An Inquiry into the Nature of Work

Like all educators, I have been adapting to the new world in which we are facilitating learning, and today, I just wanted to share some broad observations I’ve made about my students, and about my own successes and failures in creating conditions for optimal productive work through this pandemic. 

To begin, I’ve given several projects over the last few weeks (you can find a few free examples of them here), scaffolding the projects with daily practice problems that contain techniques related to those you might want to use during the projects. Furthermore, every practice problem was uploaded with a detailed review video. This might be obvious, but this is much less work than we would have been doing if we were in school. Normally, when students have a project going on, 95% of it is done outside of class time. I give very little to no homework when projects are assigned to make up for that fact, but in the 5, 75-minute classes every week, we do a LOT more than 3-4 practice problems a day. 

Well, this last round of projects – across the board from 9th grade to 12th grade – were better overall than any projects I’ve ever received from students. Dang. This is a trend worth speculating over. So I asked a few friends of mine what they were seeing; my colleagues at my school also said the work being turned in seemed to have had more attention paid to it than is typical, and an art professor I know at a local college of art and design said her student’s projects blew her away. So it’s not just my students. 

Now, I am going to begin upon some wild speculation here, but here are a few guesses that I have as to the way work happens in our modern American society: 

Even before this pandemic, I was thinking a lot about the nature of work in my school. It seemed to me that there were distinct ‘patterns of work’ that emerged from different activities, and that my big concern was that our environments generally do not allow for ‘deep work.’ By deep work, I mean the type of tasks that simply cannot be completed with distractions present. Trying to score a 1600 on the SAT could be an example of time-pressured deep work, and writing an essay that draws upon complex threads from multiple sources to create a cogent argument is an example of deep work that could take many more hours and multiple sittings. These are the sorts of tasks that you couldn’t hold a conversation while completing. The types of tasks that, when starting, can cause ‘writer’s block,’ but that when in the process and rolling, feel fantastic and make the time fly by as if we were stuck in a beautiful, productive bubble with an IV of good ideas dripping in. If we were using the analogy of ‘the brain as a computer,’ we would say these are the tasks that, to be done right, require all of the available RAM. 

All this to say that deep work is, generally, hard to come by at my school. I laugh and joke with students about it all the time – for example, there are times when a student will be using our noise blocking headphones (not music-enabled) to block out distractions and another student will come up and tap them on the shoulder to get their attention! (I find it funny because the point of the headphones is, clearly and certainly, to stop people from doing just that) This is admittedly partly because we have a collaborative environment that encourages students to figure out how to work together productively. However, I also constantly urge students to be very aware of what type of work they are completing, and to avoid partaking in those distractions. I begin every ‘groupthink’ that students complete with an ‘individual think’ of four to six minutes, and most of my students would probably, if asked about the individual think rules, roll their eyes and mock my voice: “Remember that talking during an individual think is an automatic No Evidence grade for your Character Point Average [25% of the total grade] – this is because in order to learn you have to make sure you at least try the problem yourself first before talking it over with peers to come up with collaborative solutions…” 

This is part of the reason that I tend to assign very little homework while projects are happening, and why I tend to have projects be mostly an at-home, individual type of assignment – I want projects to be about deep work*!  

So why, then, were the projects so much better? They are usually done at home! BUT, they are usually done at home after a full day of exhausting their willpower by having to do other things that are not deep work. 

And so this observation brings me back to society rather than just my classroom. Until beginning this new life of teaching remotely, I had never really slowed down and thought about what those other things that students were doing all day were… but over the past several weeks, I have noticed trends in my own work habits. At first, I was just knocking out project after project that I had wanted to do for a long time! But after one, two, three weeks, I started slowing down. It wasn’t that I wasn’t still trying to get good work done, it was just that I had a ton of different things to do, and it was like I was wasting time deciding which was most efficient and should consume my energy now. And this was what I realized: the in-school environment promotes the ability to get a lot of small tasks done efficiently, but it’s really bad at promoting deep work. The home environment is the opposite. 

I suspect that something similar happens in the big, wide work world. My wife’s experience as someone who works from home almost full-time, with the exception of traveling to her company’s headquarters once a quarter, seems to confirm this hypothesis. Every time she comes home from headquarters, she comments “I don’t know how anyone in the office gets anything done! They are just always bothering each other with dumb, unscheduled questions or going to pointless meetings… when do they just sit down and get 3 hours of uninterrupted spreadsheet analysis in?!?” Ummm… I respond… I don’t think they do. That’s why they have you. 

I’m interested to see how this pandemic and its resulting effects on school and white-collar business will interact with the nature of work in our society moving forward. In my perfect world, the average work/school-week would begin to shift to include one or two days of working-from-home as a new societal standard, and we would tacitly understand the difference between the types of work each environment supported. Now, that’s not to say that will happen – I’m not sure our society is ready for it. But it will be interesting! 

Next week, I’m going to dive into a bit more depth about my specific struggles with completing both deep and shallow work in this new world, and the experiments in major routine mix-ups that have been helpful in me overcoming those struggles. Until then, keep thinking of ways to return to the heart of education, even in this cold digital world. 


*Without going into too much depth about something I talk about a lot, I believe an incredibly important part of mathematics is being able to describe the process and logic to another person. It improves our own mathematics understanding, improves our writing, and improves our general conception of what’s needed for effective person-to-person communication. See Intro to Mathematical Literacy.

Free Math Distance-Learning Problems & Project Templates

Last week, I shared a project that I gave my students to help them both understand the realities of the coronavirus world, and to get them in the habit of being regularly informed and engaged in societal conversations. I think it’s a project that everyone who has some time on their hands right now would do well to complete themselves, so I encourage you to check it out and possibly even do the project. 

However, this week I want to share some thoughts for educators who are now learning how to continue teaching courses in an online manner. I will end by sharing a examples of mathematics projects that are NOT related to the virus (I’ve given quite a few assignments on the virus at this point, so this one is a return to ‘normal’ mathematics). Let’s jump into it! 

To begin – some considerations for educators now teaching through online learning. 

1. As a teacher, release yourself from the expectation that you have to make kids learn content. Embrace the fact that you get to provide them with the opportunity to learn, and begin convincing them that it’s worth it. 

If you’ve read this blog before, you know that I believe the job of the educator is not to make kids learn… it is to provide opportunity and access to learn, and to facilitate learning through the creation of worthy and locally-relevant tasks. I know this is a tricky nuance to comment on, but the reason I believe in this so fervently is … well, it’s got a whole host of reasons. 

On the teacher side of things, I think there is no better way to make a young, inspired, and passionate teacher absolutely HATE the profession and rapidly get out of it than to put them in a classroom with kids who don’t care and aren’t allowed to fail, and to say ‘by the end of the year, these kids must know this list of hundreds of standards’. That sort of a system makes inspired, passionate teachers (which we desperately need) feel disheartened, and it leaves behind teachers who have disconnected and emotionally-hardened themselves (aka lost the passion) in order to keep teaching. 

