Tag Archives: philosophy

Students and Working With Big Data

I happened upon this tweet today:

I hadn't heard of the Oceans of Data Institute before, but a quick look at its website revealed some interesting areas of focus:

  • Designing interfaces to let students interact with large sets of data
  • Defining the skills profile of big data scientists explicitly

As an example of their projects, the page includes a link to http://oceantracks.org, which allows students to visualize the movement of different animals in the ocean. In the image below, red is the track of an elephant seal, yellow is a blue-fin tuna, and turquoise is a white shark.
Screen Shot 2015-01-10 at 5.50.31 AM

I like the idea of students getting large data sets and learning to play with them. I agree with the idea that students need to understand the role of data in the world given how frequently it is used to guide decisions. Having students collect, manage, model, and understand data is key to the scientific method and the learning process. Feeling comfortable drawing conclusions from data is crucial to being considered quantitatively literate today. I really like that ODI is putting in the effort to make this sort of exploration possible, while also acknowledging that there is a lot of work to be done.

Here is an example of the curation they are doing to share best practices:
http://oceansofdata.org/instructional-sequences-are-thought-scaffold-students-exploration-data

All that being said, here's one quote from an executive summary about the skills profile for big data specialists that surprised me:

Unexpectedly, "soft skills" such as analytical thinking, critical thinking, and problem solving dominated the 20+ big data skill and knowledge requirements identified by the panel and endorsed by experts who completed the validation survey.

As a teacher, I find that this isn't unexpected. The skills in the profile (which can be downloaded here) include skills that I'm interested in cultivating in my students. These soft skills are the key to students being successful in any field, not just big data. These are the truly transportable skills that I hope my students have long after they have left my classroom. The executive summary also identifies "defining problems and articulating questions" as one of the key tasks that are essential to the work of data scientists. I also believe this to be a focus of my time with students, and a focus of the work of most K-12 teachers.

The site also links to this article, which suggests that the conclusions drawn in the executive summary are more declarative and alarmist than I interpret them to be:

The skills necessary for the data analytics jobs of tomorrow aren't being taught in K–12 schools today, according to a new report released by the Education Development Center, Inc.'s (EDC) Oceans of Data Institute.

I'm not sure how the Oceans of Data Institute feels about the comparison, but they do link to the article in their page about the project. I'm a big believer in teaching computational thinking skills. I acknowledge that getting more data scientists is an obvious goal for an organization with 'data' in their name. I think that using data is a nice way to tick off the 'real-world relevance' box along the way to the bigger picture skills that students need to develop.

I just don't think we need another bold statement about a skill set that is missing from today's curriculum. I want more tools that get students interacting with data, the creation of which ODI states and has demonstrated is its goal. That's certainly a better way to get educators on board.

Programming and Making Use of Structure in Math

A tweet from James Tanton caught my eye last night:

Frequent readers likely know about my obsession with playing around the borders of computational thinking and mathematical reasoning. This question from James has some richness that I think brings out the strengths of considering both approaches quite nicely. For one of the few times I can remember since starting my teaching career, I went to a computational solution before analyzing it analytically.

A computational approach is pretty simple. In Python:


sum = 0
for i in range(1,11):
for j in range(1,11):
sum += i*j
print(sum)

...and in Javascript:

sum = 0
for(i=1;i<=10;i++) { for(j = 1;j<=10;j++) { sum+=i*j } } console.log(sum)

The basic idea is the same in both languages. We iterate over each number in the first row and column of the multiplication table and add them up. From a first look, one could call this a brute force way to a solution, and therefore not elegant from a mathematical standpoint.

Taking this approach does, however, reveal some of the underlying mathematical structure that is needed to resolve this using other techniques. The sequence below is exactly how I analyzed the problem once I had written the program to solve it:

  • For a single row of the table, we are adding together the elements of that row. Instead of adding the individual elements together one by one, we could instead think about finding the sum of the elements of a single row, and then add together all of the rows. For example: 1 + 2 + 3 + ... + 10 = 55 . This is a simple arithmetic series.
  • Each row is the same as the row before it, aside from each element being multiplied by the first element in the row. Every row's sum therefore is being multiplied by the numbers in the first column of the table. 1(1+2+3...+10)+2(1+2+3...+10)+3(1+2+3+...+10)+...+10(1+2+3+...+10) .
  • Taking this one step further, this is equivalent to the sum of that first row multiplying the sum of the first column: (1 + 2 + 3 + ... + 10)(1 + 2 + 3 + ... + 10) . In other words, the answer to our problem is really the square of the sum of that first row (or column), or 55*55.

