Visibly Random Groups

Peter Liljedahl‘s work on Building Thinking Classrooms has been extremely influential in the education world. In his research, he discusses a collection of high-yield strategies that teachers can employ to help learners engage with work in the classroom and become better thinkers. They are sorted on a continuum of ease of implementation and “bluntness” (requiring less/more fine tuning).

When I first stumbled upon his research (hat tip to the Global Math Department), I immediately began to experiment with the two moves that were the easiest to implement: Visibly Random Groups and Vertical Non-Permanent Surfaces.

I created Seat Finder cards that I would hand out to my students at the beginning of every class as I greeted them at the door. The Seat Finder cards had various algebra problems on them whose solution corresponded to their group number.

Students must solve for x to find their group for the day.

The idea behind this is to help build a classroom culture where students are working collaboratively, and not just with the same people the entire year. In theory, this works great, however, I hit a few kinks along the way. The first kink was that I was not beginning my lessons with “good tasks” at the time (see here for some examples). Second, it quickly became exhausting and time consuming for students to find their seats in this manner each day. I also noticed that students would try and surreptitiously swap cards so they could sit with their friends. Rather than building that culture of community and collaboration, I was battling my students about compliance issues.

So, to alleviate some of these problems, I switched to weekly seat changes, and eventually settled with a change once per unit. Rather than assigning each student their individualized Seat Finder problem, I gave each student a playing card instead that corresponded to a group number. Rather than writing the group numbers on the tables, I opted for math problems whose solutions corresponded to group numbers instead. I created a set for different units of study that we looked at in our classes, which are meant to be solved relatively quickly.

I found that having one problem per table helped students get into their seats more quickly, and they often helped each other if some students were unsure of how to solve the problem.

Polynomials seat finder (Table Version)

Note that while students seats stayed the same for the course of a unit, they were still expected to work with others on various tasks I assigned during class. This gave them the comfort and consistency of knowing where they were sitting from day to day, but also the ability to interact with a variety of classmates.

For a link to my downloadable Seat Finders and templates, click here.

Talk Less, Ask More: My Goals and Set-Backs

This week I’ve had some great lessons, and some awful ones. Looking back at what I had done differently in the good versus not-so-good lessons, I realized that one of the biggest differences was the amount of “telling” I was doing in one class versus another. It didn’t matter that I had amazing visuals and was super enthusiastic about the content I was teaching; if I talked too much, students would start to zone out. Compound this with the fact that we are distance learning and all of my students are English language learners, we now have wi-fi/connectivity, audio, and language learning issues all thrown into the mix.

The one who does most of the talking, is doing most of the learning.

(Something I’ve heard from multiple sources throughout my teaching career)

At this point, I have to slap myself on the wrist because I know better, so I need to do better.

In yesterday’s class, I consciously made an effort to talk less and ask more questions. I also explicitly told my students that my goal as a teacher is to never tell them an answer, but to just show them the way. Classic case of easier-said-than-done.

 I realize, with a sudden mixture of nervousness, trepidation, and excitement as I’m writing this, that this might be the first time in five years of teaching that I have really made a conscious effort to take Cathy Fosnot’s advice to heart. She writes,

Don’t try to fix the mathematics; work with the mathematician. The point is not to fix the mistakes in the children’s work or to get everyone to agree with your answer, but to support your students’ development as mathematicians.

Cathy Fosnot

On the surface, I’d like to think I was doing my best to project a calm, neutral tone as I jotted down notes while students shared their thinking. I wrote everything down, regardless of whether their strategy was “right” or “wrong”. Meanwhile, it felt like Hermione Granger was living in the back of my mind jumping up and down going “Pick me, pick me!! I know the answer!!” Talk about my “rescue the student” instincts being on overdrive!

Can’t I just tell them the answer already?

In the past, I would have eventually given in to those instincts and immediately correct any mistakes that came to my attention. I tell myself that this is okay, because if I don’t, my students will continue to make those fundamental math errors, divide by zero, and initiate the end of the universe as we know it. I also think that deep down, that hidden behind this instinct is fear, fear that I can’t help them get where they need to go without just giving them the answer. Although, impatience is an equally guilty accomplice here in my crime of robbing students of a perfectly good learning moment.

This time, however, I tell myself a different story. I learn to trust myself and my students a little bit more by just letting them get where they need to go, in their own time and in their own way. This too, is a little scary.

The Lesson

To add some context, here’s a bit about how my lesson went.

The goal of today’s lesson was to introduce the idea of trigonometric identities, collect some strategies that may be helpful in identifying whether a given statement is true or false, and then work on moving towards what it means to rigorously prove the truth or falsity of a mathematical statement.

 I began the class by doing a modified version of an “Always, Sometimes, or Never True” activity with radicals (trying to introduce some interleaving here) from the Mathematics Assessment Project and called it “Truths and Lies”. I asked students to tell me which statements they believed to be truths and which they thought were lies, and to share their thinking on Padlet.

After about ten minutes of individual think time, I selected a few student strategies and had students explain them to the class. Here’s what we came up with:

Strategies Used:

  • Plug in a number for x and check to see if both sides are equal
  • Start with LS or RS, use algebra to show it is equal to the other side
  • Assume the statement is true. Square both sides of the equation. (If both sides are equal after squaring, then the statement is true). 

Next, I showed them this image about the different Levels of Convincing from Robert Kaplinsky’s site.

We then revisited each strategy and I asked students to mentally place each of these strategies fell on the spectrum of least to most convincing. Ideally, I would have given more time for students to really think this part through, but since we are doing distance learning, I was eager to get to the real meat of today’s activity, which was to prove trigonometric identities. From there, I took on the role of prosecutor and started to stir up some trouble.

For instance, in statement 1), we can demonstrate the statement is false by finding a value of x that shows LS does not equal RS, however, I argued that x = 5 worked, so wouldn’t that make the statement true? What I’m getting at here is that I want students to be able to articulate what exactly are we asking when we ask whether or not a statement is true? That it must be true all the time? Or only some of the time?

Then, I attempted to tackle the “squaring both sides” strategy… Couldn’t we also use same reasoning to show that 1 = -1?? (Can you see why?)

