How to make a course adaptive (without really trying)

Student retention (preventing students from quitting or dropping out) is a major concern for online learning. So why do students drop out in the first place? One reason is a lack of engagement. Indeed, some of our friends over at edX previously found that students tend to stop watching lecture videos that go beyond 4 or 5 minutes.

Our team has already done an incredible job of making online lessons more engaging. In a previous post, we showed how we increased student attention span up to 20 minutes with our highly interactive lessons. So the engagement problem has largely been solved.

But while we've reduced dropout rates, a fair number of students still drop out of our interactive lessons. So we asked ourselves: what's going on here, and what we can do about it?

Now imagine you're working your way through a step-by-step, interactive lesson. You know what you're doing, getting every question right, but suddenly you hit a snag. You just can't seem to get this next question. Now you could ask for a hint, or you could try a few different approaches, but sometimes you'll just quit out of frustration.

With an in-person tutor, this wouldn't happen. A live tutor can diagnose exactly where you confusion is and help you get "unstuck." So our hypothesis is this: students drop out when they're told their answer is wrong without immediate, useful feedback.

If this is true, then we should see a direct correlation between how often a student is given specific feedback (either they got the correct answer or we address their wrong answer with a separate branch in the lesson) and how often the student will stick with the lesson and not drop out. So we combed the data from millions of students' answers to the thousands of questions from our hundreds of lessons (so yes, that's a lot of data), and here's what we found:

As we expected, there is a very strong correlation between how often students are given specific feedback regarding their answer and how often they'll stick stick around for the next part of the lesson. (For you stat-heads out there, the Pearson correlation for this graph was r = 0.572.)

This suggests an obvious way to improve retention: give students more (and better) feedback! Now that's always been our goal here at School Yourself. We regularly update our lessons so that common wrong answers are met with specific, helpful feedback. For example, in the latest version of our lesson on point-slope form, when you enter a slope with an extra minus sign or you accidentally flip the fraction (common mistakes we saw students making), you now proceed down a new path in the lesson that specifically addresses your error. Here's how the lesson map has evolved (note the change from a linear sequence to a more branched structure):

Now if we were to keep doing this, adding new branches to lessons where we accept all the wrong answers students give and provide specific feedback, that could take us forever (and we don't have that kind of time!) So the strategy we've adopted is to address the common wrong answers, prioritizing where students frequently make similar mistakes.

So how much work would it be to provide feedback to 90% of students for each question? According to the first graph, that means we could expect at least 97% retention for each question, which would be outstanding. And the answer turns out to depend on the question type. Here are the results for checkbox questions, and numerical response (in which a student can enter any fraction or decimal):

It turns out that for most questions, students are only giving a handful of answers. For both of these question types, on average, more than 90% of students give either the correct answer or the two most common wrong answers. So if our team should addresses the top two wrong answers, we should achieve 97% retention for each question. It takes our team about a day to make a lesson, and it would only take another day to update the lesson so it offers specific feedback to more than 90% of students.

So we'll continue to update our lessons, making them increasingly adaptive over time. And we won't have to pull any all-nighters to get it done!

How often do students guess their way through?

When designing online courses, like AlgebraX and GeometryX, a common cause for concern is that students can just "guess their way through" lessons and assessments. They're either totally lost, or not paying much attention, but they can still advance and get credit by brute-forcing their way.

This can be very problematic for the student (who is grinding away without really learning) and the course (which loses credibility when this type of student behavior is common). So here at School Yourself, we've been asking ourselves: 1) How often does this happen in our courses, and 2) What can we do about it?

To figure out how often students guess their way through lessons, our team developed a formula we call a student's guess index. It's a number between 0 and 1, where a greater number means the student is probably guessing. A student's guess index takes into account the question type (guessing on a multiple choice question vs. an algebraic question are handled differently), number of attempts, what students do between attempts, and a few other factors. Over time (and with lots of student data) we have found that a guess index of greater than or equal to 0.4 means a student probably guessed their way through a lesson. An index of 0.2 or 0.3 means they might have struggled through parts of the lesson, but they worked throughout without resorting to guessing. And an index of 0 or 0.1 means they confidently proceeded through the lesson.

So here's the distribution for the most recent 10,000 completed lessons:

Students appear to be guess their way through lessons 2% of the time. And we can do the same analysis for reviews (assessments):

Students seem to guess their way through reviews less than 1% of the time. It's a little comforting to know that students guess less on reviews than they do in lessons, since the reviews are what hold them accountable for their knowledge.