Secondly, it does a disservice to students. When I, as a teacher, accept the fact that the great ‘low-floor, high-ceiling’ question that I asked (see below) is simply not going to capture the attention of 40% of my class, it allows me to choose to let that 40% ‘just get the question done,’ and I can focus more of my attention on the 60% who say ‘this might not have been what I chose to do on my own today, but you are excited about it and I know you always support me when I get stuck and stick with it all the way to the AHA! Moment… so yeah, let’s get into this thing!” Eventually, members of that 40% begin to see that it’s just easier to just get into a topic, especially when you have to be in the class for the next hour anyways, and when a lot of people around you are into it, the energy often brings the entire community along. Kids need communities like that around them, and a teacher can’t create that community when they are worried about forcing that 40% (or whatever you percentage is) to come along rather than coaxing them into coming along by keeping your own teacher-stoke levels high and then creating the community that actually has some fun learning. 

2. Cool it. 

Now that you’ve released the expectation that kids HAVE TO LEARN EVERYTHING THIS YEAR(!!!), cool it down. The online environment simply will not look like the classroom environment, and although we’ll comment on some of the specifics of what that looks like later on, for now understand that the amount of work that you assign can and should decrease, and the nature of the work should change correspondingly. We’ll talk about this more below, but for now understand that if your normal math class has a lesson, 20 in-class practice problems, and 30 homework problems, your current situation may now need to be 5 practice problems a day, or one problem a week depending on how you structure it. This is because while we think students have plenty of time right now, we don’t actually recognize what their family structures look like or how well parents can support them in getting work done. Sure, there may be some students who are just playing video games all day, and we want them to be using their brains instead – we have to accept that we can’t make those students learn, and there are plenty of others that are not in that situation who we can’t stress out because of wanting to force a few students change their habits. 

3. Create tasks that are ‘low-floor, high-ceiling’. 

For all the students who are currently spending their days playing video games 24/7, there are also a TON of students still pushing themselves educationally – especially if you’ve cultivated a classroom community of curiosity by giving students the opportunity to learn (with authentic failure as an option) rather than forcing students to learn. In order to have your assignments be more meaningful, shift away from questions that have one, correct response, and use questions instead that are ‘low-floor, high-ceiling’. These sorts of questions allow anyone to access the content, but provide ample space to take the solution much further by developing theories or algebraic equations to support the basic findings. The project that I have attached is intended to be an example of this sort of question, because anyone can dive into the question by physically modeling the situation and beginning to find some patterns; however, students who really want to take it further can develop a theory of mathematical structure to help them support their findings. 

4. Now is the time for writing, even in mathematics. 

This year, one of my personal projects has been integrating literacy into mathematics. I know, I haven’t written about this enough yet! However, one of the major challenges that I have faced (and that has given me a TON of respect for my colleagues in humanities that edit papers all the time) has been that I don’t have enough time to formally edit the writing that my students produce. Given the fact that (even as a mathematics teacher) I believe the one skill that everyone must gain mastery over in the modern world is editing our own writing, this lack of time was concerning to me. Now, I think I have the time to edit student writing with greater depth, and this is important because even in mathematics we learn important things about our students’ thought processes through their writing. 

For example, as a math teacher, you might try asking the question: What is multiplication? Ask for a written response, though it can include a diagram if the student desires. The responses – how your students conceive of a basic mathematical technique – may surprise you. Some students will brilliantly explain that multiplication is a way of expressing sets of items – for example, three sets of coconut milk packages which each contain four coconut milks make twelve coconut milks. Eight sets (groups) of seven people make 56 individuals, which is far too many together at once in today’s world. 

I know it feels like asking students to write an explanation about how they would solve one problem isn’t developing the same skills as asking them to factor thirty quadratics. However, you may be surprised by how much depth of understanding and appreciation for the field of mathematics kids gain through actually understanding Why the techniques exist. Here are some supporting articles on that point: Introduction to Mathematical Literacy, Mathematics and Why?, and Teaching Trigonometry for Understanding

Obviously all of the projects that I share below have a writing-heavy focus as well. I grade students as much (actually, kind of more) on the extent to which they are able to explain their process than whether or not they did the math ‘correctly.’ For examples of viable, written-response/low-floor, high-ceiling humanities questions, I refer you to the COVID-19 Analysis Project I gave my students last week. 

5. If you do assign problems with one answer, post a review of those problems with it. 

Yes, that could allow students to ‘cheat’ by simply checking out the review before ever starting on the problems. That’s OK… see tip #1 and get over it! However, as a second tip on this suggestion, I handle it by including in the instructions that I post a statement that says something to the effect of: Answer the question below! DON’T WATCH the video until AFTER you have tried the problem on your own, because as you know, if you are just handed the answers, your brain is like “oh man, cool, that didn’t require much effort, no need to keep any of that stuff around in here!” If you try it yourself first, then watch the video, your brain says the opposite “Hmm, that was a lot of effort! Well, let me see what my teacher said in his review video that he made… OHHHHH! That is a clever way to approach it! Man, I should hang on to all of these methods of thinking because it seems like the time I’m putting in makes it worth it.” I know, that was a great impression of the brain 🙂 

6. Keep connecting. 

There’s a reason (that honestly requires a longer explanation than I can give here) as to why free, online courses that provide some of the same content as is provided through very expensive, university-level courses are not being completed by a majority of the population. At least one of the reasons is because it’s hard to stay inspired when we’re at home all alone. Don’t underestimate the value of leveraging technology to inspire students – as social beings, we need to connect with and derive excitement from others. While doing this via sound-recordings or video conferences may not be perfect, when students just hear the excitement in your voice or think about a comment that someone else in the class made on an (optional!) conference call, some of them will re-engage where they weren’t before. Of course, right now, it’s not all about curriculum, so make sure to also include some updates where you are just checking in and saying ‘hey, remember that there’s someone here who cares about you and your continued growth into an awesome human!’ 


The Projects: 

Below are links to some projects that you are welcome to adapt in your own classrooms. To invoke Prince Max and Kurt Hahn here, I am proud of the fact that there is nothing original in here – this is all adapted from other geniuses like Jo Boaler, the Interactive Mathematics Program, or other ‘lore’ problems because “It is in education as in medicine, you should harvest the wisdom of a thousand years,” as Hahn said of Salem. Feel free to contact me for support if you’d like – I can make a Khan-style video for any of these projects that will get students started if they are stuck and need a little boost. 

The Light Switch Project. Number Theory. Found in Francis Su’s Mathematics for Human Flourishing. You will notice the formatting on this project – I experimented with kind of ‘forcing’ kids to get started on the project early by doing the math part of it as ‘a test,’ which they would then go back and use to finish the rest of the project. Make it your own – this one is AWESOME in terms of “why these numbers though?”

Broncos Score Dilemma. Linear Systems. I kept this from back a few years when the Broncos were good! (but you could always update for last season’s real stats). 

Jazz-Pop and Hip-Hop. Linear Systems of Inequalities. Adapt the characters in this project to be characters at your own school to make it more fun!

Squares Upon Squares. Introduction to quadratics and non-linear functions. Adapted from Jo Boaler at 

Picture Story. Number Theory / Introduction to Non-Linear Functions. Adapted from Jo Boaler at 

Longboard Design. Quadratics / Geometry / Substitution. 

Broncos Field Design. Quadratics. 

