Tag Archives: #technology

2016 - 2017 Year In Review: Technology Tools

Overview

I've always been a pretty heavy user of technology. I've been more careful in the past few years to use it for a reason, not for its own sake though. I also balance that use though with a healthy desire to try new things in a way that I would actually use them in the classroom.

This is also the first year I've been able to take advantage of the Google Tools suite since Vietnam is not subject to the limitations of China's Great Firewall. Though there have been times when the internet connection to the entire country has been subject to shark attacks, connections in general have been smooth. Seeing how effectively some folks use Google in the classroom after being unable to use it for six years make me feel seriously behind the times. Luckily, my colleagues are really eager to share what they do. I might be caught up.

  1. A Macbook Pro where I do most of my lesson planning. I connected an external widescreen monitor that mirrored all projected content. The second screen was sent through AirPlay to an Apple TV, which was then connected to the projector.
  2. Class worksheets electronically created and stored in Google Docs. These are printed out on A5 size sheets for students to tape into their notebooks for a physical record of what we did.
  3. For IB Mathematics SL and PreCalculus, I had two students per class make an additional Google Doc that was a copy of the handout. In this document, students would paste solutions to class work, homework, and whatever else they thought might be important to their classmates. The student responsibility for doing this was on a rotating schedule, similar to what I've used in my previous classes.
  4. Notability app for class notes, with a Wacom Tablet for input. I used the wireless accessory kit for around two days, because it disconnected too frequently.
  5. iPhone as a document camera for capturing student work for sharing answers or for conversation during the class. I would take pictures of student work and use AirDrop to upload them for inclusion in the notes.
  6. Moodle as a repository for all of the above documents and links. I also used it occasionally for distributing quizzes and automatic grading.
  7. My WeinbergCloud website for managing, assigning, and recording reassessments throughout the semester.
  8. PearDeck on a trial basis for first semester, and then regularly during second. I sometimes used an iPad to manage the class, but every time I regretted it, and just used my computer.
  9. Desmos Calculator usually at least once per lesson
  10. Desmos Activity Builder about once per unit per course
  11. EdPuzzle for self paced lessons, videos, and quizzes in Algebra 2. Most of the videos were produced by my colleague, Scott Hsu.
  12. Spreadsheets for building useful calculators (like discriminants for quadratics, arithmetic/geometric series sums, etc)
  13. Khan Academy for practice exercises and monitoring of student effort in reviewing material.
  14. Geogebra for checking exam questions and demonstrating its use as a work-checking tool for students.
  15. Camtasia for recording videos from time to time of solving problems, using
  16. Quizster as a way to have students submit specific homework problems for feedback.
  17. Wireless keyboard and trackpad, though these lasted about a week and a half.
  18. I dabbled with GoFormative, primarily when I was on sick leave for a while. Connection issues that were inconsistent across the class led to my abandoning it for regular use.
  19. For two units in PreCalculus, I used Trello as a way to organize units and help students organize their work for each day of class.

What worked:

  • The process of cutting and pasting images of problems or student work into Notability, and then annotating them was great for recording important information during class. These notes were then either pasted into the document created by students for each class, or exported as PDF for posting on Moodle. This felt like a good way to have a record of what went on during a given class block in case students missed a block.
  • I liked automated grading of quizzes through Google forms and Moodle. This definitely saved time, but the process of getting feedback to students in response still is awkward. When student work is analog, but answer checking is digital, where should that feedback go? Quizster offers some way of making all of this occur in the same tool, but the workflow never was smooth enough to fully commit to it.
  • The combination of PearDeck and Desmos Activity builder, along with photos of student work, made for great sources of understanding (and misunderstanding) that helped me decide how or whether to proceed with other material. These also made for great motivating elements for direct instruction when it needed to happen. The students really liked using these tools, and said they looked forward to them, according to the end of year survey results.
  • I don't think Khan Academy exercises work well for assessing students beyond a basic level. I think they can provide the practice some students need on procedural skills like factoring or evaluating trigonometric functions. It's just one tool among many to serve the needs of my students.
  • When I provided guidance on how spreadsheets could be used for more than just making charts, students appreciated it. One student went so far as to say that this instruction was "actually useful". I decided not to ask what this student thought about the rest of the class.

