Questions from middle schoolers II: New numbers

Are mathematicians looking for any new numbers?

They are always looking for the next biggest prime number. We know that there are infinitely many of them, and we know some really big ones. Every so often, it makes headlines when a supercomputer finds another, bigger one. The goal isn’t to prove that there is another one; we know that already. The goal really is to test the computational power of computers, since all the undiscovered primes have many, many digits (in 2008, mathematicians found one that has 13 million digits).

Instead of new numbers, mathematicians are always on the lookout for new categories of numbers. Whole numbers, rational numbers, irrational numbers, imaginary numbers, transcendental numbers, etc. These are all categories of numbers that mathematicians discovered (created?) in the process of their work. Surely there are more to come…

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Questions from middle schoolers I: Pi

I travel a lot to work with middle school math teachers. In one classroom recently, I was oversold as a visiting mathematician. Visiting I was. Mathematician I am not. I do think about mathematics for a living, but I do not create original mathematics.

Nonetheless, students in the class generated a number of questions they wanted to ask a mathematician. So this series is for Ms. Otto’s eighth-grade students at KMS…

Who discovered Pi?

Can you believe that an entire book has been written about this subject? It’s called A History of Pi by Petr Beckmann.

Here’s the very short version…

In order to discover pi, people needed to understand ratios. That is, they needed to not just be able to count objects, they needed to be able to see relationships between quantities. That took a long time, until about 2000 B.C.

At that time, there were two major societies doing serious mathematics-the Babylonians and the Egyptians. Each of these societies appears to have noticed that there is a relationship between the diameter of a circle and its circumference, and both societies had decent estimates for pi (although neither called it that).

The next several thousand years involved people getting better estimates of the value of pi. There are lots of interesting mathematical advances that came about as people tried to do this.

In the 1700’s, at least two mathematicians demonstrated that pi was irrational (Lambert and Legendre). In 1882, Lindemann proved that pi is not just irrational; it has an even more profound property-it is transcendental. This means that it is not the square root (nor cube root, nor…) of any rational number. It is an even more special number than the square root of 2.

So who discovered pi? That’s a bit like asking who built the Empire State Building. Pi is an achievement of human intellect more than of a single person.

Cardinal and ordinal numbers

NOTE: I get a lot of hits from Internet searches on “cardinal and ordinal numbers”. If that’s how you got here, consider going directly to an expanded and illustrated version of this post on Sophia. It is a better version of the same ideas that are here.

overview

In my math content course for future elementary and special ed teachers, we are working on the difference between cardinal and ordinal numbers. I promised them a reading on the topic so they could work on it outside of class. But the first three pages of Google results (and really, who goes deeper than that?) all return the same basic idea, which is not as nuanced as the one I need to communicate to my students. So here is my take on things. If you find it useful, link to it please so that future students (mine and others’) will find it more easily.

introduction to the topic

The basic introduction to ordinal and cardinal numbers is this: Ordinal numbers refer to the order of things. If I was fifth in a race, fifth is the ordinal number. Cardinal numbers refer to how many things there are. If there are five fingers on one hand, five is the cardinal number.

At the introductory level, the distinction can be made linguistically: first, second, third are ordinal numbers while one, two, three are cardinal numbers.

This is the distinction that is made on each and every one of the websites I looked at this afternoon. Here is a typical example.

A subtler view of the topic

What do we mean when we say that “the Minnesota Twins are number 1”? We mean that they are the best, or that if we put all baseball teams in order from best to worst, they would be first.

The claim that “the Twins are number 1” is a claim about ordinal numbers. That is, we don’t always make the linguistic distinction. Sometimes we say one when mean to refer to a cardinal number (the Twins are one team in the league), sometimes we say one when we mean to refer to an ordinal number (the Twins are number one in the league).

A problem

In class, we are reading a lovely book titled Children’s Mathematics that reports results from the Cognitively Guided Instruction (CGI) research project at University of Wisconsin, Madison. The premise of the project is that young children come to school with powerful mathematical ideas, but that these ideas may differ from an adult’s way of thinking. The better teachers understand their students’ ideas, the better basis the teachers have for making instructional decisions. CGI set out to document the ways students think about addition and subtraction problems, and to associate these ways of thinking with strategies students use to solve problems.

The book comes packaged with a CD-ROM of second-grade children solving problems of different types, and demonstrating various strategies. One major strategy is counting up/back.