Somebody’s Birdhouses. Finding the coefficients of a standard-form quadratic by solving a system of three equations for three variables. Adapted from IMP. Adapt the names to your own students for fun. 

Maetatron’s Cube. Intro to Conics / Graphing Tools. 

Have fun, get inspired to create a new cool project, and share it with us! 


Experiments in ‘Blended’ Learning Due to Coronavirus – Week One

Well, in just a few weeks our lives have all changed dramatically. In my last post, I shared another current-events and data lesson that I gave on Wednesday before Spring Break; the lesson was about the novel Coronavirus and some of the numbers that we had on it at the time. The next day – our last before Spring Break began – I followed up with a lesson on Thomas Pueyo’s first article on the virus in Medium.

Today (our first day BACK from Spring Break), we’re actually living a lot of the things that were, at the time, just extreme-case predictions from a passionate math teacher who had been doing some data digging as the threat of the virus grew closer. As educators, many of us are shaking our heads in astounded disbelief, trying to figure out what this is going to look like with no in-person school for at least the next six weeks (and likely longer).

Well, I don’t have the answers! Sorry! But I do have the first project that I gave my students today. If you’re an educator, feel free to make a copy and adapt it for your own uses. If you are NOT an educator, but DO have some time on your hands in isolation, I think it could be really fun for you to dive into it also – I am! If you do, feel free to send it my way – I can’t promise I’ll be any good at editing your work and giving feedback, but I know I’d be interested to see your thoughts!

Here it is, along with the student answer document:

All-School COVID-19 Analysis Project

COVID-19 Analysis Student Answer Doc


Data and the Coronavirus

There’s an informally-used quote that’s often incorrectly said to be an ancient Chinese curse: “May you live in interesting times.” Whatever the true source of the quote or curse may be, we are certainly living in interesting times today. 

For the past few weeks, I’ve been exploring the idea of using current events to inform my Senior curriculum focused on Economics and Data Analysis. In terms of inspiring new, current, and relevant curriculum, this week was almost too easy. In hopes of sharing ideas that other educators can use or create in their own curriculum, I’ll continue my kick today. 

First, I started the lesson with what I thought would be a quick-and-easy question, given what my seniors have been covering lately related to the Stock Market’s recent performance. I showed this clip from yesterday’s The Late Show with Stephen Colbert, 2:19 – 3:12, asking if there was ‘data bias’ involved. “Now that point drop – as I said the largest in Dow history,” Colbert reported, “larger by over 500 points, breaking the record set just eleven days ago!” Now, I absolutely love Colbert. I think he’s fantastically witty and his nightly work is incredibly impressive, so what about this clip gives me pause? 

Well, the response took a little longer than I had hoped. I asked the question: “Was this drop the worst in history?” Some students responded with “Yes, that’s what he said,” while others said that maybe that statistic wasn’t accurate. There were a whole host of other conjectures as to what the problem with this data could be, and I even pulled up the performance of the Dow by doing a quick Google search and showed students the last year’s performance and the ‘maximum’ performance:

Screenshot 2020-03-11 at 4.55.29 PMScreenshot 2020-03-11 at 4.55.40 PM

The problem with this data point is that it’s not the right metric to use in this situation. Yes, the 2,000-point drop may have been the largest in the history of the Dow Jones Industrial Average… but what would a 2,000 point drop have done to the stock market in 1995? The Dow probably wouldn’t exist anymore if that had happened! The correct metric here is not total point drop, but point drop as a percentage of the total market. Now, with that said, Colbert was certainly using data to make his point, and that’s fine… we just need to be able to recognize when data is given in incorrect metrics and take it for what it is. 

I want to move on to the data on Coronavirus; however, I should first mention that of course the total drop of the stock market over the last two weeks has certainly been precipitous (around 20% from the February high!), and I should also note that although Coronavirus is a huge component of the cause of that drop, the Saudi-Russian oil-price dilemma is certainly also contributing. This episode of The Journal was a nice overview of Monday’s events if you’d prefer a free, listening-based medium.* 

After this ‘Bellwork,’ if you will, I asked students to analyze this dataset, which was created by Elena Grewal, shared in an article by, and that I modified it for student work. All I wanted them to do was give me a brief overview of what the data ‘told’ them in terms of possible numbers of deaths from the virus, and then we compared it to what the original dataset gatherer predicted (page one of this spreadsheet). Dealing with the uncertainty surrounding the denominator of our ‘death rate for Coronavirus’ equation was the challenging part for students, but a good challenge as real-world data can have uncertainty associated with it (for a number of reasons… we’ll come back to that later). The conclusion here that Grewal made is worth restating here: “Clearly the high end of killing 1.4% of the population would be very very bad, especially with the context that in a given year only 0.87% of the population dies.”

Then, I worked through interpreting a series of diagrams and graphs with students, starting with those discussed in the article. First, the photo of a medical tent in Italy – why is this tent being used; why isn’t the man in the hospital right behind the tent? Because hospitals are full. So what we are trying to do right now is captured by this next chart from the article:


We discussed this chart, asking important questions like “Is the ‘healthcare system capacity’ line at the same height for every country?” Well, no. And another: “What are the current ‘protective measures’?” Well, essentially, they boil down to: wash your hands WAY more than you think you need to, try not to touch your face, don’t travel to places affected by the virus (or travel at all), and don’t go to places with large crowds. Those are pretty basic measures, right? Dr. Neil deGrasse Tyson said in his interview with Colbert the other night that we are currently in a huge experiment: “Will people listen to scientists?” Ha! Great question. 

Then, we analyzed the section of the article quoting Liz Specht, also found here. The challenging part for students about this was wrapping their heads around the uncertainty surrounding the percentage of cases that may require hospitalization… but no matter how it shakes out, we could be facing a major strain on our healthcare system, which would not only affect people with Coronavirus but also the 65% of the country’s hospital beds that are already filled with pre-existing patients. 

Next, we briefly discussed the analogous situation in the 1918 flu comparing the response in Philly (which did NOT cancel major parade events) to that in St. Louis (which did). I will mention that one student immediately said “wait, what were the population sizes of each city?” before realizing “oh, wait – it’s per 100,000 population.”


Instead of reading the following section of the article (quoted below), I switched sources so that students could make the same analysis on their own, if they so choose: “Part of the problem is that this has become a political issue. In particular, President Donald Trump has made it clear that he wants to see “the numbers” (that as, the number of people infected in the US) kept low. This is an example of where optimizing metrics interferes with getting good results in practice.” I figured that by showing the following chart instead of reading that passage, I may allow students to come to their own conclusions without entering a fully ‘politically charged’ situation. 

Screenshot 2020-03-11 at 9.04.03 PM

This chart came from the New York Times article called How Bad Will the Coronavirus Get? Here Are 6 Key Factors. Obviously, this brings us back to our earlier statement/question: sometimes there are extraneous factors that get in the way of being able to collect good data (e.g. lead to uncertainty in the data), and one of them is political objectives (again, I didn’t go into the roller-coaster of our own president’s experiences in this realm… but what a story! Karma, eh?) 

