An Unusual Math Lesson on Practice

We sittin’ here, I supposed to be the franchise player, and we in here talkin’ about practice. I mean, listen, we talkin’ about practice. Not a game, not a game, not a game. We talkin’ about practice. Not a game. Not a, not a, not the game that I go out there and die for, and play every game like it’s my last. Not the game. We talkin’ ‘bout practice, man. I mean how silly is that?”
-Allen Iverson 

Practice. If you’ve read some of our recent posts, you know that Vena Cava educators are big on understanding practice, which we also call Habits. In many ways, education is about teaching students how to learn, and the way to learn is to transfer more and more skills (and knowledge) from your conscious mind to your subconscious mind by doing the skill (or drawing upon the knowledge) over and over again. Right? I don’t consciously think about riding my bicycle anymore, because I’ve learned how to do it from practice. I don’t think about completing the square, because I’ve done it enough that I can just use it as one step in reasoning through a more complex logical problem. Allen Iverson, back in his heyday, didn’t think about shooting a three. He had practiced shooting three’s way too many times in his life to think about it, and that’s why he thought attending practice wasn’t important (what he missed, of course, was a team spirit. Every time we play with new people, we have to learn their habits. Check out his Career Highlights and Awards – you’ll be impressed. But then… check out how many times his team won a championship). 

Well, here’s my recent lesson on practice. Perhaps you can use the same one yourself, or take the outline and adapt it to your own purpose and style. 

I start off with an innocent question: Can you learn from just watching a YouTube video? The typical response is that some students give an immediate and defensive “YES!” – but most students are in the land of “ehhh… yes? no? I don’t like watching Khan videos…” 

I would say (as I tell students) that the proper answer to that question would be another question: What are you trying to learn? It seems to me that it is completely possible to learn from just watching a YouTube video, as long as what I am trying to learn is very basic. However, as the task begins to increase in complexity, I think you have to begin to do more than just watch. 

I then walk over to my closet that I am lucky enough to have as a teacher who rides his bike to work most days and say “for example, what if what I am trying to learn is slightly more complex? I’m not talking about rocket-surgery here. What if the learning is how to tie a bow tie?” I pull one of my bow ties out and begin mimicking a YouTube video that explains how to tie a bow tie, working through the process and explaining what I am doing until I have a perfect bow. 

I give the tie to one lucky student (because for some reason all of them want to try it) and ask them to complete the task. We watch for a minute as the repeatedly mess it up. 

In order to tie a bow tie, you have to practice along with the video. Then you have to practice some without the video. And then you have to practice some after it has been a day since you tried it. And then you have to practice some after it has been a month since you tried it. And then… you get the point. 

Of course, the next step is to relate it back to math (or whatever content you are teaching). You know how to do that. But nonetheless, I’ll describe the three ways I did it today with my three different classes. 


Freshman are studying linear systems, but we are starting slowly with a much-needed regrounding in why. A couple of days ago, I asked them to find the slope of a line that we know goes through two points – for example the (x, y) coordinate points (-1, -2) and (5, 9). Many of them unfortunately had the curse of knowledge… they said they didn’t know how to find the slope because they knew there was a formula they needed, but they didn’t remember what it was. AHHHH!!!! How frustrating as a teacher! To find the slope, just get a ruler and your graph paper notebook and draw the graph! Then, count how many spaces up the second point is from the first, and divide it by the spaces to the right – that’s the rise over run aka slope! You don’t need the equation that you can’t even remember! 

Now, imagine that you (freshman), actually did all of the practice you were supposed to do in 8th grade with graphing lines… It’s really repetitive practice, right? Drawing the same coordinate plane (WITH a ruler) over and over again, plotting the points, and counting spaces gets tedious. You would probably have developed your own ‘shortcut’ that went something like this: you just divide the difference in y-values from the difference in x-values. That would have been your own way of stating the equation “y-two minus y-one over x-two minus x-one”, and I bet you would have remembered it far better because of the practice reps you did than you remember the equation your middle school teacher told you to memorize that you now don’t remember. 

Sophomores are studying completing-the-square, and although we go through the why of completing the square quite heavily, at some point they also need to just practice doing it. Well, about a week ago I had them do a practice page in one of our main books. Yesterday, I gave them the same page over again, asking they they focus on completing the process without notes and on the clock (under 1 minute per problem). Because I teach 10th and 11th grades at the same time, I went to go do the formal lesson for 11th and left them to their work. When I came back, almost everyone had realized it was the same page they did a week ago and ceased to work on it because “we’ve already done this page.” As you can imagine, my bow tie lesson with them was about this occurrence yesterday. Without going into more details, I told them that they all had better ace this Friday’s quiz… if that means they need to do some extra practice at home on top of the new homework, then so be it. 

Juniors are studying rational functions, and I intentionally have them try the homework using only the descriptions and examples from the textbook before I teach them anything about it. We used our bow tie lesson to talk about ‘how to read’ a math text. You can’t just read – you have to read with graph paper next to the text and a pencil in hand! That is the only way you can follow along, combine your own knowledge to make sense of new knowledge, and understand. How else can you make sense of a rule like: 

Screenshot 2020-01-15 at 4.51.53 PM

Finally, seniors are learning how to do data analysis in Excel. For them, the lesson goes right along with a concept of which I am constantly trying to convince them: that part of learning Excel is just playing with it a lot in low-stakes situations until you learn the fundamentals. For example, if I wanted to program a spreadsheet to analyze how changing the number of times the loan is compounded affects the total payment, I could use the formula A=P(1+r/n)^nt. A very common mistake (after forgetting to include necessary parenthesis in the formula code) is to simply forget to add a * in-between the P and the parenthesis – because of course that is typically implied in mathematics. When programming, however, we have to tell the computer exactly what we want it to do… so catching a mistake like that can take much longer than we are used to. You know how to get good at catching it? That’s right, “We talkin’ ‘bout practice!” 


A Maths Problem on Asking Questions and Using Resources

In the last post, I presented the idea of using a Mathematical Habits Checklist to try to help students develop a skill more useful even than perhaps any mathematical technique: the skill of developing habits that expand your knowledge and resources. We discussed the fact that ‘in the real world,’ people rarely take tests – instead, they complete projects. I also mentioned that accordingly, I will sometimes give Mid-Term or Final Exams that are open-note, open-resource. This allows students who studied well and don’t need to constantly use their resources to have an advantage on this two-hour exam, which could be thought of as a project under a very quick deadline! However, it also allows students who may not hold as much information in their heads, but have an ability to keep very organized resources (notes, can use a text well, etc) to still perform well on the exam, as long as they can use the resources fast enough to complete it in the two hours available. 

The point that I want to make this week is that using resources effectively is a skill on its own. We, as educators – especially math educators – should be finding ways to teach the skill more effectively. It has been my observation that modern students often think that they should be able to just look up the answer to any problem using Google… but in practice, they kind of suck at ‘Googling.’ 

Here’s a small ‘entry-point’ lesson on the type of thing I am talking about: the other day I was teaching my statistics class, which is studying how Election Polling works, and wanted to follow-up yet again on the idea that a Binomial Distribution is different from a Normal Distribution. If it’s been a while since you taught statistics, the difference is essentially that a Binomial distribution is the probability bar chart for a specific number of outcomes, whereas a Normal Distribution looks at the generalized trend of a Binomial Distribution and makes the number of outcomes infinite. For example, if there was an election occurring in which 60% of the population was going to vote for Coretta, our made-up candidate, I could determine the percent chance that x number of people vote for Coretta out of the total number of people that I poll.


Chance that X # of People Are Voting for Coretta
The probability that 0, 1, 2, 3, 4, or 5 voters polled say they are going to vote for Coretta, given that the total population is 60% in favor of her (which we wouldn’t know in a real poll, but that we may be able to use to predict the accuracy of a real poll)
We can run the same statistics for a 100-person poll… it’s just a LOT more possible outcomes to calculate, but it’s still a Binomial Distribution because it has a discrete (finite) number of possible outcomes. 0, 1, 2, all the way to 100 people could say they were going to vote for Coretta out of all 100 people polled. Notice that this curve is approaching how a Normal Distribution looks for the same exact scenario:


In the last moments before class, I decided that instead of reading through the problem that I wanted to give students, which starts out with the words “for a discrete probability distribution, the mean and standard deviation … [blah blah blah],” I could just ask them “what does ‘discrete’ mean in the context of the math we are doing?” (the way I phrased that question was important). Then, I could ask them to “use your resources to figure it out.”

Well, I ran this little experiment, and most of the students decided they wanted to use a computer to Google the answer. Keep in mind that we have textbooks that we use regularly, and if they were to simply go to the chapter we are currently working on, they would have found the definition very quickly because the chapter is specific to probability distributions – but the computer is the preferred tool in the modern world. You may have also noticed that the question that I asked was intentionally tricky – if I had asked “what does ‘discrete’ mean in the context of a probability distribution?” they may have used very different search terms. Instead, many students just typed into the search bar “discrete math.” 

If you aren’t aware, ‘discrete math’ is a field of mathematics completely unrelated to probability distributions, and this sent many of my students ‘down the rabbit-hole,’ so to speak. My job at this point IS NOT TO JUST TELL THEM WHAT TO SEARCH FOR! It’s not about ‘hitting a standard’ right now, because the very important standard known as “Students will be able to search for and find resources online effectively” happens to not be a math standard! My job, instead, is to ask them questions that help them to become aware of the importance of search terms in getting the information that’s needed; it’s to help them become self-reflective about the fact that the question that is asked matters (this concept applies far beyond just searching Google). Rephrasing the search query to “what does discrete mean in the context of a probability distribution?” will produce different results than “discrete math.” Students may still need Tenacity in Pursuit in order to take lots of information, some of which may be tangentially related rather than directly related to their question, in order to incorporate the meaning into their own mental schemas, but that’s life! 

