When I was a young student, my mom claims I had difficulty with my public school math homework. She claims I had “weird” tasks like counting backwards before I could count forwards, and that my elementary school teachers had no power in the curriculum that had been selected for them to use – and they didn’t understand it either! I have little memory of math in those early grades (1st and 2nd). In third grade, I transferred to Catholic school, and we began learning our multiplication facts. I have little memory of this as well, but again, my mom claims that we did a lot of flash cards and practicing remembering that.
In fourth grade, I remember we learned long division – my least favorite math task EVER. I’ve never had neat handwriting, and I’ve always had trouble lining up my letters, but to keep long division organized, both of those skills can greatly aid in your ability to write each step in the accurate location (i.e. lined up directly underneath so you can keep track of which digit you just worked with). I was good at mental computation, and I loved the “shortcut” for long division, wherein you were able to just write the little remainders above the next digit, mentally multiply and subtract, and repeat. As an adult, I’ve since learned two techniques for organizing my numbers: use graph paper and let each digit take up one square OR turn your wide ruled/college ruled notebook side ways and use the lines to organize vertical alignment of your numbers. However, in my fourth grade, no one knew these techniques (nor used them with me), and I found myself hating math class. I could easily do computations in my head, but being forced to write it all out made me feel frustrated.
In fifth grade, my K-8 school had us start going upstairs to the middle school floor for math instruction. We had our “homeroom” teacher for all other subjects, but to preview middle school, where we would switch classes for each subject, we started switching in 5th grade. I don’t recall exact lessons from that middle school teacher, but she quickly became my favorite teacher and she repaired my relationship with mathematics. I loved finding patterns, seeing visual things that made me wonder (like whole numbers written upside down on the board to launch the idea of “how do you flip a whole number to create a reciprocal?”), and learning all sorts of facts and techniques. Knowing what I know now about math teaching, there was definitely an emphasis on procedures above and beyond what I prefer to do in my own teaching, but I think it was more standard at that time.
Sadly, after 7th grade, that teacher moved away (though I’ve since reconnected with her on facebook!) and she was replaced by a first year teacher who didn’t know how to engage or control the class (and my class was the best behaved class at our small Catholic School!), so I learned nothing new in all of 8th grade math. I felt bored out of my mind. In high school, I took an accelerated class in 9th grade where I did Sequential 1 and 2 in one year (since I already knew most of sequential 1 from middle school, but hadn’t had the opportunity to take the Regents Exam yet), and I wound up doing quite well in the course, but being bored and disaffected the whole year. I love to notice patterns and shortcuts and my favorite question as a student was always, “I notice this pattern/shortcut. Will that always work?” My 9th grade teacher, however, hated it because I often noticed patterns she deemed irrelevant or that she perhaps didn’t understand. At my first parent teacher conference of high school, she told me and my parents that I should “stop asking so many questions because I was confusing the other students in the class and I clearly already understood the math.” So I spent the remainder of the year doodling in my notebook instead of participating in the class discussions.
In 10th grade, I had my second experience with an awesome math teacher. I was taking honors Sequential 3 (an 11th grade course) as a sophomore, so there were only a few of us who were sophomores in the class. She was funny and would do things like, “So I said to myself, ‘Self, …'” and would make jokes. What I think I loved most about her teaching style was that she would encourage me and my observations. Instead of telling me to stop asking questions, she would let me know I was thinking a few steps ahead of where we were – but that we’d get there if I’d be patient for a day or two. I use her technique of encouraging patience when a student wants to pose a question that I don’t think the class is quite ready for yet (saying something like, “Wow, that’s a great question. We’re not quite up to that yet as a whole class – can you hold on to that question for tomorrow?”). She did make use of some mnemonics (like SOH CAH TOA), but overall, it was her sense of humor and the laughing that I did in math class that made me enjoy it again. In 11th grade, I took a class called “Accelerated Analysis” (though I’m not sure why – it was just really pre-calculus!), and she was the only teacher for it, so I got to have her two years in a row! That made me incredibly happy because I felt successful in her class, but also she used the techniques of “puzzlement” to intrigue us into the next set of problems/skills for us to learn about. I excelled in this class. When I was out for 9 days with the flu right before our midterm, I came in for the mid term and earned a 49/50 on it, despite having missed all of the in-class review. The one topic/skill I didn’t learn how to do was completing the square (because she taught it while I was absent), and it wasn’t until after I was a teacher for several years that I actually learned how to do this skill! (now it makes perfect sense, because I’ve seen it visually, but at the time, I had just routines and procedures, so I could never remember what to do).