What needs work:

  • I posted homework problems on the printed class handout, digital handout, and on a dedicated document for assignments that is an expectation across classes our high school division. Consistently updating all three documents was a challenge, despite my best efforts.
  • The student notebook entries for each class were among the least favorite element of the class, as reported by students. I've written previously about the need to have some record of what happens during class, but frustration over students that do not produce their own record through regular use of a dedicated notebook. This isn't the best solution, but I think it's the closest I've come to something that actually reaches the right balance. I just wish I could figure out how to get students to buy into its usefulness.
  • I still have not figured out the best way to bring the class back together after letting them work at their own pace through lessons. The only times it makes a lot of sense to do this is at the very beginning of class, and at the very end.
  • PearDeck, Desmos Activity Builder, and GoFormative each offer features that I really like. None of them do everything. I'm ok with this, but I wonder whether the fragmentation of activities is good for students, or a problem since their work is distributed across these tools.
  • While I liked using Trello, and some students reported that they also appreciated it, many students did not. I'm not sure if it actually is the self-paced lesson tool I'm looking for, but it was better than a static Google document.
  • At the end of the year, despite my own research and attempts to improve this, the Apple TV disconnected at least once every class period, if not more frequently.

Conclusion

My focus continues to be on using technology to free up time for the ways that I can best add value in the classroom. Many students don't need my help in making progress. Some do, and some like having me explain ideas to them. It's hard to simultaneously meet these different needs without technology, which enables me to be in multiple places at once.

Having the range of tools I describe above, and not fully committing to one, is both a blessing and a curse. The fragmentation means the residue of learning is distributed across many web addresses. The variety helps keeps students (and me) from getting into a rut. I don't know if this balance is appropriately tuned yet.

Scaling up SBG for the New Year

In my new school, the mean size of my classes has doubled. The maximum size is now 22 students, a fact about which I am not complaining. I've missed the ease of getting students to interact with simple proximity as the major factor.

I have also been given the freedom to continue with the standards based grading system that I've used over the past four years. The reality of needing to adapt my systems of assessment to these larger sizes has required me to reflect upon which aspects of my system need to be scaled, and what (if anything) needs to change.

The end result of that reflection has identified these three elements that need to remain in my system:

  • Students need to be assessed frequently through quizzes relating to one to two standards maximum.
  • These quizzes need to be graded and returned within the class period to ensure a short feedback cycle.
  • There must still be a tie between work done preparing for a reassessment and signing up for one.

Including the first element requires planning ahead. If quizzes are going to take up fifteen to twenty minutes of a class block, the rest of the block needs to be appropriately planned to ensure a balance between activities that respond to student learning needs, encourage reinforcement of old concepts, and allow interaction with new material. The second element dictates that those activities need to provide me time to grade the quizzes and enter them as standards grades before returning them to students. The third happens a bit later in the cycle as students act on their individualized needs to reassess on individual standards.

The major realization this year has been a refined need for standards that can be assessed within a twenty minute block. In the past, I've believed that a quiz that hits one or two aspects of the topic is good enough, and that an end of unit assessment will allow complete assessment on the whole topic. Now I see that a standard that has needs to have one component assessed on a quiz, and another component assessed on a test, really should be broken up into multiple standards. This has also meant that single standard quizzes are the way to go. I gave one quiz this week that tested a previously assessed standard, and then also assessed two new ones. Given how frantic I was in assessing mastery levels on three standards, I won't be doing that again.

The other part of this first element is the importance of writing efficiently targeted assessment questions. I need students to arrive at a right answer by applying their knowledge, not by accident or application of an algorithm. I need mistakes to be evidence of misunderstanding, not management of computational complexity. In short, I need assessment questions that assess what they are designed to assess. That takes time, but with my simplified schedule this year, I'm finding the time to do this important work.

My last post was about my excitement over using the Numbas web site to create and generate the quizzes. A major bottleneck in grading these quizzes quickly in the past has been not necessarily having answers to the questions I give. Numbas allows me to program and display calculated answers based on the randomized values used to generate the questions.

