Math Education

I am sure that you have all been waiting for me to land on this debate. Quite simply, educators have been experimenting with a range of protocols for teaching all subjects for the past fifty years with the enthusiastic support of text book publishers. In many cases the damage inflicted by poorly considered ideas has been limited and contained sufficiently to allow the students to progress in some form or the other.

Let me explain something else. Starting when I was nine years old, I read everything I could get my hands on until I met an actual university library and met my match. I have read hundreds of books on historical, political, sociological, anthropological, biological, agricultural, medical, physiological, geological, geographical, and astrophysical topics as well as a novel or two covering the best of classical literature for relaxation. That is what I never wrote an exam on ever and believe me I have missed a few topics. What that means is that if you hacked your way through a history degree while reading only the required material, I am hundreds of books ahead of you and a quick study on most other subjects as well.

What I did do was master Mathematics simply because it opened all plausible doors to all knowledge generally. In practice, science is a collection of data described normally with a mathematical model dumbed down enough to generate useful results within narrow bounds. Making the mathematics smarter does not necessarily improve the data, after all.

The purpose of education is to train the brain. Improved recall is one such area, but I am proof that it is barely important except to pass standard exams. Way more important is to learn and internalize working algorithms that allow you to interpret the world. That I did mostly and best through mathematics and how it was taught.

All algorithms related to arithmetic need to be first internalized, practiced to establish correct output and then be challenged to reach a high level of speed before you go forward. You must become fast with the times table and be forced to calculate in your head. Internalized properly, that is just what will happen. Better you will learn and even invent shortcuts if pressed in this way.

Now you are ready for real mathematics. This should start with a one term course in Euclid’s Elements in grade ten because it is the backbone of proof theory. Again promoting speed is wise. This is not memorization so much as reaching for and assembling the logic internally.

After that absorb all the mathematics that is thrown at you. I made my rebellious son take Grade 12 Calculus in Grade 11, not because he was ready for it but because he was not ready for it. When it came to the crunch three months in, I walked him through a demonstration proof of the differential for some convenient function. When he saw the two sides of the proof magically match up for QED, the light came on as it internalized and in two weeks he pounded through the problem sets and passed the course. He never lost the confidence boost that gave him afterwards. Confronted with a missed last class problem on a fourth year final exam, he recalled his skill, structured the most likely configuration of the necessary equations and aced it.

This is what learning mathematics is about. It is internalizing the body of work in front of you. You are training for a sports event and simple memory games will not bail you out.

Not everyone can get good at it, but everyone can become progressively better.

All education needs to be goal oriented and mathematics doubly so. I also think that high school graduation must include a sound preparation in mathematics and at least one core science at minimum. That mathematics should really include introductory calculus. It is the best true measure of a graduate’s actual mental ability.

Frustrated professors convince elementary schools to step back from ‘new math’ and go ‘back to basics’

Moira McDonald,  | 13/09/13 

http://news.nationalpost.com/2013/09/13/frustrated-professors-convince-schools-to-step-back-from-new-math-and-go-back-to-basics/

University of Winnipeg math professor Anna Stokke and two of her colleagues knew there was “a huge problem,” when they started hearing about Manitoba grade school students not being taught how to do vertical addition, carry or borrow numbers, or knowing their times tables.

Then, two years ago, she and Robert Craigen, a fellow U of W professor, and Fernando Szechtman a math professor at the University of Regina, formed WISE Math — the Western Initiative for Strengthening Math Education. They set up a website with a blog, gave lots of media interviews and started meeting with government officials to push for changes in the way math was being taught.

“Then we started hearing from a lot of parents, from all across Canada,” said Ms. Stokke, whose group has collected nearly 1,000 signatures supporting its calls for reform. “It’s a lot of work and it’s a lot of trouble to advocate for things like this … but our kids are worth it, because in the end we really need our kids to learn math.”

The group is now seeing the fruit of its efforts this fall, as Manitoba rolls out a “back to basics,” revised curriculum for kindergarten to Grade 8, one explicitly requiring students to learn times tables; have automatic recall of answers to basic problems such as 30 – 5 = 25, known as math “facts”; and standard algorithms for key math operations — and perform them without using a calculator.

It marks a step back from “new math” and “inquiry-based” teaching approaches that emphasize such things as estimating and multiple “strategies” in basic calculations — complicated methods of solving math problems in a bid to develop students’ deeper understanding of how those calculations work. Such approaches are common across Canada and are part of the Western and Northern Canadian Protocol (WNCP), a common framework, initiated in 1995 and revised in 2006, used to develop curriculum in all western provinces, Canada’s three territories, as well as in Atlantic provinces including Newfoundland, Nova Scotia and Prince Edward Island.

“It’s not perfect, but it’s definitely a step in the right direction,” said Ms. Stokke, who thinks this makes Manitoba the first province to “walk away from WNCP a bit.”