One or two guessing students out of a hundred may not sound too shabby. Indeed, these numbers are likely so low because 1) we prefer asking open-ended questions (with interactive graphs, algebraic expressions, numerical, or short answer) where possible, to ensure students understand what they're doing, and 2) our lessons are built around continuous, step-by-step user interaction, where the questions never get too far ahead of the students. But even though relatively few students are guessing, these are also the very students who need the most help! So now that we can precisely measure guessing behavior, what can we do to mitigate or prevent it?

First, it's helpful to know whether students are guessing at similar rates across all lessons and reviews (indicating a problem with the overall course design, question types, etc.), or if the guessing behavior is concentrated among certain lessons.

Half of the lessons exhibited zero guessing behavior, And guessing is concentrated among a handful of lessons, like Ambiguous case, Multiples, and Unsolvable equations. This is great news, because we can focus our attention on these lessons, see where students are guessing the most, and update the lessons (and add new branches providing feedback for common wrong answers).

When it comes to reviews, students don't guess as much:

The majority (69%) of reviews had zero guessing. And once again, it's a handful of reviews that are the most conducive to guessing. So that's where we'll focus our efforts.

So while students guessing their way through lessons doesn't seem to be a major problem for our content, it's something we can mitigate by focusing our effects on the most guess-prone lessons and reviews.

Students prefer School Yourself over Khan Academy and other MOOCs, suggest improvements

A few weeks ago we launched surveys in AlgebraX and GeometryX, asking students what they thought of the courses, how they compared to others, and what improvements they might suggest. So far, more than 400 students have responded.

A sign that we're on the right track was a series of questions on how our courses compare with other massive open online courses (MOOCs), and with Khan Academy, a popular collection of online videos and assessments.

First up, we asked "How do AlgebraX and GeometryX compare with other massive open online courses (MOOCs) you have taken?" Here are the combined responses for the two courses:

A total of 135 students said they had never taken other MOOCs. Among those who had, 106 (39%) said they had little preference between School Yourself and other MOOCs. Among those with a preference, 97% preferred School Yourself to other MOOCs. Holy moly! Now might be a good time for other MOOCs to sit up and take notice: students prefer engaging, interactive, and adaptive content. We'll continue to push the envelope, and hopefully online learning will continue to evolve and improve in the coming years.

Here at School Yourself, we're big fans of Sal Khan and all that his team is doing for millions of learners around the world. But Khan Academy continues to rely on 10-minute videos of Sal speaking and writing, without letting students interact and make sure they're following along. Khan Academy is a great resource, but we think it can be even better if it adopts our methods of content production and student engagement.

So next we asked students "How do AlgebraX and GeometryX compare with Khan Academy?" Again, here are the combined answers for the two courses:

A total of 213 students said they had never tried Khan Academy. But students who had appeared to think more highly of Khan Academy than MOOCs. Approximately 33% of students had little preference between School Yourself and Khan Academy. But among students with a preference, 80% preferred School Yourself to Khan Academy. As with MOOCs, we hope Khan Academy evolves into a more interactive, adaptive experience. With Khan's expansive library and School Yourself's adaptive, engaging style, who knows what's possible?

Here's are a few more stats from the survey:

• 90% of students would recommend AlgebraX and GeometryX to others.
• Currently, course grades are based entirely on adaptive quizzes after each topic, with ~100 quizzes in each course. 52% of students said having exams (or a final) would help them learn, 14% said they didn't want exams, and the remaining 34% were neutral. In response to this, our team is looking into adding optional adaptive exams to both AlgebraX and GeometryX. Stay tuned!
• We asked how the courses could be improved, and here were the most common responses:
• Students want more lessons on real-world applications of the material. We have a few of these, like using parallel lines and alternate interior angles to estimate the size of the Earth, and how graphs of 2D inequalities are used in machine learning. But including more examples of applications is something we're always thinking about, and we hope to add many more in the near future.
• Students want more review questions. We're always adding more questions, so we'll see how students feel again in a few months. Some students also asked for more challenge questions, in particular, feeling the courses were too easy overall (77% thought the difficulty was about right, 20% thought they were too easy, and only 3% thought they were too hard). At the moment, the challenge questions are entirely optional for students. If and when edX decides to include grades on certificates, we may look into awarding additional credit to students who master the challenge questions.