In senior year, I was disappointed that she wouldn’t be my teacher again, because I’d decided to take AP Calculus (since I was planning to be an engineer and I knew I needed to take Calculus to be competitive for admission). My teacher was not as horrible as some of my truly disastrous teachers of the past (4th, 8th & 9th grade, for example!), but she lacked the humor factor of my 10th/11th grade teacher, and she was not as good of an explainer/modeler – plus, she was absent a lot (due perhaps to a family member’s illness? I can’t recall, but it meant we often had subs who couldn’t teach us calculus, even if they were math teachers!). I remember my class doing a lot of partner work to try to teach each other and work together on math problems, but we’d never had any instruction or practice doing so at earlier grades (it was literally the first time in math I’d ever had a teacher assign a problem set and say “Work together on this”), so I don’t think I made as good use of the strategy as I could’ve. I paired up with a friend who was incredibly good at math, and she explained a lot of her procedures to me. I remember the first trimester, we all failed our first exam – the first math exam I’d failed since 4th grade! I only earned a 3 on the AP exam, so my engineering college wanted me to start over and take calculus again.
A weird fact: until I got to college, I NEVER had a math teacher who was male. Then, once I was in college, I took seven courses (four quarters in a year, so this was one course per quarter except the last quarter when I knew I was leaving engineering) in the math department at my engineering school – only two of which were taught by women! I had a male instructor for calc 1, I don’t remember my instructor for calc 2, I had a woman for calc 3, a man for calc 4, a man for differential equations, a woman for discrete math, and a man for linear algebra. I remember discovering math in college felt HARDER than anything I’d ever done before. I thought quarters would be good (since I’d always been bored in classes that went on all year, and this way I could change topics more frequently), but I realize now in retrospect that it was probably too much to learn in too little time. We had classes almost every day (4 days per week was standard, with Wednesdays for Lab), and we had tons of problem sets. There were large-style lectures, as well as smaller classes where we had a TA go over problem sets with us, and a lab where we “learned” how to use MatLab (I put quotes around learned because we mostly followed instructions from a paper about how to set up a file to do something – it felt very “demonstrate this skill” without purpose).
When I got to college, I realized that I didn’t know how to study math. Math had always come easily enough to me, and when it didn’t, I would just ask my Dad for help, and he would explain how to do a problem. This was long before the days of websites like Khan Academy, and I don’t even think it occurred to me to search the internet for an explanation at all! Mostly, I relied on informal study groups with my friends explaining math concepts and problems to me. I definitely knew the difference between copying and getting help, and I never just copied a friend’s homework, but there were definitely days where I felt like I’d just written squiggles on a paper that hadn’t meant anything to me. I recognize now, in retrospect, that most of my math classes were abstract manipulations of symbols (and there were some assumptions made about things we should’ve known from high school that I never quite felt proficient in). I remember never feeling like the Epsilon-Delta proofs were explained well, and I remember trying to read aloud the statements (For all epsilons greater than 0, there is a delta such that… blah, blah, blah!) and feeling like I had NO concept of what it meant. Even now, reading the Wolfram Alpha page on the topic feels like an exercise in reading greek: http://mathworld.wolfram.com/Epsilon-DeltaProof.html
During calc 4, at the end of my freshman year of college, I recall seeing a visual like this one in my textbook: https://i.ytimg.com/vi/hXkQqCBLRp8/maxresdefault.jpg It felt like an epiphany, and I remember demanding of a friend, “Why did no one explain it like this before?” I remember there was like one sentence that was said to me (perhaps by a teacher, TA or friend – I don’t recall!) and I was like, “WOAH! Where has that sentence been all year??” I began to realize that I learned math visually, and that mere symbol manipulation – while it was sufficient at lower levels of math – it was inadequate at the higher levels. Other than an A in Calc 1 (because it was just about derivatives and felt like a repeat of high school AP Calc), I never earned an A again in a college math course during my undergrad because I never learned the math quickly enough to keep up.
