Category: reflection

Take Time to Tech – Perspectives after a Flip


Yesterday my calculus students reaped some of the benefits of a flipped class situation – I made some videos on differentiation rules and asked that they watch the videos sometime between our last class and when we met yesterday. We spent nearly the entire period working with derivatives rules for the first time. The fact that the students were getting their first extended period of deliberate practice with peers and me around (rather than alone while doing homework later on) will hopefully result in the students developing a strong foundation what is really an important skill for the rest of calculus.

They were using Wolfram Alpha to check their work, something that I paid lip-service to doing last year but did not introduce explicitly on the first day of learning these rules last year. There was plenty of mistake-catching going on and good conversations about simplifying and equivalent answers. I needed to do very little in this process – good in that the students were teaching themselves and each other and being active in their learning.

It was also interesting doing this so soon after discussing the role of technology in helping students learn on the #mathchat Twitter discussion. There were many great points made regarding the content of technology’s effective use across grades. It made me think quite a bit about my evolution regarding technology in the classroom. Many comments were made about calculator use, teaching pencil and paper algorithms, and the role of spreadsheets and programming in developing mathematical thinking. I found a lot of connections to my own thoughts and teaching experiences and it has me buzzing now to try to explain and define my thinking in these areas. Here goes:

Developing computational and algorithmic fluency has its place.

In the context of my students learning to apply the derivative rules, I know what is coming up the road. If students can quickly use these rules to develop a derivative function, than the more interesting applications that use the derivative will involve less brain power and time in the actual mechanics of differentiation. More student energy can then be focused in figuring out how to use the derivative as a tool to describe the behavior of other functions, write equations for tangent and normal lines, and do optimization and minimization.

There was a lot of discussion during the chat about the use of calculators in place of or in addition to students knowing their arithmetic. I do think that good arithmetic ability can make a difference in how easily students can learn to solve new types of mathematical problems – in much the same way that skill in differentiation makes understanding and solving application problems easier. Giving the students the mental tools needed to do arithmetic with pencil and paper algorithms empowers them to do arithmetic in cases when a calculator is not available.

Technology allows students to explore mathematical thinking, often in spite of having skill deficiencies.

One of the initiatives my colleagues took (and I signed on since it made a lot of sense) when I first started teaching was using calculators as part of instruction in teaching students to solve single variable linear equations. There was a lot of discussion and protest regarding how the students should be able to manage arithmetic of integers in their head. It wasn’t that I disagreed with this statement – of course the students should have ideally developed these skills in middle school. The first part of the class involving evaluating algebraic expressions and doing operations on signed numbers were done without calculators in the same way it had been done before.

The truth, however, was that the incoming students were severely deficient in number sense and arithmetic ability. Spending a semester or two of remediation before moving forward to meet the benchmarks of high school did not seem to make sense, especially in the context of the fact that students could use a calculator on the state test. So we went forward and used calculators to handle the arithmetic while students needed to reason their way through solving equations of various forms. They did learn how to use the technology to check the solutions they obtained through solving the equations step-by-step using properties. There were certainly downsides to doing things this way. Students did not necessarily know if the answers the calculators gave them made sense. They would figure it out in the end when checking, but it was certainly a handicap that existed. The fact that these students were able to make progress as high school math students meant a lot to them and often gave them the confidence to push forward in their classes and, over time, develop their weaknesses in various ways.

I have seen the same thing at the higher levels of mathematics and science. I used Geogebra last year in both pre-Calculus and Calculus with students that had rather weak algebra skills to explore concepts that I was taught from an algebra standpoint when I learned them. Giving them tools that allow the computer to do what it does well (calculate) and leave student minds free to make observations, identify patterns, and test theories that describe what is happening made class visibly different for many of these students. If a computer is able to generate an infinite number of graphs for a calculus student to identify what it means for a function graph to have a zero derivative, then using that technology is worth the time and effort spent setting up those opportunities for students.

Using skill level as a prerequisite for doing interesting or applied problems in mathematics is the wrong approach.

Saying you can’t drive a car until you can demonstrate each of the involved skills separately makes no sense. Saying that students won’t appreciate proportional reasoning until they have cross-multiplied until their pencils turn blue makes no sense. Saying that learning skills through some medium makes all the other projects and applications that some of us choose to explore in class possible does not make sense. It makes mathematics elitist, which it certainly should not be.

