What my dad taught me about learning.

The first time I saw the word ‘Calculus’, I was staring at the spines of several textbooks that sat on the bookshelf at home. I didn’t think much of them; I knew they were my parents’, and that they were from their college days, but had no other awareness of what the topic actually was. I did assume that the reason there were so many of them was because my parents must have liked them so much. After further investigation, I learned that they were mostly my dad’s books. His secret was out: he must have loved Calculus. I believed this for a while.

When my older brother took Calculus, these books came off the shelf occasionally as a resource, though I don’t know if this was his decision or my dad’s. From what I knew, my brother breezed through Calculus. I know he worked hard, but it also seemed to come fairly naturally to him. I remember conversations that my parents had about not knowing where my brother got this talent from. They admitted at this point that it couldn’t have been from either of them. My dad had taken Calculus multiple times and the collection of textbooks was the evidence that hung around for no particularly good reason.

This astounded my young brain for a couple of reasons. It was mind-boggling to me that my parents ever had trouble doing anything. They always seemed to know just what to do in different situations – how could they not do well in a class designed to teach them something? It was also the first time I ever remember learning that my dad was not successful in everything he tried to do. This conflicted deeply with what I understood his capabilities to be.

As I understood it, he just knew everything.

When I was nine and my parents had bought me a keyboard to learn to play piano for the first time, there was no AC adapter in the box I had unwrapped only moments before. My dad scrounged around among his junk boxes and drawers and found one with the correct tip, but the polarity was wrong. I knew I wasn’t going to be able to start jamming that night – it was late and a trip to the store wasn’t an option. He wasn’t going to submit to that as a possibility – he took the adapter downstairs to the basement and had me follow him. There was soldering involved, and electrical tape. I had no idea what he was doing. Moments later, however, he appeared with the same adapter and a white label that said ‘modified’. We plugged it in to the keyboard and it lit up, ready for me to play and drive my parents crazy with my rendition of . I now understand that he switched the wires around to change the polarity – I did it myself with some students recently in robotics. At the time though, it seemed like magic. I just knew I had the smartest dad in the world.

His mantra has always been that if it can be fixed, it should be fixed, no matter the hilarity of the process. I watched him countless times take in the cast-off computers of other people who asked him if he knew how to fix them. Thinking back, I don’t know that he ever specifically answered that question. His usual response was (and still is) “I’ll take a look.” So he would work long hours with a vacuum, various metal tools, and a gray multimeter (that I think he still has) laid out like a surgeon investigating a patient. I rarely had the patience to sit and watch. I would see the results of his work: sheets of yellow legal pad paper filled with notes and diagrams scrawled along the way. In the end, he would inevitably find a solution, though often at this point the person who had asked him to fix the item had gone and bought a new one. I don’t recall ever believing my dad thought it was a waste.

We also worked on things together to try to get closer in my early teens. We both took tests to get amateur radio licenses. I came to really enjoy learning Morse code and got the preparation books to climb the license ladder. He commented repeatedly as I zipped through the books about memorizing the books and not understanding the underlying theory of resonant circuits and antenna diagrams. That was true – at the time I just wanted to pass the tests. I didn’t understand that the process of learning was the valuable part, not the end point. I didn’t see that. I just continued to believe that the tests were a means to an end, just as I viewed through my thirteen year old brain that his herculean efforts to fix things was a means to getting things fixed., and nothing more.

My dad is one of the smartest people I know. As I’ve grown older, however, I have come to understand that it wasn’t that about knowing everything. He instead had been continuously demonstrating what real learning is supposed to be. It was never about knowing the answer; it was about finding it. It wasn’t about fixing a computer, it was about enjoying figuring out how it can be fixed, however much frustration was involved. It wasn’t just about saving money or avoiding a trip to the store to buy an electric adapter. It was about seeing that we can understand the tools we use on a regular basis well enough to make them work for us.