At this point something really amazing happened, and that was when a student interrupted me and said, “Ms. Soo, I just thought of another way to explain this!” The student was able to connect what we were doing to our study of transformations of functions from a previous unit.

I couldn’t —

keep my poker face, that is. This was me:

For the remainder of our lesson I had students work independently on the following:

My goal for them was to use different strategies and methods to try and “prove” or “justify” which were truths and which were lies. Students always have the option of messaging me privately for hints or advice if they were stuck, very few did.

After about 15 minutes, I asked students to send me pictures of their work and we could start talking about some strategies they used. The one mistake I see students make when “proving” trigonometric identity is to start by assuming the statement is true and start manipulating both sides of the equation.

A common approach to “proving” trigonometric identities from students.

Instead of telling students WHY they can’t do that, I referenced my earlier example of how, by the same logic, we can prove that that 1 = -1 and asked them, WHERE did the mistake occur?

Let’s assume. the statement 1 = -1 to be a true statement.

Next, let’s square both sides of the equation. Doing so, I get

1=1

Therefore, it must be the case that 1 = -1. (End of proof).

Getting students to where I wanted them to be was really challenging because many were focused on the math, and not the logic of the argument itself. They focused more on the operation of ‘squaring’ and how we need to keep in mind both positive and negative square roots, which is certainly a valid piece of mathematical insight, but again not where I needed them to be.

Since we only had about 5 minutes of class left, I decided to pause the discussion there and ask students to write me a 3 – 5 sentence of the strategies we used to justify whether a statement is true or false.

What Went Well

I stuck with my goal.

Where I Need Help

Right now, students still don’t understand what a proof is. I want students to be able to articulate that while plugging in values, and graphing both sides of an equation are helpful strategies to show why a statement might be true, they don’t constitute enough rigour to show that a statement is always true.

How do I get students to this point without just handing them the answer? How can I do this effectively in an online setting? They also have a common assessment (assignment) coming up in which they will be asked to prove trigonometric identities, and the quickly approaching deadline makes me feel anxious to default to just tell students the answer.

Any tips, suggestions, or feedback would be greatly appreciated! Please leave your comments below.

Empowered Problem Solving Workshop: My Takeaways

My takeaways from #epsworkshop (April Soo)

I love it when professional development is purposeful and practical.  I’ve been following Robert Kaplinsky for some time now and finally decided to enrol in his Empowered Problem Solving Workshop.

My reflection post in the last module of the empowered problem solving workshop.
My reflection post in the last module of the workshop. Sad it’s over…

I don’t have time for problem solving in my classroom.”

TRUST me, I’ve been there. The first time I ever taught Calculus, my talk time during an 80-minute block was probably at 50-80%. It was awful, I was so dehydrated. It also didn’t help that I did not have a strong enough grasp of the material that I could deliver problem-based lessons with any sort of confidence. I was simultaneously teaching and relearning the material myself so how could I expect my students to develop these deep understandings when I was barely keeping my head above water?

No bueno…

Looking back, I realize that trying something is always better than nothing.  Problem solving isn’t something you do “if you have time for it,” like at the end of a unit. Because you know what? You’re never going to have time for it. You’ll always feel like the time could be better used for review, a project, to reinforce a skill…etc. Problem solving is not something you should “make time for”, it needs to be integrated into the content we teach. I would argue that the heart of mathematics is problem solving. The sooner we realize that math isn’t just about getting the right answer, passing a test, or even getting into university, the sooner we can teach in a way that honours what doing mathematics is truly about.

Why Problem Solving?

My students lack the basic skills and understanding to do these types of problems.”

If that is what you’re thinking, know that I too, have had this same thought. Herein lies the beauty of problem-based lessons: students don’t need to be pre-taught skills or content, they can learn them along the way.

“I’ve tried problem solving before and it doesn’t work. Students just want to be told what to do.”

Guilty. I’ve been there too. It’s not going to be perfect the first time you do this. Students WILL resist, and you need to persist. If you don’t genuinely believe that problem solving is worth the time and effort, your students won’t buy into it either.

When I first tried problem-based lessons, I did not spend enough time anticipating student responses and was taken off guard by solutions or strategies I hadn’t thought of. I tried to lead meaningful discussions about student work, but because I wasn’t getting the engagement I wanted, sometimes ended up making the connections for the students (I’m still working on scaling back my “rescue the student” instincts). Success, however, comes in small doses, like getting a student who normally never raised their hand to try a problem on the board, or maybe just seeing a decrease in off task behaviour.

Teaching problem-based lessons takes effort, from the student AND the teacher, but that is precisely why its so awesome. Students aren’t just passive recipients of knowledge, and teachers don’t need to spoon feed their students.

My Biggest Takeaways

  1. Problem-solving: Just DO IT!

2. Be deliberate about how to facilitate meaningful discussions in math. Often, we get to an answer and that’s it. Full stop. Getting to the last act of a 3-Act Math Task doesn’t mean that the learning stops there. Here is a wonderful opportunity to discuss various approaches to the problem, potential sources of error, limitations of our mathematical models, and to make connections between different solutions. This is an area where I feel I need the most practice, and it is also most difficult to implement during this time of online learning due to COVID-19. I’m limited by the fact that I cannot circulate the classroom or peep over students’ shoulders to see where they are at, but I am trying to find alternative ways to connect.

5 Practices for Orchestrating Discussion

Here’s a snapshot of me working out a selection strategy for sharing student work, and anticipating questions that might be helpful to ask:

My rough notes as I thought about how I might lead a discussion about student work.
Which responses would you pick to share? In what order would you share them?

3. You can always add information, but you can’t take it back. Dan Meyer refers to this as turning up the Math Dial. Robert Kaplinsky talks about “undercooking” our students (like you would a steak). Ask questions in a way that ranges from least helpful to most helpful to give your students a chance to make connections for themselves.

4. Ask yourself “Why” more often. Why am I doing this problem? To introduce a new concept? Get my students used to productive struggle? Problem completion?  Be intentional about the purpose of your lesson and what can be realistically achieved.