Yes, having limited math skills is a limit on the range of problem solving techniques that are available to students. A student that can’t solve an equation using algebra is destined to solve it by guess and check. Never underestimate the power that a good problem has to entice kids to want to know more about the mathematics involved. Sometimes (and I am not saying all the time) we need to work on the demand side in education, on the why, on the context of how learning to think in different ways applies to the lives of our students.

Emphasizing algorithms without providing students opportunity to develop context or some level of intuitive understanding (or both) has significant negative consequences.

I don’t mean to suggest that teaching algorithms on their own can’t result in students performing better on a type of problem. The human brain handles repetition extremely so well that learning to do one skill through repetition is not necessarily a bad way to learn to do that one thing.

One problem I see with this has to do with transferring this skill to something new, especially when the depth of available skills is not great. Toss a weak student ten one-step equations of the form x + 3 = -8, and then give them something like 0.2 x = 25, and chances are that student won’t solve it correctly without some level of intuition about the subtle differences between the two. Getting this right takes practice and feedback really good opportunity for students to be reflective of their process.

It is also far too easy when applying an algorithm to stop thinking critically about intermediate steps. I spoke to a colleague this week about his students learning long division and we both questioned the idea that the algorithm itself teaches place value. We looked at a student’s paper that was sitting on the desk and instantly found an example of how the algorithm was incorrectly applied but through a second error resulted in a correct answer. If we teach algorithms too much without giving activities that allow students to show some sort of understanding of some aspect of how the algorithm fits into their existing mathematical knowledge, it’s undercutting a real opportunity to get students to think rather than compute. I like the concepts pushed by the Computer-Based Math movement in using computers to compute as they do best, and leave the thinking (currently the strength of the human brain) to those possessing one.

As often as we can, it is important to get students to interact with the numbers they are manipulating. Teaching the algorithms for multiplying and adding large numbers does provide students with useful tools and does reinforce basic one digit arithmetic. I get worried sometimes when I hear about students going home and doing hundreds of these problems on their own for various reasons. If they enjoy doing it, that’s great, though I think we could introduce them to some other activities that they might see as equally if not more stimulating.

I do believe to some extent that full understanding is not necessary to move forward in mathematics, or any subject for that matter. I took a differential equations course in college trying to really understand things, and my first exam score was in the seventies, not what I wanted. I ended up memorizing a lot after that point and did very well for the rest of the course. It wasn’t until a systems design course I took the following year that I actually grasped many of the concepts that eluded me during the first exposure. This same thing worked for me in high school when I took my first honors track math class after being behind for a couple years. My teacher told me at one point to “memorize it if I didn’t understand it” which worked that year as I was developing my skills. Over time, I did figure out how to make it make sense for myself, but that took work on my part.

Uses of technology to apply/show/explore mathematical reasoning comprise the best public relations tool that mathematics has and desperately needs.

I really enjoyed reading Gary Rubenstein’s recent post about the difference between “math” and mathematics. I read it and agreed and have been thinking a lot along the lines of his entry since then.

Too many people say “I’m not good at math.” What they likely mean is that they aren’t good at computing. Or algorithms. Or they aren’t good at ________ where __________ is a set of steps that someone tried to teach them in school to solve a certain type of problem.

On the other end of the perceived “math” ability spectrum, parents are proud that their children come home and do hundreds of math problems during their free time. These students take the biggest numbers they can find and add them together or multiply them and then show their parents who are impressed that their normally distracted kids are able to focus on these tasks long enough to do them correctly.

It makes sense that most people, when asked to describe their experiences in math, describe pencil and paper algorithms and repetitive homework sets because that’s what their teachers spent their time doing. This, unfortunately, is the repetitive skills development process that is part of mathematical learning, but should not be the main course of any class. We show what we value by how we spend our time – if we spend our time on algorithmic thinking, then this is what students will think that we as teachers and as thinkers value as being important in mathematics.

This fact is one of the main reasons I started thinking how to change my class structure. My students were talking about not being good at a certain type of problem (“I don’t get this problem…I can’t do problems that need you to…”) rather than having difficulties with concepts (“I don’t get why linear functions have constant slope…I don’t get why x^2 + 9 is not factorable while x^2 – 9 is).

