Republic of Mathematics blog

Should we get rid of the “=” sign in mathematics?

Gary Ernest Davis — Wed, 16 May 2012 14:27:07 +0000

Should we get rid of the “=” sign in mathematics?

Robert Recorde

Probably, since it creates more confusion than it alleviates.

Robert Recorde had a lovely idea in 1557 when he introduced the “=” sign, writing “… bicause noe 2 thynges can be moare equalle.”

x
Studies have demonstrated that students have 3 problems with equality, as in .

These difficulties are:

Reflexivity: the statement that x=x for all x. Students often see this as wrong because to them it’s nonsensical: the “=” sign is an instruction to do something. What is “2=2″ asking us to do?

Symmetry: the statement that if x=y then y=x. To many students this is simply wrong. For example, 2+3=5 is correct because we did something to 2 and 3 to get 5, but 5=2+3 is wrong because it’s meaningless to them.

Transitivity: if x=y and y=z then x=z. A statement such as and so while logically correct is also seen as meaningless by many students because it does not fully involve them using “=” as a production rule – an instruction to do something.

Of course (apart from syntactic identity, such as x-1=x-1 but not x-1=-1+x, which occurs rarely) these are the only properties possessed by equality!

Another common misuse of “=” as a production rule consists of chains of equal signs between propositions, as in (x+y=2 and x-y=4) = (x+y=2 and 2x=6) = (x+y=2 and x=3) = (x=3 and y=-1)

It’s clear that some people do not see there’s a problem:

MathnasiumMiddletown ‏@MiddletownMath It’s a challenge in itself to instill a basic appreciation of #Math in younger students. Messing with = won’t change that.
MathnasiumMiddletown ‏@MiddletownMath We acknowledge the study, we just don’t see THAT as a serious problem in day-to-day interaction with kids and #Math!

For a different point of view – and one to which I subscribe – see the video below:

Programming languages use equality differently

For example, in the high level language Mathematica, (where variable types do not have to be defined), the statement

x=3

means something quite different to what it normally means in mathematics.

It means to establish a place in memory called “x” and to place “3″ into that place in memory.

The data analysis language R writes this as

x <- 3

which might, at first, take some getting used to, but is really much more explanatory: put “3″ into the place in memory called “x”.

If we give Mathematica the command “x=3″ it stores the value “3″ in a place in memory called “x”. So if we now want to check whether x is equal to 7-4, we write:

x == 7-4

The program gives the result “True”.

That is because it did two things: it performed the calculation 7-4 to get a result of 3, then looked to see if the value 3 was stored in the space in memory called “x”.

If we gave the software the command

x-1 == 5+2

it responds with”False” because it looks in memory to find “3″ in the place called “x”, performs a subtraction, performs and addition, and checks whether the result is the same in both cases.

Mathematica, like many programming languages, also has a form of equality written “===” and called “same as” or “identical”. This refers to exact syntactic identity.

So, for example, when we give Mathematica the instruction

4 === 6-2

we get a result of “False”. That is because the left hand side “4″ is not exactly identical in its written form with the left hand side “6-2″.

Similarly, the command

x-1 === -1+x

yields “False”

Why would we ever want such a pedantic meaning for exact literal equality?

On example is when we are performing a large number of calculations, some of which result in “ComplexInfinity” as an answer. If we want to delete these cases from our list of answers we cannot simply ask the software to find the cases x for which “x == ComplexInfinity”. That is because ComplexInfinity” is not a value against which the memory in x can be tested for equality. So we look, instead for cases x for which “x === ComplexInfnity”. The program looks in memory at x to see if there is a literal sting that reads “ComplexInfinity”. If there is, it deletes that string (provide you asked it to do that).

The point is that “==” allows for semantic interpretations, unlike “===” which demands exact literal syntactic equality.

I have on many occasions “corrected” students who wrote “4-2 -> 2″, but now I’m not so sure.

As a production rule sign “->” is not too bad. It has an arrow of direction, indicating that the left hand side was to be interpreted through an arithmetic procedure and the right hand side asserts this procedure led to the number 2.

This is not a bad thing, and in light of the many confusions in mathematics about “=”, and the more precise use of “=” in computer science, maybe it’s time to ditch Robert Recorde’s ingenious notation? At the very least, we should be more careful about its use.

Children aren’t born knowing about “=” and its use in mathematics. They, if not their teachers, could get used to a more careful use of equality.