Finally, we asked students what other School Yourself courses they'd like to see on edX. Here's what people said:

It looks like our team should get cracking on Algebra II, which won the vote at 82%. Of course, we hope to eventually create courses covering all of these subjects! They all build on one another -- for example, you use trigonometry in physics, and calculus in statistics. The more subjects we add to the School Yourself library, the more paths students can take through the lessons, and the more adaptive and powerful the experience becomes.

Will this be on the test?

If you've ever taught a math class, then you've probably encountered the following scenario: you're showing the class a cool proof (like why an inscribed angle always has half the measure of its arc), and you get asked: "Will this be on the test?" You're demonstrating how math is a consistent framework and that even the hardest facts are derived from simpler theorems. But your students just aren't interested. And from their perspective, this makes a lot of sense. Students have many demands on their time (some related to education, some not), and they're really just trying to optimize their schedules.

And so with our lessons, we tried something a little different. We know students aren't interested in working through proofs all the time. But sometimes they're genuinely curious, or they already passed the quiz and are now returning to see why the rule they memorized actually works. So we try to leave it up to the student to choose: if you want to work through a proof, click here; to just do practice problems, click here instead.

Here's an example from our lesson on inscribed angles:

After a student sees the rule in action and plays with an interactive, he or she can decide whether to work through a proof or skip ahead. If they opt for the proof, they'll work with general angles and prove it for themselves.

Otherwise, practice problems it is!

While it's nice that we give students the choice, we're really hoping they'll elect to work through the proof. They'll get more experience with geometric proofs, get more practice with algebra, and convince themselves that the rule truly works. So how many students decide to work through the proof?

Using our analytics dashboard (which comes with our authoring tool, QuickBranch, for you content authors out there), we found that of the 1701 students who made it to this point in the lesson, 987 (or 58% of them) decided to work through the proof. And of those, 779 (~78%) made it all the way through the proof. We found similar rates in other lessons as well, like the Pythagorean theorem. More often than not, in our self-paced learning environment, students will choose to work through the proof.

When they're asking "Will this be on the test?," it's not that students don't care or don't want to learn. It's that their time is limited, and they're trying to optimize. If you free them from these constraints, and let them come back to the proof when they feel motivated or incentivized to really learn it, they'll choose to do so.

So when it comes to material that won't be on the test, trust your students and give them the choice. They may surprise you.

Can you learn Algebra in 24 hours?

How long does it take a typical student to learn algebra? This is a tough question, for a variety of reasons. Who is a "typical" student? How do you know if they've really learned the material? And what's the best way to measure the total time a student spends learning?

To get a rough estimate, consider a typical high school or college course: a student will spend about 6 hours per week in a yearlong course (or about 12 hours per week over a semester), which add ups to about 200 hours, a figure that includes both time in the classroom and homework.

With almost 40,000 students now enrolled in AlgebraX and GeometryX, our two courses on the edX platform, we were curious to see how much time students were spending on each course, and how this compares to the 200-hour figure. Now because the lessons and assessments of our courses are adaptive, different students spend different amounts of time. So to study a "typical" student, we added up all the median times that students spend on each lesson and assessment. To estimate an upper bound of how long students might spend, we also added up the 90th percentile times of the lessons and assessments, representing the slowest 10% of students.

We were definitely surprised by the results. As you can see in the graph below, the median total time to complete AlgebraX is less than 24 hours! And the same goes for GeometryX! The slowest 10th percentile of students take about twice as long, finishing AlgebraX in about 48 hours, and GeometryX in about 36 hours.

Total time spent on lessons and assessments in AlgebraX and GeometryX. For each course, the lower curve shows median time, while the upper curve shows the slowest 10th percentile of students.

Next, we compared the median times for students with different backgrounds: 1) current high school students, 2) adult learners with no college degree, 3) learners with an Associates or Bachelors degree, 4) learners with a graduate degree. As you can see in the graph below, the surprise this time was that all four groups typically finished the coursework in the same amount of time.

Median time spent on AlgebraX and GeometryX by students with different backgrounds: 1) current high school students (red), 2) adult learners with no college degree (green), 3) learners with an Associates or Bachelors degree (blue), and 4) learners with a graduate degree (yellow).

So what does this all mean? First off, this does not mean that our students have been learning algebra or geometry from start to finish in a single day. We're talking about the total time spent within the courses, which most students complete over multiple weeks. Also, while we stratified students by educational background, edX students are probably not your "typical" students. And while students demonstrate mastery of the subject material by completing formative and summative assessments, this does not represent a rigorously designed study of precisely what a student has learned.

