“One, two, buckle my shoe. Three, four, shut the door….” Teaching children the numbers one through ten using rote memorization makes sense. I doubt you’d find a math expert who would oppose young children memorizing the basics of numeracy. But what about 12 x 12? It makes sense to memorize that, doesn’t it? Not according to the Ontario math curriculum. Memorization of multiplication tables is optional, so a child’s ability to recall multiplication facts with ease and automaticity (a skill referred to as math fact fluency) may depend upon the teacher.
Growing up, I learned that “sixteen and sixteen are thirty-two” while singing along with my mother to Doris Day’s rendition of “The Inchworm”. When my daughter was young, we listened (ad nauseum) to annoyingly perky multiplication songs while driving in the car. After a few hundred kilometres, she was rhyming off the times tables without skipping a beat. But during both my and my daughter’s early years in school, these musical memory aids were supplemented by teachers who ensured that a part of every school day was devoted to math drills, flash cards, fun games of math baseball, and other reinforcements of “the basics” that were a part of the regular curriculum at that time.
In the wake of last week’s news that half of Ontario Grade 6 students failed to meet provincial standards in math, the debate pitting old ways of teaching against the newer “discovery math” has reemerged with a vengeance. Turn on the news and you’ll find experts arguing that we need more (or less) rote learning, or that we should introduce a completely overhauled (or a slightly tweaked) curriculum. Some blame inadequate teacher training while others point to teachers, some of whom critics claim don’t even understand the math they’re teaching. The one thing that can’t be blamed in all of this, however, is the student. Any argument that we simply have a different crop of youngsters today versus 2009, when the new math curriculum was introduced, is disproved by provincial scores in Reading, which are steadily on the rise [1].
Whenever the perennial discussion reemerges about Provincial test scores and student outcomes, I am reminded how fortunate we are at Trafalgar Castle. As an independent school in Ontario, we are not limited to teaching the Ontario curriculum in our Lower School. As a result, we have greater flexibility in how and what we teach. Our approach to math instruction has yielded exceptionally strong outcomes, as evidenced by our most recent results showing that 90% of our Grade 6 students met or exceeded Provincial standards [2]. This tells me we’re doing something right in our classrooms because our assessment also shows that not all our girls come to us with a high level of proficiency in math. It’s something we work hard to develop.
More than five years ago, we supplemented the Ontario curriculum with Singapore Math and Singapore Science. In the case of Singapore Math, there is a strong focus on students’ mastery of concepts, ensuring solid understanding before automatically moving on to the next lesson. The basics are consolidated but there is no easy memorization of the answers on a test. Instead, the early grades focus on the introduction of fewer concepts but to a greater level of understanding, thereby ensuring that each student secures a solid foundation for later learning. We also adopted Singapore Math’s use of model drawing, an instructional approach that teaches children to use pictorial representations, thereby scaffolding between the concrete (count out 3 buttons and 2 buttons) and the abstract (3 + 2 = ___ ). We know our students are prepared for the study of higher math based on their ongoing achievement in the Upper School.
Student success depends on more than good curriculum. We recognize the need to provide ongoing professional development for our teachers and to ensure that our classrooms are resourced with the materials our teachers need. We also understand there is no “one size fits all” approach to teaching. As such, our teachers know how to meet students where they are, and possess the professional knowledge, resources and supports to both remediate and extend students at either end of the learning spectrum. But perhaps most important is our belief that good mathematicians are not simply born. Rather, they are taught, encouraged and nurtured. We truly believe that, and our teaching reflects it. We simply don’t give up on our girls, and we certainly don’t let them give up on themselves (even when the going gets tough).
As a child, my family moved frequently for my father’s work. I was an avid reader and loved to write, so English studies came easily to me. However, due to my changing schools so often, I ended up with significant gaps in my math education, and by the time I arrived in Miss MacDougall’s Grade 9 math class, I had little confidence (and weak ability) in math. But if I arrived thinking I would be allowed merely to scrape by with a passing grade, I soon discovered I was mistaken.
Miss MacDougall was one of those teachers you fail to appreciate at the time but look back upon with immense gratitude. She was tough, with a thick Scottish accent, ruddy complexion and a roughly cropped haircut that was a perfunctory as her wardrobe. Her approach was direct – downright frightening at times. Homework would be done, students would participate, and extra help was not only available but also mandatory if required. In short, failure was not an option in Miss MacDougall’s class.
Early in the school year, Miss MacDougall decided that I wasn’t achieving my potential, and I learned firsthand how powerful it is when a teacher believes in you, more than you believe in yourself. During my time with her, I moved from disengaged, to competent, to interested, and finally, to excited and excelling. I learned to love math, and discovered I was good at it. My confidence soared, but more importantly, I experienced what it felt like to push through a difficult problem until I arrived at the answer. That feeling, that skill, that “mindset” is what later prepared me for the often solitary, always rigourous challenge of doctoral work.
What I didn’t know then, but what I do know now, is that Miss MacDougall instilled in me a growth mindset – a belief that success can be achieved through hard work, perseverance and dedication. This theory of mind, developed by Stanford psychologist Carol Dweck [3], has immense implications on learning and life-long success. A growth mindset – “You can always incrementally improve if you keep on trying” – is the opposite of a fixed mindset that sees individuals as set in their abilities – “You’re either born good at math or you’re not.”
Dweck’s research shows how the development of a growth mindset is essential for success in mathematics, particularly amongst girls, and most certainly amongst girls who naturally excel in math in the junior grades [4]. Believing that one is either born good at math or not good at math limits the degree to which a student will persevere when encountering a difficult problem. For the student who has sailed through the early grades, only to be suddenly stumped by Grade 8 algebra, a belief that eventual success is possible through practice and perseverance is essential. Without a growth mindset, students are more likely to give up and lower their expectations for success. And the research tells us that girls are more likely to give up than boys. They tell themselves, “This used to be easy. Now it feels hard. I must not be as smart as I thought I was.” What we need to teach them is an inner dialogue that instead says, “Well, that approach didn’t work. I wonder what else I can try,” or “Gosh, this doesn’t feel easy anymore, but I know I’ll get better with practice.” It’s this type of mindset that’s necessary for success in learning and, most importantly, in life.
Every year, when the math debate reemerges and arguments about curriculum reach a fevered pitch, it seems we forget what’s at the heart of student success. It’s the relationship between teacher and student – the belief Miss MacDougall had in me. We should care about what’s in the curriculum. We should understand what good instructional practice looks like in the classroom. But let’s not forget the importance of the relationship between student and teacher, and the impact a caring community has on the development of a child’s belief in her own abilities. That’s how the seed of perseverance is planted and a child’s potential is realized.
[1] 81% of Grade 6 students met the provincial reading standard, an increase of six percentage points from the results five years prior. http://www.eqao.com/en/about_eqao/media_room/news_releases/Pages/2016-highlighted-provincial-results.aspx [2] We administer the Canadian Achievement Test (CAT4) every spring to all students in Grades 5 through 9 in order to measure the effectiveness of our classroom instruction and to identify areas for future growth and development. The CAT4 provides more detailed data on individual students than EQAO. As an independent school, we are not required to administer EQAO and, therefore, elect to use a standardized test designed to provide data that is more useful at the level of the classroom and individual student. [3] https://www.ted.com/talks/carol_dweck_the_power_of_believing_that_you_can_improve [4] https://www.psychologytoday.com/blog/the-science-success/201101/the-trouble-bright-girls