Finally, we arrive at the culminating event of this lesson. The following chart: 

Screenshot 2020-03-11 at 9.05.19 PM

Obviously we have to start with the ‘log scale’. What is a logarithm? Well, we know that it is essentially the ‘opposite’ of an exponent, just as division ‘undoes’ multiplication. That means that if we use the standard base-ten logarithm, every increment of 1 on the logarithm actually makes our ‘answer’ go up tenfold. That’s what this scale is all about! Then, we examined the rate of spread. I wanted to especially highlight the SARS datapoint – we seem to have eradicated the virus from the planet completely in 2004 through containment. With a death rate of almost 10%, that’s a BIG deal. Wow. Given that the rate of transmission seemed to be right in the middle of the possible range of Coronavirus rates of transmission, why then were we so successful in containing it? Is it a function of the fact that because the disease was more deadly, we took containment more seriously? Or is it a function of numbers of people who travel today as compared to 2004, especially globally on planes? Although 2004 doesn’t seem that long ago, the casual nature with which we approach global plane travel has certainly continued to grow exponentially… 

Ultimately, the point to be made is that the virus may not seem like a big deal to those of us who are young with healthy immune systems, but our civic duty as global citizens is to go deeper than that and think about this from the perspective of our society. If we were to get a ‘second flu’ that came back every year, killing more people than the plain-old seasonal flu, that would be a dramatic loss of human life. It doesn’t matter if you are young and healthy, you will certainly know people in your life who will be affected more drastically than you, not to mention that you will also be at risk one day, no matter who you are. As global citizens, can we make a few minor changes to our actions in service of protecting our global community, reducing the chances that this thing continues, and to ensure that we’re ready for the next one? Only time will tell. 

As I concluded the lesson, a student (who by the way had been out sick for the last two days) said to the four other students around him “anyone want some of my dried mangos?!” and started handing them out straight from the bag. Goodness, human habits are hard to change… 

Thanks for reading – obviously this lesson was ‘on the fly,’ because of the nature of planning it as events unfolded. However, I hope it inspires you to talk about some current events in your own classroom, and if this article has brought up questions like “hmm, we’ve been talking about things like our Healthcare system for a while now in our national conversation… how would various ideas about healthcare systems in circulation right now have changed the situation, or would they have?”, I’d love to hear them as well as your thoughts as to the answers. 


* Lastly in my ‘yeah but first I want to say,’ tirade, I’d like to express a strong opinion (disclaimer has been stated): those who say we need to focus on ‘boosting the economy’ in a way that makes people start buying stuff again or taking flights again, GTFO. “There’s a joke in economics that the stock market has predicted ten of the last three recessions,” says John Hilsenrath in this episode of The Journal published one week ago. Hilsenrath goes on to talk about some of the instruments we can use to measure the health of the economy that are far more accurate than the fickle stock market. All I mean to say by this is that I’m tired of hearing from a certain pro-economy crowd that seems to think the stock market is the economy, as the market benefits a small minority of the population much more than the rest of it (people who invest – draw what conclusions you may about that demographic), and who also thinks that the economy can be separated into a bubble unto itself, separate from other components of our world it has affected, like, oh, say, the environment. How much will the environment benefit from our collective choice to fly less? You know, perhaps the Coronavirus is meant to just be a challenge in which we recognize our collective need to draw back just a bit and live a little more locally and within our means…

**Update: after posting this I found tomorrow’s data lesson:

Recognizing Bias in Data

In the last few posts, we’ve discussed using data to quantitatively analyze aspects of the world around us. Often, we will not look at the data itself (due to lack of time, etc), and instead we will look at the graphical representations of data that people have created for us. This allows us to quickly take in what was previously an overwhelming amount of numbers. Today, we’ll continue that theme in a (perhaps) more down-to-earth manner. 

To begin, the Learning Target for today’s class was “I can analyze inherent bias in graphical displays of information.” In trying to give an example that had several ‘layers’ of difficulty embedded into it, I chose this chart, which comes from one of the Data Crunches on the fantastic financial education resource, It is important to note that the chart actually comes from 2017, and 2020 was based on projected numbers. 

Screenshot 2020-03-05 at 6.33.04 AM

First, I of course gave students a minute to analyze it, as well as a second minute to discuss with a turn-and-talk partner. Right at the start of this process, I already had one student in my class exclaim, upon reading the learning target, “It’s data – it can’t have a bias! It’s just the numbers, and numbers don’t have a bias!” Excellent starting place for a teacher! 

The natural first topic of conversation, based on the learning target, was the title (although I should mention a few students wanted to talk about the colors used). The questions raised were: Which is a more accurate overview of what the chart shows us – the main title, or the sub-title? Why? Is this an example of how the title of a chart can subtly alter what sorts of information people are looking for in the data? What sort of a political bias – right or left – would one assume this chart came from based on the title? 

Next, we began to analyze the graph’s trends. Now, normally I would make sure to call attention to the axes, but for two reasons I saved that until later, wondering if a student would call out the things that I was noticing about them. Obviously the percentage of federal spending used for military and defense has dropped the most out of the categories listed from 1962 until now, and health spending as a percentage of the national budget has increased the most. Where do these topics fit into the national conversation? My students unanimously agreed that military and defense spending was a traditionally right talking point, but interestingly pegged health as an issue on the left. It seems to me that health care is being discussed across the board, and there are certainly contrasting views of it, but for a quick example of some of the shifting baselines within health care, look to President Trump’s feelings on the Government’s ability to negotiate prices of pharmaceutical drugs. It seems that this is at least one issue within the broader conversation of health care on which both sides of the isle agree something needs to begin to change. I assigned the February 25th episode of The Journal – How Big Pharma Lost Its Swagger for homework. 

But back to the point! My class and I were looking for potential bias, and I felt that it was time to draw attention to the axes. What scale was chosen on the x-axis? A four-year scale. Why? Presidential election years! But what the heck happens here, because I know off the top of my head that 1964 and (especially) 1968 were BIG election years during the height of the Civil Rights Movement and the Vietnam War that eventually lead to stepping down of LBJ, the assassination of Bobby Kennedy, and the election (for the first term) of Richard Nixon. Why does the axes have 1962 as the first year? What the heck!? They missed an increment and went up by six years between 1994 and 2020! What does this tell us about the authors of the graph? Honestly, I’m not really sure… However, if that discrepancy causes us to just ask the next question – what is the Tax Foundation and what might their aims be? – then we may be able to research it to learn more. 

However, my BIG question for the class was this: Did the dollars on Military and Defense spending decrease between 1962 and 2020? 

“YES!” They all answered. 

“Hold on, listen to the question again. Did the dollar amount spent on Military and Defense decrease from ‘62 to ‘20?” 

A few ceased to answer, sensing that they needed to think more slowly. “YES!” the others said. 

“OK, I’m going to add to my question. Did the dollar amount spent – NOT the percentage of the total budget, but the dollars spent in total on Military and Defense – decrease?” 

Now they were onto it. “What this graph shows us is only the percentage of total spending… but if the total amount of government spending went up over these years as well, the amount spent on Military and Defense might be exactly the same or even more than it was, even while the spending as a percent of the total budget decreased.” Exactly. 

