“Don’t armor the places that sustained the most damage on planes that came back. By virtue of the fact that these planes came back, these parts of the planes can sustain damage.” -Abraham Wald
An Interesting Example
During World War II, Abraham Wald worked for the New York company Statistical Research Group. The American military approached him with a problem. War planes were coming back from air battles covered in bullet holes, and the military was interested in protecting future planes by using minimal armour placement. The question was where to best place this armour so as to protect the planes and pilots. The bullet holes were roughly distributed as such:Where would you choose to place the minimal armour?
If you said any section other than the engine section, then you would be absolutely… wrong. Wald’s explanation is as follows: the statistics that were presented were gathered from planes that had survived battle – the planes that didn’t survive battle were not included in the numbers. Thus, the planes that returned were not a random sample and conclusions could not be made by only reflecting on the surviving planes. Wald recommended adding armour to the sections of the plane that returned home relatively unscathed (here, the engine section) would be wise, as the surviving planes tell us the regions of the plane that can take damage and still return home safely.
Suggesting to give armour to regions of the plane that are damaged the most reflects a statistical bias known as survivorship bias. This is an error that arises when we only consider objects that “survive” a particular process, ignoring the objects that do not survive. For example, in In Search of Excellence, Peters & Waterman observe 43 “excellent” companies and determine eight traits that have propelled these companies to the top. However, as of 2016, 20 of the 35 public traded companies were under market average, and five of these companies have since gone bankrupt. So what’s going on here? Are these companies actually “not excellent”? Well, there are other factors at play here, such as luck, that may have helped these companies propel themselves to the top. And since Peters & Waterman only chose companies they deemed as “excellent”, their sample reflected survivorship bias: maybe some of the poorest-performing businesses also used these eight traits – we don’t know since only “excellent” companies were analyzed.
Survivorship Bias in Education
The above aforementioned is why I typically don’t care too much for the advice of celebrities. Why would I listen to their advice on how to be successful when they are a part of the small minority that have survived the process? How do we know there aren’t other factors such as luck or socioeconomic status that have helped advance their careers? What about all of the people who have tried to become a celebrity by following this advice, but haven’t survived the process?
So why is it that we listen to similar advice of celebrities in mathematics education?
Arguments such as “I never memorized my times tables and it’s never held me back.” reflect survivorship bias. Maybe the statement is true, and that’s fine, but what is missing are the countless number of individuals who did not memorize their times tables and have been held back in some form. We need to see and hear those stories to get a full picture on what to conclude. In addition to the previous argument, award-winning mathematicians who “think slowly” and “visually” are often given as key examples in pedagogical arguments. What about all the mathematicians who are not visual thinkers, or who don’t think slowly? Are there other factors, such as the field of study, that cause these traits? Again, more examples are needed to get a better sense of the typical traits of a mathematician.
As educators, I think we get so excited when we hear about the stories of progressive-style pedagogy that works, that we often forget that there may be just as many stories in which progressive-style pedagogy doesn’t work. Traditional-style pedagogy isn’t safe from survivorship bias either – how many times have you, within a traditional-style lesson, received a correct answer from one student and proceeded through the lesson, oblivious to the fact that there may be several students confused? You are making an assumption based on the “survivors” and your “non-survivors” are potentially being left behind. We need to be mindful of this, and not allow survivorship bias to elude us into thinking something is there when it is not.
I want to ask a favour of all the teachers out there: I want to know of a time when you utilized progressive-style pedagogy and it failed. Personally, I had one class in which I flipped my lessons and it was a disaster – with pass rates being abysmally lower than normal. This came after I was on a high from the last semester, in which I flipped my class and pass rates were much higher than normal! There were definitely differences between the two cohorts of students that I had failed to see until post-reflection.
Interestingly enough, Peters & Waterman, the authors of In Search of Excellence, later came out and said that they had made up some of their data. I firmly believe that both sides of the math debate have valuable advice to offer, which is why I am interested in hearing the different stories out there. I do get concerned, however, when teachers/researchers begin to fall into the Peters & Waterman trap: “… go find smart people who are doing cool stuff from which you can learn the most useful, cutting-edge principles … [and] worry about proving the facts later.” Rather, let’s observe all sides of our pedagogy (both the good and the bad) and use this information to perfect our craft.