Spreadsheets and big data

Gary Ernest Davis — Tue, 08 May 2012 12:11:47 +0000

Many people use spreadsheets for calculation and for storing data.

The tabular format of spreadsheets, the ability to use formulas, to search the cells, to plot charts, and to change parameters and have plots redraw are compelling features of spreadsheets and have embedded them into popular use.

Spreadsheets are widely use for storing and transmitting data: the tabular layout that allows for sorting by columns is very appealing to anyone in an organization who collects and needs to disseminate data.

The widely used data analysis and statical software R imports files directly from most spreadsheet formats, so it is very tempting for students of statistics and data analysis to store their data in a spreadsheet. For teaching purposes this does no apparent harm in the short term. However, longer term, the habit of using spreadsheets to store and disseminate data can be very problematic.

Despite the many rows and columns, a spreadsheet can effectively manipulate only a limited amount of data.

Excel 2007 has 17,179,869,184 cells. If each of these cells were filled with data, that would seem to be a large amount of data by anyone’s standards. Imagine that in each of these cells there was only a 0 or a 1. A terabyte of data is 1,000,000,000,000 (a trillion) bytes, so an Excel 2007 spreadsheet filled with o’s and 1′s would hold only about 1.7% of a terabyte of information: it would take about 58 such filled Excel spreadsheets to get a single terabyte of data.

A petabyte of data is 1,000 terabytes. To get this much data from spreadsheets filled with 0′s and 1′s we would need about 58,000 spreadsheets.

So if every single person in the town of Great Yarmouth in the UK had an Excel 2007 spreadsheet filled with 0′s and 1′s we would have about a petabyte of data.

Surely no-one could regularly want to deal with that much data?

But that is just what Big Data sets (and extremely large data sets) contain. In fields such as genomics, meteorology, internet searching, and finance informatics, petabytes of data are routine.

In fact exabytes of data are not uncommon: an exabyte is 1,000,000 terabytes - about the equivalent of every single person in a country such Italy as having an Excel 2007 spreadsheet full of data.

But wait, you say: a person in a medium size business, producing a list of employees and job descriptions, for example, doesn’t have to worry about exabytes of data. Surely they can keep on using a spreadsheet to store their data?

The answer is: of course they can and of course they will. Spreadsheets are simply too useful in everyday life to abandon.

Now we have a problem when we want to amalgamate, or consolidate, the data from many, many thousands of spreadsheets.

How do we handle such data, how do we ensure its integrity and fidelity, how and where do we store it, and how do we analyze it?

One suggestion is to store spreadsheet data in a large spreadsheet format in the cloud that is scalable to handle big data sets. Another is to develop a spreadsheet search engine that could extract semantic information from large collections of spreadsheets.

Spreadsheets are probably not going way anytime soon, because of their useful features for handling small scale data. Yet demands of Big Data steer us to thinking of effective ways of managing the accumulation and consolidation of manifold spreadsheet data sets.

Reference

Jacek Becla1, Daniel Liwei Wang, Kian-Tat Lim, REPORT FROM THE 5th WORKSHOP ON EXTREMELY LARGE DATABASES, Data Science Journal, Volume 11, 23 March 2012 [ Becla_et-al ]

Is this the sexiest mathematics job ever?

Gary Ernest Davis — Mon, 07 May 2012 00:08:25 +0000

Alexis Wajsbrot

Alexis Wajsbrot is a technical film director specializing in the simulation of movement of fluids and textiles.

Sound sexy so far?

Well, here’s some of the projects Alexis has worked on:

Here’s a Vimeo link to some of Alexis’ film effects.

Alexis approximates a moving fluid or textile by a large number of particles in a 3-dimensional co-ordinate system.

The software he uses allows him to choose how the particles move in space – controlling their speed and acceleration.

Another way Alexis simulates a fluid is to dissect the region of the fluid into a 3-dimensional grid of voxels (from volume & pixel). He then uses the mathematics of fluid dynamics to calculate the velocity in each voxel at each time step, to create a realistic fluid motion.

More details of Alexis’ work and Quicktime movies of his simulations can be found at +plus magazine, from which this information was taken.

Finally, here’s Alexis describing some of his work on Red Balloon (en Francais):

So you want to be a data scientist?

Gary Ernest Davis — Sun, 06 May 2012 21:29:01 +0000

Well, listen up.

Here are some a well known data scientists:

What do you need to know, and know how to do to be a data scientist, and hang with these cool folks?