So what does this ‘catch’ tell us about bias? Might we want to know why the authors only included percent of the total budget, and might it be important to analyze the total spending as well? Is there a way to show the graph that displays the total spending, but also still gives the reader a sense of how the proportions stack up against each other? Yes, but what would be the downsides of that graph? Well, it would be easier to see the proportions where the total government spending was highest – presumably closer to the present – and harder to see in the early part of the graph where total spending was lower. Yet, the title seemed to have a right-leaning bias, and conservatives traditionally want to minimize total government spending, so would it not also benefit a conservative to show total spending going up? Or was it simply more important to highlight the decreasing Military and Defense spending in contrast to the spending on Social Programs? These are all questions I want my kids to ask. 

My wife is a data analyst for a big software company. She’s excellent at the job, and I enjoy hearing her discuss some of the specifics of the position in informal conversations with people we meet in town or on ski chairlifts. Most people are just interested to hear about what she does, but every once in a while the person we are talking to has a background in data and business. Recently, we met one of those people, and when my wife told him what she does, he said “Oh, yeah! Cool, so you ask your boss what she wants the data to say, and then you make the data say what the company wants it to say, huh?” 

“Yes. Exactly,” she responded. 

I think our kids ought to know that data can have bias.


Economic Data Analysis, Part II

Last week, we talked about using current events – in this case the State of the Union – as a starting point for lessons in all subjects. As a mathematics teacher, I took advantage of the speech as an entry point to conversations about global and national economics, and how to ask better questions as well as actually use data to analyze trends and form your own opinions. If you haven’t already, check out some of the prompts that I used as a starting point for my class lesson the day after the speech. You may even be able to adapt some of these questions into your classes! 

Now, before we jump into things, I guess I have to give my usual disclaimer. I dislike the party system. I am not affiliated with a political party. I try to use data to make informed decisions, and those decisions are often all over the board on the left-right spectrum, if only that spectrum didn’t include so much BS that I detest – the issues that politicians use to stir up emotions in people rather than reason. As nobel-economist Paul Krugman notes in his new book Arguing with Zombies, “In 21st-century America, accepting what the evidence says about an economic question will be seen as a partisan act.” Yes, it’s easy as a teacher to avoid trying to teach any of this stuff. But now, in this day and age, we need to more than ever.

Alright, picking off from where we left off last week, the prompt for students was to download this data set and begin analyzing it in Excel. I wanted them to do several things: first, calculate the ‘rich/poor ratio’ for every data-row shown, then, choose five countries for which you would like to compare the rich/poor ratios, and finally represent the country’s rich/poor ratio over time in a visual manner (e.g. create a graph). Keep in mind that this group of seniors have worked with Excel (Google Docs) over the course of the last six months in the context of polling statistics and financial literacy, including creating an entire amortization schedule using spreadsheets. 

This assignment, however, forces them to work with more data than they have ever seen. In order to answer my question, then, they also had to make use of some tools like the new Google Docs ‘Slicer’. As is typically the case for me, I gave them the hint that the tool might be useful, but forced them to use tenacity in pursuit with internet-based learning and figure out how to use it on their own (by reading the help bar associated with the tool as well as searching for instructions or tutorial videos on Google). 

After slicing the data for the countries they want, they are still faced with a conundrum: there is too much data to efficiently graph a chart that compares the rich/poor ratio for multiple countries over the entire 34 year time period. Thus, they need to re-organize the data to make graphing more efficient. Essentially, they need to create this: 

Screenshot 2020-02-20 at 5.04.36 PM

Notice that this table is reformatted to have ONLY the rich/poor ratio for each country shown, allowing us to create the following graph: 

I asked my class about what my graph tells us, which is that in the US, the top ten percent of richest citizens were between 11 and 20 times richer than the poorest ten percent – a ratio exceeded (on my graph) only by the Russian Federation! Thus, the conclusion we draw is that the US has much higher levels of inequity than the rest of the world. 

But HOLD ON there – is that a solid conclusion? What biases might my graph hold? What other countries did I choose, and how might that have affected our interpretations? What do we know about these countries and their histories/governmental systems that might give us more depth of insights into the data we just displayed? Obviously, this discussion eventually leads to me displaying another graph:

Does this graph lead us to draw different conclusions about the amount of inequity present in the United States? What do we know about these countries that may help us explain their rich/poor ratios? 

Ultimately, the point of this exercise was to convey the idea that we need to be able to quantitatively analyze our world if we are to understand its dynamics and be informed and engaged citizens, but that for all data that we can analyze, there are still ways of displaying the data that may draw us to different conclusions. Thus, for all of objective information that data can give us, there is always still a human element to understanding our world – a dose of morality overlaid on top of the cold, hard numbers. How will we choose to use our morality to draw proper conclusions? What questions do we still need to ask that will help us better understand this data? 

Well, one potential question was brought up during the last post I wrote on this topic – we are always hearing people like Bernie or Warren talk about the ‘top 1%’ – so how rich are the top 1% of Americans vs. the top 10%? Is that piece of data important as well? This is where my class dove back into the open-source economics textbook from core-econ; we analyzed the following chart from section 1.11:


As we can see, the modern state of affairs as captured by this particular measurement appears to be much more equitable than it has been in the past; however, the top 1% still control around 20% of the wealth of each of these nations! Thus, not only is it our responsibility to analyze quantitative data in order to make our own decisions about the economy, but we must also seek out a multitude of measures that can help us break down and express global trends in economic equity on a more complete level. 

To bring all of this back to the State of the Union, our students will still need a continuing education in economics and politics in order to understand common issues like our inflation rate and the role of the Federal Reserve. However, this can at least give them a starting point for discussions. For example, some students asked ‘Why do these inequities that we have been discussing arise?’ and began to answer their own question from a logical starting point: “Well, from a business perspective, there has to be an incentive for acting the way we act, and the incentive for a lot of our elected officials is all messed up – they serve richer voters disproportionately because that’s who donates to campaigns.” They even had the wherewithal to connect this statement to Andrew Yang’s comments during the PBS NewsHour/Politico debate in December: “And the question is why am I the lone candidate of color on this stage? Fewer than 5% of Americans donate to political campaigns. You know what you need to donate to political campaigns? Disposable income.” Don’t go all American on them now. They weren’t trying to claim that the ‘Freedom Dividend’ was the greatest idea they’d ever heard; they were just connecting the dots that the governmental system, the economic system, and inequities within a country are all connected in a big feedback loop that we should be paying attention to and thinking about more deeply. 


Yes it will be ‘on the fly,’ but let’s talk Current Events

So it’s the day after the State of the Union, and of course people are going to be talking about the speech around the school. I would hope the event is still at the very least discussed in Government classes across the country, as I can remember from my youth. As a math teacher, the speech presents me with an opportunity to dive into numbers and big data a bit further – typically in the context of economics, so I love to plan some mini-lessons around the speech. 


But why, this year, am I hearing more about what happened after the speech than the contents of the speech itself!?!? I’m sorry, and we will get to the speech itself and lessons that could be run off of it in just a moment, but we need to say a few things about Pelosi’s paper-rip first to clear the air, because both sides of the isle are driving me crazy with what I will call emotional myopia aka understanding humans. 