I’ll never forget the moment that I decided I definitely didn’t want to be an aerospace engineer anymore (my dream since I was literally 3 years old). I was taking an intermediate mechanics class that was required for my major, and it was a bunch of sophomores who all had the same major as me. We would read the textbook and do practice problems from it. I often felt quite confused, especially about orbital mechanics type problems, involving centripetal acceleration. I was working as hard as I knew how to (which was not actually very effectively at the time, sadly!), attending office hours and going to see my TA for help, and I felt very confused about all of the skills involved. I recall earning grades in the 50s, and when I checked in with my professor, she reassured me that it was okay – I would still pass the class with a C. Because my college had a policy that if you didn’t earn a C in a course, you had no record of ever having taken it, she had changed the ranges for grades (A was above an 85, B was like a 70 – 85, and C was like 50 – 70 – approximately). I realized that the mechanics involved in that course were the necessary underpinnings for my entire career as an aerospace engineer, and I didn’t understand them very well at all. And rather than dealing with my confusion and my desire to learn it better, my professor just reassured me that I knew enough to pass. That terrified me, in fact, because I became convinced that I would make a mistake when designing a spaceship because of my lack of understanding, so I decided to leave engineering college and perhaps go into the liberal arts (since I had enjoyed all of my literature and philosophy courses more than any of my math, science or engineering courses thus far).
I took a year off, working at space camp for most of that year, and I researched colleges to transfer to. Ultimately, I decided to go to Sarah Lawrence College, my Mom’s Alma Mater to study writing. Once there, however, I took a science education course that changed my trajectory completely. We read Liping Ma’s Knowing and Teaching Elementary Mathematics (because my professor said there wasn’t enough literature about science education yet), and I was fascinated. I had a better understanding of the algorithms than most of the elementary school math teachers, but I definitely didn’t have as good of a vision of how all the pieces were fitted together the way that the Chinese teachers did. That, combined with my service learning placement at the American Museum of Natural History (and my previous experience working as an educator at Space Camp/Aviation Challenge) made me consider informal education. During my senior year, I took a psychology class with a professor who recommended I take two more courses in the spring and then stay for an extra year to do a masters in Child Development. I decided I would!
During the spring of my senior year, I took an awesome course with an awesome physics professor. He was definitely a geek like I was; the fall of my senior year I took a course called “Crazy Ideas in Physics” with him, where we spent the first half of the semester in a role-playing game, pretending to be different characters trying to either get funding for our research or trying to dissuade the panel from giving funding. My character (who I barely remember the details of now) was interested in breaking up the entire convention through some philosophical readings. In the second half of the course, we evaluated science ideas for accuracy – things like string theory and cold fusion – and distinguished pseudoscience from real science. Each person or partnership researched the ideas they were assigned by the professor, and we presented to the class about the topic. They voted in two ways: were we telling the truth about the theory and was the theory real? It was a very cool way to learn about science ideas! Anyway, I took a second course with him in the Spring because I was bumped from my first choice. The course was called “Rocket Science” or something to that effect, and it was about building model rockets. It was the first time I’d ever had a chance to design my own rocket – and to do so in advance, using a computer model (and not just to build the kit or to design my rocket based on the pieces that were available). It was also the first time I’d ever painted my rockets! In talking with my professor, I told him that I’d done staging (something I consider rudimentary, since I’d been doing it since I was 13 and launched my first model rocket at space camp, and my later rockets as a camp counselor). I expressed interest in creating a rocket with parallel engines that ignited simultaneously. My first design had the rocket boosters glued to the side of the rocket, with little nose cones of their very own, and vent shafts on the side. After the launch, one of the nose cones tore off because it didn’t have quite enough vent, and motor burnt through it. For my final project, I decided I wanted to build a rocket that had boosters like the space shuttle that would separate. We decided this in part because the smaller boosters didn’t have enough space for venting, but adding length to them would create extra weight for the main body of the rocket to carry. It was really quite cool. One of my three engines didn’t ignite at all, but despite that, I had a relatively good trajectory and launched quite high. I wrote a paper about it and presented a poster at my college’s science poster fair that Spring. It reignited my interest in the more mathematical side of science education (i.e. not just the fun facts!), and we did some orbital mechanics in that class too! However, the combination of having longer time to do the calculations and some direct, hands-on experiences with computer simulations that showed us the rockets (and some orbital problems, like the Hollman transfer), doing the calculations themselves felt simpler than when I’d been in engineering school.