“I am just a human being trying to make it in a world that is rapidly losing its understanding of being human.” -John Trudell
A couple days ago, I was having a chat with Barry Garelick, and I posed a challenge to the general Twitter-sphere to bust down one barrier that had been put up.
Originally, I had planned an apology-type post to Jo Boaler for past transgressions (perhaps coming later); however, some interesting things have come up over the past two days that made me think about a more important barrier that has been erected over the past two years: the barrier to myself and to my happiness.
You see, it is very easy to forget your abilities, skills and self-worth when you are constantly receiving messages that your opinions and efforts are not valued. So I want to take the time today to break down the barrier that others have erected – the barrier to my self-worth, and to my happiness. The barrier to myself.
This weekend I have been attending a holistic education conference, and have been doing some inner reflecting. One of the prompts I was asked to write about today was “Describe something positive about yourself.” To be honest, I left it blank because I didn’t think I could find anything positive about myself to write about. After answering all the other prompts, I returned back to this one, and began reflecting over my most recent vacation to Ontario. What was it that others saw within me? Why did people want to meet and talk with me? I am just a nobody from a small-town in Ontario, population: 3 cows. Eventually, my pencil started moving and I wrote about story-telling and using story-telling to create connections to others:
“I enjoy story-telling, especially hearing the stories of others. To me, I believe this helps foster true connection and empathy. Even when I don’t know what to say, I will be there to share in whatever emotional space presents itself.”
I wasn’t prepared for what happened next. Much to my dread, we had to share our written responses with someone else. Here is what I received:
“We are stories. It is who we are and at the heart of what makes us human.”
Wow. Such powerful and insightful words. But they were true. I have been so caught up in the hustle of the math wars. For example, I cared too deeply about what “side” others were perceiving me to be on. That, in turn, ended up defining my self-worth. Honestly, it was exhausting. Over the past several weeks, sharing stories with some awesome individuals, and spending much time reflecting inwards, I have come to the conclusion that I don’t want to choose a “side.” And if that makes me a traitor to some, then so be it. I have come to the conclusion that I cannot make everyone happy with my educational philosophy, but also that I cannot continue my journey at the expense of my own happiness.
I want to spend some time now sharing some of my fondest memories teaching mathematics with you. Below is a picture from when I taught multivariable calculus. If Pi Day happens to fall during class time, I will let the students bring in various pies and share. During the final few moments of this class, we took the picture below. I still keep in touch with some of these students, and am proud to say a couple are graduating from the Faculty of Education this year. One is even designing the ties for my wedding party!
Flipping my linear algebra class was one of the best experiences of my teaching career. With only about 15 students, it allowed me time to really hone in on their mathematical abilities and get to know them on a more one-on-one basis. They were one of the most high-achieving (and high-energy) classes I have ever taught. I think the picture sums up the class very well (this group had won the “Determinants Game Show” and were pretty excited). If you are curious about the E>0 reference, I called them my “epsilons” and told them to always “stay positive.”
Staying positive is something I struggle with on a daily basis, but I am beginning to realize that there are individuals who truly do value my opinions and are interested in the skills that I bring to the table. One thing I have done in my office is create a space for me to collect tokens of gratitude. This space reminds me that, no matter what challenges I am facing, there are individuals who look up to you for motivation and reassurance. It reminds me that I am doing good and that I should be proud of my accomplishments. I highly recommend everyone to do the same, as we all run into days where we doubt our abilities and sometimes we need reassurance that we are on the right path.
So, thank you for allowing me to share my struggles and accomplishments, and my stories (I have a few more logical and argumentative posts coming soon, I promise!). If you would like to reciprocate the story-telling I would honestly love that. Please feel free to share a positive moment as a teacher or educator, to remind yourself and others that we all have moments worthwhile to share and to be proud of.
Let us all break down the walls that have been put up, and let our humanity out.
“Honeydew you want to get married?” “Sorry, but I canteloupe.” -Anon
A few months ago, several teachers and administrators whom I follow on Twitter convinced me that, since I was visiting Ontario already, I should extend my visit to attend the Ontario Association for Mathematics Education (OAME) yearly conference.