I have seen time and time again how he mentors people to make them better at what they do. I have seen it in the way he mentors FIRST robotics teams as a robot inspector at the Great Lakes regional competition in Cleveland. I have seen it in the way he has spent his time since selling the company he founded with partners years ago. He chooses to do work that matters and makes sure that others are right there to learn beside him. There were times growing up when, admittedly, I just wanted him to fix things that needed to be fixed. To his credit, he insisted on involving me in the process, even when I protested or became impatient.. I didn’t see it when I was younger. Knowing how to go about solving problems is among the most important skills that everyone needs. I was getting free lessons from someone that not only was really good at it, but cared enough about me to want me to learn the joy of figuring things out.

One of my students this year was really into electronic circuits and microcontrollers. He soldered 120 LEDs into a display and wanted to use an Arduino to program it to scroll text across it. The student’s program wasn’t working and he didn’t know why. I had only been tangentially paying attention to the issues he was having, and when he was visibly frustrated, I pulled up a chair and sat next to him, and then said ‘let’s take a look.” We went through lines of code and found some missing semicolons and incorrectly indexed arrays, and I asked him to tell me what each line did. I was only a couple steps ahead of him in identifying the problem, but we laughed and tried making changes while speaking out loud what we thought the results would be. At one point, he said to me “Mr. Weinberg, you’re so smart. You just know what to do to fix the program.”

I immediately corrected him. I didn’t know what was wrong. We were able to make progress by talking to each other and experimenting. It wasn’t about knowing just what to do. It was about figuring out what to try next and having strategies to analyze what was and was not working. I learned this from a master.

On this Father’s day (that also happens to be the day before my dad’s birthday), I celebrate this truth: much of what I do as a teacher comes from trying to channel my dad’s habits while confronting big challenges. I don’t want my students to memorize steps to pass tests; I want them to understand well enough to be able to solve any challenge set before them. I don’t want to fix my students’ problems – I want to help them learn to fix problems themselves. I don’t want my students to be afraid to fail; I want them to understand through example that failure leads to finding a better way.

I am grateful for all that I have learned from him., and I try to teach my students what he has taught me about learning at every opportunity. It would be fine by me if I ever need to do Calculus for him – I’d still be in the red.

Why I’m thinking today about the Tufts class of 2012.

I had my first group of ninth grade students during my second year teaching in the Bronx. It’s a unique experience being an adult mentor to a group of students fresh out of middle school. I’ve always gotten a kick out of seeing them first test the rules in their new high school environment, and this group being my first, it was new to me as well.

It has been a while since I’ve heard from many of them. I’m proud to say that a number of students from this group will be donning caps and gowns over the next couple of weeks to celebrate their earning undergraduate degrees. There’s a whole list of superlatives that describe the magnitude of pride I feel for this group and their accomplishments. As a digital pack-rat, I’ve held on to the spreadsheets I used to keep track of grades. I took a look at them just before writing this, which prompted a slideshow of smiling faces as I went through the list, name by name. I think I could vaguely place them in their seats in the classroom, but in all likelihood, this was just as likely my brain coddling me in my hope that I could remember such minimal details.

One student in particular in this group is pushing me to assert my bragging rights.

As I’ve mentioned other times on my blog, I am a proud graduate of Tufts University, majoring in Mechanical Engineering as a member of the class of 2003. It was through my work as a resident tutor in math and physics that I discovered that I had an interest in teaching, and this prompted me to apply to alternative certification programs that would help me do this. I could have applied my engineering credentials to be one engineer in the working world. Another option was to teach students to become engineers too, in effect, multiplying my own influence on the field. Through the New York City Teaching Fellows , I joined the faculty at Herbert H. Lehman high school in the Bronx during the fall of 2003 to teach math.

The first year was a blur. It was the fastest I’ve ever needed to learn a multidimensional set of skills and the most agonizing; I knew when I wasn’t getting across to my students and had few tricks to use in managing a class. The one thing I figured out very quickly though was that the students in my classes were sharp. They were good at picking up on things presented in the right way. Their skills were not necessarily where they needed to be, but that is a work in progress that can be managed through classroom work. I saw there was tremendous opportunity to help those students that were interested to become engineers.

