the fundamentals of number

I'm still in pursuit of the fundamentals about number. That's really two sets of fundamentals: the fundamentals of number itself and the fundamentals of how children learn number.

That would seem to me to be a prerequisite for:

1. good teaching
2. writing good maths software about number

Papert helped develop some good maths software about geometry (logo) and that deserves much praise but I'm starting to think that number might be harder. For instance the very smart Greeks did more with geometry than with number, according to this excellent history of Zero:

Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid's Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines. Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed.

Well, my quest for number fundamentals led me to review several books. I ended up looking at about seven books using google books and amazon as starting points. Here are the two books I ended up deciding to buy:

Children Doing Mathematics by Terezinha Nunes and Peter Bryant (1996), 268pp (amazon link)

They are psychologists who are interested in children's reasoning ... Keith Devlin recommends this book for its treatment of multiplication, which I mentioned in an earlier blog (Multiplication is not repeated addition). This book is entirely devoted to number, which is what I wanted

Knowing and Teaching Elementary Mathematics by Liping Ma (2000), 166pp (amazon link)

This book asserts and documents the claim that maths is better taught in China than in the USA because Chinese teachers have a more profound understanding of maths knowledge. One thing that appeals to me here is that it contains concrete examples of a good way and a not so good way of teaching various maths concepts

I see a need a need to promote books about the fundamentals of learning. This has just come up again in discussion arising out of the IAEP (Its an Education Project) list, see this blog by Red Hat and Sugar Developer Greg DeKoenigsberg, promoting a book which is not about fundamentals but seems to be more of a lazy transfer of Clayton Christensen's concept of disruptive technology from the marketplace to learning. I left a comment on Greg's blog arguing that point

Summing up:

1. Read books
2. Study books about the fundamentals of knowledge and learning
3. Good learning software (eg. logo, etoys, scratch) requires a process such as this

numbers to die for

The Pythagorans wanted to believe that everything could be expressed in terms of whole numbers. I was wondering about that. For example, how could you prove that square root (2) could not be somehow expressed in terms of whole numbers?

Hippasus, one of the Pythagoreans, did prove that such an assumption did lead to a logical contradiction. This so upset his group that they killed him. This history gives the expression "irrational number" some real bite.

Here is the brilliant reasoning of Hippasus in the 5th Century BC, which cost him his life:

Start with a right angled isosceles triangle, assume all the numbers are whole:

• The ratio of the hypotenuse to an arm of an isosceles right triangle is a:b expressed in the smallest units possible.
• By the Pythagorean theorem: a² = 2b²
• 2b² must be even, since anything multiplied by 2 is even
• Hence a² is even and furthermore a must be even as the square of an odd number is odd.
• Since a:b is in its lowest terms (first assumption above), then b must be odd (otherwise the ratio could be further simplified by dividing by 2)
• Since a is even, let a = 2y
• Then a² = 4y² = 2b²
• b² = 2y² so b² must be even, therefore b is even
• However we asserted b must be odd. Here is the contradiction.

The story goes that Hippasus made this discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.”

I've taken this mainly from wikipedia where there is much more detail about irrational number but have added a few extra bits of explanation to the reasoning.

Multiplication is not repeated addition

A series of articles by Keith Devlin (Devlin's Angle) has made me realise again that my understanding of number, in this case multiplication, is fairly superficial:

It Ain't No Repeated Addition

It's Still Not Repeated Addition

Multiplication and Those Pesky British Spellings

Devlin is a good mathematician. However, he keeps on repeating that he is not a K-12 teacher and so can't offer advice as to how to teach at that level. But he does offer lots of advice about what shouldn't be taught. So in that sense he is a tease and a provocateur.

One of his recommended readings, from the third article, is more helpful for practitioners:
How do we teach children to be numerate? (pdf, 23pp) by Mike Askew and Margaret Brown

page 10:

Calculations can be identified with several different types of interpretations and contextual problems. For example 4 x 5 can be linked to:

• repeated sets (eg. 4 boxes each with 5 hats)
• multiplicative comparison (scale factor) (eg. 4 hats and 5 times as many scarves)
• rectangular arrays (eg. 4 rows of 5 hats)
• cartesian product (eg. the number of different possibilities of wearing a hat and a scarf from 4 hats and 5 scarves)

Similarly division calculations can be taught in two ways. For example, 20 / 5 can be associated with:

• measurement / grouping (quotition) (eg. 20 apples put into bags of 5, how many bags get filled?)
• sharing (partitioning) (eg. 20 apples put into 5 bags, how many apples in each bag?)

Of these possible interpretations, research has shown that multiplication as repeated addition and division as sharing appear to be widely understood by primary aged children. However, as the examples above show, understanding the meaning of multiplication is more complex (Nunes and Bryant, 1996) and difficulties with fully understanding multiplication and division persist into secondary school (Hart, 1981)

There is evidence that such early ideas - multiplication as repeated addition and division as sharing - have an enduring effect and can limit children's later understanding of these operations. For example, understanding multiplication only as repeated addition may lead to misconceptions as "multiplication make bigger" and "division makes smaller" (Hart 1981, Greer 1988) ....