I bring up this problem because I think it suggests a useful connection between a practical method of solving a problem, and what we often expect in the world of classroom mathematics. This is clearly a great application of concepts behind a traditional presentation of arithmetic series, and a teacher might give this as part of such a unit to see if students are able to see the structure of the arithmetic series formulas within it.

My question is what a teacher does if he or she presents this problem and the students don't make that connection. Is the next step a whole class discussion about how to proceed? Is it a leading question asking how arithmetic series applies here? This, by the way, zaps the whole point of the activity if the goal was to see if students see that underlying structure based on what they already know. Once this happens, it becomes yet another 'example' presented to the class.

I wonder what happens if a computer/spreadsheet solution is consistently recognized throughout the class as a viable tool to investigate problems like this. A computer solution is really nothing more than an abstraction of the process of adding the numbers together one by one. If a student did actually do this by hand, we'd groan and ask if they thought there was a better way, and the response inevitably is 'yes, but I don't know a better way'. In the way I found myself thinking about this problem last night, I started from the computational method, discovered the structure from those computations, and then found a path toward a more elegant solution using algebraic techniques.

In other words, I made use of the structure of my program to identify an analytical approach. Contrast this with a more traditional approach where we start with an abstract definition of an arithmetic series (by hand), do practice problems (by hand) and once we understand how it works, use computational shortcuts.

The consistent power that I see in approaching and developing ideas with students from a computational standpoint first is not that it often makes it easier to find an answer, though that can be a good thing when the goal is to find an answer. Computational methods can make it easy to change things around and generalize a problem - what Polya termed generalization. It's easy to change the Javascript program to this and ask what multiplication table it models:


sum = 0
for(i=5;i<=10;i++) { for(j = 5;j<=10;j++) { sum+=i*j } } console.log(sum)

Computation makes the process of finding a more elegant way seems much more natural - in the best situations, it builds intellectual need for an easier way. It is arbitrary to say that a student should be able to do a problem without a calculator. Computational tools demand that we find a more compelling reason to solve problems by hand if computers are able to do them rapidly once they are set up to solve them through programming. It is a realistic motivation to show that an easier way speeds up finding a solution to a problem by a factor of 10. It means less waiting for a web page to load or an image to post.

The language of mathematics is difficult enough to throw in the additional complications of computer language syntax. I fully acknowledge that this is a hurdle. I also think, however, that this syntax is more closely related to the concepts that we are trying to teach our students (3*x is three times x) than we sometimes think. The power of computer programming to be a bridge between the hand calculations that our students do and the abstractions of the mathematical content we teach is too great to ignore.

Selling mathematical thinking the Apple way

After reading Gizmodo's post on the recently created blog Applefied Ads, I started thinking about the relationship of good advertising and the public relations problem that learning mathematics has.

Most people think of math class as that "special" time of day when you learn step-by-step procedures on how to do something. I've posted on this before, so I don't need to go into it in detail. The common idea that math class is a time for nothing more than skill development is the reason this problem exists.

One thing that's interesting about Apple is that their advertising is always focused first on what it allows someone to do. Some companies often focus on the speed of the processor, the number of ports available, or installed memory. While these are things that Apple might mention in their ads, it isn't the first thing that is said about a product. Without exception, Apple focuses on how the product will improve things or be different from what is already possible. Despite the media rich world we live in, it does so in a strikingly minimalist fashion.

Textbook companies are faced with the task of engaging with media overloaded students and connect with the oft repeated goal of making math education focused on "the real world". In doing this, they usually stuff their pages with as many pictures as they can be found, contrived examples, and carefully crafted "investigations" that usually are nothing more than a series of guided steps to a single end. Dan Meyer does an amazing job of pointing out how Pearson has already tried to do more of the same in creating electronic textbooks in this post.

