Early maths experiences
Other than starting school learning about sets, I don't remember much about my early maths experiences; although I do remember my dad having a small box of cuisenaire rods in his study that I liked to play with, and I vaguely remember a pack of maths flash cards that were a partner pair to the sight word flash cards. I remember having some problems learning how to read, but none with maths so this can't have been an issue. (I do remember that my younger sister used to always miss 13 when she was counting until Mum reminded her that her birthday was the 13th May, so she better remember or she might miss out on presents!)
Our family is good at maths
There was a family history of being "good at maths." My mum used to work in the local TAB betting office in the days before computers or even calculators. She always claimed that she could have a conversation while adding up two rows of figures and still get the right answer at the end of the page. Someone even picked an argument with her once while she was tallying the balance sheet and not only did she get the right answer, it didn't even slow her down. (Or so the legend goes!) Mum liked to play the odds and would sometimes have a punt solely on the grounds that there was no-one else betting on a horse and if it did win, she could score a lot of money for very little outlay.
My paternal grandfather was a supervisor in a glass factory on week days, but on weekends he was a bookie's penciller at the local racetrack. It was his job to keep track of the bets and manage the risk for the bookie by continually calculating and readjusting the odds. By all accounts, he was able to do these complex calculations in his head. Dad has a disdain for gambling: his father taught him that bookmakers ensure that the odds are always in their favour.
School maths: get it right, do it fast.
My first memory of maths in Primary school is from Year 3 when Sr Pauline would put a whole stream of sums up on the board and show us how to do them. Once I'd seen where she was going, the only challenge was waiting for her to move out of the way so that I could see the numbers she was blocking with her body. I think she found it quite frustrating that she had only just finished explaining the first problem and I'd already finished the set, so she stopped setting out the board ahead of time and I had to wait for her to write each one. At least that way we finished together and she didn't need to occupy me for as long before everyone else finished.
In Year 4 we would have a maths warm-up each day where we would line up in two parallel lines down the classroom and approach the teacher in pairs. Like an elimination round in a gameshow, each pair would be asked a maths fact. The first partner to give the right answer went to the back of the line and stayed in the game, while the losing partner returned to their desk. I don't remember ever winning, but I usually made it through the first round at least. My report of that year states, "[Sally] has an accurate knowledge of tables though her delivery is a little slow." Perhaps that's why my parents got me one of the first ever electronic teaching tools for Christmas that year: The Little Professor.
It was marketed as a "backwards calculator" in that it would randomly generate a mathematical expression and the user (me!) had to provide the answer. If the answer was incorrect, it would display what I thought of as a set of "three backwards 3s" but was in fact 'E E E' for "Error Error Error." I might have misunderstood the exact reference, but I knew it meant I'd got the wrong answer. It was one of the earliest efforts at a digital teaching tool. There were three chances to get the correct answer, but if you still got it wrong, it would tell you the correct answer. At the end of each set of 10 questions there was a score: the number answered correctly on the first attempt. There was no time limit however - or not that I remember anyway - just the jolt of seeing the backwards 3 3 3 when an answer was incorrect. My nemesis was always 7 x 8. Was the answer 54? or 56? (Even today I have to think this through: it's as if the correct answer just won't 'stick' in my mind.)
My Year 5 teacher was newly minted and also used the pair strategy to regularly test us on our tables. I clearly remember a fellow student who found school difficult and was often the first person eliminated. In the way that kids do, people used to push to line up next to him because they were assured of at least making it to round two. I remember him getting very upset and frustrated one day, and our teacher telling him that anyone could be good at tables with enough practice. His parents must have heard the message, because he came back after the holidays and we knew something had changed. Darren not only avoided elimination in the first round, he won! As a class we were stunned the first time it happened, and I even remember spontaneous applause as we recognised a fellow student who had overcome a monumental personal challenge to achieve a seemingly impossible goal. From that point on, he won every time we played. The dynamic quickly changed as everyone tried to ensure that they weren't matched up with Darren, because it was a guaranteed exit in the first round.
High school maths: more of the same
I attended an all girls high school from Year 7 to Year 10 and there were about 90 students in my year. Mrs Webber taught the top maths class. She was a diminutive, soft-spoken woman who's response to classroom noise was to whisper - and it worked. In my first ever high school maths half yearly, Mrs Webber explained that she had planned the marking so that all the scores would add up to 100 to make it easy to convert our scores to a percentage. But during the marking process she had realised they added to 99. So, she had decided that our final score would be whatever mark we had plus one mark to take it up to 100. Her plan was to give us all an extra mark 'for free' to make her life easier. It's the only time it ever happened, but I can lay claim to scoring 101% on a maths test. I was happy, but it didn't win me any friends.