But what these results do suggest is that with adaptive lessons and assessments, it appears possible for the majority of online students to work through and demonstrate competence for a year's worth of course content in a fraction of the time. It seems quite plausible that a "typical" student can learn algebra or geometry in less than 24 hours.

School Yourself featured on Innovation Showcase

Last Friday, School Yourself CEO Zach Wissner-Gross sat down with Jay Sugarman, the host of Innovation Showcase on NewTV (based in Newton, MA). During the half-hour program, they discussed the origins of School Yourself, how the team personalizes learning at scale, and what's on the horizon.

In other news, we're proud to announce that the students in AlgebraX and GeometryX, our two MOOCs on edX, have now collectively solved more than 5 MILLION review questions. We're very proud of this milestone. And more students are signing up and working through these courses every day!

Finding exactly where algebra gets hard

Now that thousands of students have gone through the 88 interactive lessons of AlgebraX (not to mention the 92 lessons of GeometryX), we're starting to see trends emerge. We previously found how engaging our interactive lessons are (compared to plain video lessons), and in this post we'll dive into the content itself.

One thing we hear over and over again is how hard algebra is. It's true -- learning algebra can be quite challenging. But it's also important: every subsequent math subject (geometry, trigonometry, calculus, statistics, and so on) builds on the foundations of prior ones, starting at algebra.

With AlgebraX, we're trying to make it easier (and more fun!) to pick up algebra. So are we accomplishing this? Where should we be improving our lessons? (That's something we ask ourselves all our time, because our authoring platform is uniquely suited toward iteration and improvement.)

One way to measure the difficulty of different topics is to look at how many students completed review questions (the assessments in AlgebraX) vs. how many students attempted those questions. If a topic is hard to learn (and/or our lesson on that topic could use improvement), we'd expect to see fewer students complete the corresponding review. Now this differs from how advanced a topic might be. For example, factoring quadratic polynomials is pretty advanced stuff for Algebra I, but students zipped right through those review questions in the course.

Here's a bar graph showing the completion of all the reviews in AlgebraX. Each topic has its own bar, and the vertical axis shows the percentage of students who started a review who finished it (meaning they mastered the topic).

The first thing to notice is that it's mostly green. In fact, the average completion rate for AlgebraX reviews is 96.1%. Once a students starts a review, it's highly likely that he/she will finish it. So what about those red and yellow areas? Well, here's a list of the hardest topics in algebra -- a "hit-list," if you will, where we'll focus on improving our lessons:

1. Two equations, two unknowns (78.3% completion)
2. Fractional exponents (82.0%)
3. Two equations, with no solution (84.1%)
4. Solving for intercepts (85.0%)
5. Point-slope form (86.7%)
6. Discriminants and roots (87.5%)
7. The quadratic formula (87.9%)
8. Solving multi-step equations (88.1%)
9. Multi-variable equations (90.5%)
10. Distributing roots (91.0%)
11. Perpendicular slopes (91.3%)
12. Calculating averages (91.4%)

And what's the "easiest" topic? You might think it's something early on in the course, but then you'd be wrong. It's simplifying expressions by combining like terms, a topic that 99.8% of students have mastered.

We're already underway improving the more challenging topics in the course, both by improving the lessons themselves, as well as by adding new hints to review questions. Our goal is to turn this entire chart green. Yes, algebra will still have advanced topics, but nothing will be too hard for students to conquer!

MOOCs are getting personal, but also more engaging!

All the lessons of AlgebraX (one of our math MOOCs on edX) have now been posted. And the feedback from students has been incredible! We still have two more weeks of lessons to post for GeometryX, on surface area and transformations, and then that MOOC will be complete as well.

As edX CEO Anant Agarwal has said, our courses represent the first two adaptive MOOCs on the edX platform, where students can "choose their own adventure" through each lesson, enjoying an experience that is tailored to their individual knowledge and abilities. And we're using all the data we're collecting to improve these lessons and add more paths for different learners to follow.

Despite our MOOCs being only a few months old, we can already measure how they're being received. One important metric is engagement, or how long students' attention is focused. One-on-one learning certainly is more engaging than a lecture, but does this trend translate over to MOOCs? How do interactive, personalized lessons compare to passive video content when it comes to online learning?

Researchers have studied how long students will watch videos on edX before "dropping out," which in this case means closing the video.