During grad school at SLC, I worked at the early childhood center doing science education with the K/1 class. They do an emergent curriculum there (other than some “Beautiful Writing” lessons where they create their own dictionaries and learn to write), and a lot of games were used by the teacher. I was fascinated how she played Bingo with the kids as an assessment and she told me that she had had concerns from parents before that their kids came home and told them “We played games all day in school!” but that she was able to allay their fears by showing them how playing Bingo allowed her to assess which students could recognize the numbers by name to visual, which kids needed to see the number (because they didn’t know the name yet, but could match symbols), and which kids needed someone to point out the number that had just been called because they had no recognition for it. She had kids play games where they rolled two dice and added the pips (or counted the pips – or counted on!), and recorded which numbers rolled most frequently. This could be used with older kids to discuss probability, but with her young kids, it was designed to help them grasp one-to-one correspondence and begin to recognize strategies for adding these small numbers (as well as to begin recognizing the patterns that were always 1, 2, 3, 4, 5, and 6). She also incorporated A LOT of math into her morning meetings, everything from counting the days of school (and of course doing something cool on the 100th day!), talking about how many students were present, how many adults were present, and how many people were absent, and then tallying the total number of people present (and developing adding strategies like counting on and adding to friendly numbers as they came from students). After the alphabet, she had kids include the numbers 0 – 9 in their beautiful writing dictionaries, and she had them write “number sentences” using those numbers – asking students about what they knew how to do, and what they wanted to write. It was pretty cool to watch the students develop over the course of the year.
By the end of the year, I’d found the Math for America (MfA) Fellowship and decided to apply to become a math teacher. I had to take the Praxis Exam (two in fact!) and score well enough on them to demonstrate mastery of mathematics. I found the exams to be incredibly difficult, and while I did better than required to get certification, I barely scored high enough to get into the fellowship. During my Masters’ Degree for Education at teachers college, I had to take four math content courses, and I discovered there were whole branches of mathematics that I’d never studied! I took a probability and statistics course (where the professor read aloud to the text in class! course taught me how NOT to teach!), a number theory course that introduced me to Euclidean Algorithm, used modular arithmetic and hinted at cryptography applications (an idea that’s always fascinated me, before and since!), a course on graph theory (I think?), and a course on abstract algebra (which introduced me to groups and fields and rings). We also had to take courses on methods of teaching, as well as teaching literacy and students with disabilities of course. There were also two student teaching placements, one in high school and one in middle school.
By then, I’d discovered that I needed more support than just a TA and class time to really learn higher levels of math, so in each of my courses, I made study groups with people from my fellowship. We would get together and do homework together. And I found that by far, I knew the least amount of mathematics as anyone else there – it made me feel a bit deficient, but it also motivated me to learn from these people who had tons more experience with the math content than I did. I was able to contribute more in classes that required our knowledge of psychological theories of development, as I’d done a whole masters in that and had read many of the original theories myself (i.e. I read Piaget and Vygotsky, not JUST someone else’s summary of their writings). I didn’t earn a grade lower than a B in any of my graduate courses, and I earned some A’s as well (though I’ll be honest – I don’t think the A’s were in my math content courses, except maybe abstract algebra, which I enjoyed a lot). As I was taking these courses, I was thinking about why I struggled with math so much in college – I didn’t have any study strategies. I realized some of the things I needed to learn how to do to be prepared to take college math was to learn how to STUDY math – and to learn the language of math.
Sadly, in many ways, I felt woefully underprepared for my first year of teaching (yes, yes, I know – most people do, but I felt like I was even less successful than many, perhaps because I had no real mentor or perhaps because none of the people who did mentor me taught exactly the way I wanted to – and I hadn’t had any experience learning the way I wanted to teach, so I wound up relying on the direct instruction methods I had been taught under as a kid).
However, when I started going to PDs with MfA, I started discovering great problems to use. Additionally, my second year teaching, I had a much better mentor (at my then new, but now old school) who helped me “storify” my teaching, where I told my students stories to launch investigations, rather than just saying “Here’s some math you need to learn.” I began to think more about the ways to introduce students to problem solving tasks and challenge them to be thinkers. I lurked around the edges of the MTBoS, reading Dan Meyer’s blog about three act math, reading Fawn Nguyen’s blog and discovering her visual growth patterns, and taking Jo Boaler’s online course, How to Learn Math (in its first iteration in 2013).