I consistently topped the grade in every maths test during those years, but that was the only exam where I got full marks. When we would get our tests back, my friends couldn't understand how I could be disappointed with a final score that they yearned for; and I couldn't understand how they could be happy with just a pass. I also felt that I needed to hide my excitement when I did well. Instead of asking, "How did you go in the exam?" I eventually learned to ask, "Are you happy with how you went in the maths exam?" which allowed us all to express our joys and our disappointments on a personal level. And I stopped sharing my test scores.
My teachers figured out a way to use all the time I had left over in every maths class: I became a second teacher and would roam the class helping my classmates. We did a unit on circular geometrical proofs and I remember figuring each one out in my head, then helping others to see what I could see. I did this for the whole chapter and the teacher never checked my work once so I got away with never actually recording any written proofs.
I changed schools in Year 11 and found myself no longer on top of the heap. My Year 11 maths teacher was a former Kindergarten teacher. While I loved her, I found Mrs Tully very frustrating. I'd call her over to ask a question about one small aspect of a problem, and she'd start at the beginning and work her way slowly through the entire problem. When I complained(!) she told me that she needed to do this for herself.
I wasn't coming first but I did well enough to be allowed to take on the hardest maths course in Year 12. This meant that four out of my eleven units of study were maths. We started classes at the end of Year 11, just before the summer holidays and Mrs Rapp gave us Imaginary Numbers to learn over the holidays because it was "easy." It wasn't easy for me, and for the first time, I really struggled with maths.
A new way to learn: thinking like a mathematician
Mrs Rapp was unlike any other maths teacher I'd had before. Our maths classes were less about a demonstration followed by practise, and more an open class discussion. We derived our own methods - and then did the practise for homework. She challenged us to find clever, faster, more insightful ways of solving problems. Grinding through things was to be a last resort. Think first, then solve. In our discussions she would allow us to suggest approaches and help us work through them to find a solution. But she didn't always follow through student suggestions, especially if she knew it wouldn't go anywhere. As soon as she'd gone far enough that most people could see it was a dead end, she would stop and move to something else.
The people in my class that year were extraordinary: the girl who topped our year came 3rd in the state and 2nd in 4 unit maths. My classmates are now specialist doctors, psychiatrists, journalists and mathematicians. I did my best to keep up, I tried hard to be insightful and find a faster, better approach. Sometimes I was the person who found an interesting approach. Often my ideas would lead to dead ends that everyone else saw before I did. Much to my dismay, Mrs Rapp would move on without me seeing where I'd made a mistake. My saviour was my best friend, Nicole, who would patiently sit and teach me. She would always talk through at recess what Mrs Rapp didn't have time for in class.
Tertiary mathematics
I studied Pharmacy for a year at university. First semester was a repeat of much of the content of 4 Unit maths covered in lectures at twice the speed and devoid of discussion. It was revision for me but all new for my friends so it was my turn to tutor them.
Second semester was statistics, the only branch of maths I ever hated. There were so many terms whose meaning seemed so similar, and all these 'squared tests' that I never saw the point of. I just scraped through and finally understood what it meant to be grateful for a pass. Then I made the decision to swap pharmacy for an arts degree and fell in love with philosophy and history.
I didn't study maths again until I started my Masters of Teaching degree more than 20 years later. Our tutor was a former primary teacher who talked a lot about teaching maths. His stories were interesting and in hindsight, I learnt a lot from them - but we were very frustrated that we didn't get to do very much maths ourselves. Even the major assessment was an essay. When were we going to be asked to do some maths? Our final exam was a group practical assessment, with manipulatives all laid out around the table. We were able to discuss ideas in whispers within our group and were marked as a collective. It's hard to bow to peer pressure when the other two members of your group are convinced that their answer is right and you're equally convinced that yours is right!
Learning how to teach maths
I don't think I learnt how to teach maths until I was in the classroom. I was lucky to go to a school where the maths pedagogy continued to evolve while I was there. I learnt how to design open-ended tasks although at first they were written into a 'contract' that students worked through at their own pace. Every kid in the grade did identical pre- and post- tests to measure if students had learnt anything, rather than to modify our instruction. I knew it was hard for the kids who struggled, but that's what we had to do, right? While we did a whole class warm-up, there wasn't an expectation that we would engage students in a whole class reflection. In the Year 5 & 6 classes, kids were separated according to ability across the four classes - always a disaster for the kids in the bottom class.