Consider this problem:

Problem 1. Griffin had 10 apples. He gave 3 apples to his sister Tabitha. How many apples does Griffin have left?

Problem 1 standard solution. The standard counting technique is for a student to say, “He had ten apples. Nine, eight, seven. He has seven apples left.”

Problem 1 alternate solution. In one of the CGI videos, a student solves a similar problem in this way, “He had ten apples. Ten, nine, eight. Take away that; it’s seven.”

In the standard technique, we say “nine, eight, seven”. These are cardinal numbers. They represent how many apples are left after Griffin gave away each apple. He gave away one apple NINE, he gave away another EIGHT, he gave away another SEVEN. So after giving away three apples, there are seven left. Because we are counting back using cardinal numbers, the last number we say represents the number of apples remaining.

In the alternate solution, we say “ten, nine, eight”. I argue that these are ordinal numbers. They represent which apple Griffin is giving away at each step. He gave away apple number TEN, then he gave away apple number NINE, then he gave away apple number EIGHT. So after giving away three apples, he has given away apple number eight, there must be seven left. Because we are counting back using ordinal numbers, the last number we say is one bigger than the number of apples remaining.

A lovely question

A student asked a lovely question in class. Paraphrasing, she asked, “What if he were counting up? How would his strategy be different from the usual one?”

Surprisingly, his strategy wouldn’t be any different.

It would be highly unusual for a student to solve Problem 1 by counting up. So let’s consider another problem:

Problem 2. Tabitha had 3 apples. She picked some more. Now she has 10 apples. How many did she pick?

Problem 2 standard solution. The standard counting technique is for a student to say, “She had three apples. Four, five, six, seven, eight, nine, ten.” When the student says “four,” she puts up one finger then another for each successive number. After saying “ten,” she counts her fingers and says, “She picked seven apples.”

The numbers four, five, six, etc. are cardinal numbers. They each refer to how many apples Tabitha has after she picks another apple.

Or maybe they are ordinal numbers. Does “four” refer to Tabitha having four apples, or does it refer to the fact that the one she just picked is apple number four? Does four describe the set of apples, or does it describe the fourth apple? We can’t know from the information given. The ordinal and the cardinal counting sound identical when counting up.

And that is what makes the subtler distinction between ordinal and cardinal numbers so challenging. The easier distinction is based on the form of the words (fourth-ordinal; four-cardinal). But not all fours are cardinal.

When children are learning to count, they learn the sequence of words first: one, two, three. They do not fully understand that these words represent how many things there are. In that sense, it could be said that most children learn ordinal numbers first. The idea that the last number we say represents something about the whole group of things-that this is a cardinal number-comes later. It is not unusual to give a group of eight things to a 3-year old for her to count, to have her do so correctly, then to ask, “So how many are there?” and to have her say “twelve” or some other wrong answer. She has ordinal numbers-ascribing the correct number word to each object in the set; she does not have cardinal numbers-ascribing the correct number to the whole group.

An unnoticed rule for number language

I am coming to the end of my big place-value extravaganza in the math course I teach for future elementary and special education teachers. I had a conversation after class last week that reinforced the importance of the time I spend on place value.

Over the last two years, I have adapted a system for teaching place value in this course that comes from the ideas of JoAnn Cady and Theresa Hopkins at the University of Tennessee, Knoxville. The system, which they call “Orpda” and which I have bastardized as “Ordpa” (and two years in, it’s too late to change now!) is a base five place value system that depends on a new set of numerals-seemingly random symbols from the top row of the keyboard. But more than that, Cady and Hopkins developed an inquiry approach to using the system. As an example, the new symbols are introduced (in order in my version of the system: @, #, $, %), a symbol for 0 is introduced (!) and they ask students, “How should we represent the next number?”

From experience, very few people think of using place value. Many will invent a new symbol. Others will add the existing symbols, suggesting things like “#+$”. Still others will draw an analogy to tally marks: @@@@@. The place value answer to the question is extremely rare in my work with future teachers and in presentations I have done at state math conferences and professional development sessions. I have learned through this system that our minds are not programmed to think in terms of place value. Instead, our minds are programmed to think additively.

I have enjoyed this side of working with the Ordpa number system-it helps us to understand children’s challenges in learning and using the decimal number system. This was what Cady and Hopkins developed the system for.

But I have become even more interested in how the system highlights number language. I have written about this in other places, including a forthcoming article in for the learning of mathematics. But last week a student asked a question that I had not previously considered.