First, you need to be able to hack and scrub data – lots and lots of data, usually messy. To do this you’ll need to be familiar with a language such as Perl or Python, and keep an eye on the Julia language.

You should know how to work in the command line, in a Unix environment, to interact with APIs.

You need to know how to do a decent statistical analysis of data (again lots and lots of it). You should at least know how to carry out an exploratory data analysis, a regression (maybe even a loess regression) and design an experiment to test a hypothesis.

As part of your statistical background you should be fluent in R, and be up to speed with Python pandas.

You should know how to design and test algorithms, and be familiar with data mining and machine learning.

Database programming, of the SQL variety, should be your bread and butter.

Then you need to be very familiar with techniques of data visualization.

It would help if you knew how to carry out a simulation, and also knew something about Hadoop and Map Reduce, or – nowadays, High Performance Computer Cluster management.

If all that isn’t enough you need to be able to communicate a story really well.

If you lack some or all of these skills, you need to get up to speed by yourself or find someone, somewhere, to teach you.

This sounds like a lot, and it is – yet the work of a data scientist, is so interesting, so rewarding, and so important, that you’ll figure out how to do it.

Here’s links to the data scientists, above:

Why is -3+5=5-3?

Gary Ernest Davis — Sun, 06 May 2012 18:32:30 +0000

@MrMathsTeacher tweeted (5/6/2012) “Is 5x-3=7 easy, but -3+5x=7 hard due to the way neg numbers have been taught or because of algebra difficulties, or something else?”

@ColleenYoung responded: ”I think so many are not really familiar with the fact that say -3+5 is the same as 5-3. #mathchat“

A lot of the difficulty for kids in learning mathematics, is knowing what models to use to think about things.

For example, in thinking about functions it helps to have a model of a function as a machine with an input and an output:

This may not be strictly correct from a rigorous point of view, yet it serves as a model that is accurate, that can be later modified as students require more rigor, and does not need to be thrown overboard to do that. It is an instance of a cognitive root.

A useful cognitive root for signed numbers, such as -3 and 5, is as instructions to move from a given place on the number line.

For example, if we imagine ourselves as being at 0 on the number line, then the number -2 instructs us to move 2 places to the left:

If we are at the point 1 on the number line the number -2 still instructs us to move 2 places to the left – it just lands as at a different spot:

Similarly, a positive number such as 4 tells us to move 4 places to the right from where we are:

Now a number such as 3 has an ambiguity of interpretation: does it mean the place 3 to the right of 0 on the number line, or does it mean to move 3 places right from where we are?

The answer, of course, depends on the context. For a number to mean a place on the number line we move that many spots from 0 (left for negative numbers, right for positive numbers). Otherwise, we keep the interpretation of a number open, as meaning move to the left or right of where we currently are on the number line.

Now with addition interpreted as following on (concatenation) we have our cognitive root of signed number addition:

-3 + 5 means move 3 places to the left of where we are and then 5 places to the right. This lands us 2 places to the right of where we started.

On the other hand, 5-3 means move 5 places to the right of where we are and then move 3 places to the left. This again lands us 2 places to the right of where we started.

In both cases, if we started at 0, then we would end at 2.

So -3+5= 2 = 5-3

At least, that’s one way to think about it.

What can you do with math (except teach)?

Gary Ernest Davis — Sat, 05 May 2012 18:03:31 +0000

That’s a good question.

Let’s see – here’s what a few folks have done:

Jim McElwaine

Avalanche

Jim McElwaine is – wait for it – an avalanche researcher.

No kidding, Jim actually researches the mathematics and physics of avalanches.

He was one of a team that investigated the 1999 avalanche that killed 12 people in the Swiss alps. You can read about Jim’s career here.

Sandy Black

Tranquil vale

Sandy Black is a fashion designer.

Sandy works with textiles and knitwear, where she uses her mathematical skills.

She is Professor of Fashion and Textile Design at the London College of Fashion, and Director of the Centre for Fashion Science. You can read more about Sandy here.

Tanya Morton

Bloodhound

Tanya Morton is an application engineering manager at MathWorks, a company that produces MATLAB and Simulink software for scientific and engineering computing.

Her modeling applications include the effect of weather on demand for food in supermarkets, drug discovery and impact of drugs on the heart, modeling the speed of Formula 1 cars to improve lap times; and financial modeling to develop new economic indices. You can read more about Tanya here.