Numbas has a feature that allows students to take the exam entirely online and enter their answers to be graded automatically. In this situation, I have students pass in their work as well. While I like the speed this offers, that advantage primarily exists in cases where students answer questions correctly. If they make mistakes, I look at the written work and figure out what went wrong, and individual values require that I recalculate along the way. This isn't a huge problem, but it brings into question the need for individualized values which are (as far as I know right now) the only option for the fully online assessment. The option I like more is the printed worksheet theme that allows generation of printable quizzes. I make four versions and pass these out, and then there are only four sets of answers to have to compare student work against.

With the answers, I can grade the quizzes and give feedback where needed on wrong answers in no more than ten or fifteen minutes total. This time is divided into short intervals throughout the class block while students are working individually. The lesson and class activities need to be designed to provide this time so I can focus on grading.

The third element is still under development, but my credit system from previous years is going to make an appearance. Construction is still underway on that one. Please pardon the dust.


P.S:

If you're an ed-tech company that wants to impress me, make it easy for me to (a) generate different versions of good assessment questions with answers, (b) distribute those questions to students, (c) capture the student thinking and writing that goes with that question so that I can adjust my instruction accordingly, and (d) make it super easy to share that thinking in different ways.

That step of capturing student work is the roughest element of the UX experience of the four. At this time, nothing beats looking at a student's paper for evidence of their thinking, and then deciding what comes next based on experience. Snapping a picture with a phone is the best I've got right now. Please don't bring up using tablets and a stylus. We aren't there yet.

Right now there are solutions that hit two or three, but I'm greedy. Let me know if you know about a tool that might be what I'm looking for.

Teaching from Anywhere

I use my phone as a document camera, which is nothing new. AirDrop is an option since my school computer is now running OS X Yosemite. I was using my own Python web application to upload these to the computer last year, but that was limited to one file at a time. Now I can send a whole stack of photos of student work at once, which makes it the obvious choice.

The laptop is parked to be plugged into the projector in a spot that doesn't sacrifice student real estate, but is accessible if I need to get to it:

IMG_1409

The thing that has always bugged me is having to be in one place in the room to do, well, anything. I like sitting with students. I have interesting and useful conversations with students when I'm among them, not while standing at the front of the room. My solution in the past has been to bring the laptop around the classroom with me and sit down next to students. Two things bother me about this:

  • When move to join a table next to students, I always take up more room than any other person. This is because I'm there with a laptop, Wacom tablet, and some notes if I need them for the lesson. My students are too polite to actually object when I move in and they always consolidate their things to make room. I know the whole time, however, that they are wishing I wouldn't. This whole process repeats if I want to move during the lesson, which I always do.
  • I have an Apple TV that I've used in the past to wirelessly display my screen in this situation, but the lag between my movement and the display is enough to be uncomfortable for me, and render my handwriting into the illegible range if I'm not extremely careful. I can stream student work to the Apple TV from my phone directly, but without the ability to zoom in on what's actually important or annotate, the capability limits more than it offers.

I have had the wireless kit for my Wacom tablet since last year, so that doesn't need to be connected to the projector laptop anymore. To switch applications (which I do frequently), write more than a couple words on the screen (which is more efficiently done through typing), or upload student work, I've always needed to go back to the laptop. This additional step during class is a moment of dead time - a moment during which students have no choice but to wait and do nothing, or do worse. This moment of dead time has been an unavoidable consequence of my classroom design and configuration.

The arrangement that has minimized (if not eliminated) all of these issues for this new year is this set of devices:

IMG_1407

I already mentioned the wireless Wacom tablet for handwritten work. The wireless keyboard (picked up during RadioShack's sale of excess inventory this summer) lets me type from anywhere in the room. The Magic Trackpad lets me do the rest.

I can take all three of these anywhere in the classroom if I need to, though often one at a time will suffice. I can switch applications, write on the wall, and type from pretty much anywhere. For sharing, viewing, and cropping student work, I can use the trackpad to manage the stream of photos that I (or my students) send to the computer through Airdrop.