You may not teach statistics, so this example may not be relevant to your classroom… but the point of it is certainly applicable! How can you structure your content this week in a way that challenges students to ask better questions and understand that finding/using resources is an invaluable skill? Please get in touch to let us know what you do!


Idea: Mathematical Habits Checklist

I believe in the Ecological Model of education, wherein we celebrate natural diversity of learners and attempt to broach subjects with a more complete understanding by expressing the ideas and language of the subject in multiple ways.

That’s why in my math classroom, tests and quizzes make up slightly less than 25% of the total grade. They are one way (out of many) to show a depth of mathematical understanding – in the ‘real world,’ though, mathematicians who are able to do excellent work more slowly and thoroughly are also celebrated, usually even moreso than quick ones. Mathematicians who are able to show and make comprehensible the techniques they used and the reasoning behind the techniques are as critical to applications in the field as innovators of new branches of study. So why not allow those sorts of mathematicians to shine as well, in the form of Projects that require slow work and excellent communication and craftsmanship? After all, once we graduate from school, projects are what we do, not take tests.

When completing projects, we have resources. Yes, being able to think quickly on one’s feet is useful in determining which resources are necessary for the problem we are currently solving, but I also think practicing the process of drawing upon resources efficiently is useful, and this is why it is not uncommon for my Mid-Term or Final Exams to be open-note, open-resource (textbook, sometimes even the internet!). The amount of content being covered is voluminous – so completing a 50-question exam in two hours is no small feat, even with notes! You are forced to create some organization or mental schemas to delineate your note-seeking, or to just have studied well enough to not need very many notes (which would obviously put you at an advantage – which is what makes open-note a equitable/fair differentiator).

In order to do well on an open-note final exam, then, it is important that one has great habits throughout the year. And now that we mention them, I think habits are something we do even more often in the ‘real world’ than projects… so they should be part of math class anyways!

As part of my attempts to build habits in my students, for the past two years I have had them create a ‘Study Guide’ as the only homework assignment on Thursday nights before their weekly Friday Check-In’s (quizzes). The idea here is that at the end of every week, instead of doing more practice you are given a little time to organize your thoughts on the subject(s) of the week and place them into a document format that would be useful to you in studying for your final. Imagine at the end of the semester, using all of your previously self-created Study Guides as well as your returned quizzes to study for the final!

I’m becoming more sold on the belief that school is really just a way to force kids to adapt habits: you are forced to engage with a topic every day for a short period of time, and over time you don’t even realize that the subtle, 1% shifts over many days begin to really add up. And since I am deepening the belief in this concept, I recently created and added the ‘Mathematical Habits of Scholarship Checklist’. The idea here is that students fill out this sheet during the last 5 minutes of class every day. Now, as we all know, this is valuable time for math teachers to continue to skill-and-drill! But my pushback is twofold: first, becoming meta-cognitively aware of the things you are doing to improve your understanding of math will also improve your math! And secondly, mathematics is about making sense of this world. That means that taking the time to make sense of how various concepts relate to each other and to organize the concepts into your broader conceptions of the world is important – at least as important as continuously completing practice problems. After all, which skill do you (adult reader) use more: completing the square of a quadratic or creating a healthy daily habit of doing what needs to be done to put food on the table? The other cool thing about this is that, at the end of the week students who have filled out the back page of the tracker with fidelity will already have the outline for their Study Guide!

Accordingly, I am having students fill out this tracker for the last five minutes of each day and I collect it when giving them their Friday Check-In. I don’t judge the students on the coding they use for the checkboxes on the front page (I have them rate 1-10 their performance in each of the category boxes) – I just check to see that they actually filled them out and then check how thoughtful are their notes on the back page, and that’s it! Even if students try to ‘play the system’ at the very least we’re giving them the language to talk about it in the future.

M-HOS Self-Reflection Checklist

Try it in your classroom and tell me what you think!


‘Having the Language to Talk About It’: An Anecdote

My wife and I talk a lot of philosophy, which if you read my writing probably comes as no surprise… The other day she was talking about her brother, who is now a young adult with a couple of years of experience in the workplace. I will paraphrase her comments here:

I feel like he’s struggling a little bit right now with the same thing that I struggled with for a while – that ‘high-achiever syndrome’ where you grew up being the straight-A student and the star athlete and you get into the real-world and just want to be the best at everything and have all the best things in your life. But, I feel like he is just struggling with it in a completely different way than I did. I think yoga helped me a lot in that struggle, because yoga is a practice rather than a ‘sport’. This subtle difference is conveyed repeatedly when undergoing the practice of yoga, and just having the language to talk about it allows you to see a bit more clearly the reality of the situation. Had a bad day and didn’t get the results you wanted? Cool, what can you learn from that and do to get better at your next practice? That was a really big shift for me…

Yes! We think through our tools, and our biggest tool is Language. We talk about this all the time on this site, but my wife’s point is a poignant reminder for us: doing yoga sessions as part of our weekly routine for many years subtly shifts and expands our language, and therefore our ability to perceive situations in different ways. It expands our perspectives. 

For her, the language of the practice, starting with calling the activity a practice, allowed her to take a step back and think about work and life in ways that may not have been available to her without the shift in language. Furthermore, the practice has other language associated with it, like actually choosing our values, or choosing our level of calm/stress, or recognizing the flow of our daily energies. All of these ideas, though not revolutionary in any way, completely change our perspectives when they are part of the language that we use on a daily basis to describe our hand in this life. For my wife, these ways of ‘languaging’ made her realize that being the best in the office was an arbitrary standard, and that she wanted to choose different values. By choosing instead to see every situation as practice, as ‘an opportunity for spiritual growth, whether you like the opportunity or not,’ she was maintaining sanity, maintaining life/work balance, and also becoming the best in the office through consistent, daily improvements. So, she obviously crushes at her job… but she’s never going to be the CEO because she takes way too many vacations and spends time with family at 5pm, and because she has the language to express why being CEO is not her top priority. She has the language to express why she already has all of the best things in her life, all while maintaining the desire to continue to improve. 

So what’s the educational context?

It’s easy for us, as teachers, to feel like our consistent efforts at going ‘above and beyond’ the curriculum aren’t going anywhere. I’m talking about our preaching to color-code solution processes to math problems, or our constant re-reading of quotes from Flow or The War of Art or Roots, or my focus on the process of Mathematical Dialogue, or my constant efforts to have students understand habit formation and the value of consistency over intensity. Sometimes it feels like we aren’t changing the habits and customs these kids came to our school possessing and with which will likely leave.

But we are. We are giving them ‘the language to talk about it.’

Sure, my kids hate doing their Training Plans. But when they leave this place, they will have the language to understand more about habit formation than that which our Primary Curriculum would have provided them (yes, let’s not kid ourselves: the media/information environment of our time is the Primary Curriculum, not the school environment. And, of course, the internet/YouTube make up the media environment of our current era). And having the language to understand the situation is a tremendous start. They will not be the same people they would have been without us in their lives. Just as water flowing on rocks, the effects are difficult to notice until years down the line. So keep going beyond the curriculum my friends. 

– mmm

The Web: A Small Deviation

The other day I was in class with the 9th grade, working on Trigonometry. A table group of boys that I would categorize as ‘experienced thinkers who tend to get a bit rowdy or distracted when it comes time to practice’ stopped me. In the process of doing a deep think on the derivation of the Pythagorean Theorem using Pythagoras’s Proof, one of them had all-of-a-sudden noticed that on his Trig Ratio sheet*, there was just a blank spot for the tangent of 90°. 

Of course, we all know that 9th-graders aren’t supposed to know the answer to that question yet, so I told them all that I wanted them to get back to work and not ask questions that they aren’t supposed to know the answers to! Maybe one day they would find out, or maybe not. 

I’m kidding. That would have been the worst possible thing a teacher could do in that scenario, right? Remember that The Medium is the Message, what students learn is not the content that we teach, what they learn is the methods through which we teach it. That approach, then, would not have taught students to think about the Pythagorean Theorem more deeply. It would have taught them to not ask questions sparked from real curiosity, or that only questions asked by a teacher or a textbook are real questions (see point 2 in this post), or that the only content that matters in mathematics is the content that I have to teach you in school (after all, they don’t know that they will learn the answer to this question in 11th grade). The last lesson is one that we are constantly struggling with, even today, as high schools still have standards that, while good, still miss out on a lot of important and useful maths; accordingly, it’s much easier for current teachers, who also grew up in an educational system founded on ‘the industrial model’ or ‘linear model,’ to unknowingly support their students adoption of the linear model as well instead of the ecological metaphor of education or ‘The Web.’ 

But back to the kids: this should be the dream moment for teachers everywhere. It’s one of those rare opportunities that we get to potentially plant a seed of thought that will grow and thrive. Moments like these are what makes learning come alive for us, right?! 

So no, I didn’t tell them to not ask questions like that, or to get back to work, or that the answer was ‘above their level.’ Instead, I had fun asking questions with them and starting a mathematical dialogue

“Hmm. That’s a good question! Why wouldn’t there be a value there? I am going to have to assume it’s because the author of the table just messed it up… or do you guys think there is an actual reason?” 

“YES!” One of them said, “It’s just because 90° is the right angle! So why would you try to take sine or cosine or tangent of the right angle?” 

“That makes sense – after all, the right angle wouldn’t have an ‘opposite’ and ‘hypotenuse’, right? Because they would be the same side? So then how would you be able to do opposite over adjacent?” 

“Yeah!” They seemed satisfied with that answer. Oops – time for some contriving. 

“Hmm. One other thing that I just thought about though…” I paused. And then, I tried to trick them with one of my typical tricks. I will often say incorrect things just to see if they catch me, like: so 8 x 7 = 64, right? This time I said “You guys know how Trigonometry is really about the study of circles, right?” 