Matt Parker

Matt Parker is a mathematics communicator and stand-up mathematician. Matt’s job is to get people excited about mathematics.

He goes into schools to give mathematics talks and run workshops with students. He also runs training for teachers, university students and academics to do his job for him: to put himself out of work.

Matt writes and speaks about mathematics through radio and newspapers, and is also a stand-up comedian. You can read more about Matt and his activities here.

Virginia Pearcy

Virginia Pearcy is an attorney with the international law firm of Orrick, Herrington & Sutcliffe LLP.

She counsels clients on protection and utilization of intellectual property rights.

Virginia’s advice regarding mathematics is this: ”A mathematics education is a strong foundation for just about any professional career. It provides a challenging curriculum and helps one develop analytical reasoning skills that can be used in many different subject matter areas.” You can read more about Virginia’s law career here.

Bjorn Roche started his own music software company, XO Audio LLC in 2004. His XO Wave software is used for music recording, editing and mixing.

Bjorn says about his mathematical training: “I use my intuition about math, sound and computer programming to decide how to try tackling a problem. After drawing a few diagrams, and making notes, most of the work becomes programming work, and then listening carefully to make sure it sounds right.” You can read more about Bjorn here.

This is a small sample of the many and varied careers of people whose training was in mathematics.

You can read many more stories like these at the following links:

Enjoy! And remember – being a mathematics teacher can also be a great and rewarding career.

The Minds of Modern Mathematics – an iPad app

Gary Ernest Davis — Fri, 04 May 2012 14:47:50 +0000

“As teachers this is our paycheck. When we inspire children – that’s what we live for”

Gary Ernest Davis — Fri, 04 May 2012 12:29:51 +0000

Mathematics teacher Ben Johnson inspires his students to achieve

What’s up with pre-calculus?

Gary Ernest Davis — Wed, 02 May 2012 13:35:22 +0000

Pre-calculus is an odd mongrel of a course.

It’s name suggests it’s preparation for calculus.

The course content, commonly involving polynomials, exponential and logarithmic functions, trigonometry and circular functions, suggests the intent of pre-calculus is indeed preparation for calculus.

But there’s something rotten in the state of pre-calculus.

Several people responded recently to a tweet about pre-calculus apparently not working as calculus preparation for about 50% of students who pass it:

Julianna Stockton ‏ (@DrJStockton), April 29, 2012:

“what is the content of PreCalc? Ours (trig) more aptly named “NonCalc” than “Pre…”. Not setting up big ideas of Calculus”

Cody Coy Barlow ‏(@coach_barlow), April 28, 2012:

“math skills are watered down at the elementary school level. When kids can’t do fractions how do we teach calc?”

Jim Wolper ‏ (@DrATP), April 28, 2012:

“Many pre-calc students at my university are given undeserved high grades despite demonstrated unreadiness for Calculus”

ANOVA Learning ‏ (@anova_learning), April 28, 2012:

“more than likely, because they are inadequately prepared for pre-calculus!”

Gregory Cover ‏ (@gcmathfilm), April 28, 2012:

“precalc doesn’t do the job because basic skills are poor coming into the class. I spend half my time teaching pc reviewing”

A colleague checked on several hundred students who were placed by an Accuplacer test into college pre-calculus. Of those students who passed pre-calculus and went into Calculus I, only 50% obtained a grade of C or better.

This is like a coin toss: students place into pre-calculus, pass the course, yet 50% of them do poorly in Calculus I.

Can we say which students do poorly in Calculus I, and can we say why?

Can we figure out how to help them do better?

Is there something about the pre-calculus experience that is not adequately preparing about 50% of students for calculus?

Having discussed this with several other colleagues it seems to me that in pre-calculus classes we are facing a widespread lack of algebra skills, a lack of trigonometry skills, little to no time for preparation for ideas of calculus, a view of mathematics that is entirely procedural ( http://bit.ly/isyw5d ), poor study habits, likely failure to form study groups (http://bit.ly/IvhZ6k ) and a current lack of knowledge on our part as to who are the students that pass pre-calculus but get a D or less in calculus.