This freedom to run my class untethered from the computer and centered wherever student thinking is happening is worth every ounce of aluminum, glass, and plastic. This freedom makes a difference.

The (Un)changing Role of Teachers

I happened upon this article today from The Atlantic titled The Deconstruction of the K-12 Teacher. Here is a highlight from the article:

The relatively recent emergence of the Internet, and the ever-increasing ease of access to web, has unmistakably usurped the teacher from the former role as dictator of subject content. These days, teachers are expected to concentrate on the "facilitation" of factual knowledge that is suddenly widely accessible.

This line of reasoning inevitably comes up in my conversations with those that don't teach, including those that have children currently in the system. What is the role of the teacher in today's classroom?

My response usually pays lip-service to the idea that the role of teachers is certainly changing in response to the presence of technology. I think it's obvious that is the case. I don't believe that most of us are turning our classrooms into rows of students doing computerized lessons because of their effectiveness - that certainly isn't he case either. My arguments for there being a place for teachers in the classroom surround the social situation that exists in having learners together in one place. In the best classrooms, historically, it has never really been about transferring knowledge from the front to the back.  It has instead always been about the community.

Here are my main ideas on this concept:

  • Making the social network of the classroom into a learning resource requires careful planning and experience in managing the process.
  • Students need to learn that it is normal to make mistakes along the road to understanding. This isn't easy when done in isolation.
  • Making big picture connections is done best in conversation with others having a diversity of experiences and understandings.
  • Some skills are learned best in context with someone knowledgable in their use. 
  • Asking a question of a source you know and trust is easier than taking a shot in the dark on an online forum or through a chat window.

I'm not saying these processes can't be completed online. Our students certainly have experience communicating through online channels. They need our guidance as teachers in using these networks for learning, however, and the classroom is a great place to give them that guidance. In light of the social, emotional, and finally academic needs of teenagers, I think we will be needed for a while yet before computers can fully take over the classroom for good.

Speed of sound lab, 21st century version

I love the standard lab used to measure the speed of sound using standing waves. I love the fact that it's possible to measure physical quantities that are too fast to really visualize effectively.

This image from the 1995 Physics B exam describes the basic set-up:
Screen Shot 2013-05-16 at 3.43.30 PM

The general procedure involves holding a tuning fork at the opening of the top of the tube and then raising and lowering the tube in the graduated cylinder of water until the tube 'sings' at the frequency of the tuning fork. The shortest height at which this occurs is the fundamental frequency of vibration of the air in the tube, and this can be used to find the speed of sound waves in the air.

The problem is in the execution. A quick Google search for speed of sound labs for high school and university settings all use tuning forks as the frequency source. I have always found the same problems come up every time I have tried to do this experiment with tuning forks:

  • Not having enough tuning forks for the whole group. Sharing tuning forks is fine, but raises the lower limit required for the whole group to complete the experiment.
  • Not enough tuning forks at different frequencies for each group to measure. At one of my schools, we had tuning forks of four different frequencies available. My current school has five. Five data points for making a measurement is not the ideal, particularly for showing a linear (or other functional) relationship.
  • The challenge of simultaneously keeping the tuning fork vibrating, raising and lowering the tube, and making height measurements is frustrating. This (together with sharing tuning forks) is why this lab can take so long just to get five data points. I'm all for giving students the realistic experience of the frustration of real world data collection, but this is made arbitrarily difficult by the equipment.

So what's the solution? Obviously we don't all have access to a lab quality function generator, let alone one for every group in the classroom. I have noticed an abundance of earphones in the pockets of students during the day. Earphones that can easily play a whole bunch of frequencies through them, if only a 3.5 millimeter jack could somehow be configured to play a specific frequency waveform. Where might we get a device that has the capacity to play specific (and known) frequencies of sound?