While Manitoba continues to be a part of WNCP, its education minister, Nancy Allan, credits WISE Math’s efforts for helping to drive the change.

“We were hearing concerns from parents and we were hearing concerns from some math professionals,” said Ms. Allan, who called her province “a leader” in math reform. Besides worrying about students not learning basic math skills, parents trying to pitch in with their children’s homework, “were having difficulty helping their young people because they weren’t able to understand it either.”

Although standard algorithms “have been used in the past” by teachers, the revision explicitly states they must be taught, said Blaine Aston, vice-principal and numeracy specialist at Brandon’s Ecole New Era School.

“The clarity in what [teachers] are supposed to teach in each grade level in terms of math facts is a positive step,” said Mr. Aston.

Winnipeg parent Laura Lamont says the changes are “a huge relief,” especially after watching teachers get uncomfortable when asked why they couldn’t teach students how to add numbers in vertical columns instead of horizontally.

“The kids are bright and the teachers are dedicated, but it felt like everybody had their hands tied behind their back,” said the mother of twin nine-year-old boys who took matters into her own hands last year when she enrolled them in Archimedes Math Schools, a non-profit after-school math program developed by Ms. Stokke. The program itself dates back to the professor’s efforts to give informal remedial math help to her own daughter and some of her friends.

But this is no wholesale dumping of new math teaching. Teaching students multiple strategies in problem-solving will still be part of the mix, but the government says it is now striking “the appropriate balance,” between students’ basic math skills, conceptual understanding and problem-solving ability. The province also plans to create a mathematics education advisory committee, update high school math courses and work with university faculties of education to improve teacher training in math.

“We’re not going back to ‘kill and drill,’ that’s not what we’re trying to accomplish here,” said Ms. Allan. “But there has to be a basic foundation in regards to adding, and subtracting, and memorizing math facts [and] knowing how to do math at an early age.”

That leaves Sherry Mantyka skeptical. Although the math professor at Newfoundland’s Memorial University said she would welcome a true back-to-basics approach in her own province’s schools, an announcement touting just that in 2008 only led to Newfoundland’s adoption of WNCP.  As a result, she continues to work with hundreds of university students in remedial math programs every semester. The director of Memorial’s Mathematics Learning Centre estimates about a 20% failure rate on the university’s math placement test, required for every student wanting to take at least one math course.

“They do not know sums up to 20. They do not know multiplication products up to 100. … A question like nine into 83,209, they’ll try to do with repeated subtractions,” she said. One problem, she said, is that students who resort to using the complicated “strategies” they’ve been taught in grade school, even for simple math sums, use up their working memory and are then helpless to solve more complicated calculations.

She called the changes in Manitoba “not a bad thing, but is it going to fix the problem? I doubt it.”

If we focus on memorization, we’re not going to get there

Even calling them “back to basics” is “inaccurate,” argued education professor Ralph Mason, who participated in development of the revisions. But that’s a good thing, said the University of Manitoba specialist in math education.

“The whole idea of rolling these things back to a time when everyone learned these basic facts didn’t exist,” said Mr. Mason. Previous methods that have focused on memorization and rote performance are “strategies we know never worked,” and left some students struggling.

But Ms. Stokke complains that “there’s always this false dichotomy that gets set up where they say, ‘We want kids to learn with understanding and you want skills.’ Well, that’s ridiculous. They should have both. You don’t start neglecting one side of it in favour of the other.”

Alberta has had a key hand in developing WNCP and uses it as a framework for its own curriculum. Christine Henzel, director of mathematics, arts and communication for Alberta’s education department, and who has worked on the development of WNCP in the past, said she could not speak to whether Manitoba’s changes signify a rejection of any aspect of WNCP, but said it is expected that all provinces using it will adapt it to their local needs.

WNCP is based on research, she said, and is aimed at providing students with real world math skills so that they understand how and when to apply the math facts they know.

“If we focus on memorization, we’re not going to get there,” she said.

Another question arises, if the WNCP is so bad, how come Alberta, which uses it, remains a Canadian leader when it comes to international testing?

Ms. Stokke and, her fellow U of W professor, Mr. Craigen argue that in fact even Alberta and other provinces such as British Columbia have seen their math achievement drop in outside assessments since WNCP came on the scene.

“Every jurisdiction under WNCP has shown steadily decreasing assessment outcomes since the introduction of the WNCP curriculum,” said Mr. Craigen.

Meanwhile, there are signs parents outside of Manitoba are also searching for more help with building their children’s essential math skills, says Doretta Wilson, executive director for the Ontario-based Society for Quality Education, an advocacy group pushing for more back-to-basics approaches. The province’s student testing agency recently reported a five-year decline in Grade 3 and 6 students’ math skills and a growing number of students seeing a drop in their achievement between Grade 3 and Grade 6.

“I can tell you through our own website, our math worksheets are our highest in-demand resource,” said Ms. Wilson. “And it doesn’t look like the tutoring centres are going to go out of business any time soon in Ontario.”