Measuring engagement for passive videos on edX

This data was collected from four of edX's most popular MOOCs. One way to interpret this graph is to look at where the trend line crosses 50% (at about 4 minutes). So after about 4 minutes of a video, half the students have dropped out. With passive video content, MOOC students have an attention span of 4 minutes.

So how do interactive, personalized lessons compare? Well, here's the corresponding graph containing all the lessons from our AlgebraX and GeometryX MOOCs:

Measuring engagement for School Yourself's AlgebraX and GeometryX MOOCs on edX

You're reading that correctly -- the trend line doesn't cross 50% until about 22 minutes. So with our interactive personalized lessons, attention span is 22 minutes, a 450% increase over passive video.

Now this isn't a perfect comparison. The user interface for our lessons is a little different from the traditional YouTube scrub bar that appears in most edX videos. Also, because of the adaptive nature of our lessons, different students will spend different amounts of time on any given lesson. So to generate the above graph, we used the average lesson duration among students who completed the lesson. But despite these difficulties in comparing interactive versus passive video content, the difference is striking.

It's our hope that online learning (and MOOCs in particular) continue to become more personalized and interactive over the coming years. It seems that with this evolution, increased engagement will be an added bonus!

School Yourself Named a Verizon Powerful Answers Award Winner for 2014

Verizon’s Multi-Million Dollar Challenge Generated Thousands of Innovative Ideas from across the Globe

We're proud to announce that School Yourself has been named a 2014 Powerful Answers Award winner in the education category. Verizon’s second consecutive multi-million dollar global challenge sought powerful ideas that may leverage Verizon’s cutting-edge technology and will help deliver groundbreaking solutions and social good in four core areas: education, healthcare, sustainability, and transportation.

School Yourself is one of three winners in the education category and one of 12 winners total (three in each category). The ideas that win the top prizes of $1 million in each category will be revealed on Jan. 27, 2015 in New York City. The remaining eight winners will receive $250,000 each, for a total of $6 million awarded by Verizon.

John Doherty, senior vice president, Corporate Development for Verizon, said, "We’re blown away by this year’s winners and look forward to revealing who will take home the top prize of $1 million. All 12 winners have shared impressive ideas and solutions and we look forward to seeing them succeed and working with them to leverage our technology and help make the world a better place."

In addition to the prize money, School Yourself has the opportunity to connect with a variety of network, business development and marketing experts, and partners across Verizon to explore ongoing collaborations.

Watch School Yourself's participation in the Powerful Answers Awards Event live on January 27 via www.verizonwireless.com/news. And don’t forget to root for us via #VZPAA.

School Yourself coming to edX

We're excited to announce that our Algebra and Geometry lessons will be appearing on edX as part of their High School Initiative. You can sign up for the courses here, and you can check out the announcement by edX CEO Anant Agarwal on the edX blog.

Both courses will be available in early 2015. And of course, all the lessons will continue to be available on the School Yourself site as we produce them.

A quick shout-out to our very own Michael Fountaine, who designed the beautiful banners for our two courses. First up is the banner for AlgebraX:

If you look carefully, you'll notice the repeating pattern in the background actually consists of the graphs of y = x^0, y = x^1, y = x^2, and y = x^3. Nifty, right?

And here's the corresponding banner for GeometryX:

The background pattern for this one are 0-, 1-, 2-, and 3-dimensional hypercubes.

Happy Pi Day!

It's March 14, which means it's Pi Day!

To celebrate, we've compiled a list of our favorite interactive lessons involving pi:

Circumference and pi

Find the distance around a circle (and then eat some pi). And maybe get a head start on memorizing pi's digits?

Circle area

Using circumference to find a circle's area. If you've never worked through this proof before, you'll definitely want to check it out!

Radians

Discover another way to measure angles. There's more to angles than just degrees...

Sector area

If you're eating a slice of pie, discover a sure-fire way to find its area!

Sine, cosine, and tangent functions

We've completely reworked our introduction to trigonometry, and those infamous trig functions. Play with interactive right triangles as you learn about these functions, and discover what they can do!





In other news, our probability-based game "Beat the Odds" has been named a winner of WGBH's Education Innovation Challenge! (And they even made a video about us.)

And on a final note, we're hard at work producing lessons on trigonometry and volumes of solids, and many of those formulas also have a pi floating around! Here's a teaser of what's to come:

Practice makes perfect

Today, we're excited to announce two of our most-anticipated features, review questions and automated suggestions for our online precalculus and calculus content. Give it a try, or sign up for an account to get started!