I visited this website and generated a bunch of WAV files, which I then converted into MP3s. Here is the bundle of sound files we used:
SpeedOfSoundFrequencies

I showed the students the basics of the lab and was holding the earphone close to the top of the tube with one hand while raising the tube with the other. After getting started on their own, the students quickly found an additional improvement to the technique by using the hook shape of their earphones:
Screen Shot 2013-05-16 at 4.03.13 PM

Data collection took around 20 minutes for all students, not counting students retaking data for some of the cases at the extremes. The frequencies I used kept the heights of the tubes measurable given the rulers we had around to measure the heights. This is the plot of our data, linearized as frequency vs. 1/4L with an length correction factor of 0.4*diameter added on to the student data:
Screen Shot 2013-05-16 at 4.14.22 PM

The slope of this line is approximately 300 m/s with the best fit line allowed to have any intercept it wants, and would have a slightly higher value if the regression is constrained to pass through the origin. I'm less concerned with that, and more excited with how smoothly data collection was to make this lab much less of a headache than it has been in the past.

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.

Presenting the MVT In Calculus w/ Geogebra...tech as a game changer.

During our warm-up activity today, we looked at a function and identified critical points, relative, and absolute extrema for this function:

It was kind of neat talking about this and the extreme value theorem from last time. Since the domain is not defined over a closed interval, the EVT doesn't guarantee the existence of an absolute maximum or minimum value. The students seemed to really get the idea this year that this function specifically has no absolute maximum over the domain because it is an open interval - last year there was a lot of confused faces on this idea. There were a couple really insightful comments about whether there would be an open interval domain over which the function did have an absolute maximum, even though the hypothesis wasn't satisfied. The theorem just tells you whether or not you are guaranteed to find one, not that there isn't one at all. Really good stuff, and I'm proud of the way everyone was chiming in to talk about what they understood.

The most important thing was that this led perfectly into introducing the idea of an existence theorem. This idea is different from other theorems (especially in comparison to geometry) that students have learned because the information it gives you is not as specific as "alternate interior angles are congruent" or "the remainder of polynomial P(x) upon division by (x - c) is P(c)". All it does is tells you whether you can find what the theorem says is there. I didn't plan on having this discussion today, but it was perfect for then introducing the mean value theorem, and I will definitely repeat it in the future.

I then gave my students this geogebra applet to play with today.

Download link here.

The students understood pretty quickly what they had to do, and didn't seem to have a hard time. It was kind of interesting to watch them rediscover the concept of forming a tangent line using two points, as that concept has been a bit overshadowed by other things as we looked at derivative rules before the test they took last week. Some students moved P and Q so that they were tangent, and then adjusted the domain using C and D to find a domain over which the tangent line and line AB were parallel.

From this, I showed them what the slope of line AB represented (average rate of change over the interval) and came up with the right side of the MVT. We then talked about what the slope of the tangent line they identified represented - a couple immediately referenced the derivative of the function. What is the relationship between parallel lines? What would make it so that you couldn't find this value? Ideas of continuity and differentiability jumped out. There it was: the entire mean value theorem.

Last year I presented the students with the MVT, and then we drew graphs to represent what it was saying. They kind of got it, but it wasn't a sticky idea. I was doing all the developing. This approach today started with something visual that they were doing, that they could understand intuitively, and then that intuition was applied to develop an abstract concept out of that understanding.

I continued doing what I had done last year - answering some multiple choice questions about the MVT (See here for today's handout) analytically, and I immediately lost a couple students. So I showed them how to throw the new function into Geogebra and adjust the domain to match the problem. They could then solve the problems graphically - they immediately located the points to be able to answer the questions.

The group is a mix of AP and non-AP exam bound students. I will introduce them all to the analytic ways of identifying these points, and we did some of it today. It was really nice that the moment things got a bit too abstract, I could push students to identify how the question being asked was the same as the idea of the MVT, and they were then able to solve it.

Without the technology, these students would have been done for the rest of the period. Those that could handle the algebra, would. Those that couldn't would spend the rest of the period feeling like they were in over their heads. Introducing how to use the technology to really understand what was being said by the abstract theorem enabled many more students to get in on the game. That made me feel all warm and fuzzy inside. The rest of the class focused on definitions of increasing functions using the derivative, something that was made incredibly easy by referring back to the activity at the beginning of the period.

We'll see how well they remember the ideas moving forward, but it felt great knowing that, at least for today's lesson, everyone in the room had a way into the game.

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.