I’ve had a number of students follow this route through my courses in math, engineering, and AP Physics. They chime in from time to time to let me know what they are doing, and I am always really impressed with their work. I’ve also had the occasional graduate write me to ask if it’s alright with me to not study physics or engineering as they originally planned to do. I am, of course, fine with this! I am always telling my students to go where their passions are, and am always a bit amused when they are afraid they are letting me down with such an admission.

The special case that I am writing about today is a young man that not only followed the engineering path, but decided to go to Tufts himself after leaving Lehman. He was a member of my first group of ninth graders, and though he was quickly switched into another section that year, he joined me for physics during his senior year. He also frequently contributed to the robotics team, never shying away from tackling the big challenges of robot design or from the small tasks of sweeping the shop floor at the end of the day. He also honored me during his senior year by speaking at a ceremony at which I received an award for my work, and his very kind words have stuck with me ever since.

It isn’t a miracle that he will cross the stage to receive his Tufts diploma today. Far from it – he did the hard work to get where he is, and I can’t take credit for the great things he learned both in my presence and away from it. And his story is far from over – I hope he (like many other students I’ve told this) keeps me in mind if I ever need a job. His story, and those of the rest of his class earning degrees this month, make me incredibly proud to be a teacher.

That said, there is something special about our story. The unique way that Tufts now connects us is unlike any I’ve ever had with others, even with my own Tufts classmates in the class of 2003. I hope that he can look back fondly to his times on campus as I do from time to time. For whatever small part I served in getting him there, I am glad to have helped him out.

I have nothing but excitement and pride for the adventures that lay ahead of him and his classmates.

Congratulations, Class of 2012!

Geometric Optics – hitting complexity first

I started what may end up being the last unit in physics with the idea that I would do things differently compared to my usual approach. I taught optics as part of Physics B for a few years, and as many things end to be in that rushed curriculum, it was fairly traditional. Plane mirrors, ray diagrams, equations. Snell’s law, lenses, ray tracing, equations. This was followed by a summary lesson shamefully titled “Mirrors and lenses are both similar and different” , a tribute to the unfortunate starter sentence for many students’ answers to compare and contrast questions that always got my blood boiling.

This time, given the absence of any time pressure, there has been plenty more space to play. We played with the question of how big a plane mirror must be to see one’s whole body with diagrams and debate. We messed with a quick reflection diagram of a circular mirror I threw together in Geogebra to show that light seems to be brought to a point under certain conditions. Granted, I did make suggestions on the three rays that could be used in a ray diagram to locate an image – that was a bit of direct instruction – but today when the warm up involved just drawing some diagrams, they had an entry point to start from.

After drawing diagrams for some convex and concave mirrors, I put a set of mirrors in front of them and asked them to set up the situation described by their diagrams. They made the connection to the terms convex and concave by the labels printed on the flimsy paper envelopes they were shipped in – no big introduction of the vocabulary first was needed, and it would have broken the natural flow of their work. They observed images getting magnified and minefied, and forming inverted or upright. They gasped when I told them to hold a blank sheet of paper above a concave mirror pointed at one of the overhead lights and saw the clear edges of the fluorescent tubes projected on the paper surface. They poked and stared, mystified, while moving their faces forward and backward at the focal point to find the exact location where their face shifted upside down.

After a while with this, I took out some lenses. Each got two to play with. They instantly started holding them up to their eyes and moving them away and noticing the connections to their observations with the mirrors. One immediately noticed that one lens flipped the room when held at arms length but didn’t when it was close, and that another always made everything smaller like the convex mirror did. I asked them to use the terms virtual and real, and they were right on. They were again amazed when the view outside was clearly projected through the convex lens was held in front of a student’s notebook.

I hope I never take for granted how great this small group of students is – I appreciate their willingness to explore and humor me when I am clearly not telling them everything that they need to know to analyze a situation. That said, there is really something to the backwards model of presenting complexity up front, and using that complexity to motivate students to want to understand the basics that will help them explain what they observe. Now that my students see that the lenses are somehow acting like mirrors, it is so much easier to call upon their curiosity to motivate exploring why that is. Now there is a reason for Snell’s law to be in our classroom.