One of them nodded enthusiastically and said yes, but the other four all said “No they aren’t! It’s about triangles! You aren’t going to get that one by us!” 

“Ok, you guys are getting better at that. But seriously, consider this: there’s a guy on a big, 50-foot radius ferris wheel at a carnival, and I know that the center point of the ferris wheel is 65 feet off the ground. But what if I wanted to know how far off the ground the person was at this point?” I said, pointing to the circle that I just drew on the board, right at about the traditionally-measured 40° mark. I added in a dotted, horizontal midline to the circle and said “Yeah, this is a good question. Figure this out for me and I’ll be back.” I went and did some conferring with other students and came back a little while later when I saw that they had filled in the circle with a triangle. They were excited to report that trigonometry actually could, kind-of, be about the study of circles, like I had ‘jokingly’ said earlier! But, they would have to know some more about the situation to figure out where the guy was. I pushed them on what more they would need to know, and they came up with: the radius of the ferris wheel, the angle of the person above the midline, and eventually the time it takes the ferris wheel to do one rotation (in service of finding the angle). 

“Now, here’s an idea…” I said, “It seems like if you wanted to, you could create a graph that showed how high above the ground this person is for the entirety of the ferris wheel ride – however long that is…” And of course this is the moment that makes it all worth it. Their eyes actually got wide with realization. “Obviously we have work to do now, but that would be cool if one of you remembered that and proposed it as an alternative project when I give you the next project.” 

And then, I walked away. 

The point, obviously, is that The Web of Content metaphor has a very different end-game than the linear model. My hope is that these sorts of interactions plant seeds, not make kids into experts in trigonometric functions or even into experts on any of the other obscure mathematical topics that don’t get used anymore in a modern world dominated by Data Analysis, a world where being able to calculate a definite integral by hand is just not that impressive compared to being able to understand what integrals are and how they relate to derivatives and how to program a Differential Equation into Matlab for the purposes of modeling a complex economic situation. I want them to know that knowledge is not ‘pre-existing’ in the world – it comes from somewhere! Namely, from people who create it by observing the world and recognizing patterns that are that way because there is no other way they could possibly be! I want them to know that they are just as capable of ‘discovering’ concepts on their own as they are capable of learning those same concepts from a book. I want them to go through the process of creating their own language and symbols to keep track of how they are going to model their new discovery of ‘height above the ground while on the ferris wheel,’ and to then come to realize that all mathematical symbols, no matter how complicated-looking they are, came from people who had a similar desire to communicate their thoughts. And, I want them to then ‘discover’ that what they had ‘discovered’ has already been discovered and there is already a language and symbols through which people talk about it, and to go through the process of reconciling their symbols and language with that of the ‘standard’ way of talking about ‘height above the ground for guy on ferris wheel’ aka Trig Functions. 

This is a very different end-game than ‘I want kids to have mastery over a*a+b*b=c*c’. It will take patience and lots of moment-to-moment distinction. But I think it’s sure worth it. How are you going to plant some seeds this week? 


*Why do they use a trig ratio sheet? I’m glad I interpreted that you asked, because it’s important! I write more about it in Mathematics and Why, II.

Mathematics and Why, Part II: How to Teach Trigonometry for Understanding


Last week, I gave some background into Mathematics and Why. This week, we are picking up where we left off: with a second alternative for a better way to teach maths in the US – a way that begins the ‘Mathematics Revolution’ from inside the walls of our curriculum rather than through changing the curriculum completely.  

To begin, I must reference you to my ‘Welcome Letter’ – as I mentioned in that post, I think creating a Welcome Letter that outlines the big ‘why?’ questions for a lot of the practices we use in our classrooms on a daily basis is incredibly enlightening for a teacher and will sometimes cause us to change some of our policies for the better. However, I am also referencing the post because my own Welcome Letter explains a lot of the structural designs of my own classroom; in this post, I plan to jump far more heavily into the specific math content rather than the broad policies (like, for example, why I use a Character Point Average as 25% of the grade in my classroom). 

So let’s jump into it by starting with a broad-scale discussion of the organization of content. I teach at an Expeditionary Learning School where I am the Chair of the High School Math Department for two reasons: one, my extraordinary skill in teaching mathematics, and two, because I am the only math teacher. OK, fine, I made up the first reason, but the point is that the second reason allows me some flexibility with moving content around, and here I’d like to share the results of some experiments I have done in that realm. 

To begin, I changed the starting unit of Freshman year; where we used to jump straight into solving linear systems of equations and inequalities, we now begin the year with a unit called ‘Do Bees Build it Best?’ – a unit that derives from the Interactive Mathematics Program (IMP). As with all of my units, I do a lot of adaptation and innovation from the base unit, but the topic of this unit is an exploration of the honeycomb shape that bees create in their hives – thus the essential mathematical features of the unit are Geometry, Trigonometry, Simplifying Radicals, and a heavy focus on deriving equations that will work for specific scenarios. 

The reason I made this change is twofold: first, students are coming off of a middle school experience where they did quite a bit of Algebra I, and secondly because this suite of topics sets students up very well for understanding the idea of deriving a ‘new’ concept. The first reason is important because, let’s face it: Algebra I is abstract. Given that many mathematics programs don’t ground the topics in real-world usefulness to begin with, students are commonly feeling like math is a mysterious set of difficult-to-memorize rules that were derived for the sole purpose of inflicting misery upon middle school students. In starting these students off with ‘an exploration of the shape honeycombs take on,’ students begin to understand that math is not about arbitrary rules… it is about observing the world around us, trying to recognize patterns within it, and eventually deriving rules about how the world works based on the fact that we observe it can’t possibly be any other way. They also get a break from the ‘abstraction’ of Algebra I – they can literally see the shapes that we are working with, and start to develop their ability to ask the next question about the shapes. For example, when looking at a honeycomb, students might ask “How much area does the honeycomb ‘hole’ cover? Obviously it has to be enough space for bees to fit through it, but not more, so how much is that?” The obvious ‘next question’ is “How do we find the area of this shape?” The answer: “Well, it would probably be easier to find if we break it up into triangles first.”

animals apiary beehive beekeeping
Photo by David Hablützel on


Once the hexagon of a honeycomb has been broken up into triangles, we have to ask “What is the area of each of the triangles?” Obviously this means that we have to find the base and height of the triangles, which launches us into a full-on study of triangles and trigonometry. Now, as I mentioned previously, the second thing that this unit allows me to do is set students up for success with later concepts in the class, including deriving equations as well as attending to precision and understanding that math class in high school will be a balance of big, creative style explorations (like that of the honeycomb) and skill-development. Specifically, how this happens for the Bees Unit is that I introduce (remind) students to the three main ways to express patterns in mathematics: tables, graphs, and equations. Tables are incredibly useful tools for exploring patterns when we don’t already know what pattern exists – so I have students derive the area formula for triangles using both ‘geoboards’ and a bunch of triangles of different base/altitude lengths where they use the principles they learned on the geoboards (halfing rectangles of particular sizes) to manually calculate the area of the triangles. Yes, my students already know that A=(½)bh, so forcing them to “see” the area of the triangles is a challenge, and we spend a lot of time having me force them to prove area to themselves rather than believing the formula they had been taught. Eventually, they use their tables, which have an input of base and height of the triangles they have sketched on graph paper and an output of the area, to derive anew the area formula for triangles. “Good!” I tell them, “soon enough, you will be using the same skill to derive formulas you don’t already ‘know’!”

The next question sparked by the honeycomb area is “what angles do the triangles involved in the honeycomb have.” This launches a week-and-a-half of me instructing students to learn how to be precise with measurements – both length using a ruler and angles using a protractor. I have them eventually cut out right triangles with an angle of 55 degrees; as you may imagine, these triangles are all sorts of different sizes, but they are all right with an angle of 55 degrees. We post these shapes up on the whiteboards and start measuring the side lengths and comparing the ratios of various side lengths. After recognizing that the ratios have to be the same, I ask if students think this is a trend for all triangles or just right triangles with a 55 degree angle? They eventually determine that if the triangle is right, then the ratios of the side lengths will be the same for any triangles that have the same angle measure of the non-right angles. 


This leads me to do a bit of expert acting: if this is true for all right triangles, couldn’t we make a giant table of the side-length ratios for every angle possible for a right triangle, from 1 degree to 89 degrees? They say, “yeah, we could! But that would be hard.” I respond, “well, what if we start by deciding upon a few easier concerns – do we really need to know the ratios for every possible side length? Meaning, if we get the ratio for the side opposite of the angle and the hypotenuse, do we really need to do the reverse – hypotenuse over opposite? Wouldn’t it just be the converse?” Then we go into a discussion and test this theory. Students eventually decide that we really only need three ratios, and I then ask what we should call the ratios before we start doing all the work to create this giant table that may help us in the future. They begin arguing with each other, and then I bring them back to our full-class discussion and announce that I have an idea – perhaps somebody has already done the work of finding all of these ratios? I send them on a scavenger hunt where they can search through any book in the classroom (I have a large library of math and non-math books) to see if there’s already a version of the table we want to create. They go through the modern struggles of having me explain to them how books work (e.g. the use for an index vs. an appendix, etc) and eventually cheer with excitement when one of their classmates finds the table they need. It turns out, they realize, that we don’t even need to name the ratios we will use – they already have names! Etcetera, etcetera, etcetera. 

The downside to this approach is that it just doesn’t work with the linear or industrial model of education. It takes FOREVER before my students are even introduced to the words Sine (NOT ‘sin’ despite the unfortunate abbreviation of the term!), Cosine, or Tangent! In the linear model, we have some serious ground to cover – there’s no time for this whole progressional charade! But think about how much more students get out of this approach than out of being told to memorize these strange and abstract terms and concepts, and then being told that they need to just use the ‘sin’ or ‘cos’ buttons their calculators to find the missing side lengths. Beyond contributing to the over-reliance on technology, students would end up seeing trigonometry as a ‘magic black-box’ process rather than understanding that the field comes from somewhere, specifically from a person’s need or desire to make predicting how things will work in the natural world easier. 

