6 months later will have no clue how to do long division… This is part of the reason why schools often utilize spiral curricula. Because if you have gifted kids, it’s easy to think “master one skill and then move on to the next” is the only sensible way to approach mathematics education. But most kids need more repetition than that, and many kids don’t fully “get” something the first time, even if they appear to have “mastered” it. Steve is right that a spiral curriculum can lead to a lax attitude of “it’s ok if they don’t master this now, because they’ll see it again later” that goes on ad infinitum, and the kid never masters anything. This is clearly no good. But the solution isn’t necessarily to take away the spiraling for those who need it, IMO. The solution is to have limits — for example, it’s ok if they don’t completely “get” long multiplication when it’s previewed in 3rd grade, or even when it’s introduced more formally in 4th, but they have to get it when it’s reviewed in 5th, or they shouldn’t move on. It needs to be clear where in the spiral one is, and whether this is a preview, or core instruction, or a last chance review, and make sure that kids really “get it” before moving past that “last chance”.

Dave is advocating and providing samples of “non-routine” problems. These often considered the domain of math contests, and to be reserved for only the most gifted math students. In my opinion (and I know Dave agrees), this type of problem solving is important for math students of all levels of ability for many reasons.

First, it provides a fabulous way of helping students to appreciate the uses of the procedures and skills they have learned or are learning (or in some cases, motivates a procedure yet to be learned). It provides a great way for teachers to re-assess students’ continued mastery of multiple skills, to see which need further review or clarification. This addresses that slippery slope of mastery. Invariably, some previously “mastered” skills are shown to be weak, and must be re-visited. In this way, we avoid just “checking off” topics, and make sure that students can recognize when a particular method or procedure is called for, and
remember how to use it long after the first time that they supposedly “mastered” it.

Second, this is the kind of thing that “real mathematicians” do! If one of the goals of K-12 math education is to prime future mathematicians, this is a valuable opportunity to do so. A “mathematician” does not sit down and solve 25 ratio and percent word problems, knowing exactly which skills are required to perform the computations. Instead, she investigates “puzzles”, looks for interesting patterns makes new discoveries, generalizes results. For students who might have the inclination to pursue mathematics further at some point, this kind of experience early on may spark their interest, and excite and motivate them.

Third, it develops self esteem and confidence. This may seem surprising, since the problems are very hard for most students. But the students that I work with know that the problems I bring them are meant to be hard. That they aren’t meant to be able to solve them all on their first try. That they may need help. But, when they do solve one correctly on their own, they are so very proud of themselves, and rightly so. A student gains so much more self-esteem and confidence from struggling and succeeding at something hard than they do from breezing through something easy. (Not to say that the rest of the math curriculum is easy, of course, but there is a persistent fallacy that I’ve seen many times, that the way to develop self-esteem in kids is to make sure to give them lots of easy work that they can effortlessly succeed at.)

Fourth, it builds transferrable problem solving skills. Math class is not the only place where people will have to solve difficult problems, problems where the best approach isn’t obvious at first glance. Practice in solving problems like this transfers to many different areas of the curriculum, and to “real life” as well.

Personally, I’d love to see what Dave would come up with as a Math Curriculum. However, I don’t think it’s so easy to just write a great curriculum and have the world beat a path to your door. There are huge corporations with a lot invested in selling math curricula, that he’d be competing with.

Most kids learn, in kindergarten or first grade, to count by 2’s, 5’s, 10’s. By rote. This has its uses, but going beyond that level of skip-counting can make great numeracy practice, and it seems to be quite under-used.

In our school, we have the primary kids skip-count by 2’s starting from numbers other than 2. Then they practice skip-counting by other numbers, again starting at arbitrary places. Forward and eventually backward. This is an easy one to differentiate, because you can have some kids counting up by 2’s and others counting backward by 7’s at the same time. 🙂  Eventually skip counting is used to introduce multiplication.

Today I asked the intermediate group (approx 3rd – 5th grade math levels) to do some skip-counting for me, forward and backward by 2, 3, 5, 10, 20, 25, 50, 100 and 99! Boy that last one was a doozie! (“Can we use calculators?”) But all of it was clearly good practice even for these “big kids”. For counting (up) by 99’s, they quickly realized that they could add 100 and then subtract one. But a LOT of them got stuck crossing the 1000 boundary (going from something like 987 to 1086). I didn’t think of it at the time, but now I think that pointing out that 1000 is really the same as “ten hundred” might be helpful for a lot of them. Now backward by 99’s was really tricky! So… you first subtract 100 and then… what? Only one of the 6 kids I was working with today (only half of the usual group) really understood that you would then have to ADD 1 back. “I’m taking away a little LESS than 100, so should the answer be a little more or a little less than what we got by taking away a whole hundred?” This is not intuitive for a lot of kids!

So, if you’re working with elementary school aged kids, skip count! Regularly! I think it’s a key skill for developing number sense.