If we as teachers want students to value mathematics as more than learning a set of problems to be solved on a test, then we have to invest time into those activities that allow students to experience other types of mathematical thinking. This is where technology shines. The videos of Vi Hart, Wolfram Alpha, the antics of Dan Meyer, the Wolfram Demonstrations Project, the amazing capabilities of Geogebra – all of these offer different dimensions of what mathematical thinking really is all about.

We can share these with students and say “check these out tonight” at the end of a lesson and hope that students do so. Sometimes that works for a couple students. That isn’t enough.

I think we need to invest in technology with our students with our time. We need to deliberately use valuable class time to take them through how to use it and why it makes us excited to use it with them. It’s really the only way students will believe us. Show that it’s important, don’t just tell your students it is. That’s right – that valuable class time that we often plan out too carefully and structure so that they reach the well-defined goals we have for them – that time. Plan to use a specific amount of class time, and enough time, to let students play around with a mathematical idea using any of the amazing technology tools out there. Show them how you play with the tools yourself, but don’t make this the focus of this time – do so afterwards, perhaps.

To be clear – I am not saying do this all the time. Students need to learn algorithms, as I have already stated. Students also need to be looking at interesting problems. We should not wait to show them these problems until after students have demonstrated automaticity because it gives students the impression that the algorithms came before the thinking that went into them.

I am saying that balance is key.

The only way we are going to change the perception of what mathematical thinking really looks like is by living it and sharing it with our students.

What do you do when they don’t need you?

I’ve tried an experiment over the last two days – my advanced algebra students and geometry students each had some challenging tasks that I sort of left to them to figure out. Last year, I taught them very explicitly how to do the tasks at hand, modeled some examples along side their own work, and then gave them time in class to practice. For homework, I gave them more problems that were similar to those we did in class, giving them more chances to practice what I had assigned them.

This year I turned it around. In geometry, we are starting proofs. I gave them a couple relatively simple ones, and asked them in groups of two to construct some sort of logical reasoning to go from a starting point to proving the statement I had given them. There was a lot of struggling, difficulty stating using facts why one logical statement led to another. Over time, they did start communicating with each other and sharing what they were thinking. I did occasionally poke one group in a certain direction, but didn’t lead the whole group in that way. Eventually they were all thinking along the lines that I envisioned at the beginning. I could have modeled for them what I did last year, but I saw a lot of really good conversations along the way. By the end, they were much closer to making their own proofs than they had in the beginning. By the end, they were clearly seeing the connections between thoughts. This was only the second class period during which we had talked about proofs. While I don’t think any of them would wager large sums of money over constructing geometric proofs, I think they at least see how the system can be used to make logical statements that are irrefutable.

I did something similar with the advanced algebra group which was to figure out graphing absolute value functions during our lesson last Friday. I gave them an exploration that was, in hindsight, confusing and didn’t do much aside from frustrate them with Geogebra commands. I told them that I wanted them to use Geogebra, the textbook, Wolfram Alpha, and any other resources available to learn how to graph any arbitrary absolute value function by hand. At the end of the class I broke down and apologized for giving a poorly designed exploration. I told them I would put together a video on graphing functions, and I did – posted it on the wiki.

When we had class on Tuesday, I found out that none of the students had actually taken advantage of the video. They had looked in the textbook. They had graphed functions over and over. When I asked students to share what they had figured out, one student used a table of values and a piecewise function based on the sign of the argument of the absolute value function. Another student had graphed both the argument of the absolute value function and its opposite since that was what this student had observed, and then erased either the top half or bottom half of the graph. Another student broke down and did what the book said to do. By the end, all of them were graphing absolute value functions using their own method. I wasn’t sure about understanding, but in the end, I admit that I didn’t quite mind. They all had their own models for what was going on, and they were confident that they could use the technology to confirm whether what they were doing was right or not.

I have always wondered about what I would do in the situation where it was clear a group of students didn’t need me to be there. In a way, this is part of my fear of tools like Khan academy – if there are others out there that are more engaging, better at explaining ideas, or better at coming up with really interesting questions that got students thinking about what they were doing, and these people happened to make videos: what would I do in my classroom if students got hold of these videos? Would I be mad? Or would I figure out a way to take advantage of the fact that students had figured something out on their own and use it to do something even more interesting or impressive?