It is for this reason that I never once tell my 9th-grade students about the fact that calculators have trig functions built into them. Obviously some of them eventually find out on their own, but the process of using a physical table with trig ratios has some important advantages. For one, students begin to learn that the abstract statement “sin(55)” is actually a real number – a decimal that represents the ratio of the length of the opposite side from the 55-degree angle over the hypotenuse side. This allows them to grasp both substitution and the idea that we sometimes use abstract symbols to represent numbers, when convenient.

From this point on, we obviously go into some skill-building portions of the unit, where students are completing more traditional-style “complete these 30 practice problems” sorts of assignments; however, we always transition back to the more creative problems as well:


As you can see, we go through a similar process of ‘deriving’ the Pythagorean Theorem, which then allows us to ramp up our derivation to a ‘new’ formula: what would be the area of a regular polygon with a known perimeter?


All-in-all, I feel that this approach, despite being very slow and thus limiting the amount of content we can cover in a year, allows students to actually understand mathematical thinking. I know it’s not the large-scale curriculum redesign we discussed in the first post on this topic, but what do you think? Feasible in your mathematics classroom next year? Beneficial for student understanding? Going slow to go fast, sort of like the Degeneration Effect describes? What did I miss? 


Idea: The Welcome Letter

Here’s an idea for educators out there that I hope will help with answering ‘Why?’ – both for your students/parents and also for yourself. Instead of doing the classic ‘teacher thing’ and giving out a syllabus on day one, write your families a ‘Welcome Letter’ that outlines your most important beliefs (that families would do well to hear) on education and the reason your class is designed as it is. Really lay it out there – perhaps even choose a theme (as I did in the example below) and explain how each practice in your classroom contributes to that theme. I did this for my own class and found it very useful for my own understanding of my own classroom. The process even caused me to change some of the structures of my class for the better – hopefully you will encounter the same!

Here’s the letter I wrote to families about my classroom:

Welcome to Your 2019-20 Mathematics Journey

(A portion of the only 2019-20 school year you will ever have)

The goal of education is to help a society preserve humane values in the face of 

economic and political interests that would be better served otherwise.

Your Guide: 

Your guide on this mathematical journey is I, mmm. 

I graduated from Brown University with a degree in Biogeochemistry, and as both an environmental scientist and someone who didn’t know his multiplication tables in fourth grade, I never thought I would become a mathematics guide. But, after working in science laboratories for several years I took off on a 5,000 mile bicycle trip and learned that just talking to people was an effective way to spread environmental concerns. And, it’s fun! The natural extension of this type of communication is education.

So, I started teaching – AP Chemistry at first, and then I transitioned into mathematics because of my own experience of feeling that I was ‘bad’ at math for many years… at least, before realizing two fundamental facts. First, that anyone can learn mathematics with a knowledge of how to learn combined with requisite effort, and second, everyone – no matter what their natural aptitudes – has something to contribute to the dialogue of mathematics. It is through these two facts that I hope to invite you to learn and think about mathematics on a deeper level – not only so you can use the language and creativity of mathematics for your own personal financial benefit, but also so that you may play a part in helping to preserve humane values, whether environmental or intra-personal, in a changing world. 

The Vision

The vision of the RMSEL HS Mathematics program is described (in great detail) below through the medium of the ten Expeditionary Learning Design Principles. 

The Primacy of Self-Discovery

We believe learning happens best with emotion, challenge, and the requisite support. People discover their abilities, values, passions, and responsibilities in situations that offer adventure and the unexpected. In mathematics students undertake tasks that require perseverance, mental fitness, craftsmanship, imagination, self-discipline, and significant achievement. A teacher’s primary task is to help students overcome their fears and discover they can do more than they think they can. 

This translates into the mathematics classroom as much as it does on trips, manifesting in several ways. The first and most important way this manifests is that the classroom is at once a comfortable and uncomfortable place. This is because if the teacher’s primary task is to help students overcome fears, there must be a level of comfort for students to feel safe taking risks. Yet, on the flip side of that coin, there is no way to overcome fear without directly facing it – one step at a time. To illustrate, allow me to give a short anecdote: 

I love to rock climb. Yet, a few years ago I realized that I would never actually get better as a climber unless I pushed myself outside the comfort zone and tried new things. It’s a delicate balance, though, between finding the point of eustress – positive stress that helps you grow – and cortisol-inducing stress, which can have a negative impact on learning new skills. I realized that I needed to try that challenging route and risk falling, all while not letting my mental fear interfere with my performance. I learned how to create the proper internal environment to allow challenge without anxiety. And of course, falls happen, and in the safe environment that we create by having extremely well developed protection systems, they are a fantastic learning experience. 

The same thing happens in the mathematics classroom when a big test is coming up. Our students are playing with the balance between eustress and anxiety, and learning how to make the uncomfortable feel comfortable. They are risking failure, but learning how to create a mental environment in which they will succeed. Through this process, they are not only learning math – they are discovering how the body and mind they were blessed with works. They are in the process of self-discovery. 

The Having of Wonderful Ideas

The mathematics classroom has two physical environments which can be thought of as separate from each other: the GroupThink Tables to the South, and the Presentation Area to the North. We believe that the GroupThink area provides space for creative thought, collaborative inspiration, and time to experiment and make sense of what is observed as well as struggle with new concepts. The presentation area provides space for subjecting ideas to a broader audience for critical review as well as an area for the whole class to learn new techniques or tools that may be useful in investigating new topics. Tools that are useful for success in both of these spaces are available (clipboards for notes to be taken in the presentation area and ‘math boxes’ with compasses, protractors, rulers, etc. in the GroupThink area). 

Wonderful ideas may be had at either of these locations, or they may be had in solitude and reflection, while out on math solos (see the Solitude and Reflection section below). It is important to remember that wonderful ideas aren’t always new, they are always new to you. Sometimes the purpose of an activity is to re-create a formula that already exists – this is not a pointless endeavor! The process of innovating a formula leads to a depth of understanding unrivaled by the process of memorization! It is this depth of understanding that we seek through the curriculum used in this math course. 

To support the development of this level of understanding, we focus on two differing concepts: development of the ‘mathematical toolbelt,’ and inquiry-based learning. The mathematical toolbelt represents all of the mathematical skills that students may need to use to solve an inquiry-based problem. Tools within the toolbelt include concepts like number, variable, and polynomial operations (addition/subtraction and multiplication/division), graphing, creating a table, simplifying expressions, rationalizing denominators, substitution, elimination, etc. Just as a carpenter needs to choose the proper tool for the job of building a cabinet (wouldn’t want to use a tape measure to place a nail properly…), mathematicians need to choose the proper tools for the job when engaging in inquiry-based learning. The key difference is that mathematics is often times abstractly representing reality, making it trickier at times to know if one is using the correct tool (although this comes with practice). 

The Responsibility for Learning

Learning is both a personal process of discovery and a social activity. Everyone learns both individually and as part of a group. Every aspect of an EL Education school encourages both children and adults to become increasingly responsible for directing their own personal and collective learning.

The hope for educators at our school is that our seniors, after four years of engaging with the educative techniques that we offer, come to us and say with cheery excitement ‘Look at what I get to put in my portfolio! I chose this piece because […] !’ rather than [eye roll] ‘What to I have to have in my portfolio? Just tell me and I’ll get it done. Ughhh.” Everything that we do in math this year is hopefully working towards a comprehensive content understanding and that truly passionate engagement as described in the anecdote above. 

To break into specifics of the course itself, two main tools allow students to take ownership of their own learning. The first is Google Classroom, which provides students with access to pertinent details on assignments like homework or projects. The second is Infinite Campus, which provides students with access to feedback on their learning success. 

Announcements and Assignments for the class will be posted on Google Classroom daily (students will also have a ‘master list’ of all homework assignments), and students have the ability to check ‘done’ or ‘not done’ on various assignments, giving them an online assignment-completion tracking tool. However, students are encouraged to not rely solely on Google Classroom, as there are occasions (for example, a substitute hands out an assignment while an instructor is gone) when students will have an assignment that they are told about but that is not posted on Google Classroom. In these cases, students are still expected to complete the assignment for credit.

Grades will be posted to Infinite Campus weekly; missing assignments are also noted here. We suggest that parents and students get in the habit of checking grades together on IC weekly. Because allowing students and families to check their grades and assignments weekly encourages self-monitoring and ownership of the documentation of learning, I will not contact parents if a student’s grades are low or failing!

Once student work is graded, it is placed in a file folder on the shelf next to the sink. This allows students to have further ownership over their work as it forces them to go through the act of collecting their work. This is a small but important difference from being handed back their work! 

Students are welcome and encouraged to communicate with me in a professional manner via email. In the event that an email is unprofessional (unaddressed, grammar/spelling errors, informal language, etc.)  I will ask students to re-compose the email before returning their communication. 

If students communicate with me in person and would like the contents of our dialogue to be communicated to parents, they may write an email that details such, send it to the parents with me carbon copied, and I will be able to respond to confirm our correspondence. 

Empathy and Caring

Learning is fostered best in communities where students’ and teachers’ ideas are respected and where there is mutual trust. Learning groups are small in EL Education schools, with a caring adult looking after the progress and acting as an advocate for each child. Older students mentor younger ones, and students feel physically and emotionally safe.

