Mathematicians and parents
We, the present authors (DHB, from USA, and JMB, from Australia) are research mathematicians/ 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). 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, and while they do not always match they usually do. So what have we learned?
This blog is stimulated in part by a recent book on preparation of mathematics teachers for the classroom. While the book, entitled 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 teacher knowledge of mathematics by asking a sample of 4000 teachers in Michigan and Ohio, “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 that they routinely taught their classes. 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 example, for basic geometry topics, in one district only 1/4 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, with only 1/4 of the teachers in one district well-prepared to teach decimals, but virtually all teachers in another district.
In middle school (6-8 grade), the situation was 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 entire sample of teachers felt well-prepared.
Fortunately, high school teachers are relatively better prepared, although there are concerns here too, particularly in more specialized areas such as 3-D geometry, logarithmic and trigonometric functions, probability and calculus. For example, many U.S. 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, less than 10% have a major or minor 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 specialization in mathematics — only at 11th and 12th grade does the ratio rise to a more respectable 71%. Additional details about the authors’ study are given in this Scientific American blog.
The 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. For example, JMB lived briefly in a large U.S. city, and was greatly disappointed in the quality and indeed safety of the state schools. By comparison, the state schools that JMB experienced in five Canadian cities were all of reliable if not always outstanding quality.
DHB 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 over 1900 in average SAT score, whereas schools at the other end of the list, such as Thomas Riley High in Los Angeles and Mandela High in Oakland, scored only about 1000. 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. For example, 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, it is not the mathematical breadth of the teachers that is a problem. Pedagogical narrowness is a greater problem. Telling that there is a correct idea in a wrong solution to a problem on fractions requires unpacking elementary concepts in a way that even an expert mathematician is not usually trained to do. JMB learned this only- too-well when he first taught future elementary school teachers their final university mathematics course.
For instance, Australian teachers at an elite private school could not understand one of JMB’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. They 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 JMB’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. Another Scientific American blog emphasises the damage schools can do to the spirit of such a child as does the wonderful song Red Flowers by the late Harry Chapin.
And so what to do?
Thus, it is also crucial that mathematics teachers are pedagogically sophisticated enough so they act to encourage creativity rather than kill it. This is emphasised by Yong Zhao in his keynote at the 2012 ISTE conference (see Utube). He shows 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 are the Asian innovators like Steve Jobs? A question that sounds less racist 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. 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 freedom to experiment.
Acknowledgements. Thanks are to Kathryn Holmes for many useful comments and pointers.
[A version of this article also appeared in The Conversation.]