First – I’ll describe what happened. After the speech was over, Nancy Pelosi ripped the copy of President Trump’s speech – which he had given to her before the speech – in half. It should be noted that obviously the two do not like each other, and that Trump ignored Pelosi’s slightly late attempt to shake hands with him after getting the speech. 

Was the act of ripping the paper a good idea for promoting bipartisanship right now? Of course not. But if I practice one of the foundational concepts I try to teach (as an educator) – dialogue as defined by David Bohm – then I would be trying to step into Pelosi’s shoes. What she said about why she ripped the paper was that she was frustrated by the speech because it was a “manifesto of mistruths.” Have I ever felt like I wasn’t being heard, at all, and gotten so frustrated I did something that wasn’t helpful to my goals? Oh heck yeah! But there seems to be more than that. The act seemed only partially-spontaneous, and obviously given the nature of the situation there was at least some political calculation. However, even if you add that in, the Speaker of the House has been frustrated for years by a President that seems to blatantly lie and play stupid, was elected partially due to interference by a foreign power and an antiquated electoral system, has attempted to further meddle (albeit in a far less effective way than Russia did) in future elections, and was at the moment undergoing an impeachment trial in which the Senate Majority Leader (as well as others) said before swearing an oath to exercise “impartial justice,” that they in no way intended to be impartial jurors. Come on now…  I mean, are the Democrats shining examples of bipartisanship? Fuck no. But give me a break if you think that the state of affairs is not frustrating, or that if she just hadn’t ripped the paper, we’d maybe be mending partisan relations right now. 

So that’s what I think happened. Some serious frustration led to a politically unproductive act that won’t really have any impact in a party system that is already as separated as it can possibly get. So can we get the fuck over it? Can we talk about the contents of the speech? Did people read the transcript of it, with fact-checking footnotes attached? Need I remind us of Postman’s prophecy: “When a population becomes distracted by trivia, when cultural life is redefined as a perpetual round of entertainments, when serious public conversation becomes a form of baby-talk, when, in short, a people become an audience, and their public business a vaudeville act, then a nation finds itself at risk; culture-death is a clear possibility.”

OK, so let’s talk about the speech. Again, I will remind you that I am not affiliated with a political party, and I find myself all over the board on issues that are traditionally conceived of as right or left. I try to find myself on a constant pursuit of truth and logic. I don’t do it perfectly, but I’m sure as Hell going to try. 

Here’s my analysis of the speech: it was characterized by a) made-for T.V. moments (and I must say, our President sure knows how to create those moments and put on a show), and b) an attempt to use statistics (not super-accurate statistics) to position the president as the lead engineer of a complete economic turn-around/revival. Now, if you are a student in the U.S. right now, you are probably hearing the speech on the one hand, and the democratic debates on the other hand where we’re hearing a lot about inequity in our economic system, and you’re wondering what’s actually happening. So let’s take the opportunity to allow them to both get an introduction to economics as a field of study, and to have our students look at some global data. No, we’re not going to be teaching a thorough representation of all economic theory – I’m a high school teacher, so I don’t have the capacity to understand all of that myself! But we can at least plant seeds of self-sufficiency in answering one’s own questions using data.

First of all, check out this awesome new open-source economics textbook, available online. Obviously it would be a lot to process for a teacher if you are just looking at it or reading it for the first time, so I will direct you to some of the first activities within it, which deal with this set of data. You could start there – let’s analyze this chart. After all, it is right up our alley as teachers – it’s complex, requires data analysis, and also requires us to understand the calculations that go into a decile. In the process of analyzing this chart, one may give an example problem to show how we might calculate the top 10% – say you make this dataset up for the income of a population, listed in dollars: 1, 1, 2, 2, 2.5, 3, 3.5, 5, 6, 7, 9, 9.5, 10, 10, 13, 15, 16, 20, 21, 22, 33, 38, 40, 44, 45, 48, 51, 53, 55, 1000. Since there are thirty data points, the poorest ten percent of this population would be made up of the average of a $1, a $1, and a $2 income. The top ten percent, on the other hand, would be made up of the average of a $53, 55, and 1000 dollar income. Wait… we keep hearing Bernie talk about “the 1%” – how would it affect the average income of the richest ten percent of a population if that population was made up of 9% of people with high incomes that are swamped by 1% with just ridiculously high incomes? Would the average seem high or low? These are all questions that students can and should discuss in the process of trying to analyze this graph. Then, students should actually get this data – it’s available for download in xlsx format just below the graph! Have them complete some exercises with the data – like the book’s suggested activity of calculating a rich/poor ratio for several countries in order to compare. Hopefully they also begin to ask some other questions… like ‘man, some of these numbers seem strange… what’s PPP and how is it calculated?” or “hmm, I wonder what the 2005 dollar means as compared to the 2020 dollar…” Great!

I’ll leave you with these possibilities for discussion points today and let you play with these resources. Next week, I’ll dive back into how you may have interpreted some of these resources, as well as how we can bring them back to some basic points about the State of the Union address in order to be able to at least use some data to begin to form our own opinions about the economy and whether or not it can or should exist in a bubble all to itself, elevated in importance above every other endeavor we may undertake as a nation. Until next week, have fun!


Having Some Fun with Online Loan Calculators

* I, personally, do not have a political party affiliation. I think the two-party system, as currently conceived, is one of the biggest impediments to Democracy in the modern world. With that said, I recognize that I live in a largely Blue community and thus, my implicit availability bias is altered. Nonetheless, if you do have a party affiliation (ehem… especially GOP) and a tendency to get irrationally butthurt, please skip this article. 

Last week, I wrote about part of the reason why I give my Amortization Project that seniors are working on currently. As part of that project, students read the Wall Street Journal article called Beware Online Loan Calculators; however, earlier this week I realized that some of them were struggling with the concept of why, exactly, the calculators are misleading. 

The reason is not because the calculators are giving answers that are wrong. The reason is because they are using (to their advantage) a combination of human psychology and the presumption that most humans in the modern world won’t take the time to understand semi-complex information. This occurs by programming the calculator to have a ‘default’ set of inputs that are advantageous to the lender and not the borrower. Potential borrowers who don’t understand the system that they are working in, then, will tend to just agree with the default settings and not vary their inputs very much from the that setting, leading them to take loans that may ultimately be more expensive than if they had looked at more than just the monthly payment. 

In the spirit of having some fun, I used media in my class (a rare occurrence indeed) by showing them the recent Jordan Klepper segment on The Daily Show. Obviously this comparison misses one of the major topics I invest time in – Kahneman’s difference between ‘thinking fast and slow’ – but the exaggeration of the technique allows students to see the effect that online calculators have on people in more slow, subtle ways. 

In the segment, Klepper goes to a Trump rally and asks supporters about their take on the impeachment (which obviously ended today, though I gave the lesson a few days ago). Of the full ridiculousness of the segment, some highlights that we discussed were these: 


Klepper: “Do you think John Bolton should testify?” 

Interviewee: “No.” 

Klepper: “Why not?” 

Interviewee: “Well he could testify, but I think he’s vengeful for getting fired from his job. I think he’s a liar.” 

Klepper: “You think John Bolton’s a liar?”

Interviewee: “Absolutely.” 

Klepper: “There should be a system set up where he takes an oath, and then under oath he tells the truth, otherwise he’s punished.” 