Dan has also done incredible work in getting the mathematical problem to jump off the page or screen, but in an understated and minimalist way through the power of multimedia. If done correctly, you don't need a bunch of fluffy text or pictures to explain a math problem to a student. A question and a picture or video, and often just a picture, is all that is required to set a student off investigating and developing problem solving skills. In math, these are the skills that will have lasting power and utility for a student beyond a single school year, not the steps of an algorithm.

I wonder what happens if we make a concerted effort to sell math (or any subject for that matter) in the same way that Apple does. What does it enable us to do? How does it let us look at the world in a new way? How can its elegance and beauty be captured through a picture and a few carefully chosen words? How do we get students to think about it as a philosophy?

My purpose is NOT to dress mathematics and mathematical thinking as a ruse to fool students into being engaged by it. This is what many of the textbooks do already. I'd love to see what draws students in and gets them thinking mathematically without our having to mess it up by talking or explaining it further. Less is more.

How would you sell the classes you teach in a way that engages students without tricking them? How would you show what your course is about on the first day of class? Can you do this with a picture and a few words? Try it and share what you create.

Can ideas and a little money be a bad thing?

I was having a conversation with someone recently about technology in education. I brought up Udacity as an interesting model for using video and interactivity for learning. My example was (expectedly) countered with Khan Academy. I shared my opinions about its strengths and weaknesses and we had a really great discussion about what its existence means. I felt good about being able to share some of the 'other side' of the argument that Time and CBS haven't really covered.

One point that was brought up has stuck with me, and I want to explore it a bit. Here's the basic idea with my own paraphrasing.

Here's a smart guy with an idea. He sees a problem and wants to help, so he puts his own time and resources into solving that problem. Other people noticed what he was doing and thought it was a good idea, so they put money into his project. What could it hurt?

What could it hurt?

My focus has nothing to do with the fact that many people have benefited from Khan's resources and his website. Many teachers have used the site as a tool to help their students in skill development. I also don't want to focus on the fact that many learning professionals have questioned the pedagogy of Khan's videos given the fact he has no teaching background. Many others have already fleshed out this line of reasoning pretty effectively.

The big problem I see comes from applying how business investment works - a business starting up needs investors so it can start getting what it needs to generate revenue. If the business is actually fulfilling a real need in the market, it will increase in value through the income earned and the equipment and intellectual property generated or acquired by the business. Venture capitalists research companies and their ideas to see which ones have potential to be successful in the market and then select those that, based on their experience in the field, are most likely to succeed. Often these capitalists invest in a number of different ideas to maximize the potential that one will be a real money maker - they understand that not every investment will actually be a winner.

Here's why I see the hubbub about Khan Academy as an indication of a bigger problem: We get things backward when we see a major investment as a measure of its value, whether in an idea or a business.

So much has been made out of the fact that Bill Gates and Google have invested in the Khan Academy that people might thing it's a good idea specifically because Gates and Google have done so. Don't get me wrong - Google has invested in many really great causes (FIRST being one of my favorites) but they don't always get things right. As I said before, this is the nature of investment though. Not everything works out. I challenge anyone to defend the content of the video below as really, honestly, being truly deserving of a major investment to help it be implemented in schools:

This is the Explicit Direct Instruction initiative that Google recently supported in the Mountain View school district. The manager of community affairs says in the linked article that EDI "...seemed like a really successful program that we want to continue to support." Google wants to help solve the complex problems of the educational problem, and based on the manager's assessment, it will continue to do so.

Why might this situation hurt rather than help?

The news story was all about Google's generous donation in support of the initiative. A person reading this article might make the mental connection that if Google is supporting the idea, then the idea must be one that will effectively address an educational issue. Google's donation (and subsequent coverage of the donation) have consequently turned it from a company that just wants to help, into a 'player' in the educational reform world. What was this originally based on? A manager that saw kids 'engaged' because they were compliant, which to her meant that learning was going on in the classroom. What does she know about education?

Many people think they are experts in education because they went to school and they know what worked for them. Salman Khan has been reported to be good at explaining concepts through solving problems - he has said himself that procedures are how he and everyone he knew learned. He makes videos that primarily show procedures, though there are some exceptions. Investors see this and contribute millions. The media picks up on this and says that because of these investments, school has been "rebooted" and education has been revolutionized by his contribution.