5. Ask better questions. Shallow questions tend to lead to false positives. A student may appear to have procedural knowledge, fluency, and conceptual understanding, when in reality they are just good at replicating the work that you do (me in school…). You might be asking, “How do I really know if my students have the components of mathematical rigor?” Check out Robert Kaplinsky’s Open Middle problems and Depth of Knowledge Matrices.

Depth of knowledge matrix (credit to Robert Kaplinsky)

What do you do when students submit low quality/low effort work?

I’ve been taking an online workshop to learn more about practical ways we can implement problem-based lessons in our math classrooms called Empowered Problem Solving by Robert Kaplinsky (#mathhero #teachercrush). In one of the workshop modules we troubleshoot various issues that may arise throughout the process of teaching a problem based lesson, for instance:

  • What happens if students don’t ask for information that they need to solve the problem?
  • What do you do if a student shares a strategy that you don’t understand or did not anticipate?
  • What do you do when students submit low quality or low effort work?

That last question really had me thinking a lot about assessment. When students submit low quality work it is often because they don’t know what the expectations are. Something I do quite often in my classes is share student work samples after an assignment or test to address common errors or mathematical practices. Here’s a brief overview of my journey in providing feedback for my students:

“What Should My Answer Look Like” Posters from MathEqualsLove,
Examples are from my class 🙂

I don’t make enough time for level 3 work, and I should. Within a single semester, my goal is to give students at least two opportunities to do meaningful peer assessment. Of course, I anticipate this to be a gradual process, and it might take some time to get to a point where students can comfortably and confidently do peer assessment.

Assessment is difficult; even with a simplified assessment scheme one two teachers may assess the same student work slightly differently depending on their interpretation of what is “correct” or what qualifies as “sufficient reasoning.” Unfortunately these discrepancies will arise no matter what, but I think there is a lot of value in putting the students in our shoes and giving them opportunities to assess each other’s work.

Inspired by the Empowered Problem Solving Workshop, I’ve created a Mathematical Peer Editing Checklist and Feedback Form with practices I value and that I think is general enough to be used with most and/or all problem-based lessons. I’ve also incorporated an “overall feedback” section in the form based on Kaplinsky’s Levels of Convincing (originally inspired by Jo Boaler #mathhero #teachercrush) that asks students to rate each other’s mathematical writing based on how convincing they think their argument/work is.

  • Do you think this framework would work with your students?
  • How would you modify it to make it better?
  • Anyone have suggestions for a more concise title, as opposed to “Mathematical Peer Editing Checklist and Feedback Form”?
  • Thoughts on my use of the word “writer” to describe the student who’s work is being critiqued?
  • Other thoughts?

How Old is the Shepherd?

We asked 101 high schoolers the following question: 

There are 125 sheep in a flock and 25 dogs.
How old is the shepherd?

The question is an invitation to take a closer look at the kinds of mathematics that we are asking students to engage with in our maths classrooms today. What does it mean for us as educators when students give responses like 130 because 125 + 5 = 130 or 25 because 125/5 = 25? Moreover, what does it mean for us as educators when we expect these responses from students? 

I first heard about the shepherd question through Robert Kaplinsky though the question has its origins based on research by Professor Kurt Reusser from 1986, possibly sooner

Context

The data was collected via an online survey (on account of school closures due to COVID-19) and was given to students in China earlier this year. Our initial goals were: 

  • To collect some data regarding sense making in mathematics amongst our high schoolers (grades 10 – 12)
  • To analyze the data and assess what this means for us as educators. Are there differences in responses amongst the different grade levels? Are there gaps in student learning that we need to address? If so, how might we begin to address those gaps? 
  • To use this activity as a way to begin a dialogue with students and teachers about sense-making in mathematics 
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The survey was conducted via Microsoft forms.

Results

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Of the 101 student who were surveyed: 

  • 25 stated there was not enough information to answer the problem, or did not supply a numerical response based on the constraints identified in the problem
  • 74 gave numeric responses
  • 2 students did not answer the question (e.g. one student wrote “nice question”) 

At first, this data seems to be consistent with the results from Kaplinsky’s experiment with the 32 eight graders, in which 75% of them gave numerical responses by using random addition, subtraction, division, or multiplication of 125 and 5. Upon closer examination, however, we see that of the 73 that gave numeric responses, 28 used random math procedures, thus not making sense of the problem, but 45 of those students gave some sort of reasoning independent of the problem to support their numeric responses. 

Sample Responses

Students that gave numeric responses by combining 125 and 5 via random math operations (not making sense): 
Note that a couple of students pointed out that there seemed to be an issue with the problem, but proceeded to give an answer anyway.

Students that did not provide a numeric response (making sense): 

One response in particular really blew me away (click to expand the image): 
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This superstar response really blew me away.

​Not only did this student state that the question did not give enough information to provide a specific answer, they used what information was presented in the problem, along with sources to support their thinking, to deduce an age range for the shepherd! Wow. How can I get the rest of my students here?? 

This is the point where I began to see another category emerge… Students who provided a numeric response, but justified their answers outside the range of expected range numeric responses such as:

  • 125 + 5 = 130
  • 125 – 5 = 120
  • 125/5 = 25 
  • Or other such random combinations of numbers. Note that no one responded with 125*5 = 625  as they seemed to realize this may be ridiculous age for a shepherd to be. Although one student did provide a response of 956000 through a series of random additions and subtractions.

I referred to this new category of responses as “supported guesses.” To be honest, I had difficulties categorizing some of these responses as “making sense” after seeing the superstar example from above, but ultimately decided that anything other than random math was a step in the right direction, although you will probably agree with me that some responses seem to employ more evidence of reasoning than others: 

​Results

​My Take-Aways

I was quite blown away by the number of students that treated this as a “trick” question and thus gave a wide range of responses, which ranged in creativity and depth of thinking. Like I mentioned, I found it difficult categorizing some of these responses, and found that after reading that Superstar response from above, my expectations rose (not necessarily a bad thing, but definitely made categorizing more difficult). 

Some factors worth considering: 

  • The responses were collected via an online survey. Would results have differed if this was done face to face? Did students try looking up the problem before attempting to solve it? 
  • Students feeling like they did not have intellectual autonomy; not wanting to question the questioner due to respect for authority. 
  • The old “my math teacher is asking me this, so I must calculate something” trick. 
  • Cultural factors may be at play here. Perhaps students have seen some version of this problem before, thus accounting for the varied supported guesses observed. 