Another colleague, Jim Soden, adds:

“I tell my students that if they don’t earn in the mid 70s in Calc I, it doesn’t project well for their success in Calc II. And it doesn’t, by my observation. I am doing them no favors by skirting the issue, and the hard conversations have to be held. But the kids have to take Calc II so they press on. Many of these are committed students who do other things very well. Sometimes they get religion and do better in Calc II, but as a general rule, they struggle. The issue then, is what if anything we can do about the situation or is it, say, an Engineering Dept. decision? ”

Our observations on the parlious state of pre-calculus are not the first (and hopefully not the last):

“The failure rate in … precalculus courses can be so bad that as many as 50 percent of students need to take the class a second time. Ms. Thille and her colleagues hope to improve on that record while developing materials of such quality that they’re used by perhaps 100,000 students each year.” http://bit.ly/xPy7bI
“… data on UIC students confirmed that passing calculus was a major stumbling block for African-American and Hispanic students in entering into mathematics, science and engineering majors. For many UIC minority students with aspirations for a mathematics or science degree, the obstacle came even earlier, in pre-calculus courses. The staggering percentage of failing minority students exceeded 55% in pre-calculus courses.” http://bit.ly/IvhZ6k
“Success and Failures of a Precalculus Reform Project … College algebra is often taken as a terminal course or as a prerequisite for precalculus or business calculus. Failure rates in both courses are high, and neither course has lasting value for those students who pass, but choose not to take calculus. The size of the institution and the need for transferability of credits between campuses make it difficult for an individual instructor to depart radically from prescribed course content. This limitation led us to conclude that we could improve the courses most effectively by providing innovative approaches to traditional topics.” http://bit.ly/InKXL4

Postscript:

Comments seem currently to have a glitch so I’ve posted the following comment from @suburbanlion as a postscript to the post:

“In high school, I skipped Pre-Calc and went straight into Calculus. I think that doing so was actually beneficial for me in the long term. I had a lot to catch up on about trigonometry and logarithms, but learning them in the context of Calculus seemed to give me a different perspective on them. While most of my classmates would relate the trigonometric functions to right triangles, I would relate them to the unit circle or differential equations. Perhaps the problem with Pre-Calc is that it requires a rudimentary understanding of Calculus to begin with?”

5 great things about being a maths teacher

Gary Ernest Davis — Tue, 01 May 2012 16:17:18 +0000

This is a guest post written by Kimberley McCosh (@spyanki_apso on Twitter)

Kim McCosh

________________________________________________

5 great things about being a maths teacher

Kimberley McCosh

I love maths. I have had a few jobs before becoming a maths teacher but the urge to teach was always there. I am a self confessed maths geek and I love nothing more than converting some of my students to math lovers too! I teach 12 to 18 year olds in a secondary school in Scotland.

1. The interaction with pupils and knowing when you’ve really got through to them with maths. One particular highlight was when my class cut out triangles then stuck the angles from this down in a line to prove that the angles in a triangle sum to 180 degrees. The next day one of the boys (aprox 13 years old) was so eager to tell me that after the class he went home and searched the internet and found that all the angles did in fact always add up to 180 degrees. I know I had got through to him since he was choosing to look up maths in his own time.

2. Getting pupils interested in maths. I always try not to give “just a maths lesson” but also giving some background too. Ask any of my S3 class and they will be able to tell you more interesting facts about Pythagoras and his life than they can about the latest boy band! I always try to make my lessons interesting, different but still always relevant. When the pupils are interested, they are engaged and I have achieved my goal of sparking their interest in maths.

3. Helping pupils to think for themselves. Whether it be problem solving or applications of maths, whenever the pupils make the links for themselves it is always a real fantastic moment for me as a teacher. They have learned the building blocks and are piecing them together and starting to see the big picture.

4. The feeling of achievement when the penny drops and the class “get it”. It’s all in that moment when the pupils say “Ahhh! So that means…”. Or even better, when the pupil who has been struggling but working hard turns round and says “This is really easy!”. To know you have taught something which the pupils can now use in future years is what it’s all about.

5. Although not specific to maths, it is fabulous to make a difference in someone’s life. As a teacher you have daily interaction with pupils who may not always have the perfect home life but when they come into your class they are praised, encouraged, challenged and motivated to be the best they can be. To see a whole class strive to be the very best they can is the biggest reward you can ever receive.

I could go on – I just love my job! As a maths teacher you really make a difference. From teaching basic numeracy skills to complicated calculus, each lesson is important. I always try to remember that we are preparing pupils for jobs that haven’t been invented yet so who knows what level of maths they will require in later life. As a teacher, you can get an amazing high from something as simple as a pupil finally mastering percentages or cracking vector calculus. Each pupil, each class, and each lesson has highlights and I wouldn’t change my career for anything!