Money talks. If the money goes toward Khan, EDI, and other flavors of the week, a few things can happen. The media pounces and says that these ideas that have attracted investors must be what will revolutionize education. Not the genuine ways that many teachers have individually used technology to improve instruction in their classrooms. Not the ways teachers are able to have improved communication with parents and students about their progress. Administrators looking for quick solutions to achievement deficiencies in their districts might sink considerable resources into these ideas without consulting with the teachers responsible for implementing them. Parents can demand that teachers spend less time creating rich explorations and applications of topics in their classrooms in order to focus on this 'innovative' idea they learned about on TV or the internet. As I have said before, there is no silver bullet in education, not any one piece of technology, not a single pedagogical technique, nor a single textbook. Solutions to problems in a learning community must be influenced primarily by the parents, teachers, students, and administrators in that community, and not by what the news says is innovative because of a million dollar donation.

It is possible that the involved players can act with a bit more restraint. I know there are many administrators that do. We are so reactionary these days. We want a quick fix. The media's tendency to hype, the power of the internet to exponentially transmit ideas, and the ability money has to set our priorities: these form a dangerous formula that could lead us to rapidly pursue options that really don't resolve the issues we face. Hopefully we do not lose sight of what we really value in education. I don't believe it is too late.

Planning for instruction: Not just for humans!

My wife and I welcomed a new member to our family a couple months ago. Meet Mileaux:


His name is a play on the more standard Milo, with the end spelled in the Cajun way as a tribute to Josie (my wife's) roots. He's now around six months old. We're not exactly sure what he is - the current theory is a mix of a Pekinese and a Pomeranian, but there are hints of a whole bunch of other dogs in his behavior. His hobbies include chewing on towels and begging on command with his paws clenched together like an Italian soccer player trying to get out of a yellow card call. You have to see it to understand how spot on this description is.

Training him has been really interesting. As with every other part of my life since I started teaching, it serves as yet one more source of data on how learning occurs naturally. A disclaimer:

Yes, I know that my students are not dogs. I am saying, for the purposes of understanding the learning process, that outside of the supremely unnatural structure currently called 'school', that some aspects of learning are universal. As another comparison with my students, I can say for sure that Mileaux doesn't like when I lecture him either.

Mileaux shows a lot of behavior that makes sense when thinking about how learning really should happen. He responds more strongly to positive reinforcement than negative, and the negative (when we do resort to it) has the consequence of sometimes leaving him confused rather than corrected. He sometimes gets tired of learning when he's had enough. Sometimes he just needs to take a break in order to get it the next time.

One command we hadn't tried until today was to lay down. We hadn't really figured out the best way to do it. Yes, there are videos with suggestions on how to do it, but it's fun to try to figure out how to communicate what we want him to do. I went for a quick 20-minute run to think of how I wanted to approach it. Here was my process:

  • I knew what he already knew how to do - specifically to sit. That seemed like a good entry point into getting him to lay down.
  • He just had his Lepto shot yesterday and was consequently a bit stiff and sore today. I didn't want to use a leash or pressure to urge him into the down position. I wanted him to be able to figure out what we wanted him to do, and do it on his own.
  • There would, of course, be treats involved in the process when he did exactly what I wanted him to do.

Since he knew how to sit, I could put a treat within his reach laying down on the floor in my fingers. Any time he got up to move toward the treat, I would again give the sitting command. After around five minutes of doing this, he figured out that he needed to stay seated, and chose to stretch out into an awkward leaning position with his head arched down toward the ground. Then came strained reaching and pawing toward the treat on the floor. Soon after, he realized that laying down was a much more comfortable option for getting the treat, and started doing that every time. Copious petting, treats, and praise followed.

The connections to teaching content?