I think I do a good job of engaging students – today we were talking about differentiability and we were joking like crazy about whether functions would be differentiable at a given point or not. Really? Were we really joking about this? It seemed like everyone was minimally entertained, but based on the questions I was bouncing around from person to person, it seemed like they also understood the concept. I know teachers that are more effective though at making kids understand and be entertained and do problem after problem until concepts are so painfully clear that they become automatic. These teachers could easily make a career in stand-up or television based on their comedic brilliance and presence – what if they decide to make videos?

What then? What do I do? If I get to that point, is it the end of my usefulness as a teacher?

Or is that just the beginning? Maybe that’s the point where I can assume my students have a certain level of basic knowledge and I can then build off of that level to do even cooler stuff. Maybe that’s where I can assume, for once, that my students have a base level of skills, and can then rise above to analyze bridges or patterns in nature or create a mathematical model that inspires a student to choose to be a doctor or engineer where he or she never would have if I hadn’t done the right project or group activity or lesson. Maybe the fact that I entrusted my students to try to figure out something on their own is enough that they feel empowered to try things even if failing again and again is a possibility. Make mistakes and come back asking questions about why their theories were incorrect. I remember worrying at one point in my career what horrible things would happen if I somehow introduced students to a concept in a way that it caused them get a question wrong and cause them just one more failure in a line of failures. If I could teach in a way that makes students feel OK coming in the next day saying “I didn’t get it, but this is what I tried” I’d feel pretty good about myself. That is, ultimately, the sort of resilience that a person needs to survive in this world.

If the students show us that they don’t need us to show them how to solve a specific problem, then we as teachers should honestly accept this fact. Our goals, if they can do the problem in a way that is mathematically correct, should shift to applying that ability to doing something more profound and relevant, be it communicating that solution or applying the solution to a new situation that is different but connected in a subtle way. Our job puts us in contact with some really amazing minds that are eager to do what we say in some circumstances. In the cases that they demonstrate that they don’t need us – that is when we must apply our professional judgment and teach them to expand their knowledge to something bigger than themselves.

Not sure why I’m waxing so philosophical today, but I’ve been really impressed with my students this week, and it’s only Wednesday! After these few days, it feels like what I’m doing with this group is like using a super computer to do word processing. I only hope they are enjoying the process as much as I am.

Your students might not be cursing at you…

One of the students I had the pleasure of teaching in AP physics in the Bronx started with quite a reputation. As a student that spoke Chinese and little English in the 9th grade, he was placed in the entry level math class. It took only a short time for his teacher to notice that, given his background and obvious mathematical skills, this probably wasn’t the right place for him. He was quickly moved up the sequence of courses until he ended up in a Math B course that included trigonometry as I recall.

This was not just a case of this student having memorized mathematical concepts from his time in China, though he had seen a lot of math by the time he arrived at Lehman. In his junior and senior years, the quality of his insights and ability to predict, comprehend, and connect ideas in both math and physics were truly impressive and indicative of a strong talent. As his teacher in physics, the greatest challenge I had was not in teaching him how to solve a physics problem, but to write down his line of reasoning that scattered together with frightening speed in his head. My favorite teaching moments with him came on the rare occasion when he had an actual misunderstanding and I witnessed the exact moment of his realization of what he did not get; the physical change in his face was unforgettable.

I was brought back to a story I heard a while back from colleagues about his early times in the classroom. He had a tendency to mutter to himself during class. On an occasion when a student made a comment that was an oversimplification of a concept, this student started saying at a noticeable volume something that sounded like ‘bull-shit’.

The teacher, clearly shocked by this, reacted softly with a word after class. Given the student’s limited English ability, the message had little chance of making it across. The outburst happened again under more unlucky circumstances when the assistant principal and principal were both in the room observing the teacher – this time, the consequences were a bit more serious. The fact was that, given his personality and the directness associated with translation into a second language, it didn’t seem completely out of character for him to call out a teacher on glossing over a math concept. He saw past the simplification for the sake of his classmates. Calling a teacher out publicly like that, though clearly inappropriate to all of us, might have just been a side effect of being in a new place with new people.

If math was the only language he understood well, and he witnessed math being communicated in an way that was not fully clear to him, of course those moments would attract such a reaction. Over time, we learned to react constructively to these reactions and counsel him into more appropriate ways to ask questions or address his usually correct abstractions of the ideas presented in class.