After securing a place to stay with some friends, I signed up and had the utmost joy of having to select my sessions from this massive list. Fortunately, I had some idea of what I wanted to see and think about, so it only took me two hours to select my sessions. I made a few excellent selections, but perhaps I will discuss those on another day. For now, there is one sessions and one Ignite talk that I want to focus on for the time being.
I did something a little wild.
I signed up for a Marian Small talk.
For those of you who don’t know of Small’s work, I will let you go explore here. In short, I had predispositions about some of the philosophies that she stood for, but I wanted to go to see what the hype was about. I liken wanting to see her session to my relationship with cantaloupe – I think cantaloupe is gross, but I insist on trying it every few years to ensure that I still dislike it.
So here I am, sitting mid-row with a teacher whom I absolutely adore, and I am trying to keep my cool: (1) because this teacher literally has pet a tiger, and (2) because Marian Small’s talk is starting to be hit or miss for me, with a larger portion of the swings being misses. Aside from the misspelling of Pythagorus (always spell check your slides), I jotted down a few notes and questions from the session:
- Is there any evidence for visuals leading to increased test scores and fact recall? I don’t disagree that visuals/manipulatives are excellent tools to start the learning process; however, this was a big claim that didn’t seem to have any articles attached to it – I want to read the sources!
- I thought it was interesting to hear that students may move the visuals, manipulatives from the concrete world into their minds. I think there is some weight to this, as I use the number line to visualize addition and subtraction of integers in order to determine the overall sign. A few interesting visuals were given that I hadn’t played around with before, such as one for n^2 – 1 = (n+1)(n-1) shown below. Personally, I find these visuals appealing, as they give opportunity to introduce the concept in a tangible way and facilitate the discussion of the symbolic. Of course, my philosophy is that the visual serves the symbolic – the visual is never a means to an end.
- Is one conversation enough for the ideas to “stick”? In my opinion, no – it might be enough to open up the conversation, but more consolidation is needed for movement to long term memory. This is part of a larger conversation I was having with many attendees that skills practice needs to be an integral part of our teaching. The bigger questions for us to think about are “What does this practice look like?” and “How much practice is sufficient?” Hopefully, I will write more on spaced and mixed practice soon, as I think this gives teachers a concrete way of thinking about what this might look like in practice. A few teachers from the Halton Board presented on their journey with these ideas, and it was an excellent and insightful presentation. If you are looking for more information on these ideas, I recommend following The Learning Scientists (@AceThatTest) blog, as they often discuss these ideas.
- In relation to the third point, there was a time in the session in which Marian Small mentioned that there should be more learning questions as opposed to more assessment questions. I’m not sure if this was a hint to more assessment for learning as opposed to more assessment of learning? Either way, let’s say we do agree that we should have more questioning in the classroom (in the “progressive” sense of the word). I am not against this, as we need to believe in the judgement of our teachers to select the most appropriate tools for classroom instruction – and some teachers will be excellent with more progressive types of instruction. To restrict them by telling them how they should teach is unfair. This said, I would argue that teachers using this style of instruction require vast mathematics content knowledge, and quite a bit of pedagogical content knowledge as well. (Here, pedagogical content knowledge refers to items such as knowing/understanding the multiple representations your students have, knowledge of where the learning outcomes are coming from and where they are going, common misconceptions, ability to guide students towards common goal despite various strategies, and so on.) Are we absolutely sure that teachers at the elementary level are obtaining this knowledge base during teacher training or while in-service?
Marian Small then really lost me. A few things were said that I was really at odds with. First, she mentioned that some students find the mechanical operations difficult and that open questioning can be used as a way for these students to see success without looking at the mechanical. I do agree with this statement, and I do agree that using open questions does allow students who struggle to be a bit more active and engaged. Provided we create good questions, it also has the potential benefit of reducing working memory load and ensuring that we are observing/critiquing a student’s knowledge on a particular skill where the mechanical may convolute our observations. For example, observe the completion problem:
I don’t want my students to be punished in part (b) because they were unable to add/subtract rational expressions. If their answer in part (a) is incorrect, then this has potential to convolute my assessment of their skills in part (b). So I understand why we might be interested in using particular types of questions.
