Wednesday, 18 November 2015

Maths in Science - an interview with Don Knuth


In March this year I published interviews with some very prominent scientists, asking them about their experience of maths education and how they used mathematics in their field of science.

At that time I received a reply from the legendary computer scientist and mathematician Professor Don Knuth, a reply which I overlooked and failed to open. 

Today I received a polite e-mail from him wondering what had happened to his responses and why I had not acknowledged his participation in the "Maths in Science" project. When I checked my in-box, it was still sitting there from March and I had not opened it or read it.

I am deeply embarrassed and publicly offer my apology to Professor Knuth for my oversight. I have also replied to his e-mail and offered sincere apologies to him directly.

So please, read below the responses to my 10 questions provided so generously by Professor Knuth, Professor Emeritus at Stanford University, "the father of the analysis of algorithms", creator of several computer programming systems, creator of METAFONT and the author of "The Art of Computer Programming" - the bible for computer programmers everywhere.


Here is the note to the e-mail that Professor Knuth sent me in March this year. His kind comments make me feel even worse that I failed to open his e-mail:

Hi Bruce,

Ten answers are below!

One of the most important things for students to learn is how to ask good questions. You evidently have learned that well.

Best wishes, Don Knuth


1. Describe what maths lessons were like for you at school.

People of my generation (in Wisconsin, USA) learned multiplication tables in grade 2, fractions in grade 5, algebra in grade 9, two-dimensional geometry in grade 10, complex arithmetic in grade 11, three-dimensional geometry in grade 12. I came up with lots of questions that my teachers couldn't answer; so I spent most of my time thinking about other subjects (English, Latin, physics, chemistry, biology, music). But at home my father had a mechanical adding/multiplying machine, and I enjoyed playing with that. I spent hundreds of hours plotting the graphs of functions like 

$\sqrt{x+a} - \sqrt{x+b}$ for different values of $a$ and $b$, using colored pencils so that I could put several graphs on the same page.

2. Was the maths that you learned at school useful to you later in life?

Absolutely; I can't think anything from those classes that I have NOT used repeatedly! For example, the geometry classes not only taught me how to prove things rigorously, they also gave me the ideas needed to create the METAFONT language, with which many fonts of type have been designed; those fonts are now used by millions of people all over the world.

3. How good do you need to be at mental arithmetic to do calculations in your head?

I'm glad that I memorized multiplication tables up to 12x12. But I think going any further (like up to 99x99) would have been a waste of time. Calculations in my head are important only on problems that are fairly simple, or on problems that involve symbols instead of numbers. When I'm working on a research problem I generally begin by filling dozens of sheets of scratch paper with partial calculations. When I eventually get to a point where I can think about the problem while swimming, then I'm often ready to solve it.

4. Mathematics teaches us that you can put two things together to make a new thing. Is this important in what you do?

Complicated structures are made up of simple structures that are combined in simple ways. I think computer scientists understand this even better than mathematicians do, because we've learned how to represent many kinds of data inside a machine.

5. Mathematics is about finding patterns. Do you need to look for patterns, or exceptions to patterns, in your research?

Yes, I like to think that mathematics is in fact the science of patterns. The patterns that I work with daily are usually some regularities in relationships between objects, not between numbers. But numerical patterns are important too: Like the facts that:

$1=1^2$, $1+3=2^2$, $1+3+5=3^2$, $1+3+5+7=4^2$, etc., 
and that
$1^3=1^2$, $1^3+2^3=(1+2)^2$, $1^3+2^3+3^3=(1+2+3)^2$, etc.

6. Mathematics also teaches us about balance and equality. Is this idea useful in your research?

In the METAFONT language referred to earlier, we express the shape of the letter A by giving equations that should be satisfied by key points in the lines being drawn. "The left stem runs from the baseline, half a unit from the left edge of the enclosing box, up to the cap-height. Its slope equals the negative of the slope of the right stem." 

[Reference: Computer Modern Typefaces, page 369.]

