Education rankings

This CBC feature, though focussed primarily on Manitoba’s declining PISA scores, reveals some interesting statistics about each province (dropout rates, expenditures per student, student teacher ratios, etc.). In the Manitoba rankings graph, the blue and yellow lines overlap after 2006; that is Manitoba went from fifth place in math in 2006, when compared with other Canadian provinces, to 9th place in 2009.

A world without numbers

Since we have been discussing the relevance of mathematics, I thought I’d post a “relevant” video.  I saw it for the first time during an excellent public lecture last week.  It is appropriate for all ages.

Who needs algebra?

Why should kids learn algebra, trigonometry, geometry when they may never actively use these concepts in their adult lives? For that matter, why study Shakespeare, history, music, languages, psychology? Certainly, there are many things I studied in school, and in university, that I have never explicitly used in my daily life. Does this mean that all of these learning experiences were a complete waste of my time?

Certainly, if taught properly, algebra and trigonometry help to develop abstract thought, critical thinking and problem-solving skills. Beyond that, these are beautiful developments in mathematics, handed down to us by our ancestors. These, like the great works in literature, music and art are remarkable accomplishments of the human race.

It is unfortunate that mathematics must constantly be pushed as a contextual, hands-on subject. Yes, it is extremely useful. Without mathematics, we would not have many of the modern luxuries to which we are accustomed. The fact that mathematics is a beautiful, elegant and mysterious art is something that is often missed, or perhaps not even realized.

As an illustration, consider one of my favourite irrational numbers.   Find a few circles of varying sizes. Measure the circumference of each circle, and the diameter of each circle using string and a ruler. Divide the circumference by the diameter (using, umm, long division or – gasp – a calculator). What you will find is that, for your experimental examples, the result seems to be the same, regardless of the size of the circle.   It is a fact that this ratio is the same (or constant) whether you consider a tiny circle, like the circles in the left hand side of a notebook, or a gigantic circle, like the earth’s equator (which is approximately circular). We call this ratio pi. The Ancient Egyptians and Mesopotamians essentially knew that this ratio was constant, although they did not prove it; the proof of this theorem is due to Archimedes. This really is amazing! Is pi useful? Absolutely! We need an approximation of pi to determine the area of any circle, the volume of a sphere or cylinder, among many other things.

Why must questions always take this form: “What is this good for?”, “How will I use this in my life?”, “What’s the point?”?  In my opinion, these questions are often imposed on children by educators and other adults.  Children then become accustomed to asking these questions and, if they do not receive satisfactory answers, they become frustrated and discouraged with mathematics.  We do not hear these types questions in regards to music, art or literature.

With my own students, I certainly discuss applications of calculus and linear algebra (there are many), but I try to instil an appreciation of the subject in students that goes beyond this constant push for practicality.  Students can enjoy doing mathematics for the sake of doing mathematics, just like they can enjoy doing crossword puzzles, playing the piano or reading.  Instilling an appreciation of the intrinsic beauty of mathematics, for its own sake, is just as likely (if not more likely) to inspire a lifelong love of mathematics in students as illustrating its practical applications.  In turn, a student who has a strong appreciation of mathematics will be much more likely to continue studying the subject and later apply it in his or her career of choice.

Knowledge and skills are important!

Parents often tell me that it seems as though their kids are not learning much mathematics at school.  Their kids have limited knowledge when it comes to mathematics.   It is often claimed that, while it may seem that children have few skills and limited knowledge, they are nonetheless developing a deep understanding of math at school.

Who could argue against understanding?  Certainly not me.  I am a mathematician, after all, and understanding mathematics is the bloodline of my profession.   However, should we be content with the claim that our children are developing a deep understanding of mathematics even though we can clearly see that they are lacking knowledge and skills?

My husband recently wrote a short note regarding this claim and I’ve included a slightly edited version of it below.

Developing competency in mathematics involves developing an understanding of the concepts involved but it also involves mastering the skills involved.  It is not possible to develop proper understanding of mathematics without developing a mastery of basic skills.  As with sports or music, this will not be achieved without practice.  Furthermore, it is not sufficient to merely be able to eventually demonstrate the needed skills: Taking two minutes to skate from the goal line to centre ice is not sufficient for a ten year old hockey player just as taking 30 seconds in the fourth grade to determine the product of 7 and 9 is not sufficient.  Would anyone find it acceptable for a grade three student to take more than 3 seconds to read the word “dangerous“? It should similarly be unacceptable for a grade three student to take longer than 3 seconds to provide the exact value of 7 times 8.

Knowledge and understanding go hand-in-hand: Just as it is difficult to walk with only one leg, it is difficult to do mathematics without the pairing of some basic skills/knowledge with understanding.

Why must parents constantly be lectured on the virtues of (conceptual) understanding?  I suspect that the explanation is that we, the parents, are being told that although our children may not appear to “know” very much math they are nevertheless developing a deep understanding of various concepts in math.  It is certainly true that while it is very easy to identify when an individual knows something, it can be a trickier matter to identify deeper levels of understanding. However, in mathematics, it is very difficult to understand something well without some basic knowledge. Hence, if someone is lacking basic knowledge, they almost certainly also lack understanding. Therefore, parents should not accept “understanding” over “skills” arguments and parents whose children cannot multiply (for example) with speed and efficiency should be similarly concerned about their children’s level of mathematical understanding.

It is also often argued that the skills needed by students change over time.  However, the topics in  mathematics needed by today’s children are, essentially, exactly the same as they were when our mothers and fathers were children. Moreover, I don’t believe that the human brain has evolved significantly for the worse over the last fifty years, so these children are equally as capable of learning this math as we (and our mothers and fathers) were.