  • There is no paragraph in the textbook introducing the concept of laying down. Mileaux and I didn't read it together and then do a share-out. I just needed to clearly define what I wanted him to learn, and this didn't involve words.
  • While it is true that the skill of 'sitting' is one that he needed to have beforehand for my method to work, if he didn't, I would have chosen another entry point to the activity. He lays down every day. He knows what it is. My goal for him was to make the connection between this skill of laying down with the verbal command. The knowledge he already had was really useful in helping him understand what he needed to do, but the background knowledge was not necessarily a prerequisite for the task we were doing.
  • I posed the problem in a way that had constraints that he figured out on his own. I couldn't tell him not to move his hind legs. That limitation needed to be obvious to him as part of the activity. Managing this limitation as part of getting the delicious snack was what led him to learn the command as quickly as he did.
  • I had him go through this activity from a number of different starting points - standing up in the kitchen, sitting next to the couch, begging in the doorway - because I needed him to see that in these different contexts, the one skill I wanted him to learn was to lay down on command. He figured out that it was the common thread, and not any of the other simpler cues or tricks he could have used as a crutch or shortcut.
  • He didn't do exactly what I wanted him to do, and felt alright about that. He knew it was just fine to get things wrong. The key to his getting it right in the end was clearly communicating when he did what he was supposed to do.

Granted, this may be strained. I accept that this may not be immediately be applicable to everyone's classrooms. I do think it's important to think about what we are asking our students to do, how we are communicating those objectives, and how we are helping them develop a healthy mindset toward learning along the way. We need to be thinking about knowledge in the context of figuring out problems. Solving them is an innate part of living in the world, whether as a snail, a dog, or as a human. The more we can create learning experiences that connect to this need to challenge and interact with our world, the more effective these experiences can be for our students.

Topic for #mathchat: Do we need students to reach automaticity?

I was honored when asked recently to offer a topic for discussion on #mathchat.

My suggested topic:

Is it necessary for students to develop automaticity in their pencil and paper mathematics skills? Why or why not?

First some definitions and examples to clarify the intent of the question.

By automaticity, I also mean procedural fluency. A student that has developed automaticity is familiar enough with the mechanics of a particular task to not have to devote substantial thought to how to do it. It also is connected to retention over time - how well do the details stick with a student as more information is learned over time?

In an Algebra class, for example, do the details of arithmetic need to be automatic so that the student can focus on applying algebra knowledge to solving an equation? In Calculus, should students be able to apply the product and quotient rules efficiently when working on optimization or related rates? Or is it reasonable for them to figure out the derivative using basic principles or use a computer algebra system to take care of this step when it comes up?

I also refer specifically to pencil and paper skills because, for what I would guess is a majority of us that teach math, we tend to assess students by pencil and paper at the end of the day. A student can use a graphing calculator, Geogebra, or other piece of technology to explore a concept and check her/his work. The thing I often wonder about is how the use of activities and technologies help students perform mathematical tasks when these technologies are not available.

Is it necessary to do these tasks when these tools are not available? I don't know. I think that's open to interpretation and individual opinion. There are some cases, however, when that choice is not up to us. Standardized tests are one example. Given that they do exist (and independent of whether or not we agree with their content/quality/use), standardized tests are not typically electronic and are timed. These are often posed as opportunities for students to choose an appropriate method of finding answers to questions and then find those answers with a limited set of resources available.

Let me be clear - I am wildly inconsistent on this, because I don't have a good answer to the question. I emphasize understanding through the activities I do in my classes - very rarely will I directly tell students how to solve a problem, have them practice the skills with me, and then send them home to practice those skills in isolation from others. I really appreciate Conrad Wolfram's point about using computers to handle the calculating, and leave the thinking to us and our students. I have decided on occasion not to assign #1-30 for students to practice differentiation because my feeling at that time is that if they can apply it correctly several times, they get the point, and are ready to apply that knowledge to more interesting contexts.

But when these same students that complete the short assignment, later struggle in finding anti-derivatives, I wonder if I should have drilled them more. My decision not to burden them with repetitive exercises because they are repetitive often has implications for the future of the students in class. Do I need to drill this to automaticity so that next year's teacher doesn't come complaining to me about how "your old students can't XXXXXXXXXX" where XXXXXXXXXX = [arbitrary math skill that either (a) will mean the difference between getting into a top choice school during Senior year or (b)won't matter at all ten years after leaving the classroom]?

So I call upon the collective brilliance of the #mathchat community to help find an answer.

For those unaware, #mathchat is a Twitter based chat held every Thursday night at 8PM in which all respondents use the hashtag #mathchat in their post so that everyone else following that hashtag is updated with the latest responses. If you aren't up on using Twitter for professional development, you need to be. It completely changed my perception of how Twitter is useful and has put me in contact with some pretty amazing folks from around the world.