7. Mathematics helps us to represent quantities and measurements numerically. Do you do this in your work?

In fact my program that draws the Greek letter $\pi$ actually uses the number 3.14159 in two places. [Computer Modern Typefaces, page 159.]

8. Is estimation good enough or do you need to measure things accurately?

A computer scientist must be especially careful, because tiny errors can easily be magnified --- with catastrophic consequences.

9. How do you use statistics to analyze your results?

Much of my work involves comparing different computer methods, to see which one is fastest. Basic statistics such as the maximum, mean, and median running time, together with the variance, are crucial in this analysis. More broadly, concepts of random numbers and probability are absolutely essential ingredients in most of the best computer methods known today.

10. Do you have any other insights to offer into how you use maths in your work?

For instance, when I brush my teeth I've got eight areas to cover, namely Left and right, upper and lower, inside and outside. It's most efficient to follow a "Hamiltonian path" or "Gray code":
  left upper outside
  right upper outside
  right upper inside
  left upper inside
  left lower inside
  right lower inside
  right lower outside
  left lower outside


 Thank you so much Professor Knuth for answering these questions for me and for being a part of the "Maths in Science" project. You have given me, and hopefully many others, lots to reflect on, including great advice on dental hygiene.

And once again, I apologise for my error in not including your thoughts in the initial project back in March.

Sunday, 8 November 2015

I Quit

Last week, I quit.

Not something I do often and I'm still feeling a bit funny about it.

It had been my New Year's Resolution in 2014 to learn to play the bagpipes. I have been learning for 2 years now. 

Well, I had been until last week.

It was the hardest thing I have ever done in my life. I play a few musical instruments but this was the hardest I have ever attempted. 

I spent 12 months on the practice chanter before I got unleashed on the pipes. 

I gave it a go but last week I realised I had to give it up.


Or indefinitely suspend input of time and energy in this endeavour.

Maybe I'll come back to the pipes one day.

We'll see…

What have I learned?

Reflecting on this experience I have learnt a few things:

1. I can't do everything. Thought I could. Must have been mistaken. My resources of time, commitment, patience, enthusiasm etc are finite.

2. It doesn't feel totally satisfying to quit something. In fact, I was a bit disoriented. I can no longer think of myself as a "student of the bagpipes" - I am now "a former dabbler in the noble pipes".

3. My family is probably celebrating the peace and quiet.

4. It's not that big a deal - it was only the bagpipes.

5. It's okay to quit sometimes. It doesn't make me a loser. In fact, maybe there's a bit of "growth mindset" here for me.

6. There are some positives in giving up - it reduces my stress levels, gives me back some precious time and opens up a few possibilities for the future. I wonder how the family would feel about trumpet lessons?

Friday, 25 September 2015

Why do we teach multiple ways of doing the same thing?

This morning I got a message from a colleague on Twitter.

@JasonGraham99 asked:

Why are there so many different math strategies that exist today that (didn't)? exist when I grew up? And more importantly why do we need the different strategies - a Devil's advocate question…sorry but I wonder why. Sometimes we teach 5 different ways to get to the same answer (lattice, partial products etc..) Why? - again Devil’s advoc) But I am serious. Does having multiple ways of learning math confuse learners? Or open minds?

This is our last day of Term 3 - but we work up to the last bell of the day so I am keen to give this some thought.

Or better still, get the kids to do the thinking.

Here is what my Grade 2 class had to say about why we learn to do the same thing in multiple ways:

So we can make things easier. Say everyone’s thought of one way but someone thinks of a new way.

Having different ways is what we need to do because in maths you need different techniques.

You’ll learn more.

You don’t want to just get the same answer everytime.

Because it’s really fun to have different ways. It can make it easier for little kids. 

So people get more knowledge.

If you don’t know one strategy you can go on to the next one. One maths question can help you with another one.

Pictures can be used to help you if you can’t read numbers.

There could be simpler ways so we need to keep looking for them.

And then I asked them if having multiple methods was confusing:

I think it would be really confusing if you had lots of different answers.

It gets confusing if the same question is asked in lots of different ways.