One of the things that I was slow to learn about was the value of talk in mathematics. I knew (theoretically, from Vygotsky) that learning was a social activity, and that talking about our thinking helped to clarify it, but I had rarely felt that way in mathematics classes as a kid. I knew talk was important in my ELA, science and even history courses, and certainly going to SLC for college (where the normal class style is a small seminar!) drove home the learning from seminars, but I didn’t quite know how to apply it to math classes yet.
Two things happened to me that changed my mind forever. My last principal encouraged me to read an article about the turn-and-talk procedure (which I don’t recall learning much about in grad school, though it’s certainly possible I did). After reading it, I told her that I didn’t quite understand what types of questions to use with students or when to use it. She sent me to visit another school as part of a PD cycle, and I met an amazing teacher who collaborated with me for another two years, corresponding via email and sharing her knowledge and resources generously. I saw the power of the turn-and-talk in her classroom (and her colleagues in the same school), and I listened to the students talk to each other. I began to use turn-and-talks sporadically, but I still thought they had to be used for big math ideas or main focus points.
Then I attended a Summer Institute with Metamorphosis (Lucy West’s company) a few years ago. We were given some math tasks to solve ourselves and we were directed a few times to do turn-and-talks about specific things – including such basic ones as “tell your partner what you think the problem is asking about and ask any clarifying questions you might have.” I remember on the third day of the training, we had a guest lecturer who was teaching us about geometer’s sketch pad, and he gave us confusing instructions about how to start the problem, and I raised my hand, and for the first time in my life, asked, “Can we do a turn and talk to make sure we understand what to do?” He looked a bit baffled, but the facilitators agreed, and suddenly, after talking to my partner, I completely understood how to begin (and realized that the instructions weren’t quite as confusing as I thought). After that training, I began incorporating turn-and-talks into my classroom practice daily, and I sometimes would give more or less direct prompts to the students (i.e. “Share one thing you notice and one thing you wonder with your neighbor” vs. “Ask any clarifying questions you might have” vs. “Compare answers and strategies, and see how they’re similar or different and if you need to convince each other to change answers or find mistakes, try to do so.”
During my fourth year teaching, I applied to become a Master Teacher with MfA, and they required me to retake the Praxis Exam because I hadn’t scored well enough the first time to be a Master Teacher. In preparing to study for it, I bought a test prep book with different types of questions, and I asked a friend of mine who was strong mathematically to meet with me weekly and support me when I had questions about problems I didn’t know how to solve. She’s the one who taught me to finally complete the square! I also laughed because I realized that topics I was teaching were helping me with math problems (i.e. I hadn’t remembered anything about angle relationships, and while I couldn’t quite remember all of the facts about quadrilaterals – which opposite angles are equal? – I knew enough about two parallel lines cut by a transversal to deduce the correct answer). I improved my score on the Praxis by 35 points, going from a 153 to a 188 (on a scale from 100 to 200).
My experiences as a learner have driven some of the messages I like to send my students off to high school with. The number one most important message I found is that the only way to STUDY math is to DO math. Unlike other subjects, where reviewing flash cards can help you remember key concepts or vocabulary, math is primarily a skill-based, application of key concepts or vocabulary, so flash cards are much less useful. I also try to reduce memorization and push the idea that math should make sense to students. I love the idea of convincing yourself, a friend, and a skeptic, and I want to do better at having students “play the skeptic” in class and push for better explanations from students who think they understand. I know the biggest things that hurt me
I also recently read an article that supported why I left engineering. Apparently, women (which I was raised as, and identified with in part at the time) have lower confidence in their math skills when the topics get harder (and I definitely did NOT have a growth mindset during undergrad, and certainly not about math!) and thus think of themselves as less proficient and are more likely to leave when they face failure because they perceive it as a lack in themselves. I also had no study skills when it came to math – reading the textbook wasn’t helpful, and taking notes didn’t seem to be helpful – what’s been helpful was practice solving the same types of problems I would see and then getting feedback from other students or sharing ideas with them.