Fast forward eight yearsto when I was with my ninth graders on our class trip to Shandong province a week ago. As a reward for a hike up thousands of stairs the day before, we spent the final night of the trip visiting a hot springs pool. While the students were splashing around, our tour guide was having a conversation with one of the other tourists in the pool. I was relaxing my eyes staring out at the rocks around the pool when I heard something strangely familiar in their conversation.

“Bu shi…Bu shi…”

I knew both of these words now with my limited experience, but had never thought of them together before. The character bu (不) negates whatever comes after it, and shi (是)is essentially the verb ‘to be’. Putting it together in my head while getting prune fingers at the time, I realized that the phrase bu shi must then mean ‘isn’t’. I confirmed my reasoning with the guide: she was saying that something the tourist was saying wasn’t true.

There I was, seven thousand miles away, realizing long after the fact that this student we all came to admire was probably not cursing at us. He was just saying he thought something he was being taught wasn’t entirely true. It’s the sort of thing we hope our students are thinking about during lessons, questioning their understanding of the content of a lesson. I’ve had students do this in English and never felt threatened by it.

There are many different lessons to take from this. I have been cursed at as a teacher, and I knew it was happening when it was happening because, well, it’s pretty hard to ignore it when it’s happening to you. The fact that this student was having a fairly normal reaction when something wasn’t making sense to him was overshadowed by our misunderstanding of what HE was saying. We assumed he was being out of line. He was innocently saying what was on his mind.

How often do we assume we know what our students are saying without really listening? I’m guilty of wanting to hear an answer that moves a lesson along, but it’s not right, especially when the understanding isn’t there. My students in the Chinese student’s physics class would say an answer they thought was right, and I would on occasion fill in the gaps and go on as if I had heard the correct answer I wanted to hear, even though what the students actually said wasn’t even close to what I wanted. Over the years since they called me out on that, I’ve worked to make that not happen.

In an international school like the one at which I am now teaching, there are languages on top of ideas on top of personalities in my classroom that mix together every day. It is incredibly important to make sure that with such a complex mix of factors, you really know what your students are saying to you and each other.

How China Keeps Me Learning: Part I

Ever since moving to Hangzhou, China in August of 2010, I’ve been amazed at the number of ways it has forced me to use my own problem solving and critical thinking skills. I’ve remarked inwards that talking about these experiences would help greatly in describing the sorts of experiences I want my own students to have, as well as the factors that have helped me be successful as I’ve explored. Now that I am taking the time to write about my experiences, I think this theme is a good one to return to from time to time to describe how these experiences I have relate to my classroom.

Hangzhou has a number of truly incredible places within its city limits. Some are incredibly beautiful. A few of them, however, are incredible for how they address my geeky-tinkerer side.

This building is one of two that sit on opposite sides of the road in the North-east section of Hangzhou. Inside are rows and rows of little booths that each sell electronic parts. Some specialize in motors or solar cells. Others have all different electronic components from resistors to circuit boards to jumper wires, all on display.

I’ve been to this place several times to get parts, other times just to wander around and gawk at the amazing quantity of raw materials there for projects not yet materialized. This week I returned for a different reason. My parents decided to take a big step and visit my wife (Josie) and I here in China, so they have been on numerous adventures with us for the past week. Another post on that is imminent, so stay tuned.

My dad is an engineer and was the first person I thought of when I walked into the building for the first time and saw what was there, so I knew I had to take my dad there for a visit. I also had a vague goal for what I wanted to get while I was there: sensors. Whether for robots or for upcoming units in physics, I knew it would be good to see what was available there so I had more available for experimentation in the classroom and to think ahead.

One other thing to be aware of: I don’t speak Mandarin. I know some basic greetings and scattered vocabulary, but don’t know ‘sensor’, ‘resistor’, or even ‘electric’ either in symbolic or spoken Mandarin. On every visit to the market, I have always had to resort to sketches and diagrams to communicate. This, however, is the most entertaining and enriching part of these trips to the market – figuring out how to say what I am looking for. This was my first visit to the market since my summer acquisition of an iPad, which together with Google Translate, tended to improve the quality of my communication with the dealers to an extent this time. It was, however, still a challenge.

After some wandering around and some awkward interactions with parts dealers that weren’t sure why we were there, my dad and I ended up in a booth with a pair of women intrigued by the site of us in their store. I get the impression on every visit that foreigners don’t enter the building with any regularity, so I’m used to it. I pulled out the iPad and entered ‘gas sensors’ , showing the translation to the women. They pointed to a column of plastic containers beneath a glass counter, gesturing and pointing while saying (in Mandarin) what each one was. Eventually with Translate’s help, they ended up identifying the various gases that they had sensors for, and I came to the conclusion that I needed to do more research before making any purchases. Bottom line – they had some great stuff, much of it exactly what I was looking for.

I went through a similar process in getting some platinum temperature sensors and aluminum blocks with strain gauges for measuring a cantilevered force.

Needless to say, the whole experience was a good one. We all left happy and having had a good time. Here’s just a start of what’s bouncing around in my head for how this experience connects to set up learning opportunities for my students:


I felt free to experiment and play in my learning environment.

I loosely defined goals for my time at the market, but there was no pressure for me to buy anything if I didn’t want to. If my attempts to communicate and find what I was looking for were unsuccessful, I would have other chances to figure it out later on. I wasn’t being evaluated on my time at the market – I was instead free to have fun and try my best to achieve the goals I set for myself.

How much time do we give our students to experiment and play with the material we want to teach them? How are we making the most of the tools we have available to let them do this?


I had the tools I needed to make up for my weaknesses.

The iPad translating capability really made it possible for me to communicate in the way I needed to communicate to achieve my goals. I do want to learn more Mandarin, but I don’t see it necessary that I learn Mandarin completely before I visit the market for my other learning goals. Since my goal had nothing to do with learning the language, but instead to use the tools I had (iPad, electronics market, seemingly amused dad looking on) to reach a desired outcome, I felt free to be creative in how I used the tools to have success.

I speak enough Spanish to be able to have been able to joke and shoot the breeze with cab drivers, store clerks, etc. in the Latin American countries that Josie and I have visited. I have really missed that ability here in China, though I am getting better. The technology lets me be comfortable and interact in a way that makes the entire process enjoyable rather than frustrating. Some frustration is to be expected when trying something new, but not so much to be uncomfortable throughout the process.

How much do the learning goals we set for our students require students have acquired previous skills? How do we address deficiencies in these skills when they arise? Do we give them the tools so they can reach the goals we set for them, or do we modify the goals themselves for these students?


I accepted that I was going to make mistakes, and felt comfortable changing my approach in response to these mistakes.

There were many times when even Google Translate failed to communicate exactly what I was saying (or what the parts dealers were saying) not to mention the challenges that arose in figuring out what I wanted to ask. There were times when I used the Mandarin I did have to confirm that I understood what they were saying, and many times they showed me that I did not. In either case, the dealers were incredibly patient and supportive in figuring out how to help me. It was clear that they were enjoying the process as much as I was, which made me appreciate the time they were willing to take to get me what I wanted. I knew instantly from their reactions to my translated questions whether I had communicated clearly to them, and we were both gesturing and checking that we understood each other as often as possible.

How do we encourage and acknowledge mistake-making as part of the learning process? How do our students feel about making mistakes? How do we develop an environment in which students feel comfortable experimenting and getting things wrong along the way to getting them right?

I love these trips to the market because the feeling of exhilaration and achievement I get when I succeed is worth every moment of frustration. The worst thing that can happen is I walk away empty handed. What usually happens is a scene like the one below:

Somewhere along the line in my classroom, however, students get the feeling that there’s a lot more at stake, that others (unfortunately including me) must be judging their abilities when they don’t get a question right the first time. Students get the feeling that they shouldn’t need to use the tools they have in front of them (graphing calculator, laptop, Geogebra, etc) to learn if they are smart enough. How do I show them that it isn’t about being smart, it is about working hard to get it right in the end? Is it enough to value the mistakes they make? Do I need to share my own mistakes in doing things? (This is part of my plan, at the moment, and is partly why I made the decision to commit time to blogging about what I do in the classroom.)

If I can turn my lessons into explorations and activities in which students feel safe experimenting with concepts, sharing their ideas and helping each other learn, it would make every other goal I have for what I want my students to achieve possible. I’m all ears if you have ideas on how to make this happen!

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