Interviewee: “I think there should be, yes.”

Klepper: “And then maybe there’s a judge that’s put in charge. Like, the highest judge in the land!” 

Interviewee: “Right.” 

Klepper: “Appointed by a Republican! And then we could all see what he has to say, would you be for a system like that?” 

Interviewee: “Sure!”


Another one that we had a good laugh at, but also compared to online calculators: 


Klepper: “Let’s say it happened tomorrow, Trump beats impeachment. Trump can get onto running the country.”

Interviewee: “Exactly, like he has been doing for the past three and a half years now.”

Klepper: “And now, with like no impediments. No checks, no balances?”

Interviewee: “Exactly.” 




Klepper: “He’s evolved his Presidency into a Dictatorship that we can all understand.” 

Interviewee: “Exactly. Yeah.” 

Now, I must put the disclaimer in here – of course there are people from all walks of life and of all political persuasions that make similarly erroneous lapses in judgement on a daily basis, despite the fact that these interviewees were all of one political persuasion. That’s important to recognize. And, of course, Klepper has a talent for making people react in strange ways. End of disclaimer. 

As I discussed in the introduction to my personal statement last week, I teach math. But the reason we use mathematics is to actually be able to analyze quantitative components of our world in order to understand it better, and to hopefully take responsibility for our small share of creating a more just society. Anytime I can link mathematics to understanding the broader scale issues of our times, and even add a bit of having fun with my students on top of it while encouraging them to research, analyze, and think, I call it a win. 

Next week I’ll tell the story of a more serious lesson that spanned mathematics to global economic policy, sparked by last night’s State of the Union. But alas, I still have to write that one – so get out there and connect daily, consistent mathematics to broader political conversations, and try to have some fun with it in the meantime. 



Amortization and Social Justice

This year, I am completing an EL ‘Portfolio’ along with my students. One of the sections of a portfolio is the Personal Statement. Here is Part I of mine – the nitty-gritty of what I hope to accomplish through teaching mathematics. As you may be able to tell, in the next section the ‘bird’s-eye view’ that I begin here will take an even higher vantage point, but I consider this to be pretty important stuff on its own. Tell me what you think!

Personal Statement – Perspective

As a human being – sorry, I mean educator – it’s easy to get caught up in the day-to-day tasks and interactions that make up our hourly existence; after all, we’re here, on the ground, not looking down on ourselves at ‘the Big Picture’ of our lives from a distant satellite. However, ‘the Big Picture’ is actually why we are here – it’s what grounds us in purpose, gives us perspective, and colors our perceptions of what daily tasks are important to achieve the end-goals we desire. 

What I mean to say is perhaps better answered by asking myself the important question: Why do I teach? The reason I teach is certainly not because I believe quadratics are an all-important topic of the universe. They are interesting, sure! But I teach so as to affect what I believe to be right in the world. For me, that boils down to three realms: 

  • Pursuing human sustainability
  • Creating a truly just society
  • Embettering ourselves as human beings

Although I do still use classes like Crew, Internship, and the Outward Bound Trip as times to teach ideas and principles of sustainability, in this piece I will concentrate on the other two driving forces for my practice, as they are where I spend the majority of my time focusing in mathematics classes, and where I believe the majority of perspective expansion – aka ‘growth’ – occurs. I find it important to repeat and emphasize a duality here – I am hoping to expand my students’ perspectives of the world, but I am also – through teaching – expanding my own. When I get frustrated in my craft, it often boils down to me just needing to reground myself in the broader perspective of why I am here. These two objectives happen and evolve simultaneously. 

Creating a Truly Just Society

Alright, so why does mathematics relate to pursuing a just society? Because as of 2020, capitalism is by far the best social system we have come up with, but that doesn’t mean there isn’t still room for improvement. Let’s be honest, capitalism is grossly inequitable. People in positions of power and wealth find ways, intentionally or unintentionally, to take advantage of and profit from people in positions of less power. I’m sorry, but that’s not a controversial statement – there will be no debate about it. Look to the subprime mortgage issues of the 2008 recession or subprime auto loan industry that began to spring up in the following years for quick and easy examples of this inequity – no matter how you slice it, calling those practices ‘legal’ does not mean they are fair, especially when you consider the Huxlian ‘Brave New World’ we live in today. I’m going to stop myself now before I go down the rabbit-hole on how myopic and greedy people can be in the name of ‘pro-economy’. 

Now, I am not here to provide any more reasons as to why capitalism, in its currently-conceived state, is inequitable – I’m here to propose my small contribution to the list of potential solutions to the problem. I am also not here to complain about inequity – I happen to be one of the lucky few in our society who was born into unbelievable privilege. Instead, I am here to share that I fundamentally believe with great privilege comes great responsibility. 

Allow me to discuss my privilege with you. On my first birthday, my hard-working, depression-surviving grandparents gifted unto me 10 shares of 3M stock (my initials are 3 M’s), and they continued to do the same (with different companies) every birthday and Christmas until they passed. Think about how lucky that is – a one-year old who didn’t get a cool toy from his grandparents for any of his birthdays, just got a stock portfolio. When I was sixteen and wanted to buy a car – because of course that is the most important thing a high school student can possibly have – my dad contrived a deal in which he would help me pay for the car if I invested my saved money that would have gone into the car into a stock instead. At the time, I was playing high school football and using some great new stretchy clothing under my pads, so after doing some research with him we put all $7,000 into the IPO for that company – Under Armour. I sold that stock several years ago to diversify, and the selling price was an order of magnitude higher than buying. To me, that sort of privilege without the responsibility to help others to understand their financial lives would be criminal. 

Thus, one of the major contributions that I have decided I want to make to the world is to – at least on my local level – teach financial literacy to our youth. I believe education is one potential pathway to overcoming inequity – when people have the baseline of understanding and the skill to be able to quantitatively analyze their financial lives, they can be less likely to be taken advantage of, or less likely to miss out on long-term wealth management strategies. 

One of the most important trends that I have noticed in my four years of teaching Financial Literacy is the strong correlation between students succeeding in ‘normal’ math class for 3.5 years and succeeding in quantitatively analyzing their finances. I know – that seems intuitive. If a student is good at math, shouldn’t they should be good at finances also? But let me assure you that the types of math are not necessarily directly correlated. Quadratics and cubics don’t have much use in personal finance, nor do rational functions. If I became the Secretary of Education right now, I’d be pushing to eliminate those topics from the curriculum and replacing them with a complete Data Literacy curriculum; however, that’s not the point. The point is that although the current mathematical standards in the U.S. seem to have low relevance to the types of maths that people actually use in the real world, if students understand the required mathematics well, they will have a much higher probability of being able to truly learn and understand financial mathematics… so long as you are exposed to it at some point. 