What I do worry about is the over-use of this open question methodology. If we constantly are using questions that remove the computational portions of mathematics, then when are students going to learn the computational part of mathematics? Do we value computational skill? If so, we should definitely reflect on our methodology – are we building in time for students to consolidate the mechanical as well as the conceptual? To me, this seems to harken back to points 3. and 4. above.
The next thing that I didn’t agree with was the idea that pictures are more elegant than algebra. While I do value an elegant picture, I can’t look past beautiful advancements in our mathematics history. Algebra is perhaps one of the most elegant recent advancements in mathematics. There is a reason why the language of mathematics rests on the pillar of algebra! It allows us to succinctly write our ideas in a universal language, so that anyone who knows the language can understand our message. Seriously, how cool is that?! Aside from my fan-girling about algebra, the message Marian Small may want to be sending here is that a visual is often helpful in allowing students to see connections and opening up the initial conversations about the workings of algebra. To me, this harkens back to point 2., where I stated that I see the visual serving the symbolic, not being a means to an end in of itself.
Here is where things got a bit interesting for me. One thing I have really been trying to do on my journeys is to hear everyone’s stories. My philosophy on this is that by understanding and sharing our stories we foster connection. Putting ourselves in a place of vulnerability, and seeing that vulnerability reciprocated leads to empathy, a part of our humanity that is often missing within the context of the “math wars”. I have been so honored to share in many of your stories, and for this I am truly grateful.
If you are unsure of the format of an Ignite talk, the premise is that the presenter has five minutes and twenty slides, each slide progressing after fifteen seconds. It is fast and furious and awesome. Marian Small was a part of the Ignite session and did something that I absolutely have to throw respect at. Speaking from the viewpoint of a parent, she laid out her beliefs of what an excellent education system might look like.
It was honest.
It was passionate.
It was the starting point for important conversations.
And that’s when it all clicked for me. Marian Small is respected in the educational community because of the way she is involved. She talks with teachers, not to teachers. She collaborates with teachers in classrooms instead of demanding to teachers about classrooms. Rather than connecting to teachers, she connects with teachers. And while I don’t agree with every message that she speaks, I respect this.
I wonder if this means it is time I give cantaloupe another shot?
This post is the second in a three-part series on teacher training. The first, found here, explored evidence-based practices in teacher training programs. The third, found here, discusses how re-structuring mathematics content and pedagogical content courses to include deep content knowledge and evidence-based practices may be of benefit for our future teachers.
In this post, I want to further explore pre-service teacher education by breaking-down a few key items that I think would strengthen our current programs. Specifically, I will look at K-8 mathematics training, using Brock University’s Concurrent Education Program as a reference point (although other university programs will be brought up as well).
HOW MATHEMATICS DEPARTMENTS CAN HELP: ENTRANCE REQUIREMENTS INTO EDUCATIONAL PROGRAMS
I believe that if we desire excellent teachers of mathematics, then our entrance requirements should be high. To get a sense of what I am looking for, let’s explore Brock’s entrance requirements for their Primary/Junior and Junior/Intermediate programs for 2016.
For entry into the Primary/Junior stream (BA in Child & Youth Studies), students are required to have one Math 4U (grade 12 math) credit, and Data Management is specifically stated in the academic calendar as being preferred. To me, this raises some concerns. Why is Data Management preferred over Advanced Functions or Calculus & Vectors? I am not stating that Advanced Functions or Calculus & Vectors are necessarily better choices, but Brock goes out of the way to state Data Management is preferred and this strikes me as odd. Another concern I have is over the use of calculators. Can we be assured that all high schools in Ontario aren’t using calculators and that our potential future teachers are getting practice with calculations involving fractions, decimals, percent and whole numbers in Data Management? Those in the Primary/Junior stream will be teaching K-6, which includes topics such as comparing fractions, operations with decimal numbers and percent, and operations with multi-digit whole numbers. Does Data Management truly prepare them to master this content? (Go check it out yourself, the curriculum documents are here.)