Language is important here as different expressions will greatly influence children's solution methods. For example, interpreting 52 x 3 as "52 times 3" or "52 lots of 3" may lead to a less efficient calculation method then reading the symbols as "52 multiplied by 3" or "3 fifty-twos"

References:
Hart K (1981). Hierarchies in mathematics education. Educational Studies in Mathematics, 12, 205-218.
Nunes T and Bryant P. Children Doing Mathematics (google books, amazon)

thoughts about OLPC cutbacks

Some thoughts about the OLPC project cutting back on 50% of their staff (refocusing on our mission)

When this happens the prophets of doom, such as wayan vota (olpc just got gutted) and arstechnia (OLPC downsizes ...) publish quickly and with a tendency to sensationalise and catastrophise. We are all familiar with this style of journalism.

This really is a 40 year story going back to when alan kay first envisaged the dynabook. We need to take a long term view. It is also a good time to revisit Walter Bender’s 23 questions which is another insightful way of viewing the big picture

The Surface Mission is to bring education to the children of the Developing world using the XO as a vehicle. This is a serious and realistic although a very hard to achieve goal.

The XO-1 has been a partial success. However, it did not achieve its initial (problematic) goal of selling to the Developing world governments in millions. Clearly, the stakeholders are not satisfied and cutbacks have now occurred. Also the 2008 give one, get one was a relative flop, with only 7% of the sales of the G1G1 in 2007 (source). But Negroponte keeps pushing forward. The dual screen XO-2 (article) may achieve the scaling that was hoped for but did not eventuate with XO-1.

Sugar, the new OS, has been a partial success. It is diverse, free (FLOSS) educational software with unique built in (when sugarized) collaboration features. But some / many aspects of Sugar are buggy and / or unfinished. There is still a lot of work to do.

All of this does spin off in multiple directions – Walter’s question categories are Computer Science, Engineering, Education, Economics and Social Sciences. There was some discussion about his 23 questions at OLPC_news (comments). It would be good to see more.

Different people bring different skills and expertise to the Project. Everyone who has been involved with it has been broadened and deepened in some way. No regrets. I'm not aware of anyone who has been involved who wishes that they hadn't been.

There may or may not be ultimate success. But there is ongoing partial success as the consciousness of every participant, adult or child, of the social possibilities of disruptive technology is increased.

One aspect is that the Projects are in more or less constant turmoil. I see this as inevitable.

We have cutting edge, complicated, disruptive technology – hardware and software.

We have a difficult, hard to answer social question: How best to help the impoverished children of the Developing world?

We have conflict between FLOSS and Proprietary pathways. Can FLOSS alone deliver the goods?

We have division between expert and non expert enthusiasts in different areas. Some fields demand democracy and transparency, eg. FLOSS. But even FLOSS has benevolent dictators as a concession to expertise. But in some other areas democracy is frankly a waste of time. For example, there is little point in Mary Lou Jepson consulting with non experts about the latest in screen technology. This can be extrapolated, more or less, to many other parts of the Development process.

We have conflict over the best methods to educate children. Constructionism is not well understood and is not the only path.

How much should local issues influence global development? (refer to the Bryan Berry thread on Its An Education Project)

How do hardware experts, software experts and educators work together effectively to produce a better product?

There is limited money and people to carry out this project. Even though a Keynsian approach – spend money on public works to increase employment - in economic crisis would suggest generous funding for this project. Sounds like a good idea but I doubt that it will happen.

We have entrenched, powerful interests who are threatened by these developments (Intel, Microsoft, educational bureaucracies, existing NGOs)

How could there not be continual turmoil?

At any rate, the XO / Sugar Labs projects continue to be excellent objects to think with and to act through. In the words of David Farning:

"I would like to offer a _heart_felt_ thanks to everyone at One Laptop Per Child who has made Sugar and the XO the products that they are today.

You still are at the forefront of a revolution in learning such as the world has not seen since the invention of the printing press.

But, this revolution, as with all revolutions, is hard to plan; there are no maps, there is no rule book, there is no gray bearded sage to guide your way"

The xo and sugar labs will continue to transform the world for the better. By just how much remains to be seen.

Numbers

Should Children Learn Math by Starting with Counting? by Keith Devlin

Thanks to Rob (blog) for putting me onto the Devlin's Angle essays. Still reading. He is well researched, discusses fundamental issues of maths pedagogy and is making me think.

This particular essay made me think about Number and how it is taught in Schools. To tell the truth I had to go back and review my own foundational understanding of number. The way that it is taught in our Schools is like a drip feed but for most students and teachers the "big picture" is never put together. It's like a jigsaw puzzle that is seldom completed.