Without planting a hint of why anyone aside from over excited physics teachers would give a flying fish about normals and indices of refraction, it becomes yet one more fact to remember. There’s no mystery. To demand that students go through the entire process of developing physics from basic principles betrays the reality that reverse engineering a finished product can be just as enlightening. I would wager that few people read an instruction manual anymore. Even the design of help in software has changed from a linear list of features in one menu after another to a web of wiki-style tidbits of information on how to do things. Our students are used to managing complexity to do things that are not school related, things that are a lot more real world to them. There is no reason school world has to be different from real world in how we explore and approach learning new things.

The TacoCopter? – a gimmick for integration review

I received an email sending me to this site yesterday about the TacoCopter, which of course was spot on given my interest in all things robotic. I also had PID control on the brain thanks to my course on driving a robot car from Udacity. Bits of python code were in my head already, and I had a strong need to put it all together. Given that it was also Sunday (a workday for most teachers) I had to plan for classes tomorrow, specifically Calculus and Physics.

All of this was in the context of the beautiful afternoon I spent on the balcony of the apartment looking out at the warmest, bluest Hangzhou skies of the year so far. It put me in the mood to do something a bit different for tomorrow’s Calculus class. The AP students will be reviewing related rates and implicit differentiation, but the regular students…they get to have a bit more fun.

This is the activity we will be looking at tomorrow in class: CW – TacoCopter Project

The full wiki page that students will be following is located here: http://wiki.hischina.org/groups/gealgerobophysiculus/wiki/42712/Calculus_Unit_8__The_TacoCopter.html

Some python code for simulating the TacoCopter rising to altitude, which can be found here at github.

Then Geogebra for plotting the data, which shows the lovely simulated accelerometer data with noise:

I don’t really know how it will go. At least students will have an excuse to grin as they review.

A tale of two classrooms – which is yours?

Consider the two scenarios below in the context of your own classroom, or if you are an administrator, in the context of how you might react to the following situations occurring in your teacher’s classrooms. Assume the class skill level is normally distributed from weak to strong.

Situation one:

You are teaching a lesson in a mathematics class on a skills-heavy topic – perhaps solving a quadratic equation with rational roots. You have a lesson completely planned, a great intuitive hook problem at the beginning, and plenty of pivotal questions to shape student understanding around the process. Perhaps you have a carefully crafted exploration that guides students to figure out for themselves exactly how the procedure works. You have students work in groups to create a set of procedures to follow, and then students individually solve practice problems and compare to each other to check their work and help each other.

Situation two:

You are again teaching students to solve a quadratic equation with rational roots. You give them the set of practice problems at the beginning of the class and briefly review what it means to solve an equation – what should your final answer look like on your paper? You then give them textbooks, laptops with internet access, Geogebra, graphing calculators, whiteboards – all sorts of materials and tell the students your expectation is that they learn using whatever method works for them how to solve the equation. Some look on Youtube for hints. Some students might already know how to solve the equation – those students quickly tell their friends how to do so. Some decide to graph the quadratic function, get the solutions to the equation first, and then try to get those answers algebraically. You find that some students are struggling, so you are able to give additional help to those students, and they do seem to understand the general procedure after getting some help from online videos and their peers in the class. By the end, everyone has solved at least a couple of these types of problems on their own.

Suppose also that the next day you give the students a quiz with two of these problems, the second with an additional layer of difficulty. The strongest students get both questions correct, and the rest get at least the simplest question correct, with some fundamental flaw in reasoning or procedure for the second. In other words, I want the measured outcome of both situations to be roughly the same.

Before I go on, let me be clear about my own background here. When I was first trained to teach in New York City public schools, I was expected to teach lessons fitting the mould of the first scenario. The “I-do, we-do, you-do” model or the developmental lesson were the names often given to this type of classroom. The principal expected teachers to stick to a well defined structure for each lesson, and he was in and out of classrooms frequently to ensure that this was the case. The idea was that the structure helped with classroom management, made learning objectives clear to students, and made it easy for students to take notes and keep track of what they learned. Another part of doing things this way was that there was some level of control over how students were guided to an answer. If the activities or examples are shrewdly selected, a lesson doesn’t devolve into situations in which it is necessary to say “Yes, [generic shortcut that students will find if it exist] works in this case, but it won’t always do so.”

