Those that can’t do, teach – or so goes the famous saying. But what of those who want to do teaching. What of those who do maths teaching? Can we be sure the job they are doing is the best one for our children, or the training they are getting as teachers is adequate? Sadly, we cannot.
We, the present authors (Jon, from Australia, and David, from the USA) are research mathematicians and computer scientists. We are also the proud fathers of seven adult daughters, and a gamut of grandchildren of whom the the oldest is starting school.
Together with our spouses, we have attended a multitude of PTA meetings, sports games, concerts and science fairs. We have read almost as many report cards (and not all of them have been glowing). But, at the end of the day, our daughters include PhDs, veterinary doctors, lawyers, teachers, web designers, postgraduate students and one senior undergraduate. We have also acquired four sons-in-law.
We have firm opinions, both as professionals and as parents. So what have we learned about teaching – and specifically about maths teaching?
This article was stimulated in part by a recent book on preparation of mathematics teachers for the classroom. While the book – Inequality for All: The Challenge of Unequal Opportunity in American Schools – deals with schools in North America, its message of uneven educational quality and uneven preparation rings true worldwide, albeit to differing degrees.
The authors of the book, William H. Schmidt and Curtis C. McKnight, approached the issue of teachers' knowledge of mathematics by asking a sample of 4,000 teachers in Michigan and Ohio the following question:
How well prepared academically do you feel you are – that is, you feel you have the necessary disciplinary coursework and understanding – to teach each of the following?
Teachers in primary school (grades 1-3) judged themselves to be well qualified only in mathematics topics they routinely taught their pupils. For even moderately more sophisticated topics, such as geometry, proportionality, and the beginnings of algebra, only 50% to 60% felt well-prepared.
What’s more, the coverage was surprisingly uneven. For basic geometry topics in one district, only a quarter of the teachers felt well-prepared, but in another 90% felt well-prepared.
In upper elementary school (grades 4-5), where topics such as decimals, percentages and geometry, variability across districts was even more pronounced. Only a quarter of the teachers in one district felt well-prepared to teach decimals, compared with virtually all teachers in another district.
In middle school (6-8 grade), the situation was even grimmer. The topics the authors chose (most of which are in the Michigan and Ohio standards for these grades) included negative numbers, rationals and reals, exponents, roots and radicals, elementary number theory, polygons and circles, congruences, proportionality, simple equations, linear equalities and inequalities. Here, only 50% of the teachers questioned felt well-prepared.
Fortunately, high school teachers are relatively better prepared, although there are concerns here too, particularly in more specialised areas such as 3-D geometry, logarithmic and trigonometric functions, probability and calculus.
Many US states are pressing to include probability and statistics in high school, yet less than half of the teachers surveyed regarded themselves as adequately prepared to teach the topic.
So why is teacher preparation lacking? The authors found that in grades 1-4, fewer than 10% of teachers have a major or minor degree in mathematics. This might be understandable, given the basic nature of the material. But this ratio remains even among 6th grade teachers! Even for 7th and 8th grade, only 35% to 40% had a major or minor in mathematics.
And in high school, only about half of 9th and 10th grade teachers had a specialisation in mathematics — only at 11th and 12th grade does the ratio rise to a more respectable 71%. Additional details about the Inequality for All study is given in this Scientific American blog.
Your present authors have been rather fortunate in living in school districts that, for the most part, offered relatively high-quality education, including high-quality mathematics education. But even here there have been lapses, and we question whether some of the material currently being taught is truly relevant in the 21st century economy.
From our experience, unevenness in quality is definitely an issue. One of us – Jon – lived briefly in a large US city, and was greatly disappointed in the quality and indeed safety of the state schools.
By comparison, the state schools he experienced in five Canadian cities were all of reliable if not always outstanding quality.
The other one of us – David – found it necessary to study rankings of schools, based for example on published SAT scores for California high schools, in considerable detail before choosing an area to select a home when he moved a few years ago.
Top California schools, such as Mission San Jose High in Fremont, Palo Alto High in Palo Alto, and Lynbrook High in San Jose, achieved far greater scores than schools at the other end of the list, such as Thomas Riley High in Los Angeles and Mandela High in Oakland. Few first world nations have such a wide disparity in educational quality.
Pedagogy and mathematics
It is undeniably important that mathematics teachers have mastered the topics they need to teach. The new Australian national curriculum is misguidedly increasing the amount of “statistics” of the school mathematics curriculum from less that 10% to as much as 40%. Many teachers are far from ready for the change.
But more often than not, the problem is not the mathematical expertise of the teachers. Pedagogical narrowness is a greater problem. Telling that there is a correct idea in a wrong solution to a problem on fractions requires unpacking of elementary concepts in a way that even an expert mathematician is not usually trained to do.
One of us – Jon – learned this only too well when he first taught future elementary school teachers their final university mathematics course.
Australian teachers at an elite private school could not understand one of Jon’s daughter’s Canadian long-division method nor her solution techniques for many advanced school topics. She got mediocre marks during the year because of this.
The school also scheduled advanced mathematics at 7:30am and 4:30pm. Despite, or perhaps because of this, she was the only female at the school to complete state-wide advanced mathematics school leaving exams, and did so with distinction.
One of Jon’s grandsons, who had learned to read by the whole word route, was classified as “slow” by a phonics-based teacher in his new country. The experience demolished the confidence of a previously robust little boy.
And so what to do?
It’s crucial that mathematics teachers are pedagogically sophisticated enough to encourage creativity rather than kill it. This was emphasised by the scholar and speaker Yong Zhao in his keynote speech at the 2012 ISTE conference – which you can see in the video below.
Yong Zhao showed data indicating a negative correlation between countries’ mathematics test scores and their entrepreneurialness. Indeed, Asian countries that top PISA and TIMSS ratings are beginning to discover that training good test takers does not assure creative citizens.
He asks where the Asian innovators like Steve Jobs are – a question that sounds less xenophobic when posed by a Chinese expatriate.
None of the crises in mathematical or other education have easy solutions. Even phonics needs some rote whole-word learning, and advanced mathematical knowledge does not remove the need for pedagogical skills.
There is a growing body of serious opinion that some appropriate form of mathematics instruction should be compulsory throughout school.
But, for certain, no two kids are the same and no one teacher can cope with everything that is thrown at him or her.
One thing seems clear: more, better trained, better paid, and better respected teachers are a big part of the solution. As is the time and freedom to experiment.
A version of this article first appeared on Math Drudge.