Hence my need for an overview even though this terminology is not definitive, eg. see Wolfram for a more systematic approach:

natural number (also called counting number) can mean either an element of the set {1, 2, 3, ...} (the positive integers) or an element of the set {0, 1, 2, 3, ...} (the non-negative integers)

whole numbers: 0, 1, 2, 3 ...

integers ... -2, -1, 0, 1, 2 ...

algebraic numbers: integers, roots of positive numbers, fractions

rational numbers: 42, -23/129 (integers, fractions)

irrational numbers: pi, sqrt (2) infinite decimals

real numbers are the numbers that can be written in decimal notation, including those that require an infinite decimal expansion ... OR points on a infinitely long number line

imaginary numbers (unreal): i, 4th root(-9)

complex numbers (mixture of real and unreal): 2 + 3i


At any rate, Devlin points out that in the USSR due to the influence of Vygotsky and his followers, such as Davydov, that the curriculum from the beginning focuses on real numbers, it does not begin with natural numbers, as we do in Australia (or in Devlin's case the USA). He gives a broad overview of that curriculum in his article.

Here is a quote from Devlin which provides a rationale for the validity of this approach:

Humans have not only a natural ability to abstract discrete counting numbers from our everyday experience (sizes of collections of discrete objects) but also have a natural sense of continuous quantities such as length and volume (area seems less natural), and abstraction in that domain leads to positive real numbers.

In other words, from a cognitive viewpoint (as opposed to a mathematical one), the natural numbers are neither more fundamental nor more natural than the real numbers. They both arise directly from our experiences in the everyday world. Moreover, they appear to arise in parallel, out of different cognitive processes, used for different purposes, with neither dependent on the other. In fact, what little evidence there is from present-day brain research suggests that from a neurophysiological viewpoint, the real numbers - our sense of continuous number - is more basic than the natural numbers, which appear to build upon the continuous number sense by way of our language capacity

Fascinating. We need to explore fundamental knowledge deeply. This is probably the most important part of improving education.

Chinese teachers better at foundational maths than US teachers

Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning.)
by Liping Ma

google books URL

amazon books URL

This looks fantastic. I read the Contents page, Forward and Introduction from the google books URL

From the Forward by Lee Schulman:

Chinese teachers are far more likely to have developed "profound understanding of fundamental mathematics." To say that they "know more" or "understand more" is to make a deeply theoretical claim. They actually may have studied far less mathematics, but what they know they know more profoundly, more flexibly, more adaptively"

foundational knowledge: one and three quarters divided by a half

** Make up a good story or model to represent that problem **

From the Introduction by Liping Ma, reporting from the USA:

I was particularly stuck by the answers to this question. Very few teachers gave a correct response. Most, more than 100 preservice, new and experienced teachers, made up a story that represented one and three quarters multiplied by a half, or, one and three quarters divided by two. Many other teachers were not able to make up a story

Mary Laycock

Some maths resources by Mary Laycock, recommended by alan kay

"mathematicians turned great math teachers (such as Mary Laycocke) who have spent decades learning how to get young children to learn "real math" "

http://www.activityresources.com/store/catalog/
The hands on manipulatives and other resources look great:

• Fraction Tiles,
• Weaving your way from arithmetic to mathematics,
• Graphiti,
• Puzzling your way into Algebra

http://openlibrary.org/a/OL3631937A
Books by Mary Laycock with intriguing titles such as Skateboard Practice, Tapestry of Mathematics, Straw Polyhedra, Correlation of Activity-Centred Science and Mathematics

I wish some of these old books could be digitised and put into the public domain, they might get a new lease of life that way.

I looked up one of them, Skateboard Practice, in google books search and found other books that refer to them, they are obviously great resources
eg.
http://books.google.com/books?id=rHYmAAAACAAJ&source=gbs_ViewAPI
cites another book that uses Skateboard Practice as a resource
http://books.google.com/books?id=jDXAj-ym6xYC&pg=PA54&vq=%22Skateboard+Practice%22&source=gbs_book_citations_r&cad=0_2
cites some of the activities from this book and how this teacher used them

It would be good if young(er) educational software developers took some time out to check out these works by an old master

the drama and humour of numbers

The Story of 1 (60 minutes)

I just saw this excellent TV show about the history of numbers (ABC review) and, for joy, it's available on the internet too :-)

Some Australian aboriginal tribes did not have a number system, just one and many. Arithmetic evolved in cities which had more complexity which required calculations. The first writing was with numbers.

3000 BC: The Egyptians conceived of 1 million. Also they invented the cubit, a unit of measurement, required for the buildings they constructed

Pythagoras invented odd and even numbers, things such as magic triangles (1, 2, 3, 4) and explored the relationship between music and the size of containers (the music of the spheres). But his dogmatic idealism about number led to tragedy. One of his disciples discovered irrational numbers and was drowned.

The Romans murdered Archimedes and then imposed their crummy numerals onto the world. They were so useless for doing calculations that the abacus was used instead.

Our decimal system and most notably the number zero wasn't thought of until 500 AD by someone in India. From there it was passed onto the Arabic Muslim world. Then the decimal system was brought to Europe by Fibonacci.

There ensued a struggle between the Roman numerals and the decimals system which lasted for hundreds of years. Eventually the decimal system won out because of the need for capitalism to calculate compound interest accurately.

Finally, Liebnitz invented the binary system but we had to wait another 200 years for the computer

This video is very enlightening and funny being narrated by Terry Jones of Monty Python fame. The simulated battles between our modern sprightly numbers and clunky Roman numerals are fabulous.