It is also worth noting that in a follow-up reflection activity with students, some pointed out the need for teachers to ask less ambiguous problems, a few attributed their responses to poor understanding of the problem due to language barriers, while a fair number mentioned the importance of practicing different kinds of problem solving to develop critical thinking skills. This, I think, is a step in the right direction.

Descriptive Statistics and Deceptive Description

Teachers are getting a bad rap these days. To put it in perspective, my own mother — to whom I am and shall always remain eternally grateful for —  expressed her annoyance at the fact that Ontario teachers were, yet again, going on strike. (I will also add here that she is also very supportive of the fact that her own daughter chose teaching as a profession). Like others, she felt that the strikes are an unnecessary waste of time, making teachers appear selfish and lazy. When asked what information she had to support her claim, the figure “100K” came up in conversation. WHATHow much are teachers making a year?

$100 000. One hundred THOUSAND Canadian dollars. The supposed “average” salary Ontario teachers make a year.

Reported source? “The government.”

Had it not been for the fact that a) my mother has a tendency to exaggerate the truth and b) I am a teacher myself, I may have been inclined to side with her claim. To add a bit of context: I have been teaching for five years internationally and making nowhere near that figure. For me to be earning 100K a year, I would need to have my masters degree and an additional fifteen years of full time experience. 

Let’s Talk About Averages

“Average” is a misleading term; it can refer to the mean, median or mode. In statistics, we call these “measures of central tendency.” Let me borrow an example from Wheelan’s book (Naked Statistics) to make a point.

Suppose five people are at a bar, each earning a salary of $35k a year. Undisputedly, the average salary (by all counts) of the group would be $35k. Typically, when we hear the word average, we equate it with the mean, which is the sum of all the points in a data set, and divided by the total number of values within the set. 

Suppose Bill Gates walks into the bar,  with a salary of $1 billion a year, bringing the average (mean) salary to $160 million. The reported figure, while still accurate, is not a fair representation of the average earnings of the majority of individuals in the group. 

In this case, the knowing the median (middle value when all values are arranged from smallest to greatest)  provides a bit of context. After all, the difference between 35 thousand and 160 million is no small sum. 
This is a classic example of how precision can mask accuracy. Think about any time you’ve heard a number or figure reported in the news, consider the following statements, for instance: 

Statement 1:  “99% of statistics are made up” (Ha!)

Statement 2: “I have here in my hand a list of 205 — a list of names that were made known to the Secretary of State as being members of the Communist Party and who nevertheless are still working and shaping policy in the State Department” – Joseph McCarthy, a US previous senator (1950) 

Don’t these it seem to bring credibility to whatever claim the person or organization is trying to assert? The first statement is, of course, made up. As for the second statement, it turns out that the paper had no names on it at all. Statistics is a tool that helps us bring meaning to data, but can be abused for nefarious purposes if wielded irresponsibly.

We should be cautious

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While math may be infallible, we are not. No matter how convincing the data may be, there is always more than one way to interpret it. It’s a little like telling your friends and family that the guy you just met “has a great personality,” which almost always implies that there is some other flaw or red flag that has not been said (Wheelan 37). 

So, back to the this 100k salary I’m supposed to be making… How did they get this data? What are the demographics of the teachers being surveyed? (It makes a difference if the majority of teachers who have been working full time in Ontario have at least 15 years of experience under their belt). Are they including retired teachers? Teachers who have recently been laid off?

I tried to trace the origins of where this figure of 100k came from. After a bit of digging, I think its likely that my mother mis-reported the figure she heard from sources that gave out misinformation.  


​[NOTE FROM THE AUTHOR: I purchased Naked Statistics by Charles Wheelan many years ago, thinking its an important book to add to any Math Teacher’s arsenal (and it is!) but had only gotten through the first three chapters before dismissing it for another read. It is not a boring book – quite the opposite in fact – but I felt that mere passive reading was not enough for me to really retain the important ideas and intuition that Wheelan is trying to impart to his readers. This time, I’m giving it another chance and plan to summarize material I am learning, relate it to my own experiences, and share that learning here on my blog.]  

Map Projection

We’ve been looking at map projections for my masters course and I continue to be blown away by how embedded mathematics truly is in our every day lives. As a self-identified directionally-challenged individual, geography and anything like it is to be avoided at all costs. I find myself at my wit’s end now and have to admit that even maps hold a lot of mathematically intriguing ideas that are worth exploring. The course I’m taking now is called “Math for Global Citizens”, offered at the University of Waterloo to MMT students, taught by Judith Koeller and is arguably one of my favourites in the program. 

 

The problem of the “flat earth” has been around for centuries. It is believed that as early as the year 354, pre-medieval scholars asserted that the earth was in fact spherical (University of Waterloo). The problem for map-makers, then, is to find a way to depict a spherical object on a 2D surface, and this is turns out to be an impossible task. Take a look at the animation below for what’s called a “Myriahedral projection” developed by Jack Van Wijk from the Netherlands. 

 

The idea behind a “Myriahedral projection” is to split the earth up into polygons, thousands of them, in order to preserve both shape and size of major land masses or bodies of waters (see article here). Map projections have not always been so advanced however. 

 

In trying to depict a spherical surface onto a 2D plane, one can try to preserve distances, shape, areas, or shortest distances between points by straight lines. It is impossible to have all these desirable properties in one map. For instance, the Mercator projection map is the one that we are probably all most familiar with as it preserves angular distances, making it easy for navigation, but it drastically skews areas the further away the land masses are from the equator. See this true size (thetruesize.com) comparison below, showing how large the continent of Africa actually is compared to  the US, China and India:

On that note, I would highly recommend checking out thetruesize.com and just playing around. 

 

Here’s another great video explaining “Why all world maps are wrong” that was recommended to me by Mr. Schwartz, a geography teacher and the humanities Department Head at my school. 

 

Ch 1. What’s the Point?

What is the point? The point is to do math, or to dazzle friends and colleagues with advanced statistical techniques. The point is to learn things that inform our lives.