At any time, you can ask our system for a short quiz to test your newly-gained knowledge. We will choose questions from our large (and still growing) database of review questions that are best-suited for you based on how you've interacted with other lessons and questions. We'll make sure to choose questions on topics that you need to review the most, and at an appropriate level of difficulty.

As you progress through the course, you'll be able to see which topics you excel at and which ones you struggle on. Our system notices these things too, and can automatically highlight lessons you ought to review if you are having trouble.

With automated suggestions and review questions, there is now no need to have to browse through endless lists of videos or tables of contents just to find what you should look at next. School Yourself can automatically find out what your weak points are and help you master them!

How much could you expect to win from the Mega Millions lottery?

In case you haven't heard, the Mega Millions lottery reached an astonishing $648 million this week. There were two winning tickets, but only one winner has come forward so far.

Here we'll look at the question of how much money you could expect to win from this lottery. In other words, for every dollar you spend playing the Mega Millions lottery, how much money could you expect to get back?

For reference, here's how some other investments stack up:

• If you invested $1.00 in the stock market a year ago, you would have $1.23 today. So for every dollar you invested, you would have made an additional 23 cents.
• If you play roulette at a casino, for every $1.00 you bet, you make, on average, $0.95 back. So for every dollar you bet, you would lose 5 cents.

So it looks like investing in the stock market is a pretty good way to spend a dollar, while playing roulette will lose you some money. How does playing the lottery stack up?

First, we need to know how likely it is for you to win different prizes. In each Mega Millions game you play, you select 5 numbers between 1 and 75, and then one addition number between 1 and 15. Here's what five games look like:

We could calculate the probability of getting all 6 numbers right (using a branch of math called combinatorics), but we'll skip that step for now. The Mega Millions lottery is very up-front with the probabilities of winning. Here's the chart they have on their website, where the left-most column is how many of the five numbers you match (form 1 to 75), and the next column is whether you match that sixth number:

They advertise at the bottom that your chances of "winning any prize" are 1 in 14.7, or about 6.8%, which sounds pretty good. Unfortunately, just about all of that 6.8% is taken up by prizes of $5 or less.

Anyway, let's figure this out: for every $1 you spend on the Mega Millions, how much could you expect to win? In other words, we want to find the expected value of your cash winnings.

Here's a simpler example: suppose you roll a fair die (numbered 1-6), and you win a number of dollars equal to the number that comes up on a roll. So you have a 1 in 6 chance of getting $1, $2, $3, $4, $5, or $6. What's the average amount (or expected value) of money you'd make from this game? To find out, you can add up the probabilities of each event by the outcome of that event. In other words, you can expect to make:

which equals $3.50.

The expected value of the Mega Millions lottery without the jackpot is only about $0.18. But if we include a jackpot of $648 million, the calculation becomes a little more challenging, because that prize can be split if there's more than one winning ticket.

As far as we know, the Mega Millions lottery doesn't publicly announce how many tickets are sold, but it's probably in the many hundreds of millions for big jackpots like this one. As more tickets get sold, it's more likely that there are multiple winners. Let's say enough tickets were sold so that we can expect (there's that word again) about 2 winners.

With these numbers, you can expect to make $1.42 off of every dollar you put in the lottery. Those are better results than the stock market! Any time the lottery exceeds about $250,000,000, your expected winnings are greater than $1.00, so it seems like a good idea to play.

But not so fast. While you're definitely spending a full dollar to play, your winnings are taxable. The government may not tax you much when you win $50 for matching a few numbers, but you can bet they'll tax your jackpot prize (or the $1 million prize for matching 6 numbers). Assuming a 40% tax rate, for every $1.00 you spend, you'll now make only about $0.90, meaning you'll lose 10 cents. And you actually lose at lot more (closer to 40 cents) if you decide to take your winnings all at once. The 10-cent loss is only if you let the lottery make smaller payouts to you over the course of 20 years (and they don't account for inflation).

So to summarize, for every $1.00 you spent on Mega Millions, you could expect to lose about 40 cents. You're better off playing roulette.

What is Bitcoin, anyway?

From Bitcoin's own website:

Bitcoin has the characteristics of money (durability, portability, fungibility, scarcity, divisibility, and recognizability) based on the properties of mathematics rather than relying on physical properties (like gold and silver) or trust in central authorities (like fiat currencies). In short, Bitcoin is backed by mathematics.