The norms that we encourage in this class are based on some of Jo Boaler’s work and are listed below: 

  1. Everyone Can Learn Math to the Highest Levels. Students are encouraged to believe in themselves. There is no such thing as a ‘math’ person. Everyone can reach the highest levels if they want to, with hard work. 
  2. Mistakes are Valuable. Mistakes grow your brain – it is good to struggle and make mistakes. Go for it! 
  3. Questions are Really Important. Always ask questions. Always answer questions to your highest capacity. Ask yourself: why does that make sense? 
  4. Math is About Creativity and Making Sense. Math is a very creative subject that is, at its core, about visualizing patterns and creating solution paths that others can see, discuss, and critique. 
  5. Math is About Connections and Communicating. Math is a subject that is intricately interconnected to all of reality. Math is also simultaneously a form of communication. Strive to represent math in different formats (like a picture, words, a graph, an equation, a table, a flip-page cartoon, etc.) and link all of the formats. 
  6. Depth is Much More Important Than Speed. Top mathematicians, like Laurent Schwartz, think slowly and deeply. It may be impressive to solve a known problem quickly, but it’s not indicative of your aptitude. 
  7. Math Class is About Learning Not Performing. Math is a growth subject (like learning a language – think about how long it took you to learn to speak English as a baby… and you are still learning it!). It takes time to learn, and it’s all about effort. 

Success and Failure

All students need to be successful if they are to build the confidence and capacity to take risks and meet increasingly difficult challenges. But it is also important for students to learn from their failures, to persevere when things are hard, and to learn to turn disabilities into opportunities.

Of course, success and failure requires a very fine balance. Countless times I have had students ask for help in tasks as varied as putting up a tent to solving a quadratic equation, and I have helped them by only encouraging them that they can complete the task on their own! After all, students don’t build confidence from success not earned on their own, yet, they also need the encouragement that risking a mistake is worth it. There exists a parable of a butterfly struggling to escape a cocoon who has a man help him out by slightly tearing the edges of the cocoon. It turns out, little to the man’s knowledge, that the butterfly was permanently crippled because the intense struggle of his escape was the necessary prerequisite to developing the strength to fly. We must remember this with our students, avoid crippling amounts of help, and provide lots of encouragement in tough times. 

Collaboration and Competition

Individual development and group development are integrated so that the value of friendship, trust, and group action is clear. Students are encouraged to compete, not against each other, but with their own personal best and with rigorous standards of excellence.

The modality we use for discourse and collaboration in this class is Dialogue, as described by David Bohm. Although the practice of Dialogue is quite extensive, it is differentiated from discussion in that during a discussion, we are (sometimes subconsciously) seeking evidence that corroborates our current perceptions of the world or listening in order to create an effective response to our speaker. Dialogue, on the other hand, requires the listener to suspend assumptions, step into the listener’s shoes, and truly attempt to understand the speaker’s point of view. 

Of course, just as in life, students are not entitled to be leaders or even members of a GroupThink no matter the actions they take. Instead, other members of the group are entitled to ask members of the group not practicing Dialogue to not participate. “He who cooperates because he sees the truth as the truth, the false as the false, and the truth in the false, will also know when not to cooperate – which is equally important,” J. Krishnamurti states in Life Ahead. The development of effective individual thought within a group in dialogue is included in the Dimensions of Observable Growth (DOG) rubrics in order to track changes and progress. 

Diversity and Inclusion

Both diversity and inclusion increase the richness of ideas, creative power, problem-solving ability, and respect for others. In EL Education schools, students investigate and value their different histories and talents as well as those of other communities and cultures. Schools and learning groups are heterogeneous.

Students who learn in different ways are encouraged, but usually not forced, to work together. During this time, the Dimensions of Observable Growth rubric serves as a way to track metacognition of GroupThink. 

In order to address a greater range of the issues surrounding Diversity and Inclusion, I will reference the ideas of Neil Postman, who recognized that the discrepancy between reality and the human brain’s way of processing reality may give way to the idea of ‘prejudice,’ in what he refers to as ‘the photographic effect’ of language. 

We live in a universe of constant process. Everything is changing in the physical world around us. We ourselves, physically at least, are always changing. Out of the maelstrom of happenings we abstract certain bits to attend to. We snapshot these bits by naming them. Then we begin responding to the names as if they are the bits that we have named, thus obscuring the effects of change. The names we use tend to “fix” that which is named, particularly if the names also carry emotional connotations.

A variation of the “photographic” effect of language consists of how blurred the photograph is. “Blurring” occurs as a result of general class names, rendering distinctions among members of the class less visible. One of the most common manifestations of the lack of this kind of semantic awareness can be found in what is called “prejudice”: a response to an individual is predetermined because the name of the class in which the person is included is prejudiced negatively. The most obvious and ordinary remark made in cases of this kind, “They are all alike,” makes the point clear. 

One way that we teach this in the mathematics classroom is through GroupThink sessions (and the accompanying DOG) – we relate these somewhat philosophical and esoteric ideas to relatable situations. To paraphrase Postman’s thoughts again, we teach the fact that human biology clearly dictates that you cannot avoid making judgements, but that you can indeed become more conscious of the way in which you make them. This is critically important because once we judge someone or something we tend to stop thinking about them or it. Which means, among other things, that we behave in response to our judgements rather than to that to which is being judged. People and things are processes. Judgements convert them into fixed states. This is one reason that judgements are often self-fulfilling. The relatable context for students, of course, comes from a classroom setting. If a boy, for example, is judged as being “dumb” and a “nonreader” early in his school career, that judgement sets into motion a series of teacher behaviors that cause the judgement to become self-fulfilling. What teachers need to do then, if they are seriously interested in helping students to become good learners, is to suspend or delay judgements about students. Our students, then, can ask themselves if they have been exposed to judgements that created self-fulfilling prophecies, and can have dialogue about it. They can also practice suspending judgement themselves. 

The Natural World

A direct and respectful relationship with the natural world refreshes the human spirit and teaches the important ideas of recurring cycles and cause and effect. Students learn to become stewards of the earth and of future generations.

In math, students will get outside into the natural world in several ways. Math solos (described below) are a chance for students to experience solitude and reflection while diving into a new and challenging problem. Field work is designed to capture rich and engaging problems from the world around us… which serves as the foundation for the language of mathematics. 

Because the environment is our first teacher, and because much of our time is spent in the classroom environment, we hope to provide students with a practice of being environmental stewards even within the context of the indoor educational environment. By developing a sensitivity to disturbances (a declining organization of the classroom) and the responsibility to restore the environment (for example, cleaning trash from the room at the end of class), we hope to develop a transference to our outdoor pursuits. 

Solitude and Reflection

Students and teachers need time alone to explore their own thoughts, make their own connections, and create their own ideas. They also need to exchange their reflections with other students and with adults in the form of Dialogue. 

The following is an excerpt from Roots, ed. Emily Cousins: 

If students are to tap into their own creativity, personal renewal, and thoughtfulness, schools must structure time for meaningful reflection to be valued. “I don’t retreat from the world to escape,” Robert Frost said, “but to return stronger.” Solitude is cocoon time. It helps develop powers of concentration. It requires silence, commitment, and an imaginative use of existing space. It does not cost any money; it can happen every day. Scientists and artists alike attest to the ‘click,’ the unanticipated connections they make when constructively immersed in solitude – an experience virtually unknown today in public schools. 

Essential for character development, solitude and reflection also enhance academic learning. David Kolb suggests that learning requires explicit time set aside for reflecting on experience (Kolb, 1984). His work describes a learning cycle that has four key elements: concrete experience and observation, considered reflection, synthesis and abstract conceptualization, and testing of concepts in new situations. Kolb’s model is especially pertinent to the experience-based field work that makes up much of each learning expedition. 

Solitude and reflection in the mathematics classroom comes partially through ‘Math Solos,’ which occur at or near the introduction of a challenging Project. Math Solos are a time where students find a peaceful spot in the park, by themselves, and begin to tackle a challenging problem on their own. This gives students the chance to develop their own ideas before engaging in collaboration with a group. 

Service and Compassion

We are crew, not passengers. Students and teachers are strengthened by acts of consequential service to others, and one of an EL Education school’s primary functions is to prepare students with the attitudes and skills to learn from and be of service.

In my mind, crew is often confused with ‘family,’ when the premise of the idea derives from ships at sea. Out on the water, every crew member has a specific task that – if not completed – may cause the ship to sink. If and when one crew member is feeling sick or weak and cannot complete their task, in order for the ship to keep moving the rest of the crew must step in to support the necessary work. It is this commitment that creates familial bonds. 

In the mathematics classroom, we (as a ship) have to keep moving, and it is up to the attitudes and actions of our crew members to keep us from sinking. 

Components of The Vision

A Summary 

As you may be able to sense from the information above, you will be challenged this year to push your thinking beyond what you even knew possible! However, because the information listed above was largely the essence of how math class will run, the precise details are given below. 

Class structure and physical space:

Class begins upon the ringing of the tuning rod, at which point students begin on the MindBender of the day. MindBenders consist of several problems that they may encounter on a standardized test – this is great practice for the SAT! The content of the MindBender is usually related to the content of the unit we are covering, but may occasionally be related to skills that are under-practiced. Every other day, students are in charge of creating the problem for the day’s MindBender, and after solving the problem, other students have a dialogue to understand the problem and critique the question itself. This practice serves to help students understand that all standardized tests are written by human beings with an agenda, and understanding the agenda of a test-creator can help in solving questions on the exam.  

Next, the class will break into the activity for the day. Most often, the activities consist of a formal Lesson which occurs in the Presentation Area of the classroom followed by collaborative GroupThink sessions. My expectation is that students are in the Presentation Area during presentations; however, I will only invite them to this area – the choice to engage or not is ultimately theirs (see On Freedom and Discipline below). 


The mathematical work you will be producing this year is what is kept track of in the ‘grading system.’ The work is a three-legged stool comprised of: 

  1. Unit Work (Homework) 
  2. Projects / Presentation Grades
  3. Check-ins

Unit Work refers to the practice that you complete inside and outside of the classroom that serves as the necessary prerequisite to be a valuable member of the Community of Inquiry. You can think of this work as ‘Homework;’ however, the focus of the work will also be on quality and craftsmanship rather than just having completed the work ‘just to get it done.’ 