– Charles Wheelan

[PREFACE: I purchased Naked Statistics by Charles Wheelan many years ago, thinking its an important book to add to any Math Teacher’s arsenal (and it is!) but had only gotten through the first three chapters before dismissing it for another read. It is not a boring book – quite the opposite in fact – but I felt that mere passive reading was not enough for me to really retain the important ideas and intuition that Wheelan is trying to impart to his readers. This time, I’m giving it another chance and plan to summarize material I am learning, relate it to my own experiences, and share that learning here on my blog.]  

I wrote about why statistics matters in a previous post. Here, I continue to elaborate on the point as I summarize my biggest takeaways from the first chapter. This chapter provided an overview of big ideas in statistics that we’ll be learning about throughout the book. 

Description and Comparison 
Descriptive statistics is like creating a zip file, it takes a large amount of information and compresses it into a single figure. This figure can be informative and yet completely striped of any nuance. Like any statistical tool, one must be careful of how and when we employ such figures and the implications it might have on the audience. 

So a descriptive statistic is a summary statistic. Let’s start with one that many of you may already be familiar with – GPA. Let’s say a student graduates from university with a GPA of 3.9. What can we make of this? Well, we might be able to discern that on a scale from 0 – 4.0 a GPA of 3.9 is pretty darn high. But some universities grade on a scale of 0 – 4.3, accounting for a grade of A+. What this simple statistic doesn’t tell us is what program did the student graduate from? Which school did they attend? Did they take courses that were relatively easy or difficult? How does this grade compare with others in the same program? Wheelan writes, “Descriptive statistics exist to simplify, which implies some loss of nuance or detail (6).”

Inference 
We can use statistics to draw conclusions about the “unknown world”  from the “known world.” More on that later. 

Assessing Risk and Other Probability Related Events
Examples here include using probability to predict stock market changes, car crashes or house fires (think insurance companies), or catch cheating in standardized tests. 

Identifying Important Relationships
Wheelan describes the work of identifying important relationships as “Statistical Detective Work” which is as much an art as it is a science. That is, two statisticians may look at the same data set and draw different conclusions from it. Let’s say you were asked to determine whether or not smoking causes cancer? How would you do it? Ethically speaking, running controlled experiments on people may prove unfeasible, for obvious reasons. 

An example Wheelan offers here goes something like this:
Let’s say you decide to take a few shortcuts and rather than expending time and energy into looking for a random sample, you survey the people at your 20th high school reunion and look at cancer rates of those who have smoked since graduation. The problem is that there may be other factors distinguishing smokers and nonsmokers other than smoking behaviour. For example, smokers may tend to have other habits like drinking or eating poorly that affect their health. Smokers who are ill from cancer are less likely to show up at high school reunions. Thus, the conclusions you draw from such a data set may not be adequate to properly answer your question. 

In short, statistics offers a way to bring meaning to raw data (or information). More specifically, it can also help with the following:

  • To summarize huge quantities of data
  • To make better decisions
  • To recognize patterns that can refine how we do everything from selling diapers to catching criminals
  • To catch cheaters and prosecute criminals 
  • To evaluate the effectiveness of policies, programs, drugs, medical procedures, and other innovations
  • To spot the scoundrels who use these very same powerful tools for nefarious ends 

(Wheelan 14)

Lies, damned lies, and statistics.

 – Mark Twain

Why We Should Care About Statistics

It’s easy to lie with statistics, but it’s hard to tell the truth without them.

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​[PREFACE: I purchased Naked Statistics by Charles Wheelan many years ago, thinking its an important book to add to any Math Teacher’s arsenal (and it is!) but had only gotten through the first three chapters before dismissing it for another read. It is not a boring book – quite the opposite in fact – but I felt that mere passive reading was not enough for me to really retain the important ideas and intuition that Wheelan is trying to impart to his readers. This time, I’m giving it another chance and plan to summarize material I am learning, relate it to my own experiences, and share that learning here on my blog.]  

A couple of days ago, my younger brother, who just started his first year in university in the Fall, was complaining to me about the woes of student life; in particular, the obsession with grades and the paradoxical lack of willpower to work for them. Having taken an accounting class together, his friend recounted to him that it was, “The sketchiest 90 I ever received.” Let’s break that down for a moment. Humble brag? Yes, but what he really meant was that his friend was blindly memorizing formulas, plugging and chugging without any idea how they were derived and why they are meaningful. 

Does that sound familiar? How many of you have had similar experiences in math class? I know I have. Not just math, but in science, language arts, history… sometimes it feels like we are just memorizing facts in isolation without an understanding of their greater purpose. To be fair, I’ve taken statistics classes that feel no different, a series of formulas that need to be applied to raw data. What makes statistics inherently different, however, is that unlike calculus or algebra courses, which often teach skills in isolation of their applications (to which I will argue there is intrinsic value in knowing and learning, another post perhaps) statistics IS applied mathematics. Every formula, number, distribution test…etc. is meant to clarify and add meaning to everyday phenomena (though, when wielded improperly, can have the opposite effect).

Statistics are everywhere – from which are the most influential YouTubers, to presidential polling to free throw percentages. What I love about this book is that it focuses on building intuition and making statistics accessible to the everyday reader. A quote by Andrejs Dunkels shared by the author, “It’s easy to lie with statistics, but it’s hard to tell the truth without them.”


Teaching for a Math Mindset: A Not Yet Successful Study

So I was lucky enough to have the opportunity to teach in the Head Start summer program at my international school here in China. The program is intended to help students going into high school to gain exposure to full English immersion classes in Math, Science, Socials, and Language Arts. I taught four blocks a day for 70 minutes each. Each class had anywhere between 12 – 16 students. Ten days straight; on the one hand, no break (kinda brutal), and on the other, open curriculum (YES! Free reign).

I had lofty plans. I’d been refreshing myself on Jo Boaler’s work about mathematical mindsets (see my previous ramblings here). I was going to do a little study.  Please note that I do not have any experience whatsoever doing educational research. While I have a general understanding of the scientific method, I was mostly doing this out of pure curiosity and a desire to become a better teacher.

Like all good mathematicians and in the name of good science, it was perhaps inevitable that first time was not the charm, and rather than have a very successful, replicable study, I instead gained some knowledge about how I might proceed in the future. Nice.