So what is Bitcoin, anyway?

Bitcoin is a relatively new currency (think dollars, euros, or yen), and was introduced by anonymous creators in 2009. First, how much is one bitcoin worth? As with all things: however much someone is willing to pay for it. Here's a graph of how much people have been willing to pay for one bitcoin over the last few months.

Bitcoin price has been fluctuating recently, and hit a maximum of over $1200 on December 4.

Unlike other currencies, Bitcoin is digital. You can't hold a Bitcoin in your hand. So how do you know how many Bitcoins (or "BTCs") you have? Well, it's a matter of public record! Every time someone pays someone else in Bitcoin, the transaction gets logged in a digital ledger (essentially a list of every single transaction) known as the "block chain."

So the list of who has how many BTCs is publicly available. But many Bitcoin addresses (long chains of numbers and letters) are anonymous -- everyone knows how many BTCs they have, but almost no one knows who owns the address.

In addition to an address, each Bitcoin user also has a private key, which he/she needs when making Bitcoin payments. These keys are additional long chains of numbers and letters, but they are not publicly available.

When Person A makes a payment to Person B in Bitcoin, the transaction gets added to the block chain. But how does Person B (and everyone else) know this is a real payment, as opposed to some fraud who's only pretending to be Person A (perhaps using their Bitcoin address)? Just like a bank verifies the authenticity of checks, Bitcoin transactions get verified as well, but mathematically.

Many individuals, known as Bitcoin "miners," verify transactions. Miners do this using cryptographic hash functions, which are functions for which:

• Given an input, it's easy to calculate an output
• But given an output, it's really hard to find any input.

The miners get paid for verifying transactions with -- you guessed it -- more Bitcoin! By verifying transactions, they're "mining" for Bitcoin, much like a 49er would mine for gold. But the rate at which miners are rewarded exponentially decays over time, so that the total number of BTCs in circulation is tightly controlled.

So Bitcoin involves mathematics and cryptography rather than centralized institutions (like mints and banks) to create currency and prevent fraudulent transactions. Over the next few years, we'll see if digital currencies like Bitcoin catch on.

What shape is the moon in the sky?

Have you ever looked up at the night sky and seen a full moon? A full moon looks pretty much like a perfect circle:

So if a full moon looks like a circle, how would describe the shape of a crescent moon? Sure, it's a "crescent," but let's try to be more specific. Throughout each month, the shape of the moon changes, but the shape actually follows a specific mathematical pattern.

Here are sketches of three different "crescents," but only one of these is an accurate sketch of a crescent moon as it might appear in the sky. Which one of these do you think the moon could look like? Leave your vote in the comments below, and we'll reveal the correct answer in a later post!

Here's a little more background on the three sketches: each one starts with a circle, and then part of the circle is removed. In each sketch, the shape of the removed piece is another conic section. Conic sections are a group of mathematical functions including circles, ellipses, parabolas, and hyperbolas.

In sketch A, an ellipse was removed, producing two crescents (as shown in the image below). Perhaps the moon could look like one of these crescents...

In sketch B, another circle of equal radius was removed (as shown below). Perhaps the moon looks like this crescent...

In sketch C, a parabola was drawn inside the circle, and everything to the right of the parabola was removed. Does the moon look like this crescent?

Again, vote below!

How much money will "Hunger Games: Catching Fire" make?

The new "Hunger Games" movie is opening this Friday. Get ready for more Jennifer Lawrence killing everything in sight:

Credit: Lionsgate

The first "Hunger Games" film made $408 million at the US box office. Almost half of that total was made in just the opening week! Let's take a look at the graph of how much money the first film made on a weekly basis, to see if we can predict how the second film will do.

This looks a lot like a decaying exponential function, which can be written in the form Ce^-t/τ, where C is the amplitude of the function, and τ is the time constant. Another way to think about this is that each week, the film makes a fraction of the money it made the week before. That means we can treat the weekly data as a geometric series!

The formula for the sum of a geometric series is A/(1-r), where A is the first term in the series (i.e., how much money the film makes in the first week), and r is the ratio from one week to the next. For example, if a film makes $100 million in its first week, and $60 million in its second week, then r = 0.6, which is pretty high for a blockbuster release. Larger values of r mean that a film has a longer "lifetime" in the theaters -- people are still buying tickets later into the film's run. After "Catching Fire" is out for a week, we'll know exactly what A is. But how can we determine what r is?