For entry into the Junior/Intermediate stream, students may choose one of three sub-options: a Bachelor of Arts in Integrated Studies (BA-IS), a Bachelor of Science in Integrated Studies (BSc-IS), and Bachelor of Physical Education (BPhEd). For entry into the BSc-IS stream, students must have one of Advanced Functions or Calculus & Vectors at 70%. For entry into the BA-IS and BPhEd streams, students are required to have one Math 4U credit (one of: Data Management, Advanced Functions or Calculus & Vectors). Note that both the BA-IS and BPhEd do not require a university level mathematics course whatsoever for degree requirement. Personally, if I was a student interested in teaching, but weak in mathematics, I would take the path of least resistance: BPhEd with Data Management as my math prerequisite, then avoid math at the university level altogether.
We are doing these future teachers a disservice by allowing them to actively avoid mathematics. Teacher training programs need to instill the message that mathematics matters. It is also important for our future elementary teachers to be fluent with the major number sets: whole numbers, integers, rational numbers and real numbers. Why is this important? If we have math-avoiding teachers at the elementary level, then our teachers at the high school level have an uphill and potentially impossible battle of teaching both basic numeracy and more advanced mathematical techniques.
To help give the message that mathematics is important, and that our future teachers should have a base-level of mathematical knowledge, I recommend teacher training programs offer mathematics competency exams. To ensure that future teachers are competent in calculations, I recommend that either (1) calculators not be used, or (2) that the exam is divided into two portions: a calculator-allowed portion that tests knowledge of the device rather than computation, and a non-calculator portion that tests computation. Mathematics Departments, working in conjunction with Education Departments, could offer their time to help with test creation and marking. In addition to this, Mathematics Departments could offer a skills-based mathematics course for students who do not pass the competency exam on their first attempt.
Lakehead University has a math competency exam already set up for their teacher training program. Primary education students must show competency up to a grade 7/8 level, and intermediate education students must show competency up to a grade 8/9 level. Since I have not seen a copy of the competency exams, and I do not know the regulations, I cannot comment on how effective they would be; however, I commend Lakehead for having something like this already in place.
HOW EDUCATION DEPARTMENTS CAN HELP: FINE-TUNING CERTIFICATION
As an instructor of mathematics for pre-service teachers, there is nothing worse to hear from a student than “Well, I will just go into Senior Years Education (high school) because then I don’t have to worry about math.” The main issue in Manitoba, is that when teachers get their certification, they are actually allowed to teach at any grade level (see section A1. of this document). So we have students who enter Senior Years training (in, say Arts), get certification, then land a job in an elementary school – effectively avoiding mathematics altogether. How does this help our teachers, and how does this benefit our students? This may not be the case in all provinces, but if it is, then this loophole needs to be shut.
Personally, I am fine with teachers not wanting to teach mathematics (not everyone loves math as much as I do). Maybe they will be an excellent English, History or Art teacher at the high school level – great! Then let’s ensure that, when hired, they don’t teach mathematics. At the high school level, I believe this would be fairly simple – ensure certified teachers are only permitted to obtain positions in high school in their teachable subjects. Perhaps this could be fine-tuned a bit more, but I haven’t put as much thought toward this level as I have for the K-8 level. At the moment, all pre-service teachers entering K-8 teaching certification are in a “one size fits all” kind of training. That is, we generally want all pre-service teachers to master mathematics up to at least a grade 8 level. Is a teacher who obtains a position in K-2 ever going to use mathematical knowledge from grade 8? Probably not. Mind you, it is not a bad thing for this teacher to have this knowledge. And currently, due to the certification process, someone who wants to teach K-2 may end up teaching grade 8, so I don’t think there is a way around a “one size fits all” training at the moment.
I think we can do better, though. What if we broke up teaching certification at the K-8 level into three streams: Primary (K-2), Junior (3-5) and Intermediate (6-8)? This would allow us to fine-tune what kind of content knowledge and pedagogical content knowledge we teach to our pre-service teachers (see diagram below). For example, this would allow a teacher entering certification for K-2 to learn deeply the ideas of cardinality, place value, operations with whole numbers, and methods of how to best teach these concepts to our younger students. Best-practices for teaching students aged 4-7 may include discussion of the importance of rote memorization and mnemonics, repeated and spaced out practice, learning through games or play, which manipulatives are best to use for explaining concepts, common misconceptions students at this age group have, and how mathematical concepts connect to future grade levels. If a pre-service teacher decided to go this route, their certification would allow them to obtain a job at the elementary level teaching K-2, and nowhere else. Similar certification programs could be created for the Junior and Intermediate levels, specifically focusing on the content and pedagogical content knowledge required for those age groups. I believe, if this route of certification is taken, in-service teachers should have the opportunity to upgrade their certification if they choose by taking appropriate content and pedagogical content courses. That is, if they are certified in K-2, they could upgrade to be able to teach K-5 by taking the Junior content and pedagogical content courses. This would allow in-service teachers more flexibility in searching and applying for future positions within their school board, and also ensure that both the Mathematics and Education Departments are happy with their content knowledge.