Therefore, 10% of the unit work grade is composed of completing the work, and additional points will be added to the Academic Content component of the grade based on the quality of the work. Quality is assessed in two ways: through Entrance or Exit Tickets from class that quiz you on a question from the homework, or by turning in a homework assignment for me to review. Upon my review of the assignment, I will return the document to the ‘Graded’ folder by the sink. 

During each unit, notebook checks will be conducted and added as a component of the Unit Work grade. Notes are expected to be neat, organized, and complete (everything discussed in class copied to the notebook). 

Projects / Presentations are challenging, yet creativity-oriented problems or paradoxes that you work on individually and complete a write up summary to effectively communicate your findings. Presentations are given for each of the projects by students as a modality for communicating our ideas – each student will play a part in giving one presentation per unit. 

Check-in’s are a chance for students to be honest with themselves and get a grip on how well they alone understand the content of the unit. Check-in’s may come in both Standardized Practice format or in Creative Depth format. Standardized Practice format is a multiple choice exam that requires both fundamental skills and creative solutions – the purpose of this format is to help students learn how to ‘play the game,’ if you will, on exams like the SAT (whatever your thoughts on the value of the SAT are, learning how to play the game can open more doors for our students’ futures). Creative Depth format is a free-response question that will present students with a problem that needs to be solved with creative use of tools from the Math Toolbelt. Students may or may not have seen similar problems in the past and must take their time to achieve a depth of explanation on these problems. 

A comprehensive final check-in is given at the end of the year and counts only as a 3rd trimester grade.

Character Point Average (CPA): 

You may have noticed that the mathematical products you will create only comprise 75% of the total grade in the course. The remaining 25% of the grade comes from your Character Point Average, as measured by both you and I according to the rubric developed for the Character Point Average (CPA) as well as your weekly work for tracking your Habits of Scholarship using the provided tracking sheet. 

The reason that we use a CPA grade – which some people would think of as a ‘participation’ grade – is because we believe the Character of a co-worker or employee matters in the real-world. If I am hiring employees, who would I choose between two candidates who have the same GPA and SAT scores – the one who shows up on-time to work everyday with a smile and jumps in and engages? Or the one who shows up late and spends the entire day complaining about annoyances that he could easily solve himself with some better habits? We’d like to help our students grow into the former.  

There are a few components of the CPA that are so basic as to not be listed in the CPA Rubric, but that I will mention here because they are non-negotiable. The first is simple – professionalism. The rules surrounding professionalism are in service to a greater purpose of creating a classroom like that described in The Primacy of Self Discovery section of this document – a place that is at once a comfortable and uncomfortable, allowing students to face and overcome fears. The teacher of course plays a role in creating this environment, but so does the community (in many ways even more so than the teacher). I have seen entire communities begin using profanity when only a few students used it to begin with! We are social beings, after all. 

So, the three rules of professionalism that I enforce strongly are: 1. Don’t use profanity. 2. Don’t criticize another person. 3. Respect and be sensitive to the educational environment (don’t sit on tables, leave trash, not return items to their proper place, etc.). The first two rules result in an immediate NE for the week’s CPA grade upon one offense, and a Zero upon second offense. The third rule results in an NE to the day’s CPA grade upon one offense, and a Zero upon second. 

A few other components of the CPA that are specific to the mathematics classroom are listed here: 

  • The class will start the year practicing the proper response to the chime. After this has been practiced, the class as a whole will be expected to quiet down for an announcement after hearing the chime, with each crew member making sure to be responsible for other members. In the event that this does not happen, the whole crew’s CPA grade will be lowered by one letter grade for the day. 
  • Students are expected to sit or stand in the presentation area during class reviews. Failure to do so without prior permission results in an NE for the day’s CPA grade. 
  • Every GroupThink begins with a few minutes of Individual Think designed to allow all sorts of thinkers to gather thoughts before discussion begins. A failure to respect the silence of Individual Think time results in an NE for the day (it’s only a few minutes!). Students are reminded of this fact almost every class period. 

This course is founded on a community that comes together around the work. The best way to have fun learning a TON is by making the explicit learning become implicit through practice. Creating SMART goals will be useful in this course, as well as critically assessing your work habits – do you do some amount of math work every day? The Power of Habit is an excellent resource (available in the classroom) for designing your work habits such that you enjoy math and stay up-to-date. 

On Freedom and Discipline

“The paradox seems to be, as Socrates demonstrated long ago, the truly free individual is free only to the extent of his own self-mastery. While those who will not govern themselves are condemned to find masters to govern over them.”     – Steven Pressfield

I would like to address a special consideration for students that attend our school which differentiates it from other schools – the idea of Freedom and Discipline. Allow me to take a moment to get a bit philosophical. 

The teacher’s primary task is to help the student overcome fears [see The Primacy of Self-Discovery]. Yet, where do these fears come from? I believe at their roots, they stem from social conditioning. “Wealth, status, and power have become in our culture all too powerful symbols of happiness,” wrote Mihaly Czikszentmihalyi, the neuroscientist in charge of the world’s largest ‘happiness’ study to date. “And we assume that if only we could acquire some of those same symbols, we would be much happier.” But what if we aren’t able to acquire them? And so begins our fear of not being adequate, or having enough ambition – so we work our lives away, comparing ourselves to others around us and the Jones’s, hoping to prove to ourselves our significance, when the truth of the matter is each of us is already significant. That fear blocks intelligence and makes us dependent on social controls, which is the entire purpose of socialization, to have people respond predictably to rewards and punishments. This is precisely what the Titans of Industry wanted when they helped to create modern education during the industrial revolution – they wanted a malleable workforce that was educated in certain skills, but that would not question. “And the most effective form of socialization is achieved when people identify so thoroughly with the social order that they no longer can imagine themselves breaking any of its rules,” writes Czikszentmihalyi, “[we become] dependent on a social system that exploits our energies for its own purposes.” The implications here are massive – they challenge our entire tradition-rooted manner of thinking. 

“A free human being can never feel that he belongs to any particular country, class, or type of thinking,” wrote J. Krishnamurti. “Freedom means freedom at every level, right through, and to think only along a particular line is not freedom.” So what can we do? Czikszentmihalyi answers: 

To overcome the anxieties and depressions of contemporary life, individuals must become independent of the social environment to the degree that they no longer respond exclusively in terms of rewards and punishments. To achieve such autonomy, a person has to learn to provide rewards to herself. She has to develop the ability to find enjoyment and purpose regardless of external circumstances. If a person learns to enjoy and find meaning in the ongoing stream of experience, in the process of living itself, the burden of social controls automatically falls from one’s shoulders. 

Many of the components of my class listed above are scaffolding a path through which students may achieve such autonomy, to be the master of themselves. This is not without difficulty and will certainly require a different social dynamic of Dialogue rather than Discussion; however, I believe that until all people are able to self-govern and unlock their own unique intelligence, and the world ceases to have ambitious people who seek to gain power through stripping others of their power, humanity is faced with great challenge, strife, and potentially a crisis (be it environmental, conflict-based, or other). 

There are several structures available to students to help them learn self-discipline and communication. One of them, the DOG (Dimensions of Observable Growth) is a regular component of the class Habits of Scholarship grade. A second structure available is the RMSEL Goals structure, which takes the framework of ‘goals’ so often used in our society, acknowledges them, and then transforms the process of achieving those goals into one that works more harmoniously with human biology and teaches self-autonomy. Finally, for students who have found their freedom within societal structures and are ready to engage in effective communication with others, On Dialogue and the associated resources and rubrics can provide a framework for those ambitions. 

The Mathematics

During the 9th Grade year, students will explore the concepts of Geometry, Trigonometry, Systems of Linear Equations, and Exponential Functions. The main text that we use is the Interactive Mathematics Program, Year 2; the specific units that we will be covering (in order) are Do Bees Build it Best?, Cookies, and All About Alice. A great resource for parents to refresh themselves on the concepts in order to provide help to students is A secondary resource we use is Algebra I from Pearson. Students will receive an online login that allows access to a copy of this text to take home with them. 



Mathematics and ‘Why?’

Let’s spend some time today talking about mathematics and why. To me, this is one of the most overlooked components of the modern American curriculum, and the stakes are high. What is at risk is not whether our future citizens will be able to calculate tips in their heads… why would anyone think that is important in a world where we walk around with computers in our pockets? No, what is at risk is the intellectual orientation of our democracy: will citizens be able to analyze large datasets well enough to understand outcomes of policy and affect change in a world that is increasingly disparate in net-worth distributions and political power holdings?*

But let’s take a huge step back here. The goal is to further the depth of understanding** of the questions: Why do we teach mathematics? And why do we teach it in the way that we teach it? And why the specific topics that we ‘choose’ to cover? 

I know… the first question seems pretty obvious. We teach mathematics because it is a language, and humans think through the language and tools that we use. Through the study of math, we are able to greatly enhance our pattern-recognition abilities – it’s like the equivalent of being able to put on infrared goggles! Yes, it’s true that in a lot of situations out there, the goggles aren’t going to get me any further than if I didn’t have them; on the flip side, though, there are quite a few situations where having them is pretty freaking cool if not absolutely outcome-changing. We could of course bring the metaphor into the real-world in myriad ways, but to name just one: the ability to program an Excel spreadsheet with the mathematics that make up an amortization schedule for a mortgage probably won’t be all that useful when choosing who to vote for in your local School Board Elections… but it could be a complete game-changer when you are deciding how you want to approach choosing and paying off your own personal mortgage to be able to analyze savings over renting, to get away from using corporate calculators (that have business biases built-in), and to recognize whether investing extra monthly money in pre-payments would be better than in an Index Fund like the S&P. 