Early in my teaching career our school was involved in a research project using the DS Brain Training program. In the beginning, the kids were given a page of simple equations using all four operations. They did them as quickly as possible and recorded the time taken to finish the page. Then we were given a class set of Nintendo DS machines. Every morning, the kids would spend five minutes on the DS playing Brain Training. At the end of the year the kids redid the paper test. The results? They were definitely faster at finishing - but still made the same mistakes at the end of the year that they had at the beginning. Their speed had improved, but they didn't know any more than they had in the beginning. So much for Brain Training!
The next year, I planned my first family maths task: a plastic jar full of small chocolate easter eggs with the lid super-glued shut, another identical jar with only ten eggs, and a journal for students to record their guesses. Kids would take the kit home overnight and figure out a way to use the loose eggs and the empty jar to estimate how many eggs were in the full jar. Each kid (and their family) tried to find a different way to solve the problem OR checked someone else's thinking. Our goal was to find an increasingly more accurate estimate, or to find multiple ways of getting approximately the same answer. Kids would share their approach every morning. I still remember the story of the family that went to the local supermarket in their pyjamas to use the scales in the fresh produce department because they didn't have scales at home! At the end of the term we cut to top off the jar, counted the eggs and the kids figured out how to share them equitably.In the years that followed my professional practice developed in partnership with my colleagues. We started looking for authentic contexts and created PBL-style maths tasks. For example, we did a unit at the end of the year which involved students needing to develop their understanding of multiplication, division and mass. They had to figure out if it was possible for one teacher to carry plastic shopping bags containing all the materials for making tiny gingerbread houses for every person in our Year 3/Year 4 classes (104 people, including teachers.)
Another time we planned a class party. The kids had to do a whole range of tasks related to volume and capacity to ensure that everyone got a cup of jelly, half a cup of soft drink and a treat box of lollies. In true PBL style, they developed a list of "Need to Knows" that included such questions as:
- What's a reasonable volume of jelly for one person?
- How many packets of jelly would we need to make enough for everyone?
- What might a treat box of 120 cubic cm look like?
- How many servings of 125mL of soft drink are in a 2L bottle?
- What does 125mL look like in a cup? Is this enough?
It became obvious that finding the right context could have a huge impact on kids' willingness to engage with the maths. Because we did these tasks during maths time, the kids had no problem thinking of these tasks as "doing maths" however as a teaching team we had to make sure we took the time to guide them reflect on what they had learnt each day, or the maths got lost in the excitement of the task. Parents were surprised and delighted that it was possible for their kids to have fun learning maths.
In my final three years at this school, I was given the role of an EMU teacher (Extending Mathematical Understanding). This meant that every day for 30 minutes at a time, I worked with the same group of three kids who were considered 'vulnerable' in different areas of maths. I had another 15 minutes to reflect on the learning and plan for the following day. It was such a privilege to help them make sense of their thinking and to develop their confidence and understanding. It also made me realise the importance of exploring the same concepts in multiple contexts and using a range of different materials. For example, just because a kid knows that a bundle of ten popsticks and 3 more is 13, doesn't mean that they can rerepresent this using MAB, or that the quantity should be written as '13' and not '31' or even remember that the quantity is called, 'thirteen' and not 'thirty.'
#MTBoS: Math Twitter Blog-o-Sphere
In recent years, the people who have had the most influence on my experience of doing and teaching mathematics are people I know on Twitter. David Butler's mathematical challenges reminded me of how much I liked to solve maths problems before I encountered statistics and was scared away; and playing with David, Paula Beardell Kreig and others to explore geometrical concepts is so much fun! Tracy Zager's blog posts and her most excellent book inspired me to be a better teacher of mathematics. Christopher Danielson's ideas, books and maths toys have been shared and enjoyed by family and friends as well as students and colleagues. Amie Albrecht is the applied mathematician I secretly wish I'd been brave enough to be, and her insights into maths, teaching, people and books inform and challenge my thinking. Working with Simon Gregg on our #PatternBlockProject gave my students an authentic audience for their thinking and helped them find their voices as mathematicians. Charles T. Gray's journey towards world domination of R-stats and ongoing encouragement has made me wonder whether I could possibly go back and master statistics after all.
There are so many people who make up the MTBoS community who have shared their ideas, insights, resources, fears, beliefs, pedagogical approaches, successes and amazing mathematical creations. You have helped me see myself as not only a maths educator, but an amateur mathematician. Thank you.