Yes and no. You know that having lots of answers is good because if you don't get one then you might get another.

It’s confusing when there are no pictures.

It’s confusing if the extra ways of doing it are harder or have more steps.

And finally I asked if they thought if multiple strategies actually helped open their minds:

Depends on how hard the question is.

Having more ideas of different ways of doing things can help you do other things.

Knowing how to do a little sum can help you in future learn how to do a big sum. A little mouse can one day help a big lion.

If one way is easy for you to do it opens your mind up and helps you understand other questions.

It’s helping your brain because your brain gets smarter.

It helps because you see those different ways because you can choose which one to do. If you look at a question and you’re stuck on it you could choose one of those ways.

I think it’s confusing because you don’t know which one is the best. One way could take too long.

I was impressed with what they had to say. It was certainly an interesting question for me to ask them.

I also have a few ideas of my own:

1. Golf

I always like to compare life to a game of golf. The more clubs I have in my bag, the better equipped I am for the many problems and hazards I will face out there. I might be pretty good with the driver off the tee, but the driver will be no use to me when I end up in the sand. The more clubs I have, the better chance I have.

I think the same is true for maths. If I know multiple strategies, I can select the most appropriate for the context. Do I want a quick method? - use mental calculation.
Do I want a detailed result to many decimal points? - use a calculator.
Do I want to show my thinking to someone else? - use a picture or diagram.

2. Why?

Many of us will remember being taught a method or process in mathematics, such as for multiplication or division, but few of us will have been shown WHY we do it that way. This teaches us to be "superstitious" mathematicians - we do it this way just because we always have. 

Learning multiple strategies for an operation without explaining the WHY and developing the connections simply compounds confusion for kids. I agree with Jason's question - it is confusing - if there is no explanation. 

But it is also liberating and "mind-opening" when students are able to make connections, to join up their learning and find relationships. And they can do this if we are able to help them understand a variety of different strategies and how they are related.

If a student understands the distributive law, they can appreciate a number of different methods of multiplication. Vertical algorithms, lattice method, arrays - all have a link to the distributive law. However, memorising a list of different strategies is not much help by itself -  it really does get back to knowing WHY we do something. 

3. Fun

Finally, it can be fun to know different ways to do the same thing. It can be boring doing the same thing all the time - variety of the spice of life.

So I would encourage teachers and students to play with maths, to have fun and find new ways to do things just because they can. 

Hopefully this makes some sense and answers a few of those questions from Jason.

And what a great way to finish the term!

Now I've got to get home and dust off the golf clubs...

Thursday, 27 August 2015

Patterns That Grow

As Marilyn Burns says, pattern is the password of mathematics.

I like playing with patterns. I use them to get the kids interested in finding relationships.

So I gave them this provocation:

Here's a pattern - 1, 5, 9, 13….

Based on this, can you tell me if 21 is going to be in this pattern? And then will 45 be in it as well?

So we got the blocks out and started making some patterns. Here is what the kids came up with:

"The pattern makes a cross shape. And 21 makes the same shape so it must be in the pattern."

"And 45 can make a really big cross too."

From talking as a group we were able to work out that yes, indeed, 45 is going to be in our pattern.

In fact one student pointed out that it was like the 4x pattern (4, 8, 12, 16…) but just one more.

It was a good starting point. The conversation was never going to end there.

What other patterns can you make? Can you explain your pattern using numbers? Can you tell me what the next number will be without making the model of it?

Here's a nice pattern - just like the cross but missing a leg.
CAn you see a link to the 3x pattern?

This one adds on a leg and goes 3D.
And now we were thinking about the 5x pattern.
Is Grade 2 too young to start talking about y = 5x + 1?

We had a good talk about this one. Is there a step missing somewhere? Should there be something between 1 and 8? Is 1 part of the pattern?

Another pattern based on squares - ah ha! Square numbers!

And then taking the square into the third dimension!

This is not the end of work with patterns for the year. We will revisit the concept many times but I feel that the kids have a good grounding now in identifying, making and explaining patterns.

And we had a lot of fun.