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...

How We Organise Ourselves

We have started a new inquiry in the transdisciplinary theme of "How We Organise Ourselves". I find this theme really interesting in terms of mathematical thinking so I got a bit of a provocation going for the class.

I got a bucket load of what we call "paddle pop sticks" - probably have a different name in your town - do Queenslanders really call them by-jingo sticks?

The task was to find out how many sticks there were. The task itself was pretty meaningless - my prime interest was in how the kids would approach the challenge. What evidence would they show of mathematical thinking? How would they organise themselves?

Well, first off they fragmented into friendship groups and started grabbing for as many sticks as they could get - they are 7 years old after all. 

Anyway, they split up and started to count the sticks. They showed several different methods of getting organised. Here is what it looked like:

One group started counting the sticks and making a pile of the ones they had counted.

They were a bit stunned when I asked how they were going to check for accuracy.

It's a bit easier to see if they are spread out a bit.

But you have to do a lot of counting.

And turn a corner when you hit the wall.

Another group thought of lining up the sticks - in groups of 6...

…or groups of 14 - beautifully colour-coordinated.

Other groups had the idea of making bundles of 10.

 Some appeared more organised than others.

I had to drag them back to the original question…so, how many sticks are there?

Each group wrote up their individual totals

 - but it took insight to decide that they needed to add them all together.

An answer! We got 2637 sticks!

And then I threw them a wobbly - can you check that?

What? they exclaimed.

But, I continued, did you get any ideas from the other groups about good ways to organise yourselves?

Groups of 10 was the consensus - but I was still a bit worried about the way they were laid out on the floor. When we hit the wall we had a problem... 

...some chose to go around the corner and others decided to start new rows.

Finally we got the idea that maybe we can organise these sticks into:
a) bundles of 10
b) rows of 100
c) blocks of 1000

And that made it pretty easy to see 2 groups of 1000, 3 rows of 100, 2 bundles of 10 and 2 left over = 2322 sticks.

And you can see it just by looking.

What a good idea.

Maths in Science - That's All There Is

Well, that's all there is from the Maths in Science project. A bit brief, I'm sorry, but there were some real gems in there to consider.

A Few Reflections:

Of the 39 scientists I contacted, I was able to post 6 interviews. Many of the scientists that I contacted were obviously very busy and not available to participate. There were several other scientists who said they would help out but didn't get back to me in time to get it posted. 

I had promised 2 Nobel Prize winners but the second one has gone overseas and is not available any more to share his ideas with us.

I do note the absence of female scientists in my line up. I did contact several but was not successful in getting any to write for the project which is disappointing.

I did also contact several Fields Medal winners. The Fields Medal is like the Mathematics equivalent of a Nobel Prize (there is no Nobel Prize for Maths - it's a long story). Sadly, I could not convince any of them to write for the project.

Highlights

Here are a few comments that stood out to me from the interviews:

Mathematics is simply the language of science.
- Andrew Knoll

I believe we need to rethink how we teach math and not use it to separate students.
- Don Kinard

Learning more maths, even if you don't know exactly how you'll use it, will always turn out to be helpful.
- Jack Szostak

I, most scientists, and all engineers, use math all the time. The more you know, the more you can do.
- George Whitesides 

I hope that you too found some gems to think about.




Once again, thank you to the scientists who participated in the Maths in Science project. It has been an interesting opportunity for us all to hear how these two fields overlap and interact.



 



  



 

Maths in Science - Andrew Knoll

Professor Andrew Knoll is one of the world's leading paleontologists and works at Harvard University. He has numerous awards and achievements to his credit including Paleontological Society Medal, the Wollaston Medal of the Geological Society (London), the Moore Medal of the Society for Sedimentary Geology, the Oparin Medal of the International Society for the Study of the Origin of Life, and both the Walcott and Thompson medals of the US National Academy of Sciences.  

Professor Knoll has done extensive research on the early life on earth, specifically the first 3 billion years, which is the focus of one of his more significant books. 

He was also a member of the NASA Mars Rover project team, using his skills and knowledge of early evolutionary life to support this significant endeavour.

Professor Knoll kindly gave some time to answer a few questions about is use and application of maths in his work.

Here is what he had to say:

The Questions:   

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

I grew up in US schools much influenced by Sputnik.  The “new math” taught in primary schools was heavy on Venn diagrams, number sets and the like.  Thankfully, we still learned basic arithmetic and, as we got older, algebra and geometry.

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

Beyond a basic understanding of numbers and what they mean, the maths I learned in school provided a framework for the calculus and statistics I learned later at university.  And THOSE I use every day.

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

One of the great skills that many people (including scientists) need to learn is how to approximate an answer.  To make a quick evaluation of a problem, you commonly don’t need to calculate the precise answer, but rather need to know whether the answer is close to ten, close to one hundred, or close to one thousand.  Thus, quick, approximate mental arithmetic is truly useful.

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

In a sense, yes.  Commonly, I  would like to know the relationship between two sets of data, say, body length and weight.   If I plot length on the X axis and weight on the Y axis of a graph, the spread of points on the graph really does make a “new thing.”  For example, in the case mentioned, it may tell us about how the organisms in question grow.

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

I do research in geology and paleontology and I spend my life looking for patterns.   Mathematical pattern may show how the chemistry of sediments changes along a depth gradient, or how biological diversity increased through time.

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

Balance and equality are important  -- I’m not sure how I would apply either to my research.

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

All the time.  We describe fossils, using numbers to describe length, width, and many other features.,  And we characterize the chemistry of ancient sedimentary rocks, again using numbers of indicate how much carbon, or calcium, or iron is present in a sample.

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

As I noted earlier, estimation is a useful skill.  But sometimes I really need make precise measurements or do exact calculations.  In a sense, estimation is the quick and dirty, whereas precise measurements and calculations enable the final product.

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

Again, nearly every day.  As a paleontologist, I commonly need to know whether two fossil populations are different or not, and statistics allows me to draw this kind of conclusion.

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

Scientists need maths to do research, whether that research involves measurement, comparison of different types of data, or modeling.  Mathematics is simply the language of science.

What a great line to end on - "Mathematics is simply the language of science."

Thanks you so much Professor Knoll for your support of the Maths in Science project.