We have named our first two-digit number: @! as flop and we have named our first three-digit number as flip. Today, we were considering the difference between flip flop and flop flip as a way of understanding a video we have watched in which a young girl says “six hundred plus four hundred equals ten hundred” and then writes “110”.

My student asked after class, “So how can flip flop mean flip plus flop, but flop flip means flop times flip or flop groups of flip?”

What a wonderful insight! Consider the construction two-thousand two. We don’t think about the fact that the first two is multiplied by the thousand (“two thousands”), while the second two is added. And we certainly don’t think about the implicit rule that when a smaller number word precedes a larger one, we multiply and when the smaller number word comes second, we add.

Other examples that highlight this rule include 1100 and 100,000.

So (Wump) hats off to Cady and Hopkins for sending me down a path on which I continue to learn and which has clearly gotten my students thinking harder about elementary mathematics.

Incidentally, I’ll be presenting on the system, and on children’s and adults’ learning of place value at the National Council of Teachers of Mathematics Regional Conference in Minneapolis in November.

A new algorithm for finding Least Common Multiple?

A student in a developmental algebra course was struggling with problems involving least common multiples. The lesson she was working on involved finding the least common multiple of numbers first, and then using that process as an analogy for finding least common multiples for variable expressions. Surprisingly, she felt confident with the variable expressions and was struggling with the numbers.

When I sat down with her, Karen (not her real name) could not understand why the technique she was using for variables was not working with numbers. She was able to correctly find the prime factors of numbers, so she wrote:

24=2*2*2*3 and

36=2*2*3*3

Then, drawing on her experience with fractions, she cancelled the common factors and used what was left:

24=2*2*2*3 and

36=2*2*3*3

She ended up with only a 2 and a 3 remaining, which is six and she knew this was not correct. Six is not even a multiple of these numbers, never mind the least common multiple.

I restated our textbook’s approach, to wit: We want to use each factor the same number of times as it appears in the number in which it appears most often. There really is no simple way to state this and I worked a couple of examples for her.

But I was intrigued by her ‘cancelling’ approach. In addition I had offered a strategy that, while supported by our textbook, bore very little relation to her way of thinking about the problem. This is not a recipe for success. We need to help our students refine their ways of thinking, not give them yet another rule to remember. So I explored her idea of cancelling and suggested this:

When we use each factor the greatest number of times as it appears most often, we can think of this as gathering all of the prime factors, then getting rid of the ones that come from numbers where they appear less often. When we have:

24=2*2*2*3 and

36=2*2*3*3

We want to “cancel” the 2’s in 36 and the 3 in 24-we don’t need those. So Karen’s Cancelling Algorithm-perhaps new to the world, and perhaps new only to her and to me is this:

Cancel the common factors only in one of the two numbers:

24=2*2*2*3 and

36=2*2*3*3

We cancel two of the 2’s in 24 because they match up with the two 2’s in 36. And we cancel the one 3 in 24 because it matches up with one of the 3’s in 36. We could have cancelled the two 2’s in 36 instead-that’s not important. What is important is that we cancel them only once.

Karen loved this algorithm and was very, very pleased to have her thinking changed into an algorithm that works.

The interaction was partly satisfying for me and partly disturbing. Karen began very frustrated and ended feeling successful and bright. That is always satisfying. But what had she really learned? She is not studying least common multiples for their interesting mathematical properties. Instead she is studying them in order to be able to add, subtract, simplify and solve rational expressions. Without questioning the larger goal of the whole enterprise of developmental college mathematics, it is still reasonable to ask how important least common multiple is for operating on rational expressions.

The only argument for finding least common multiple in this context is that it gives us a simpler form of the resulting rational expression than any other multiple will. If I am working with numbers, I can use the least common multiple to add:

1/24+1/36=3/72+2/72=5/72

But I can use any common multiple:

1/24+1/36=6/144+4/144=10/144

The least common multiple results in a simpler fraction, but it’s the same answer either way. Indeed, I can always use the common multiple that results from multiplying the two denominators:

1/24+1/36=36/864+ 24/864=60/864

And the same is true of the rational expressions Karen will be working with shortly. So why do we induce this stress in our students? If the only reason to find least common multiple is to work with rational expressions, and if at the same time any common multiple will do, why do put this artificial barrier in front of our students? And why do we, as teachers, allow ourselves to work as though these barriers were real?