Of course, this is where we come to a conundrum, an impasse in our exploration of why we study math. The example that I just gave is all well and good – I mean, who can argue with receiving a personal benefit from understanding mathematics, right? But what if I were to keep giving examples of ‘mathematics as infrared goggles’? Without a doubt, my examples would soon exit the realm of personal benefits and lead into the realm of professional benefits – ways in which people could not do or be as effective at their jobs without having the pattern-recognition capacities that math gives us. Now, I don’t mean to say that this is a problem… if society is being furthered as a whole by teaching mathematics in school I call that a great thing! But what I do mean to imply is that therein lies a potential answer to the question ‘why do we cover the topics that we cover in mathematics?’ This answer may also lead into an explanation for the second question ‘why do we teach mathematics the way that we do?’ 

Now, this -ehem- ‘short’ post is not intended to be a treatise on the history of education in America, so I won’t be making it into one. Instead, we’ll be sticking to the surface level here: the reason we cover some of the topics we do has to have a historical basis. That historical basis is at least partially rooted in the idea that one purpose of schools is economic productivity (see the second quote under note *). Ken Robinson has argued that this economic reason for schooling is also related to the way in which schools are set up: with ‘The Industrial Model’ in mind (I’ve argued for other models elsewhere). Certainly some of the topics that we still cover today were originally part of the curriculum for the express purpose of giving industrial workers the skills that would make all of America’s Industry run smoother and progress faster. However, our current mathematics curriculum seems (to me) to arise from cultural dynamics of the 1940’s – 1970’s. Think about the backdrop: World War II giving way to the Cold War, the Space Race, The Vietnam War, etc. Political power and a country’s skill in physical technology was intertwined. Yes, more of the Country’s jobs were transitioning from industrial labor to blue-collar work that must have required skill with numeric manipulation… but I don’t think that was the reason for curriculum at the time. I think the reason was that as a Nation, we were hoping that if even just 1% of students took hold of these foundational physical mathematical topics early on, they would be doing things like programming missile paths or developing The Bomb by the time they were adults. I mean, seriously: consider the topics we cover! Quadratics as a Freshman or Sophomore in high school?! Why? Because quadratics are foundational to projectile motion. I mean, sure, quadratics can be used to illustrate a lot of advanced mathematical concepts at a basic level… but don’t tell me that even 10% of students studying this very topic right now understand why we ‘complete-the-square,’ and much less understand how you could apply a similar creative trick in order to be able to ‘discover’ (re-discover) how to calculate a definite integral by hand, which will then allow them to invent a completely new technique. They don’t understand why – they just do it. Or more likely, they have been taught to memorize the Quadratic Formula and they just use that, without ever understanding where that formula came from (just complete-the-square with a general quadratic with constants a, b, and c). 

Now, I don’t mean to say that quadratics (as the example I have chosen – it could have been ‘rational functions’ or … oh don’t even get me started … ‘conic sections’) are only in the curriculum because for 30 years or so our Country was hoping to gain political power through physical technologies. They are also fantastic examples that show us the history of mathematics. The use of quadratics as an illustration of Newtonian Principles of projectile motion is an amazingly awesome unbelievable happy way to illustrate why and how derivatives and integrals are important and tie our world together (e.g. relate an acceleration to a velocity to a distance traveled and vice-versa). To study how Newton was so amazingly perceptive and precocious is a 100% worthwhile endeavor, and makes studying quadratics worth it on its own! But…. suffice it to say that’s not why or how we teach quadratics. 

How we teach quadratics is similar to how we teach most maths content, unfortunately, in the United States. We don’t really teach the history of its development, nor do we teach why for most of what we do. Instead, we try to fly through lots of content at a very shallow level in order to “cover it all.” WHY!?!? 


I’d like to propose a different way of teaching mathematics. Now, because my first way involves a proposal that is unlikely to be feasible for most math teachers by next year, I’ll only be commenting on it briefly and will also be providing a more extensive secondary way, introduced in this post and finished in the next. 

The first alternative I’d like to propose is that many topics – like quadratics – are outdated in today’s world and no longer need to be taught to the level they are currently. For example, quadratics could be taught at a more shallow level than we teach them now in order to introduce students to the idea that not all curves (trends) in the world are linear. After students grasp that on the x-scale of “Amount that Runners Run,” the optimum y-value of “Mile Time” or “Fitness Level” is probably not at either the “0 days a week” side or the “24/7” side of the x-scale, then they don’t need to learn the logistics of completing-the-square. Instead, use ALL of that extra class time taking a deep-dive into one line of study: Data Analysis. That’s what people do at jobs in today’s world, and it just makes sense given that nowadays political power is more tied to the development of 5G than it is to the development of a physical weapon. Furthermore, the study of data analysis will further student’s quadratic skills too – just in a different way. A way that is, I might add, much more appropriate in a world where people don’t actually ever need to calculate a definite integral by hand, so long as they understand what an integral is (I know I already said this, but I will say it again: right now our students understand the exact opposite of that). A fantastic way to begin a Data Analysis course is through the study of Financial Literacy, as most all Data courses I can imagine have to have some ‘base substrate’ on which to base the study. 

Now, as I stated previously, this proposal is unlikely for many educators who read this blog. It’s unlikely for me also – I’ll be teaching quadratics, trig functions, etcetera again next year and the year after that. So how are we to structure our mathematics revolution from within the content we have to teach now? Excellent question! Wait for Mathematics and Why, Part II and I’ll describe it in more detail than you care to hear. 🙂 


*Of course, I recognize my bias in this statement – some would argue, perfectly reasonably, that what is at stake is the future of our economy. To that I respond that the economy is a terribly important system that certainly holds up our modern way of life. Yet, I respond with my Jefferson-ian bias: “If a nation expects to be ignorant and free, in a state of civilization, it expects what never was and never will be.”
To give a brief example of why this is important, I will refer you to this post.
For further support of ole’ TJ, I’ll revert to my Postman-ian bias: “Thomas Jefferson […] knew what schools were for – to ensure citizens would know when and how to protect their liberty. This is a man who wrote an essay that could have cost him his life,” wrote Postman in The End of Education, “It would not have come easily to the mind of such a man, as it does to political leaders today, that the young should be taught to read exclusively for the purpose of increasing their economic productivity.”

** Notice that I want to further our understanding of the questions. I did not say we will ‘answer’ them, whatever that means.

The Web: A Large Deviation

Last week, I wrote about ‘The Web of Content’ metaphor. In short, schools should be wary of making the ‘industrial model’ of education, where products are passed in a linear arrangement through one checkpoint after another which ensure conformity to a narrow band of tolerances, the focus of modern curriculum. This includes 1) restricting the number of ‘outcomes’ through which a student can ‘display mastery’ to just pen-and-paper tests as well as it includes 2) teaching mathematics or history as a journey from point A to B with questions along the way that you have to answer and get ‘right.’ 

  1. After all, the key characteristic of a healthy ecosystem – like a school – is diversity. Students should of course be pushed to overcome their weaknesses, but even more so their strengths should be fostered. Why can’t a particular project have one student turning in a typed paper, one turning in a well-crafted podcast response, one turning in a YouTube video, one turning in a live presentation, and one turning in a ‘museum exhibit’? Obviously most students would choose the paper, because it seems easiest when you really get down to creating quality in each of those other modalities; however, I will now immediately go back on my statement because I would argue that really good writing should be just as challenging. But I seem to be a tougher grader on writing assignments than others… 
  2. I had a new-to-me student this year that asked a question that almost knocked me off my feet, and not in a good way. I made a comment at some point (and I wish I could remember what it was now!) to another student in the class about how they had a good question – one that was so good, in fact, that “you could probably follow that rabbit-hole for a long ways – even write a book on it! It could be a new subject to study in math!” My new student, Mark, laughed out loud. A few of us looked at him, puzzled. “It’s a funny idea!” he said, “Because it’s like, where would you get all the questions from for this new class? What would you even study without having problems to do?”
    I am very sad to say that I am serious right now. This is a ninth-grade student whose entire conception of a ‘subject’ is that someone wrote a textbook that you have to study on the subject. The idea that ‘subjects’ are actually ideas created by people who are curiously observing the world and begin to ask testable questions about it, create theories based on their observations, and then analyze/scrutinize the accuracy of those theories based on continued observation was completely beyond him. I need not say this, but I will anyways: SOMETHING WENT VERY, VERY WRONG IN THIS YOUNG MAN’S EDUCATION. 

Well, here’s a story from ‘The Web’ that highlighted just how far I ‘stretch’ its silky threads sometimes. I think high school math educators everywhere can find ways to stretch the threads similarly. 

My seniors are currently working on a unit called “Pollster’s Dilemma,” wherein they explore the statistics of Election Polling. It’s a super-fun unit, and obviously we talk a lot about topic related to voting and elections during the course of the unit. Well, last week was an election week in Denver, and we had a couple of interesting topics on the Ballot (as well as candidates). I wanted the students who were 18 to make sure they engaged in their first voting experience, but in the process of asking some of them if they were going to vote and what their research processes had been, I realized that there was an opportunity here (check out the footnotes of this post). 

One of the major topics was “Proposition CC,” which despite the language it used, was essentially about TABOR. Now, talk about some funny use of language, TABOR itself is kind of hilarious. But I won’t comment to much on that – as I did with my students, I’ll leave that to you to do the research. The point, though, is that TABOR is a classic example of a bill that has real-world implications, but that it seems the majority of people out there don’t have either the intellectual capacities or the wills to actually take the time to understand. It’s difficult, certainly! But I wanted my students to experience a bit of that difficulty and come to understand that ‘difficult’ does not mean ‘impossible;’ that ‘complex’ does not mean ‘not worth it.’ Their assignment for the entire class was to research the propositions out there as well as the candidates, and to write me a description of who they would vote for and why. To preface the assignment, we talked about the challenge of trying to place quantitative value on various issues in order to make a decision that is black-or-white when you would prefer shades from each side, and I offered an alternative assignment to only describe the history of TABOR if students felt uncomfortable sharing voting ideas. 