Since leaving that school, I’ve taught in environments in which I’ve been able to experiment a bit more and try new instructional methods. In my current school, I am supported to use whichever methods I choose to help my students learn. I find, however, that since my mind is not really made up, I go back and forth. I am more likely to use the first situation in Calculus and geometry, and the second in physics and algebra two, but there are exceptions.

Which of these classrooms is yours? What are the advantages and disadvantages of each? Since I’m the one writing, I get to share first.

Situation one has always been my go-to model for helping students that are weak in arithmetic, algebraic skills, or overall organization. These students benefit from seeing clear examples of what to do, and then from getting opportunities to practice either with guidance through whole class, pair, or independent work. In many cases these students are not sure how best they learn, so they follow steps they are given and trust that the path their teacher has selected will be one that will eventually lead them to success. In addition, my presentation and activities can be carefully chosen to make it so that students are not just memorizing a procedure, but are required to go through thinking to understand the mathematical thinking involved.

In the larger context of teaching mathematical thinking, however, this method can lead to students expecting or relying on the teacher to provide the frame work for learning. It can (though does not necessarily, depending on the group) lead to a mindset on the part of students that it’s the teacher’s job to explain everything and make it easy to learn. I do believe in my responsibility to know how to explain or present material in many different ways to help students, but there are some concepts that just aren’t easy. They may take work, practice, and interaction with me and the other students to understand and apply.

Situation two offers a bit more in terms of empowering the students to take control of their learning. It lets the students choose how they learn a concept best, whether by direct instruction, watching a video, reading example problems, or working with peers. If students learn the material on their own, have seen it before, or grasp the concept quickly, this offers many opportunities for using that knowledge to help other students or challenge them with more difficult questions. It does not require that material be presented in a linear fashion, from simple to complex, because it offers opportunities to jump back and forth, working backwards and from different representations to eventually come to an understanding.

In many cases, this offers the opportunity for the teacher to show what it looks like when figuring something out or learning something for the first time. I have read many people that refer to this position as the ‘learner-in-chief’, a concept I really like because I think students need to see that learning is non-linear, filled with mistakes and the testing of theories. Getting it right the first time, while nice when it happens, is not the norm. Sharing this fact with students can be a valuable learning experience. While it is nice to see a concept presented perfectly, it contrasts with the real learning process that is a lot more messy.

I have seen a couple negative factors that need to be considered in implementation, the first concerning the weak students. These are often the students that perhaps lack the background knowledge to figure out a mathematical procedure, or the self control to sit and figure something out on their own. What is nice in the second situation, assuming the other students know they must complete the assigned lesson and work for the day, is that the group of these students is a smaller one than the entire class. It is an example of differentiation in action – the students that need direct instruction to learn, get it. Those that do not, are able to reinforce and apply their learning habits by learning on their own. This situation also presumes the students are motivated to learn the concepts, though being able to do so in their own way and being held accountable for their learning may improve how some students react to the prospect of coming to your classroom each day.

Another downside that I’ve seen in practice is also a downside of students teaching each other mathematical processes. Students will often teach ‘just the steps’ and none of the understanding. While this is not the end of the world, it is something that teachers must reinforce with their students. The idea that mathematics is not just a list of problems, but a way of thinking, is strengthened by the arrangement in situation two. If the arrangement of resources available to students is sufficiently broad, the students will be able to piece together the overall concepts as a group. This entire process needs to be modeled, however, early on in the year to teach students both how to do it and what the expectations are.

For administrators, I imagine that walking into a classroom like this can result in an initial feeling of chaos or disorder, and might therefore lead to the feeling that this is less ideal than that presented in situation one. To be clear – it is possible to run a classroom poorly in both situations, and classroom management is essential to maximize the student learning occurring in both. Ultimately, a classroom filled with students that are all learning in their own way to reach a given set of learning standards, is the holy grail. It is important to be given the opportunity, training, time to interact with colleagues, and the necessary resources to make this feasible in every classroom. The important part, chaos or no chaos, is to determine whether (or not) learning is happening in the classroom. My main point is that there is fundamental difference in the philosophy of learning between the two classrooms.