Content that I had planned to cover in 10 days would have taken closer to 18. The students had an incredible range of English speaking ability, with drastically varied dynamics between groups of students. The schedule did not operate on a cycle, so I saw the same group of students at the same time each day, which definitely influenced their learning experience. For instance, Group C who were absolute angels and ready to learn each day in my first period class were exhausted by the time they got to third period, which led to more behavioural problems in the classroom.

STUDENT DYNAMICS
Group A: A challenging group. I saw them the period right before lunch each day and there was a group of four students who were unable to sit still and wandered the class during inappropriate times, such as in the middle of me giving instructions. I lost my cool on this group; shame on me because I wasn’t able to regulate my emotions and respond calmly to the situation. Just to clarify, a “losing my cool” moment for me doesn’t mean shouting or yelling, which is neither helpful nor productive. I simply raised my voice to get the students attention. But, in that moment,  I had lost my cool because I let the students dictate my response rather than carefully assess the situation and respond calmly and accordingly.

Group B: Did absolutely anything in their power to NOT pay attention. Would whine anytime I introduced a new activity. Would put their heads down and sleep in class. I saw this group after lunch each day, they were my last and perhaps most challenging class because of the incredible amount of sleepers and students who wanted to do absolutely nothing. There were definitely some gems in this class that would have benefitted from being in a group with other, more responsive students. Lots of patience and flexible teaching strategies required.

Group C: The first group I saw each day and by far the best group. Students had a decent command of English and I rarely had to repeat myself. They would listen and follow instructions the first time. Students would always do as they were asked. The challenge with this group was pushing them to work slightly beyond their zone of proximal development.

Group D: A diverse group with students who always wanted to be two steps ahead, students who needed a lot of personal assistance, students who got distracted easily, and students who were happy with just coasting along.

HOW I COLLECTED DATA
I used Boaler’s Mathematical Mindset Teaching Guide as a self assessment tool for how I was and was not strengthening growth mindset culture in my math classroom. I wanted to focus on changing students’ inclinations towards math learning, challenging those who believe math is a subject that defies creativity and passion, and pushing those who already saw themselves as “math” students to expand their definition of what math is. With the help of my math mentor, I settled on collecting data through a mindset survey.

Students took a before and after survey. I added two prompts on the after survey that required students to provide written answers to the following:
– What I think math is…
– How math class makes me feel… 

A source of error here is that for students with low English level, they may not have fully understood the meaning of the statements they were agreeing or disagreeing with. Another possible source of error (though unavoidable) are those students who “did” the survey by randomly clicking boxes just to appease their dear teacher.

HOW I TAUGHT
I chose content from YouCubed’s Week of Inspirational Math. I chose these tasks because they were all low-floor, high-ceiling tasks and were designed to build good mathematical habits of mind. For example, on day 1, we did an activity called “Four 4’s” which encouraged students to think creatively and work collaboratively to come up with as many expressions as they can that equal the numbers 1 – 20 using only four 4’s and any mathematical operation of their choice (see picture below).

Other activities we did:

  • Escape Room Challenge: A mixture of math puzzles, grade 9/10 content from trigonometry, polynomials, and simplifying expressions. Designed by me and was meant to last one period, ended up taking two.
  • Number Visuals: Students examined visual representations of numbers 1 – 36 and were asked to identify and describe patterns (prime v composite numbers, factorization…etc.).
  • Paper Folding: An activity from YouCubed that challenges students to slow down and justify their answers. (Meaning that, anybody who claimed they were “finished” after five minutes clearly did not understand the activity…)
  • Movie: Students complete an agree/disagree questionnaire and watched The Man Who Knew Infinity about an Indian mathematician named S. Ramanujan making waves in England. Great movie starring Dev Patel. We did a discussion circle afterwards that touched base on prompts from the questionnaire that students were interested in exploring. (E.g. “Math is creative”)
  • Pascal’s Triangle: Find and describe patterns hidden in Pascal’s triangle.

In terms of assessment, I wanted to stay as far away from tests or quizzes as possible. Instead, I focused on providing students with specific, written feedback on their journal entries, group quizzes, and one final presentation at the end. I wasn’t concerned so much with what they knew, but rather the process through which they were learning and engaging with the material.

The Four 4s Activity

Students working on the escape room activity.

Looking for patterns in the Visual Numbers activity

That time a puppy wandered into my classroom. Oops.

RESULTS

Before

After

Select responses to “What I think math is”
Interesting”
“The most important things we need to learn”
-“Have unlimited creativity”
“Magic”
“Subject between creative and and teamwork”
“is very interesting. make my brain growing”
“beautiful”
“fantastic”
“Math makes me hate and love”

Select responses to “How math class makes me feel”
“Better”
“Moer interesting than chinese class”
“It may not very interesting, but OK”
“happy that I learned a lot”
“I feel very good, I meet very good teacher also know the very good friend in the math class”
“exciting”
“I feel happy when I fiand the ancer”
“free”
​”Good! make me more confedent”

WHAT I LEARNED
​A majority of students already had tendencies towards a growth mindset in mathematics, perhaps as a result of the general high regard Chinese people hold for mathematics as a subject. For the most part, students liked math and saw themselves as capable of achieving if they worked hard enough. Of the 59 students I taught, a small number of students (three or four) were of the opinion that they were “just not math people” and were extremely hesitant in trying.

In the end, I can’t really say definitively which factors of my teaching influenced (or failed to influence) a stronger growth mindset towards maths. What I do know is that the switch to low-floor, high-ceiling tasks was extremely freeing — for me and for the students. It allowed us to take a concept or idea as far as we wanted to go. There was no script or prescribed problem set that the students had to work through in increasing levels of difficulty, but rather a greater depth of thinking, and the time and space for that thinking to happen. Despite (or maybe thanks to?) the lack of testing (there were none), students still engaged with the tasks and content at high levels, drawing conclusions they might never have done with a pre-made worksheet of the skills they were supposed to practice.

By building a stronger focus on increased depth of knowledge, it then follows that a necessary norm to advocate would be that math isn’t about speed. When people refer to themselves as not “math people”, that’s usually what they refer to, the fact that they aren’t fast at mental arithmetic. But math is so much more than that.