I, like many other mathematicians, have given much of my life to understanding, creating, and teaching mathematics. It is a beautiful art and a powerful tool.  It is interesting that some people feel that one must choose between understanding and knowledge; perhaps it is human nature to attempt to create dichotomies even when none exist.  I personally am always wary of extreme views based on ideology and strive for harmony and balance in my own teaching.

STEM Kids

I just watched this excellent TED talk. The speaker does an incredible job of outlining some of the difficulties, and unintended consequences, of the pendulum swings in math education. Having two children in school and witnessing the low mathematical expectations and the constant group work which are a consequence of some of the current teaching philosophies, much of this really hits home for me. Though somewhat slow-going at first, it’s worth watching the entire 25 minutes.

Helping your kids with math

Parents often ask me what they can do to help their kids with math. A few helpful resources are listed below.

Grades K-8: 

JUMP Math is a charity that was founded by Canadian mathematician John Mighton. You can order thorough workbooks for your child’s grade level at their website.   I use this program with my own kids and with 11 of my daughter’s friends and I highly recommend it. Additionally, JUMP Math publishes a series that was designed specifically for parents to help their children with math titled JUMP at Home .

Singapore Math is also an excellent program. I absolutely love the word problems, which become incrementally more difficult as kids progress through the levels. (Kids like them too!) Some of the Singapore Math workbooks can be ordered through Chapters or Amazon. The more thorough textbooks and accompanying workbooks can be ordered from Heritage Resources.

Free math instructional videos: You can find instructional videos on most K-12 math topics at Khan Academy.

Grades 3 – 12 enrichment: The Math Kangaroo Contest is a math contest for kids. The sample tests from previous years have lots of fun problems and are an excellent source of enrichment. They can be accessed here.

Grades 6 – 12 enrichment: You can find math books for kids, which include a variety of problems and math contest problems, as well as various enrichment materials at The Centre for Education in Mathematics Computing. Past math contests for grades 7-12 are available FOR FREE at this website.

Extra practice: If your child needs extra practice with specific skills, Society for Quality Education provides free math practice worksheets at Society for Quality Education (click on remedial programs). You can also find free math worksheets at Math-drills.com.

An unsolved math problem for kids

My husband (another math professor) and I run a Math Club for some kids in Grade 4.  Yesterday we had our Math Club Christmas party.  Before celebrating with movies and pizza, we showed the kids an unsolved math problem.  This is an interesting problem,  not just for kids in grade 4, but for older kids and adults.  I thought I’d post it here, in case there are some parents or teachers reading who might like to use this lesson.  Here is what your kids should know to understand the problem:

1.  What even and odd numbers are (this can easily be taught).

2.  How to divide even counting numbers by 2.

3.  How to multiply counting numbers by 3.  (Yes, we taught the kids the vertical algorithm for multiplication – with understanding.)

4.  Some concept of the fact that there are infinitely many counting numbers.  (How do we know this???  Well, suppose there are not infinitely many counting numbers. Then there must be a largest counting number, right?  Now take that largest counting number and add 1.  This gives a counting number larger than the largest counting number which is impossible!!  So…there cannot be a largest counting number and so there must be infinitely many counting numbers.)

I came across this problem in Professor Stewart’s Hoard of Mathematical Treasures, by Ian Stewart, and I wrote up a lesson plan for the Math Club kids which incorporates this problem.  The problem is called “The Collatz-Syracuse-Ulam Problem”.

This is meant to be done with an adult’s guidance and pictures should be drawn when introducing even/odd numbers.  Also, the adult should do many examples for the children.

Here is the lesson plan.

Now I am going to take a break from mathematics until January.  Happy holidays everyone!!

Britain bans calculators in primary school

I had to take a minute to post a link to the article Calculator ban on on young pupils, which appeared in the London Evening Standard, December 1. It seems that Britain is facing a nation-wide crisis in basic math skills.

Almost 17 million adults have the maths skills of a nine-year-old…are so bad with numbers they cannot pay household bills or understand price labels.

At least two million have the maths and literacy skills of a five-year-old. According to the Skills for Life survey, which questioned 7,000 adults, the problem is getting worse.

From Schools Minister Mr. Gibb:

Without a solid grounding in arithmetic and early maths in primary school, children go on to struggle with basic maths skills throughout their school careers.

Exactly, Mr. Gibb.

Children can become too dependent on calculators if they use them at too young an age. They shouldn’t be reaching for a gadget every time they need to do a simple sum.

Congratulations, Britain! (On the calculator ban, that is.)

Manitoba’s failing grade

Unfortunately I do not have a lot of time to post at the moment.  Below is a link to a radio interview I did on CBC Information Radio on Tuesday where I discussed some of the issues in math education.

 Manitoba Students Fail At Reading, Writing and Arithmetic

Introducing WISE Math

Please visit WISE Math (Western Initiative for Strengthening Math Education) to learn about a newly formed initiative, established to work towards improving math education for children in western Canada.

WISE Math was set up by Fernando Szechtman, Robert Craigen and Anna Stokke, who are math professors at the University of Regina, University of Manitoba and University of Winnipeg. The mission statement and some key objectives can be seen on the website. Parents, teachers, educators and citizens who care about these issues can lend their support by signing their name under “Join WISE Math” on the WISE Math website.

Please forward the URL, wisemath.org, to everyone you know – colleagues, parents and friends – and encourage them to join. (They will not be put to work – they need only sign their name.) Post the link on your facebook page and help to spread the word in any way you can.