The resulting assignments and discussions were incredibly interesting. Without commenting on TABOR too much, one quick ‘did you ask this question or not?’ moment came from the idea of ‘how much does each taxpayer actually get back as a refund? Is that amount worth saying that education and transportation shouldn’t get the ~310 million dollars that they would have?’ It turns out that the amount most people would be getting back this year was about $39-150, and by even just next year people who make over a quarter-of-a-million dollars a year (the highest tax bracket) would be getting a $79 refund. These numbers began to change some students minds on whether or not they wanted their refund over education and transportation getting more tax dollars. 

Students who really got into the trends within that hour were able to discover that a second bill was passed years after the original TABOR bill passed that turned out to be drastically affected by the economic situation of 07-08. Those students especially seemed to realize by the end of the session that we cannot rely on what we hear on the news or from our friends to give us good understandings of even the most localized politics. We have to learn to be astute observers and be able to analyze big data in order to understand situations on our own. 

The point I want to make is this: don’t be afraid to deviate from curriculum in service of providing lifelong lessons that kids need to understand. Sometimes, these lessons become the most long-lasting. Obviously, you can’t do it every day. These moments are special only when they buck the trend on an otherwise fairly rigid devotion to ‘math.’ But when they happen, they will make those 1 or 6 days a year completely worth it. 


‘Why?’ and The Web of Content

One of the foundational rules for VC educators states that students must understand the purpose of their education. They have to be provided with an inspired reason for learning; otherwise, why would they choose to expend the effort on the suite of seemingly arbitrary tasks they are asked to complete on a daily basis in school? 

Neil Postman wrote a book on this very topic; I see it as a primer into education – one that should be required reading in Teacher Development/Prep courses everywhere, as it only gives suggestions as to the types of purposes teachers could instill in their own classrooms, and the conversation is clearly meant to be continued from there. “Without a narrative,” Postman wrote, “life has no meaning. Without meaning, learning has no purpose. Without a purpose, schools are houses of detention, not attention. This is what my book is about.” 

Postman passed a torch to all future generations of educators with The End of Education, and my purpose today is not to review the contents of that message nor even to add ideas to the mixing pot of plausible educational narratives to pitch to students. Instead, I mean to call ourselves out, as educators, for not following our own advice with ourselves. 

What am I talking about? Teacher Preparatory Courses, Professional Development Courses, and texts on Best Educational Practices. Every week, it seems, I find evidence of educators not giving reasons to other educators as to why we might want to do what we do. What is our purpose? What is our inspired reason for working? I get it: reasons are messy. They can vary from one educator to the next, they can be disagreed upon and argued about, they take time to analyze and dissect relative validities, and they can, of course, have many ‘correct answers.’  

Let’s stop being so theoretical and abstract and allow me to give some examples. I was reading through EL’s Core Practices the other day, a fantastic reference text for all educators (not just those within ‘EL’), and came across this statement in Core Practice 22 – Creating Quality Assessments, Section A. Aligning Standards, Learning Targets, and Assessments: 

5. Teachers identify assessments for each set of learning targets. They almost always develop the assessments/assessment tools before each chunk of instruction begins. They often use preassessments aligned to learning targets to inform instruction and differentiation.

Why?!? Before we jump right into trying to answer that question, let’s take a moment to recognize and understand some of the metaphors at play here. To borrow from Postman again, he writes in Teaching as a Conserving Activity “I do not see how it is possible for a subject to be understood in the absence of any insight into the metaphors on which it is constructed.” In education, for example, “if you believe the mind is like a dark cavern, you will suggest activities that are quite different from those suggested by people who believe the mind is like a muscle or an empty vessel. […] Do you conceptualize the mind as a kind of computer? Or a garden? Or a lump of clay?” Well, I conceive of ‘Teacher Education’ as a vast and complex web of knowledge; this knowledge is made up of fundamental truths and paradoxes, created narratives or purposes, and tactics and actions that serve those purposes. Notice that this ‘web’ metaphor is in stark contrast to the ‘linear story’ metaphor that we could use, whereby a teacher might progress from one topic onto the next without any deviation (aka jumping around to other parts of the web, if necessary). Sir Ken Robinson has called this linear model ‘the Industrial Model of Education,’ whereby products (students) are run through an assembly line that ensures they fit within the narrow band of tolerances (standards) necessary for the product (students) to go out to the consumer (the economy). The point is not that students shouldn’t have basic skills and knowledge – they should. The point is that the ways in which we show our disparate skills can and should vary widely – they should be as the web is: diverse. 

The process of conveying information in the field of education, then, becomes a process of efficiency exchanges similar to the way a human body or mind makes efficiency exchanges. No teacher can ‘know it all’ – consider every single aspect of their craft at once – so there’s always room for improvement. Each principle, instructional coach, or pedagogical book must then do the job of sensing where a developing teacher is within this web, and decide to either ‘rubric-ify’ (create heuristics) or really process and ask ‘why?’. They have to make the decision of how much are we going to exchange efficiency for effectiveness, how much we just do something that will work passably well vs. how much time we spend giving narrative or purpose to what we’re doing. This is much the same process teachers go through with students! For example, a good math teacher decides deliberately whether or not, during the course of teaching quadratics, to structure her approach by just teaching the kids the quadratic formula or to go through the much slower – yet vastly more effective for content understanding – approach that really helps kids understand why we complete the square to solve for the roots of an un-factorable quadratic, and to have them derive the quadratic formula as an efficient shortcut to the process by completing the square using constants*. And I will pause my philosophical explanation here to say that this is largely the point of this post – that PD-providers and principles can and should treat teacher development in the same way teachers treat student development, but we don’t do that. 

Instead, we go through the process of ‘rubric-ification’ and come up with immutable statements like “[teachers] almost always develop the assessments/assessment tools before each chunk of instruction begins.” Why? Well, to be honest with you, I don’t know why. The book doesn’t discuss it, nor did my PD Instructors comment on it when I asked about it in class after they had us read Core Practice 22. So I am left to guess. 

My guess would be that when the EL Core Practices manual was originally written, every statement in the text was scrutinized and debated. However, in order to promote efficiency they did not go into detail in explaining the why behind the statements for teachers to get a better understanding. They taught the quadratic formula without giving any indication of where it came from and why it might be a great, time-saving device that has real meanings behind it. 

In my thinking, the why behind this particular statement is that by writing the assessments beforehand, teachers can engage in a ‘backwards-planning’ process. But why might backwards-planning be useful!?!? Well, if I have a specific end-goal that requires understanding of multiple concepts in order to execute, breaking that end-goal down into steps that can be achieved on a daily basis is a very useful planning strategy! I use backwards-planning all the time; however, I only use backwards-planning for projects. I know, I know – why? Because projects are a creative process (in the true sense of the word) – we are trying to create something! It has an end-goal. Learning, on the other hand, doesn’t. If it did, then every school in the world wouldn’t have a mission statement that says something to the effect of ‘to create lifelong learners…’ We hope that students get a taste of how fun it is to explore this “web of content” and want to keep doing it throughout their lives – or at the very least to know how to explore the web when it comes time to exercise their democratic powers: to really be able to understand the complex issues at stake when they go to the polls as members of a Representative Republic. 

The point is that when we begin to conceive of education as a web of possibilities, we begin to realize that the entire concept of ‘not having time to go slower’ becomes ridiculous! We want students to learn how to think and learn, and there is an infinitely large web of things we could be teaching them, not a list that we have to ‘get through!’ Why in the world would I need to limit the infinite potential of my students by saying what we’re going to cover and then commit to that only? If the class is interested in something, we follow that rabbit-hole! And, then I decide to put that on the quiz because that was the strand of the web that I didn’t necessarily plan to follow, but that we decided to anyways! Putting that content on the quiz, then, is not a way to say “Do you know the content well enough to pass the state exam?” – it’s a way of saying “Here’s a chance to check-in with yourself on your own learning process – this quiz is not only a way for you to revisit the ideas again and grow more neural pathways, but also another way of you asking yourself, in a way that forces you to be honest with yourself, how deeply you really thought about the topics at play.” You may be sensing why my class doesn’t have “Quizzes,” it only has “Check-In’s,” which are the functional equivalent to quizzes. The language (and metaphor) matters. 

If you’ve made it this far into this post, then you are somebody who understands how important it is to have a ‘why?’ – or else you would have read another blog that gave you a one-line how and went on about your merry way. You understand that the methods of good instruction can (and probably should) look very different between different teachers, depending on the why behind their practices**. So please, if you happen to find yourself in a position of teaching how to teach (Professional Development, Instructional Coaching, whatever), slow down. Ask ‘why?’ Adapt a ‘Web Metaphor,’ by abandoning the ‘Assembly Line’ metaphor and recognizing that healthy systems have end-products that display a ton of diversity. Explore and learn together, with your students; that’s how we’ll Return to the Heart of Education. 


*I recognize the irony in using an example from quadratics to describe the ‘Web of Mathematics’. Quadratics, while incredibly interesting from a developmental/historical perspective, are pretty freakin’ stupid to be teaching in 2019, especially in the context through which we are teaching them where students go out into the real-world and never really recognize why quadratics are connected and cool, and even more stupid in a world of big data where we don’t have requirements to teach statistics or Excel! More on this in a coming post, though. 

**And conversely, bad teaching actually often shares similar characteristic flaws… and thus in some ways PD should also be about identifying and avoiding certain flaws of bad teaching, because isn’t it much more efficient to analyze the smaller number of practices that are ineffective than to promote the vast sea of practices that are effective? But that’s a story for another day.