This week I tried a couple of ideas I got from MathNotations on my middle school group.

First I presented this problem:

### How much greater is the sum of 51+52+53+…+100 than the sum of 1+2+3+…+50?

Because I had predicted that they would all solve this by summing each series and subtracting, I explicitly asked them to find two ways, showing their work each time.

As predicted, they all tried summing each series using Gauss’ method (pairing up first and last, second and next-to-last, etc. gives 25 pairs, each with a sum of either 51 or 151, depending on which sequence you’re working on, so the sums are 51 x 25 and 151 x 25), with limited success (arithmetic errors, and multiplying the “pair totals” by 50 for the number of entries rather 25 for the number of pairs were the major stumbling blocks).

Only 2 (of 8 ) students came up with a second way. They were all pretty distressed that I wanted them to find another way. I hinted that it was possible to find the difference without adding up either series, which was totally mystifying to most.

One student came up with the method of comparing corresponding terms (each term in the larger sequence is 50 more than the corresponding term in the smaller sequence, so the total difference is 50 x 50), and another noticed that when using the Gauss method, the “pair sum” (151) for the larger sequence was 100 more than the pair sum (51) for the smaller sequence, so the total difference would be 100 x 25.

However, when I tried to show them that they could figure out (151 x 25) – (51 x 25) without multiplying out those terms, they were mystified. Note to the regular math teacher: review the distributive law!

Then we moved on to the fascinating facts activity. We went over the examples for 17 in class, then I had them work in pairs on 97. I assigned 153 for homework:

Some of the fun properties they found for 97:

• It is prime
• it is odd
• it can be represented as the sum of consecutive positive integers: 48 + 49
• The sum of the digits is a square (4^2), a 4th power (2^4) and unsummable!
• its palimage (79) is also prime
• take 97, subtract its palimage (79) to get 18, then sum the digits to get 9, which is the same as the sum of the digits in the product of its digits (63)
• it can be represented as a sum of squares: 9^2 + 4^2
• it can be represented as a sum of 4th powers: 3^4 + 2^4

Some of the properties they found for 153:

• it is odd
• it is composite
• all its digits are odd, as are the sum and product of its digits
• the product of the digits (15) can be written as 5^2 – 3^2 – 1^2 (notice use of the square of each digit)
• the product of the digits can be written as a difference of squares: 4^2 – 1^2
• the pairwise differences between digits are all even
• it can be written as the sum of 17 consecutive integers: 1 + 2 + 3 + … + 17
• it can be written as the sum of 3 consecutive integers: 50 + 51 + 52
• it can be written as the sum of 3 squares: 4^2 + 4^2 + 11^2 or 10^2 + 7^2 + 2^2

No one commented on my brief post about a simple but interesting geometric probability problem.

There are a lot of interesting things to discuss about probabilities when the sample space is infinite. Such as:

1. What is the probability that if a student guesses randomly on a math problem whose answer could be any rational number, he will get the problem correct? (See my post on cheating on the Math Olympiad to see why that problem comes to mind.) 😦
2. What is the probability that if an integer is chosen at random, it will be even?
3. Divide a rectangle into two parts, one with twice the area of the other. Pick a point randomly within the rectangle. What is the probability that the point is inside the smaller part?

Problem 1. exemplifies is something that bothers “beginners” about probability — that something that is “possible” can have probability zero.

Problem 2. raises many interesting questions. We know that the cardinality of the set of even integers is equal to the cardinality of all the integers. (For more on this, check the links here.) Yet, intuitively we know that the answer must be 1/2. There’s a fascinating discussion of this problem and more complicated versions at The Math Forum‘s Ask Dr. Math archive. In particular, I was surprised to read the following: “If you’re talking about the entire infinite set of integers, there is no way to do this [choose an integer at random] without some sort of a distribution function over the integers, and there is no such function that gives an “equal probability” for all integers.” (emphasis mine) The problem is solvable, of course, by using limits. And the answer, as we expect it must be, is indeed 1/2.

Problem 3 is a “basic” geometric probability problem. It was raised by another Ask Dr. Math reader (follow the link in the question) who argued that since the cardinality of each of the spaces is the same, the probability of landing in each must be the same. Doctor Tom replied, in part that “the cardinality of the set does not determine its measure” and that it is “measure” and not “cardinality” that is important in determining probabilities.

Then we come back to my original question:

• ABCD is a square with side length 10. P is a point chosen at random inside ABCD. What is the probability that P lies on one of the diagonals of ABCD?

What I think is cool about this question is that the number of points on the diagonals is infinite, but it it’s size/area/measure (corrected in response to Pseudonym’s comment) is infinitesimal in comparison to the size of the area of the square. I suppose in that way it’s related to the problem below about choosing a rational number at random out of the set of all real numbers.

Some other interesting and related questions from Ask Dr. Math:

And my favorite: