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Ends and Beginnings


Today, I bid farewell to my home away from home for the past six years.
When I first moved away from New York, I had shed all doubts that the teaching career was for me. I knew that learning and exploring were important elements of a meaningful existence on this planet, both for me and my students. I knew that few things were more satisfying than spending time with good people around plates of food. I knew that not knowing the local language or the location of the nearest supermarket was a cause for excitement, not fear. Purposely putting one’s self into situations with unknown outcomes is not a reckless act. It is precisely these challenges that define and refine who we are so that we are better prepared for those events that we do not expect.

I knew these things already. And yet, I leave China today as a changed teacher. I met students from all around the world. I made connections not just with new people in the same building as me, but with teachers in many distributed time zones. People that I respected and admired for their ideas humbled me as they invited me to join in their conversations and explore ideas with me. I found opportunities to present at conferences and get to know others that had also fallen in love with the international teaching lifestyle. I started this blog, and surprisingly, had people read it with thoughts of their own to share.

I also learned to accept the reality that life continues in twenty four time zones. News from home made it seem more foreign and paradoxically more connected to my own experiences here. When opening my eyes and my various devices in the morning to see what had happened while I slept, I again never knew what to expect. I lost family members both suddenly and over stretches of time. Kids grew up. Our parents sold their houses and apartments. Friends put prestigious letters at the end of their names.

Our world changed as well. We added new countries to our passports and got lost in cities that refused to abide by a grid system. We fell in love with our dog and his aggressive sneezing at harmless bystanders. We tried to address the life and work balance through weeknight dinners and mini vacations. We repeatedly overcommitted to traveling during our summers off and time went too quickly. We became parents.                                                         

I write this not because anything I’m saying is especially new. The ‘time marches on’ canon is well established. That does not invalidate the reality that we’re all experiencing life and its passage for the first time ourselves. This is the magic that we, as teachers, witness between the end of one year and the beginning of the next. We tweak our lessons from the previous year with the hope that they prompt more questions and productive confusion on the next iteration. Our students do experience some of the ideas we introduce for the first time in our classrooms, and it is unique that we get to design those experiences ourselves. 

The best way to understand the rich range of emotions that our students experience while in our care is to live deeply and richly in our own lives. We need to learn to know and love others, explore and make mistakes, and be ready to move forward even when the future is uncertain. My time abroad thus far has given me numerous journeys through these human experiences. I would not give them up for the world, and luckily, I do not have to do so.
I’ll write more about my next move in a future post. 

Until then, I wish you all a summer full of good times with good people. 

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Visiting the Museum of Math in NYC


I was reminded by John Burk that the Museum of Math had been opened since I was last in New York. A good friend was in town on a weekend getaway from summer coursework in preparation for starting teaching math this fall. It was the perfect motivation to make time for visiting the museum sooner rather than later.

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I was really impressed with the museum when I first walked in. The message that mathematics can be a language of play and exploration is emphasized from the experiential exhibits on the top floor. From pulling a cart that rolled across various solids to working to tesselate a hyperbolic surface with polygons, there is lots to touch, pull, and do to play with mathematical concepts. Without a doubt, these activities “stimulate inquiry, spark curiosity, and reveal the wonders of mathematics” as the mission statement aspires to do. The general organizing idea of the activities on the top floor is to provide a really interesting, perplexing object or concept to play with, and then dip into the mathematics surrounding that play for those that are interested. One could miss the kiosks that explain the underlying concepts and still feel satisfied with the overall experience.

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The museum had a good mix of activities from different fields of mathematics. Most exhibits were built around a visually defined task that required little explanation in order to start playing with it and understanding its rules through that play. For me, the activities with the largest initial investment/perplexity ratio were the line of lasers that made it possible to see the cross section of a shape and the wall that generates a fractal tree from images of you and a neighbor. Building objects that roll along a particular path was also incredibly engaging for me.

The challenge for a museum like this is to be intriguing without being tricky or elitist. Given that many people experience anxiety about mathematics for all sorts of reasons, I am absolutely sure that the museum’s exhibit designers worked extremely hard to make the bar for entry for these activities as low as possible with a high ceiling. To this end, I think the museum has done a fabulous job. Most exhibits make clear what they are all about and give visible feedback on a visitor’s progress toward reaching the goal. The only area for growth that jumped out at me was an expansion of the role of the staff circulating among the exhibits. They were extremely knowledgeable of the content of the exhibits and were really excited to share what they knew about them. Even as someone that enjoys mathematics, however, there were some exhibits that left me wondering whether I was playing with it correctly. I saw the Shape Ranger exhibit as a puzzle to figure out, for example. I could see others leaving it when the activity didn’t clearly define visually what the score of the activity represented. I envision an expanded role of staff members helping nudge visitors to understand the premises of the more abstract exhibits through careful questioning and good examples of how to play.

This was not a deal breaker for me, however, and didn’t seem to be for the crowd of people in attendance. The number of smiling kids and adults enjoying themselves is clearly the best indication that the museum is doing a great job of fulfilling its mission. This museum has the same sort of open-ended atmosphere that the Exploratorium in San Francisco creates for its visitors, and that puts it in very respectable company.

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2012-2013 Year In Review – Learning Standards


This is the second post reflecting on this past year and I what I did with my students.

My first post is located here. I wrote about this year being the first time I went with standards based grading. One of the most important aspects of this process was creating the learning standards that focused the work of each unit.

What did I do?

I set out to create learning standards for each unit of my courses: Geometry, Advanced Algebra (not my title – this was an Algebra 2 sans trig), Calculus, and Physics. While I wanted to be able to do this for the entire semester at the beginning of the semester, I ended up doing it unit by unit due to time constraints. The content of my courses didn’t change relative to what I had done in previous years though, so it was more of a matter of deciding what themes existed in the content that could be distilled into standards. This involved some combination of concepts into one to prevent the situation of having too many. In some ways, this was a neat exercise to see that two separate concepts really weren’t that different. For example, seeing absolute value equations and inequalities as the same standard led to both a presentation and an assessment process that emphasized the common application of the absolute value definition to both situations.

What worked:

  • The most powerful payoff in creating the standards came at the end of the semester. Students were used to referring to the standards and knew that they were the first place to look for what they needed to study. Students would often ask for a review sheet for the entire semester. Having the standards document available made it easy to ask the students to find problems relating to each standard. This enabled them to then make their own review sheet and ask directed questions related to the standards they did not understand.
  • The standards focus on what students should be able to do. I tried to keep this focus so that students could simultaneously recognize the connection between the content (definitions, theorems, problem types) and what I would ask them to do with that content. My courses don’t involve much recall of facts and instead focus on applying concepts in a number of different situations. The standards helped me show that I valued this application.
  • Writing problems and assessing students was always in the context of the standards. I could give big picture, open-ended problems that required a bit more synthesis on the part of students than before. I could require that students write, read, and look up information needed for a problem and be creative in their presentation as they felt was appropriate. My focus was on seeing how well their work presented and demonstrated proficiency on these standards. They got experience and got feedback on their work (misspelling words in student videos was one) but my focus was on their understanding.
  • The number standards per unit was limited to 4-6 each…eventually. I quickly realized that 7 was on the edge of being too many, but had trouble cutting them down in some cases. In particular, I had trouble doing this with the differentiation unit in Calculus. To make it so that the unit wasn’t any more important than the others, each standard for that unit was weighted 80%, a fact that turned out not to be very important to students.

What needs work:

  • The vocabulary of the standards needs to be more precise and clearly communicated. I tried (and didn’t always succeed) to make it possible for a student to read a standard and understand what they had to be able to do. I realize now, looking back over them all, that I use certain words over and over again but have never specifically said what it means. What does it mean to ‘apply’ a concept? What about ‘relate’ a definition? These explanations don’t need to be in the standards themselves, but it is important that they be somewhere and be explained in some way so students can better understand them.
  • Example problems and references for each standard would be helpful in communicating their content. I wrote about this in my last post. Students generally understood the standards, but wanted specific problems that they were sure related to a particular standard.
  • Some of the specific content needs to be adjusted. This was my first year being much more deliberate in following the Modeling Physics curriculum. I haven’t, unfortunately, been able to attend a training workshop that would probably help me understand how to implement the curriculum more effectively. The unbalanced force unit was crammed in at the end of the first semester and worked through in a fairly superficial way. Not good, Weinberg.
  • Standards for non-content related skills need to be worked in to the scheme. I wanted to have some standards for year or semester long skills standards. For example, unit 5 in Geometry included a standard (not listed in my document below) on creating a presenting a multimedia proof. This was to provide students opportunities to learn to create a video in which they clearly communicate the steps and content of a geometric proof. They could create their video, submit it to me, and get feedback to make it better over time. I also would love to include some programming or computational thinking standards as well that students can work on long term. These standards need to be communicated and cultivated over a long period of time. They will otherwise be just like the others in terms of the rush at the end of the semester. I’ll think about these this summer.

You can see my standards in this Google document:
2012-2013 – Learning Standards

I’d love to hear your comments on these standards or on the post – comment away please!

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Editing Khan


Let’s be clear – I don’t have a problem with most of the content on Khan Academy. Yes, there are mistakes. Yes, there are pedagogical choices that many educators don’t like. I don’t like how it has been sold as the solution to the educational ills of our world, but that isn’t my biggest objection to it.

I sat and watched his series on currency trading not too long ago. Given that his analogies and explanations are correct (which some colleagues have confirmed they are) he does a pretty good job of explaining the concepts in a way that I could understand. I guess that’s the thing that he is known for. I don’t have a problem with this – it’s always good to have good explainers out there.

The biggest issue I have with his videos is that they need an editor.

He repeats himself a lot. He will start explaining something, realize that he needs to back up, and then finishes a sentence that hadn’t really started. He will say something important and then slowly repeat it as he writes each word on the screen.

This is more than just an annoyance. Here’s why:

  • One of the major advantages to using video is that it can be good instruction distilled into great instruction. You can plan ahead with the examples you want to use. You can figure out how to say exactly what you need to say and nothing more, and either practice until you get it right, or just edit out the bad takes.
  • I have written and read definitions word by word on the board during direct instruction in my classes. I have watched my students faces as I do it. It’s clearly excruciating. Seeing that has forced me to resist the urge to speak as I write during class, and instead write the entire thing out before reading it. Even that doesn’t feel right as part of a solid presentation because I hate being read to, and so do my students. This doesn’t need to happen in videos.
  • If the goal of moving direct instruction to videos is to be as efficient as possible and minimize the time students spend sitting and watching rather than interacting with the content, the videos should be as short and efficient as possible. I’m not saying they should be void of personality or emotion. Khan’s conversational style is one of the high points of his material. I’m just saying that the ‘less is more’ principle applies here.

I spent an hour this morning editing one of the videos I watched on currency exchange to show what I mean. The initial length of the video was 12:03, and taking out the parts I mentioned earlier reduced it to 8:15. I think the result respects Khan’s presentation, but makes it a bit tighter and focused on what he is saying. Check it out:

The main reason I haven’t made more videos for my own classes (much to the dismay of my students, who really like them) is my insistence that the videos be efficient and short. I don’t want ten minute videos for my students to watch. I want two minutes of watching, and then two or three minutes of answering questions, discussing with other students, or applying the skills that they learned. My ratio is still about five minutes of editing time for every minute of the final video I make – this is roughly what it took this morning on the Khan Academy video too. This is too long of a process, but it’s a detail on using video that I care too much about to overlook.

What do you think?

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Building a need for math – similar polygons & mobile devices


The focus of some of my out-of-classroom obsessions right now is on building the need for mathematical tools. I’m digging into the fact that many people do well on a daily basis without doing what they think is mathematical thinking. That’s not even my claim – it’s a fact. It’s why people also claim the irrelevance of math because what they see as math (school math) almost never enters the scene in one’s day-to-day interactions with the world.

The human brain is pretty darn good at estimating size or shape or eyeballing when it is safe to cross the street – there’s no arithmetic computation there, so one could argue that there’s no math either. The group of people feeling this way includes many adults, and a good number of my own students.

What interests me these days is spending time with them hovering around the boundary of the capabilities of the brain to do this sort of reasoning. What if the gut can’t do a good enough job of answering a question? This is when measurement, arithmetic, and other skills usually deemed mathematical come into play.

We spend a lot of time looking at our electronic devices. I posed this question to my Geometry and Algebra 2 classes on Monday:
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The votes were five for A, 5 for B, and 14 for C. There was some pretty solid debate about why they felt one way or another. They made sure to note that the corners of the phone were not portrayed accurately, but aside from that, they immediately saw that additional information was needed.

Some students took the image and made measurements in Geogebra. Some measured an actual 4S. Others used the engineering drawing I posted on the class blog. I had them post a quick explanation of their answers on their personal math blogs as part of the homework. The results revealed their reasoning which was often right on. It also showed some examples of flawed reasoning that I didn’t expect – something I now know I need to address in a future class.

At the end of class today when I had the Geometry class vote again, the results were a bit more consistent:
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The students know these devices. Even those that don’t have them know what they look like. It required them to make measurements and some calculations to know which was correct. The need for the mathematics was built in to the activity. It was so simple to get them to make a guess in the beginning based on their intuition, and then figure out what they needed to do, measure, or calculate to confirm their intuition through the idea of similarity. As another chance at understanding this sort of task, I ended today’s class with a similar challenge:

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My students spend much of their time staring at a Macbook screen that is dimensioned slightly off from standard television screen. (8:5 vs. 4:3). They do see the Smartboard in the classroom that has this shape, and I know they have seen it before. I am curious to see what happens.

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Computational modeling & projectile motion, EPISODE IV


I’ve always wondered how I might assess student understanding of projectile motion separately from the algebra. I’ve tried in the past to do this, but since my presentation always started with algebra, it was really hard to separate the two. In my last three posts about this, I’ve detailed my computational approach this time. A review:

    • We used Tracker to manually follow a ball tossed in the air. It generated graphs of position vs. time for both x and y components of position. We recognized these models as constant velocity (horizontal) and constant acceleration particle models (vertical).
    • We matched graphical models to a given projectile motion problem and visually identified solutions. We saw the limitations of this method – a major one being the difficulty finding the final answer accurately from a graph. This included a standards quiz on adapting a Geogebra model to solve a traditional projectile motion problem.
    • We looked at how to create a table of values using the algebraic models. We identified key points in the motion of the projectile (maximum height, range of the projectile) directly from the tables or graphs of position and velocity versus time. This was followed with the following assessment
    • We looked at using goal seek in the spreadsheet to find these values more accurately than was possible from reading the tables.

After this, I gave a quiz to assess their abilities – the same set of questions, but asked first using a table…
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… and then using a graph:
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The following data describes a can of soup thrown from a window of a building.

  • How long is the can in the air?
  • What is the maximum height of the can?
  • How high above the ground is the window?
  • Is the can thrown horizontally? Explain your answer.
  • How far from the base of the building does the can hit the ground?
  • What is the speed of the can just before it hits the ground?</li

I was really happy with the results class wide. They really understood what they were looking at and answered the questions correctly. They have also been pretty good at using goal seek to find these values fairly easily.

I did a lesson that last day on solving the problems algebraically. It felt really strange going through the process – students already knew how to set up a problem solution in the spreadsheet, and there really wasn’t much that we gained from obtaining an algebraic solution by hand, at least in my presentation. Admittedly, I could have swung too far in the opposite direction selling the computational methods and not enough driving the need for algebra.

The real need for algebra, however, comes from exploring general cases and identifying the existence of solutions to a problem. I realized that these really deep questions are not typical of high school physics treatments of projectile motion. This is part of the reason physics gets the reputation of a subject full of ‘plug and chug’ problems and equations that need to be memorized – there aren’t enough problems that demand students match their understanding of how the equations describe real objects that move around to actual objects that are moving around.

I’m not giving a unit assessment this time – the students are demonstrating their proficiency at the standards for this unit by answering the questions in this handout:
Projectile Motion – Assessment Questions

These are problems that are not pulled directly out of the textbook – they all require the students to figure out what information they need for building and adapting their computer models to solve them. Today they got to work going outside, making measurements, and helping each other start the modeling process. This is the sort of problem solving I’ve always wanted students to see as a natural application of learning, but it has never happened so easily as it did today. I will have to see how it turns out, of course, when they submit their responses, but I am really looking forward to getting a chance to do so.

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Grouping Problems in 1st Grade


Grouping Problems in 1st Grade

My wife (Josie) was showing me the work she is doing with her first grade students in math. They are talking about grouping tens and ones, ultimately looking to explore place value. Her activity was to have students imagine situations involving collecting groups of items, and then looking at the mathematical structure behind those groups. One wrote about how a thief had a container that could only carry 10 ice cream cones at a time, which meant that he had to leave some of the ice cream cones he was stealing from a house behind. Another talked about the Grinch stealing twenty Christmas trees at a time from a forest that had 255.

There are two things that I really like about the approach. One is that it doesn’t do the common backwards approach I have seen in elementary math programs where the math problem comes first. It seems off to asking students to add 3 + 4 = 7, and then ‘make up a story problem’ that matches this abstract idea. Here, the students are coming up with problems that matter to them, and creating organization (groups) that make sense to them. Going from abstract to concrete works marginally well at best at the high school level for developing understanding, let alone for six and seven year olds that are wrapping their heads around abstract ideas like place value.

I also really like that Josie didn’t push the students to consider only even groupings. (255 trees into groups of 20? There’s a remainder. THERE’S A REMAINDER!) Word problems are often contrived to have even numbers only to make them ‘easier to understand’ and consequently even less real world. I just thought it was neat to see that she is making her students manage that messiness from the beginning.

This is clearly different from the higher level courses that I usually concern myself with in high school, but the idea still transfers well, regardless of the level. It’s always great to see things being done right by the younger students as well.

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January 31, 2013 · 1:05 am

Simulations, Models, and the 2012 US Election


After the elections last night, I found I was looking back at Nate Silver’s blog at the New York Times, Five Thirty Eight.

Here was his predicted electoral college map:

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…and here was what ended up happening (from CNN.com):

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I’ve spent some time reading through Nate Silver’s methodology throughout the election season. It’s detailed enough to get a good idea of how far he and his team  have gone to construct a good model for simulating the election results. There is plenty of description of how he has used available information to construct the models used to predict election results, and last night was an incredible validation of his model. His popular vote percentage for Romney was predicted to be 48.4%, with the actual at 48.3 %. Considering all of the variables associated with human emotion, the complex factors involved in individuals making their decisions on how to vote, the fact that the Five Thirty Eight model worked so well is a testament to what a really good model can do with large amounts of data.

My fear is that the post-election analysis of such a tool over emphasizes the hand-waving and black box nature of what simulation can do. I see this as a real opportunity for us to pick up real world analyses like these, share them with students, and use it as an opportunity to get them involved in understanding what goes into a good model. How is it constructed? How does it accommodate new information? There is a lot of really smart thinking that went into this, but it isn’t necessarily beyond our students to at a minimum understand aspects of it. At its best, this is a chance to model something that is truly complex and see how good such a model can be.

I see this as another piece of evidence that computational thinking is a necessary skill for students to learn today. Seeing how to create a computational model of something in the real world, or minimally seeing it as an comprehensible process, gives them the power to understand how to ask and answer their own questions about the world. This is really interesting mathematics, and is just about the least contrived real world problem out there. It screams out to us to use it to get our students excited about what is possible with the tools we give them.

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First Day Activities: Robotics


For our orientation activities this year, we are focusing on getting to know each other, and discussing/interacting/performing skits around digital citizenship and our school’s computer use policy. To make sure we have enough time for these activities, but find problems with the schedule, we are going through each day of the block schedule with abbreviated classes. This means that each class gets around twenty minutes to meet.

I love the limited time for the sole reason that I’m not even tempted to talk about policies. It’s a chance to do something interesting with the students and whet their appetites for what the class is going to be about. I’m going to share what I did as an opportunity to record what I did on the first day for the future (which I always plan to do, but rarely do) but also to give others ideas.

Robotics

The students came in to find three full Ziploc bags of LEGO pieces – the last of my collection from back home that was previously in storage. I asked them to build a tower as tall as they could build it using the pieces from the bags. I asked that they keep track of the number of bricks they used in their design.

They quickly got to work – I was impressed with how quickly they jumped into team-oriented roles. Some created a base for the tower. Others started stacking bricks. Another occasionally pantomimed a general shape they should try to follow.

After around eight minutes had passed, we measured the tower. I then said they needed to build a tower that reached at least the same height, but used half the number of bricks they used in the first one. Again, they quickly got to work. They scrapped the nice base they had built and did some interesting sideways building vertically to make up for the pieces they knew they had to remove. Eventually the height reached more than twice the height of their original tower – I was impressed.

We then had three minutes left – just enough to ask students to compare this task to building and designing something in the real world. I mentioned the word constraints to describe what they came up with, but they got the idea. They also mentioned that it would have helped if they knew what all of the constraints and requirements were at the beginning. I agreed, with a smirk.

Then it was off to the next class.

It’s good to be back! Expect more posts as I can fit them in.

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A Response to Slate: How the recent article on technology misses the point.


Ah, summer. A great time to kick back, relax, and have time to write reactions to things that bug me.

I read through the article on Slate titled ‘Why Johnny Can’t Add Without a Calculator’ and found it to be a rehashing of a whole slew of arguments that drive me nuts about technology in education. It also does a pretty good job of glossing over a number of issues relative to learning math.

The problem isn’t that Johnny can’t add without a calculator. It’s that we sometimes focus too much about turning our brain into one.

This was the sub-heading underneath the title of the article:

Technology is doing to math education what industrial agriculture did to food: making it efficient, monotonous, and low-quality.

The author then describes some ancedotes describing technology use and implementation:

  • An experienced teacher forced to give up his preferred blackboard in favor of an interactive whiteboard, or IWB.
  • A teacher unable to demonstrate the merits of an IWB beyond showing a video and completing a demo of an electric circuit.
  • The author trying one piece of software and finding it would not accept an answer without sufficient accuracy.

I agree with the author’s implication that blindly throwing technology into the classroom is a bad idea. I’ve said many times that technology is only really useful for teaching when it is used in ways that enhance the classroom experience. Simply using technology for its own sake is a waste.

These statements are true about many tools though. The mere presence of one tool or another doesn’t make the difference – it is all about how the tool is used. A skilled teacher can make the most of any textbook – whether recently published or decades old – for the purposes of helping a student learn. Conversely, just having an interactive whiteboard in the classroom does not make students learn more. It is all about the teacher and how he or she uses the tools in the room. The author acknowledges this fact briefly at the end in arguing that the “shortfall in math and science education can be solved not by software or gadgets but by better teachers.” He also makes the point that there is no “technological substitute for a teacher who cares.” I don’t disagree with this point at all.

The most damaging statements in the article surround how the author’s misunderstanding of good mathematical education and learning through technology.

Statement 1: “Educational researchers often present a false dichotomy between fluency and conceptual reasoning. But as in basketball, where shooting foul shots helps you learn how to take a fancier shot, computational fluency is the path to conceptual understanding. There is no way around it.”

This statement gets to the heart of what the author views as learning math. I’ve argued in previous posts on how my own view of the relationship between conceptual understanding and learning algorithms has evolved. I won’t delve too much here on this issue since there are bigger fish to fry, but the idea that math is nothing more than learning procedures that will someday be used and understood does the whole subject a disservice. This is a piece of the criticism of Khan Academy, but I’ll leave the bulk of that argument to the experts.

I will say that I’m really tired of the sports skills analogy for arguing why drilling in math is important. I’m not saying drills aren’t useful, just that they are never the point. You go through drills in basketball not just to be able to do a fancier shot (as he says) but to be able to play and succeed in a game. This analogy also falls short in other subjects, a fact not usually brought up by those using this argument. You spend time learning grammar and analysis in English classes (drills), but eventually students are also asked to write essays (the game). Musicians practice scales and fingering (drills), but also get opportunities to play pieces of music and perform in front of audiences (the game).

The general view of learning procedures as the end goal in math class is probably the most destructive reason why people view math as something acceptable not to be good at. Learning math this way can be low-quality because it is “monotonous [and] efficient”, which is not technology’s fault.

One hundred percent of class time can’t be spent on computational fluency with the expectation that one hundred percent of understanding can come later. The two are intimately entwined, particularly in the best math classrooms with the best teachers.

Statement 2: “Despite the lack of empirical evidence, the National Council of Teachers of Mathematics takes the beneficial effects of technology as dogma.”

If you visit the link the author includes in his article, you will see that what NCTM actually says is this:

“Calculators and other technological tools, such as computer algebra systems, interactive geometry software, applets, spreadsheets, and interactive presentation devices, are vital components of a high-quality mathematics education.”

…and then this:

“The use of technology cannot replace conceptual understanding, computational fluency, or problem-solving skills.”

In short, the National Council for Teachers of Mathematics wants both understanding and computational fluency. It really isn’t one or the other, as the author suggests.

The author’s view of what “technology” entails in the classroom seems to be the mere presence of an interactive whiteboard, new textbooks, calculators in the classroom, and software that teaches mathematical procedures. This is not what the NCTM intends the use of technology to be. Instead the use of technology allows exploration of concepts in ways that cannot be done using just a blackboard and chalk, or pencil and paper. The “and other technological tools next to calculators in the quote has become much more significant over the past five years, as Geometers Sketchpad, Geogebra, Wolfram Alpha, and Desmos have become available.

Teachers must know how to use these tools for the nature of math class to change to one that emphasizes mathematical thinking over rote procedure. If they don’t, then math continues as it has been for many years: a set of procedures that students may understand and use some day in the future. This might be just fine for students that are planning to study math, science, or engineering high school. What about the rest of them? (They are the majority, by the way.)

Statement 3: “…the new Common Core standards for math…fall short. They fetishize “data analysis” without giving students a sufficient grounding to meaningfully analyze data. Though not as wishy-washy as they might have been, they are of a piece with the runaway adaption of technology: The new is given preference over the rigorous.”

If “sufficient grounding” here means students doing calculations done by hand, I completely disagree. Ask a student to add 20 numbers by hand to calculate an average, and you’ll know what I mean. If calculation is the point of a lesson, I’ll have students calculate. The point of data analysis is not computation. Just because the tools take the rigor out of calculation does not diminish the mathematical thinking involved.

Statement 4: “Computer technology, while great for many things, is just not much good for teaching, yet. Paradoxically, using technology can inhibit understanding how it works. If you learn how to multiply 37 by 41 using a calculator, you only understand the black box. You’ll never learn how to build a better calculator that way.”

For my high school students, I am not focused on students understanding how to multiply 37 by 41 by hand. I do expect them to be able to do it. Usually when my students do get it wrong, it is because they feel compelled to do it by hand because they are taught (in my view incorrectly) that doing so is somehow better, even when a calculator sits in front of them. As with Statement 3, I am not usually interested in students focusing on the details of computation when we are learning difference quotients and derivatives. This is where technology comes in.

I tweeted a request to the author to check out Conrad Wolfram’s TED Talk on using computers to teach math, and asked for a response. I still haven’t heard back. I think it would be really revealing for him to listen to Wolfram’s points about computation, the traditional arguments against computation, and the reasons why computers offer students new opportunities to explore concepts in ways they could not with mere pencil and paper. His statement that math is much more than computation has really changed the way I think about teaching my students math in my classroom.

Statement 5: “Technology is bad at dealing with poorly structured concepts. One question leads to another leads to another, and the rigid structure of computer software has no way of dealing with this. Software is especially bad for smart kids, who are held back by its inflexibility.”

Looking at computers used purely as rote instruction tools, I completely agree. That is a fairly narrow view of what learning mathematics can be about.

In reality, technology tools are perfectly suited for exploring poorly structured concepts because they let a student explore the patterns of the big picture. The situation in which “one question leads to another” is exactly what we want students to feel comfortable exploring in our classroom! Finally, software that is designed for this type of exploration is good for the smart students (who might quickly make connections between different graphical, algebraic, and numerical representations of functions, for example) and for the weaker students that might need different approaches to a topic to engage with a concept.

The truly inflexible applications of technology are, sadly, the ones that are also associated with easily measured outcomes. If technology is only used to pass lectures and exercises to students so they can perform well on standardized tests, it will be “efficient, monotonous, and low quality” as the author states at the beginning.

The hope that throwing calculators or computers in the classroom will “fix” problems of engagement and achievement without the right people in the room to use those tools is a false one, as the author suggests. The move to portray mathematics as more than a set of repetitive, monotonous processes, however, is a really good thing. We want schools to produce students that can think independently and analytically, and there are many ways that true mathematical thinking contributes to this sort of development. Technology enables students to do mathematical thinking even when their computation skills are not up to par. It offers a different way for students to explore mathematical ideas when these ideas don’t make sense presented on a static blackboard. In the end, this gets more students into the game.

This should be our goal. We shouldn’t going back to the most basic textbooks and rote teaching methods because it has always worked for the strongest math students. There must have been a form of mathematical Darwinism at work there – the students that went on historically were the ones that could manage the methods. This is why we must be wary of the argument often made that since a pedagogical method “worked for one person” that that method should be continued for all students. We should instead be making the most of resources that are available to reach as many students as possible and give them a rich experience that exposes them to the depth and variety associated with true mathematical thinking.

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