In this way, I am hoping to make an impact on the massive problem of social and financial inequity in the United States through my on-the-ground financial approach. Consider just one question that I ask students to calculate during their ‘big project’ senior year: What would be the wiser financial decision, given a mortgage of $400,000 over 30 years at 4% interest – making a $1,000 prepayment every month towards the mortgage, or investing the same extra $1,000 a month in the S&P 500 which has an annual return of around 7%? Well, consider the fact that the amount of interest paid on over the lifetime of that loan comes out to $287,478.03. That’s almost ¾ the cost of the freakin’ loan! Putting $1,000 extra towards the house every month would reduce the cost by over $150,000. I don’t know about you, but that’s a lot of money to me! The very observant student will realize that the correct answer to the original question, though, is not choosing prepayments or the S&P option, but to have a mix of both – prepayments up front that transition to investing as the loan ages. Honestly, if my students never get to that point but are at least thinking for themselves about smart ways to pay off loans (like the advantages or disadvantages of prepayments versus a shorter loan term), I am happy and feel that they have earned back a bit more freedom in their lives.


Introduction to Mathematical Literacy

I was graciously invited on Monday to give a ten-minute lesson on literacy practices for part of our Professional Development session yesterday. Here’s my best attempt to capture this lesson in writing! I hope you find it informative.

Always in big woods when you leave familiar ground and step off alone into a new place there will be, along with the feelings of curiosity and excitement, a little nagging of dread. It is the ancient fear of the unknown, and it is your first bond with the wilderness you are going into. You are undertaking the first experience, not of the place, but of yourself in that place. It is an experience of our essential loneliness, for nobody can discover the world for anybody else. It is only after we have discovered it for ourselves that it becomes a common ground and a common bond, and we cease to be alone.”
-Wendell Berry,
The Unforeseen Wilderness: Kentucky’s Red River Gorge

Today we are faced with the task of answering the question “How does one effectively teach literacy?” in the next ten minutes. So let’s hop-to. 

To discuss literacy, let’s start with the foundations: it’s questions and it’s metaphors. First, the question: What is literacy? Well, in one sense it is being able to read and write. But that’s obviously not what we’re talking about here. We’re talking about the other definition: competence in a specified field. Media literacy. Financial Literacy. Mathematics literacy. In order to understand a specified field, we are faced with a conundrum as learners… but a catastrophe as educators. The catastrophe is this: in order to understand a specified field, we must be able to ‘speak the language’ of that field, if you will. But, of course, to speak the language of a particular field is to share a common bond with another… and this requires that you have discovered the world for yourself, and then expressed that through language. To demonstrate, let’s start where one would naturally when exploring literacy: mathematics. 

I’d like to teach you how to multiply and divide. Sorry, I messed up with my language: I’d like to teach you how to play with numbers. It’s fun! 

Post-It Notes come in packs of three stacks each. I want 24 of them. How many packs should I buy? HOLD ON. No answers here yet. You are in second grade and will be doing some discovering. [Pass out the manipulatives to count it out]. I’m now making you all go through the act of counting these out to discover the world for yourself: that 24 divided into groups of 3 makes 8 packs. And then I’m making you do similar types of problems for the next couple weeks, at least. 

AND THEN, and only then, I stop our whole little crew of math learners, and I say… hey, guys, we’ve been doing this thing where we want/have a lot of something and want to get enough of it / spread it evenly among a few people for a while now, and it takes a long time to say that. Should we just decide to call it something? Like … how about “division”? 

And this is the conundrum, right? In order to teach literacy, we must in some way experience doing the thing, and then get to the point of understanding the thing well enough to want to save time and name the thing, aka ‘to abstract it.’ And THEN, we have to understand that the people who ‘invented’ the field abstracted it by naming it or symbolizing it differently than we might have, so we have to learn their language! And just naming something is a first-order abstraction, right? Calling it “x” might be considered a second-order abstraction, but the point remains that mathematics is nonetheless just a language on it’s own: a way of expressing (aka communicating) patterns! 

Like this simple pattern: 

My friend Eric says not to drink soda, so instead I’ve been following my friend Chad’s advice and drinking beer! My favorite beer comes in 16 fluid-ounce tall-boy cans – packs of six – whereas Chad’s favorite beer comes in 12 fluid-ounce cans. How many fluid ounces do I get when buying a six pack of Chad’s vs. my beers? 

Well, if I assume that we are now adults again and we have learned the language of mathematics through “multiplication,” but perhaps still don’t really like to do multiplication, we’re not happy about these big numbers. That’s probably because there is an underlying problem with two things: our underlying metaphor of what multiplication is (memorized rules vs. sets of items), and the amount we’ve discovered aka “played with numbers” over time. But 16 times 6 is like 10 times 6 – 60 – plus 6 times 6 – 36, which is like 60 plus 30 – 90 – plus 6 – 96. 12 times 6 is like 10 times 6, 60, plus 2 times 6, 12, so 72. 96 fluid ounces is 72 plus ten (82) plus ten (92) plus 4, so 24 more fluid ounces in my 6-pack than Chad’s, and then from there we begin to compare the costs vs. ounces. 

What about this, no magic black boxes that produce the right answer of any sort – only logic please: 

Screenshot 2020-01-23 at 8.08.11 PM

What I find often is that most people know what this means. Ask them to discuss and they start conversing about the ‘right answer’. The conversation tends to be the interesting part. “OK take this number, and blah blah blah” or they use the number itself to talk about it. Well, what happens when we want to create a mental schema for this situation, aka “a rule”? Well, then it might be useful for us to ‘name’ parts of this statement – the numerator and the denominator; the dividend and the divisor… why do we have so many names for the same concepts/processes? Because just as we named ‘division’ before out of a need to do one thing, we then had a different need later and this is the dumb system we ended up with. Again – nobody can discover the world for anybody else – we must first discover it for ourselves, and then we may understand the common bond of developed conventions, and why they are necessary even if they are confusing. Anyways, back to this process of playing with numbers – if we were to simply think of ‘moving’ the decimal to the right in both the numerator and denominator of our first mental math problem, we can see that it’s just 56 divided by 8, which is 7. If we have our mental schema of division down (from earlier), then I can also use my understanding that division is asking how many times can 0.8 go into 5.6, we can use fingers to confirm that our answer of 7 is actually correct: 1 is 0.8, plus a 0.8 is 1.6, then 2.4, 3.2, and we can tell that indeed 7 is a reasonable answer. We can posit a rule now: if I ‘move’ the decimal the same number of spaces right in the numerator and denominator, my answer does not need a decimal-place adjustment. Following the same process for the next two mental math equations, we can discover that 70 and 0.7 are the answers and posit the corresponding rules about which way to move the decimal place on the resulting answer. In this way, we rely on a confidence that we can logic our way to the right processes in math, avoiding the constant fear that we have simply forgotten the rule, or memorized it backwards, and thus are doomed to be failures.

All this to say that trying to teach multiplication by first calling it multiplication is something we should be discussing… despite the fact that it takes far longer to develop this sort of knowledge, I personally think it is far more effective, and I would go so far as to argue that it’s the niche of Expeditionary Learning… one of the principles we are founded on… OH, and also what “literacy” means. Thus, my metaphor for literacy is ‘discovering the world for oneself first (in an abbreviated manner – that’s the job of the educator), and then learning the language through which to discuss it. 

Homework: Chapter 8 of Postman’s Teaching as a Conserving Activity

After reading, answer the reflection questions: What can you do during lessons in the next week that will help students to better understand and have mastery over “the language of” a particular subject? How does your ‘to-do’ item help to support the ‘discovering the world for ourselves first’ metaphor?