A big change like this would effectively remove the need for a “one size fits all” approach to elementary teaching certification. This way, our future teachers could develop deep content and pedagogical content knowledge for a specific age-group, rather than broad knowledge for all age-groups. In my opinion, this task would need to be taken up by the Education Departments – effectively lobbying governmental bodies and teachers’ unions to get behind such a radical change. Of course, before such a radical change is taken up, more discussion should be had, as I am simply offering an opinion on the matter.
In closing, I want to mention that successful and meaningful change to teacher training programs cannot be accomplished unless both Mathematics and Education Departments are ready and willing to communicate, collaborate and critically analyze current programs. Remember that this is part of an opinion piece regarding the current state of teacher training in Canada. As such, I welcome constructive feedback on items you think are good and items you think are not so good. Please feel free to offer alternate suggestions and reference any sources you think I might find beneficial. In the future, I would like to spend more time flushing out my ideas and backing them up with appropriate sources. Let me know if you would be interested in reading such a reference!
“Most people believe vulnerability is weakness. But really, vulnerability is courage. We must ask ourselves… are we willing to show up and be seen?” -Brene Brown
It finally happened this semester. I quietly closed the door to my office. I held my head in my hands. And I cried.
It started innocently. I was marking assignments for one of my classes. One of my students recently lost a grandparent, so I was being watchful to see how this student was doing. It seemed that the student was doing fine, receiving credit for all but the last question. However, at the bottom of the page was a message for me scrawled out in pencil… A message stating that times have been exceedingly difficult. A message reflecting that time was moving too fast. A message of apology. And a message of vulnerability.
Instinctively, I began writing. I shared my story of how I lost my grandmothers last year and how challenging it was for me and my family. I shared that I understood. I shared that I was willing to sit in the uncomfortable space of sometimes life sucks and there is nothing we can do about it. I shared my own vulnerability. In that moment of silent connection, that’s when the built-up sadness, stress, frustration, and anxiety over the past year came.
Empathy – feeling with people – is something that I feel is often missing from the current back-and-forth debate in education. How good are we at perspective-taking? When I first entered the educational sphere, I argued with a firm passion. I believed in what I said, and I still do to a certain extent, but what did I have to show for it? I am not convinced that the people who really mattered – our nation’s teachers – ever listened.
Over the past year I have been trying more diligently to hear all sides of the educational debate – I have tried my hand at perspective-taking. I try to fit the shoes. What have I found? I have found hard-working and dedicated educators looking for strategies to help their students learn in a world that is continually buzzing. I have found passionate individuals who believe that there are gaps that need to be filled and are having a go at improving aspects of education and teacher-training. I have found students who respect you and value your opinions when you aren’t afraid to listen.
I have found connection.
I have found disconnection.
I have found individuals who silence communication with verbal attack. I have found friends who, at my most vulnerable moments, erected unscalable walls. I have found vested interests with the primary goal of money-making and dehumanizing teachers. I have found groupthink and perpetuation of the other.
Take a moment to think about someone whom you really dislike. Have you verbally attacked them? Do you speak words behind their back? Next, think of someone whom you cherish. How often do you verbally attack them? How often do you speak words behind their back? Probably less when compared to the individual you dislike, right? In general, the less connection we have to someone, the easier it is for us to attack that individual. Be mindful of this the next time you open dialogue with someone.
Let’s return back to the beginning of the post. Is it OK to cry? Yes. Be proud of your vulnerability and use it to make genuine connection. Don’t be afraid to say, “Hey, I am overwhelmed and need help with this.” Because it is these moments, when reciprocated, that lead us to something that is most valuable: our humanity.