In all, while it is hard to say from the students’ perspective whether or not they appreciated a stronger switch to teaching with mathematical mindsets in mind, I know that for me it resonates as a noble endeavour. Yes, it is much easier to write a test and spend 70 minutes of your life making sure no one cheats. But take that same test, rip it up, and replace it with a diagram, an equation, a single question, a blank sheet… and possibilities begin to emerge. Some groups may reach a higher level of understanding and some may not, but then again, we teach students, not subjects.

Visual Patterns and Mathematical Mindsets

This summer I enrolled in a course called, “How to Learn Math for Teachers,” taught by Professor Jo Boaler, a Professor of Mathematics Education at Standford University. The course brings together best practices from research on brain growth and classroom techniques for anyone who’s curious about engaging students in mathematics education.

One of the course modules talks about creating or giving students tasks with a growth mindset framework, which has the following components:
1. Openness
2. Different ways of seeing
3. Multiple entry points
4. Multiple paths/strategies
5. Clear learning goals and opportunities for feedback

​The example that is given from the course is as follows:

Without any numbers or formulas, describe how you see this shape growing.

A teacher might ask, “There are more squares in case 2 than in 1, where are they? There are more squares in case 3 than in 2, where are they? Describe what you see.”

Go ahead and try this task on your own first. Watch the video to see examples of different responses (skip to 3:50).

This type of task is referred to as a “low entry, high ceiling” task, as anyone, regardless of their skill level can engage with the question, “How do you SEE this pattern growing?” and the question can be extended to higher levels. youcubed.org has  tons of videos, teaching resources, and research papers that challenge the status quo on what it means to be “mathematically minded”. Check them out!

I decided to try a similar task with my Pre-Calculus students in China, and picked a pattern from Fawn Nyugen’s site visualpatterns.org

Based on my students with Chinese students thus far, many of them are quite baffled whenever they get an open task like this. They are used to the typical, “how many squares are in the next case? The 100th case? The nth case?”  type questions and so my challenge was really to get them to train their brains to operate different ways with respect to math. This took time. Two classes in fact, but it was worthwhile.

Here are some answers that students came up with (I posted 6 copies of the same image and challenged my classes to fill all 6 with different representations).

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(From top to bottom, left to right) 1. “Raindrop” method. Squares fill in from the top. 2. “Bowling Alley”. Squares being pushed up from bottom. 3. Squares pushed in from the left. 4. L-shape 5. Rotating Left/Bottom 6. “Negative Space” the missing squares form the same number of squares as the previous case.
After, and only after students have had a chance to visualize the problem, and see other representations of the same pattern in multiple ways did I have them attempt to come up with a formula for the n-th term.

Most students were able to set up a table and saw that the difference from one case to the next increased by 1 each time:

But only a few students were able to break it down further. A message I kept telling my students, “If you’re going to fail, fail differently each time!”
It turns out that most of these students had been exposed to Gauss’ summation before. Those that did were able to find a formula for 1 + 2 + … + n, but the challenge with this pattern is that we start at 3.

Another student used the “square” representation as a part of his proof but isolated the last row.
Looking at the diagram below, we see that the total number of squares can be represented by (n+1)^2.

Ignoring the last row, we see that the number of actual squares and “negative space” squares are equal. The total number of squares (excluding the bottom row) is therefore given by [(n)(n+1)]/2.

Putting both these parts together, we get that the total number of squares for case n is:

My favorite proof thus far, though, is this one:
-Take the square representation, ADD another layer
-Now we have a rectangle with equal amounts of actual squares and “negative space” squares
-The resulting formula is just the area of the rectangle divided by 2
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Wondering if there’s a way to embed LaTex code into my blog… (AKA me trying to be more tech saavy)
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(Please excuse the crappy graphics, I did what I could with Powerpoint…)
Even though this material isn’t explicitly stated in the curriculum documents for this course, it was a valuable exercise to have done with my students. I had a few students approach me after class, eager to show me their proofs and what they had discovered. Throughout our whole discussion, I never gave students any answers, but focused on process. This is a message I want all students to internalize when they leave my classroom.

First Days of Calculus: Grapher-Explainer Activity

 Never in my life did I ever imagine myself teaching in China, and yet, here I am for a second year at that! Below are images of welcome packages I put together for the members in the Math Department this year, which includes:
– A door sign with the teacher’s name, room number, and teaching schedule
-Stickers, ‘cuz duh
-Coffee, a key element in sustaining the life force of a teacher
-A pack of cards, essential in any math teacher starter kit 
-A math puzzle, fuel for the brain 

I’m super happy with the way they turned out, and I’m looking forward to a good year ahead!  

This year I’ll be teaching Pre-Calculus 11 and Calculus 12, which I’m both excited and nervous about! It’s been years since I’ve taken Calculus and this will be my first year working with twelfth grade students (I’ve been doing a lot of review this summer on Khan Academy). Here’s a fun activity that I found on Kate Owen‘s blog that I plan on using this week with my Calculus 12 students. It’s a great way to review concepts and vocabulary from Pre-Calculus to see what students already know and remember from the course. 

I’ve added some modifications and created an accompanying PPT that’s a full lesson, all ready to go. Scroll down below to access this resource 🙂 I’m a big believer in sharing teaching resources for free, and this is my way of giving back to the online teaching community that has given so much to me. Huge shout out to everyone in the #MTBoS, I love this community. 

The activity works as follows:  

1.Students it with a partner, shoulder to shoulder.
2.One person faces the board, the other person faces away.
3.The person facing the board will be the explainer.
4.The person facing away will be the grapher.

Warm Up: Teacher does warm up round with the students, describing a basic graph (ex. linear function) and students attempt to draw it in their notebooks. Discuss: What prompts were useful? Is there something the teacher said that could have made it easier? 

The Activity: (see above)

Exit Ticket: Given a picture of a graph, students are to write a description that matches it in as much detail as possible.

Extension: Students draw a graph and write a corresponding description. Scramble the results and have students match them!  

grapher-explainer_activity.pptx
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Polynomials

Oh Polynomials. My least favourite unit by far in the Foundations of Math and Pre-Calculus 10 course I am teaching. Find the greatest common factor, least common multiple, factor these trinomials, collect and simplify like terms, the swimming pool has a width of 5x + 1 and a length of x + 2… YAWN.