Which is better? I’m not sure. I go back and forth between the two, depending on the concepts we are exploring on a particular day, or he problems we are looking at. Some of the most fulfilling lessons I have taught have involved giving the students a challenging problem and letting them figure it out in their own way. Yesterday in Calculus we did a number of activities that led to the Fundamental Theorem, but I was guiding the way. I think keeping it balanced is the way to go, but that’s partly because I haven’t structured my courses to be taught completely one way or the other. Maybe, in moving to Standards Based Grading, it might make it more natural to move toward more of situation two.

What do you think?

Are there too many people on this thing? (#anyqs)

Two images I want to share for the purposes of #anyqs. For those unaware, this hashtag means I want you to look at these images and let me know what mathematical questions jump out at you right away.

Once you’ve taken a moment to think about these, please send me a tweet (@emwdx) with the hashtag #anyqs letting me know what you think this is all about. That would be great if you could take a moment.

What would be even cooler, however, is if you could also take pictures of your own of these signs in elevators (or other places you might find them) as well as a little information about where it is being taken. For example, the left picture is taken in the elevator in my apartment building while the right is from the gondola that the 9th graders and I took to get down from Mount Tai on the rainy second-to-last day of the China trip.

I am especially curious to see how these signs vary between buildings, countries, and even between elevator banks within a building. I’d like to share these with students tomorrow, so please snap a quick picture and send it (or a link) my way. It hopefully goes without saying that I will give proper credit for photos that make it into the materials I use with students, and will share what comes of it with you all.

So, for the sake of an interesting idea that I think will start some cool discussions with students – skip the stairs today. Snap a picture in the elevator, and then reward yourself for your generosity (and for decision to postpone exercise for unselfish reasons) by eating a doughnut or other craved food of your choice.

Climbing Mount Tai – #wcydwt edition

I am spending an amazing few days with students on this year’s class trip to Shandong province in China. We spent a couple days wandering around Qufu, the home of Confucius, and the location of the temple and mansion constructed for his relatives. There were some cool opportunities to think about mathematical thinking in Chinese architecture (more on that later) but nothing ready for prime time.

Today’s trek led us to the foot of Mount Tai, China’s #1 mountain for it’s cultural significance (not due to it’s height.) we decided as a group to trek up the mountain from the Heaven’s Gate which reduced the climb somewhat, but will descend the full height of the mountain in the morning after watching the sunrise.

From Wikipedia (to be replaced by my own pics when I get home, I promise.)


The realization that I might be able to do something really cool with this came after regretting that I had decided to leave two of my favorite data collection devices (heart rate monitor and hiker’s GPS) at home being unsure during packing if they would really be worth bringing. I had done this hike in March and had several conflicting reports of the exact height we climbed up and down. The students were asking me how many steps there were, and I vaguely recalled something around 7,000, but I wasn’t sure. This question actually popped out from a few different students as we passed the first set of steps. It got me thinking. Is it possible to take either one of these numbers (height or number of steps) and try to calculate or estimate the other? If the students were asking it standing at the bottom looking up, there might be a possibility they would be interested in answering it on their own if posed the right way.

I grabbed my camera and grabbed the best standard length measure I had on me: my iPhone.


(It probably isn’t necessary to say this, but this is just an example I took in the hotel.)

I took a number of pictures like the one above on way up the steps, trying to come up with a fairly random sampling of the size of stairs compared to the phone along the entire height. Through some combination of Geogebra, pencil & paper calculations, and some group discussion, I can see some height calculations for the climb coming out of this.

On the way up, there was also a perfect “answer” to this challenge posted in the form of a placard fixed to the wall that says both the vertical height and the number of steps – again, I will include a picture of this when I can transfer photos from the camera I used to take the good photos. I could see cropping this photo in a way that hides the answer, though I’m sure there is a more dramatic Act 3 to this challenge out there.

I think there is some potential here for some fun, as well as for good student discussion and writing about how close the number actually gets to the right answer. This is the second time in a week that I’ve been able to find something good that could work for a class activity, and I wanted to get the details out while still buzzed about its prospects.