The Challenge

How can I frame a boring, completely algorithmic and skill-based unit into something that’s relevant and meaningful for my students? I am borrowing Dan Meyer’s definition for relevance here.

It Begins with a Question… 

A colleague asked me today, “How much time do you have for homework at the end of class?” This was a surprising question to me, and as I thought back over the 10 day unit, my answer was almost none. The question sparked a great dialogue between us about our approach to teaching the same content in our respective classrooms. It really made me think. I realized that while I still dreaded teaching polynomials, I had found a way to improve the way I taught it from first semester that required less rote work and more thinking.

One thing that has not changed, however, is that I avoid teaching FOIL method like the plague. It only  works for expanding binomials and does not apply for polynomials with more than two terms. After I read this article I was convinced I would never need FOIL in my classroom:
http://www.makesenseofmath.com/2016/11/why-i-will-never-teach-foil.html
For a good laugh:
https://saravanderwerf.com/2017/04/01/why-ive-started-teaching-the-foil-method-again/

Some things that came up in our discussion:

  1. Algebra tiles – benefits and fall backs
  2. Picture talks
  3. Factoring method (criss cross or sum-product?)
  4. WODB
  5. Progress checks
  6. Taboo
  7. Human Bingo

1. Algebra Tiles 

Definitely a hate-hate relationship. As a math teacher, I am obligated to entertain this idea and I do admit it has its benefits, especially in lower level math classes when students are initially being exposed to distributive property and the like. The problem, however, was that my students were already armed with the skills and knowledge of multiplying and factoring polynomials. Moreover, the limitations of using tiles far exceeded the benefits, in my opinion. Algebra tiles do NOT work for: polynomials higher than degree two, multiplying more than two polynomials, and multiplying polynomials with more than three terms. This meant that it took more effort for students to understand how and why it works.

Nevertheless, we spent a few classes examining algebra tiles and their usefulness. Rather than approach it from the typical standpoint of using algebra tiles as a manipulative, I wanted students to see the link between the algebraic and pictorial representations of polynomials. This took work and was not as straightforward as it seemed. A big takeaway for me was that students gained much more out of the experience when they were able to physically manipulate the tiles and arrange them into their “factored forms.” Last semester, I “taught” algebra tiles by merely showing them examples and drawing them on the board.  It took a bit more prep, but this semester I printed eight sets of tiles (positive and negative) in my classroom and had students manipulate them instead.

If we were to spend any more time on the unit, or if this was a lower grade level, as an enrichment activity I would have students discuss the limitations of algebra tiles and look for ways to address them.

2Picture Talks

I like to use Sarah VanDerWerf’s Stand and Talks as a format for students to discuss picture prompts. I find that the buy in for engagement is much higher when the prompt is linked to physical movement. My favourite questions for photo prompts are: “What do you notice?” and “What do you wonder?”

What are my photo prompts, you ask?

That’s right. Algebra tiles.

Goals for students:

  • Make observations and ask questions
  • Use math vocabulary
  • Share ideas with their peers

​I like this activity because it is easy to differentiate and works well as a “minds on” for any topic. Asking students a general question like what they notice/wonder means that lower ability students can comment on ANY aspect of the photo (e.g. “there are blue and green rectangles”) while higher ability students can be pushed towards making observations based on any mathematical patterns or relationships they observe (e.g. “the green tiles represent positive polynomials and red tiles are negative”).

3. Factoring Method – Criss Cross or Sum Product?

I’ve had the great fortune of only having one prep and a spare block this semester (for friends and readers who don’t teach, that’s teacher jargon for FREE TIME, kinda. The details are not important). Anyway, I’ve been making drop-in’s to my fellow colleagues classrooms with my new-found “free time” and one thing I picked up was the importance of proper SEQUENCING. For instance, a natural progression for factoring trinomials might look as follows:

  • Common factors
    • ​Ex:    12xy + 3x
  • Trinomials with leading coefficient of 1
    • ​Ex:     x^2 + 4x + 4
  • Factoring when the GCF is a binomial
    • Ex:      x(x + 1) – 2(x + 1)
  • Trinomials with leading coefficient not equal to 1
    • ​Ex:     3x^2 + 8x + 4


That, together with a quick exercise on sum/products, helped me push students towards seeing the relationship between the factored form of  a trinomial, and the sum/product method.

I prefer this method over the traditional “criss cross” method for a few reasons

  • Focus is on noticing patterns
  • Less trial and error work
  • Allows them to answer questions like this:

Which One Doesn’t Belong? (WODB)

Fantastic activity for building up thinking skills and vocabulary. Each student picks one of the expressions and must argue why that one doesn’t belong.

Benefits:

  • Everyone speaks (I always encourage full sentences and proper use of math vocabulary)
  • More than one argument may arise for each expression

Sample answers:
“27x^2 doesn’t belong because it is the only expression that has a coefficient with a perfect cube”
“45x^2 doesn’t belong because it is the only expression that has a coefficient with 5 as one of its prime factors”

More WODB prompts can be found here.

Taboo 

How it works: One student is chosen to stand/sit at the front of class facing the audience, they are in the “hot seat”. Behind them, a vocabulary term is shown for the rest of class to see. Students in the audience must help the student in the hot seat guess the vocabulary word by miming, explaining the definition, or giving examples. They may not use any part of the word in their explanation.

Modifications: Differentiate by giving students the option of bringing a “cheat sheet” of vocabulary terms with them. Prepare students for the activity by giving them cross word or fill in the blank exercise reviewing the vocabulary words for the unit. An “expert round” can include vocabulary not on the cheat sheet. “Challenge round” can be facing a peer or the teacher. Can play in teams or as a class.

Human Bingo 

chapter_5_human_bingo.docx
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This activity was shared by a good colleague of mine. To get BINGO, students must find one “expert” in the classroom to answer each question on the bingo card until all the questions have been answered. The student who answers the question must sign their name. A student may not be asked more than once to answer the same Bingo card.

Benefits:

  • Gets students moving
  • Students can pick the question they want to answer
  • Fun review activity for a test or quiz!