To estimate r for "Catching Fire," let's look at how some similar movies did in the past. Let's specifically take the total amount of money each film made in its first week and divide that by how much it made in total, and let's call this ratio w (for opening week). So then:

So using a film's opening week and total intake (or "gross"), we can estimate r. Let's see how a few different film franchises compared:

For each of these graphs, there seems to be a downward trend, meaning that later films in each franchise have shorter lifetimes than earlier films. But each of these franchises behaves a little differently. For example, The Harry Potter franchise steadily declined over its 8 films, while the Twilight films plummeted after the first one. Maybe after the first film, only the "Twihards" saw the remaining movies, often in the first few weeks of release (resulting in a low value of r).

With more study and more data from other films, we could generate a probability distribution for the value of r for "Catching Fire." But for now, we can guess that it'll probably be less than r for the first "Hunger Games" film, which was ~0.53. Given the positive buzz the new movie is getting, let's say r is a relatively healthy 0.5. So to figure out the total amount of money this second film makes, take the gross from its opening week, and divide it by (1-0.5). In other words, double it!

Here's our best guess for "Catching Fire":

Opening weekend (first 3 days) gross: $160 million
Opening week (first 7 days) gross: $220 million
Total gross: $440 million (double the opening week)

How to fix educational testing

Educational testing is currently a hot debate in this country. Here's a sampling of the discussion:

• In Texas, some schools are attempting to phase out the use of standardized tests for measuring accountability
• In New Mexico, hundreds of people protested the use of testing in evaluating teachers.
• In New York, 80% of parents at one elementary school opted out of a standardized test, forcing the school to cancel the exam.

Parents, educators, and students have all pointed out several key problems caused by standardized tests:

1. Testing uses time that could be better spent elsewhere.
2. Teachers are often uncomfortable with the idea of having months or years of their work evaluated by a brief exam.
3. Because of the pressure imposed by these tests, teachers feel constrained to teach to the tests.
4. Most students don't like taking the tests.

Instead of administering aggravating tests, at School Yourself we're combining learning and evaluation into a single, seamless experience.

What's the purpose of educational testing, anyway? It's a scalability issue. States can't afford to evaluate every second of a teacher's work in the classroom, but they can afford to give all the students a test once a year to see what they've learned.

Scalability is the same reason we have any tests in the classroom at all. Teachers don't have the time to sit down with every student and objectively explore everything the student does or does not know. So teachers give tests, which may not do as good a job. But tests are "good enough," and take up a lot less time than interviews.

With online learning platforms like School Yourself, we have the opportunity to do away with testing once and for all. With highly modular content and detailed analytics, we can assess what students know in real time. This eliminates the need for separate testing, and evaluation becomes part of the learning experience.

On the School Yourself platform, students are interacting every 30 seconds or so (whether it be answering a question, playing with an open-ended sandbox, etc.). Between our dynamic lessons and the practice problems we'll be adding in a few weeks, we can take high-resolution snapshots of student knowledge that are much better than what any standardized test can do.

As we fully integrate evaluation into the learning process, a challenge that digital learning platforms like ours are uniquely suited to, standardized tests will become a thing of the past.

Seeing who "hit the wall" in the New York City Marathon

A special congratulations to our very own John Lee, who ran the New York City Marathon this past weekend (his first full marathon)!

Photo credit: Jake Park

John trained for months, and finished the race in a time of 3:21:11 (3 hours, 21 minutes, and 11 seconds). Given that every marathon is about 26.2 miles long, we can calculate that his average pace was 7:41 (7 minutes and 41 seconds) per mile.

Smart runners will pace themselves, and run at about the same speed the entire race. But runners who don't prepare as well will often "hit the wall" around mile 20, and will slow down toward the end of the race.

Let's look at graphs of how a few runners did. We'll specifically look at John ("The Real John Lee"), another person who happened to be named John Lee and who also ran the marathon ("Fake John Lee"), and the fastest woman to run the race, Kenya's Priscah Jeptoo. Here's how they did:

The dots in the graphs represent checkpoints along the race where the runners' times were precisely measured. We don't know exactly how fast the runners traveled between the dots, but let's assume they ran at a steady pace between consecutive dots.

Of the three runners, Priscah Jeptoo (in red) finished in the least amount of time, so she was the fastest (with a time of 2:25:07). You can also see that Priscah ran a smart race, without hitting the wall, because her graph is very close to a straight line. She ran at about the same speed the entire race.

The Real John Lee (in blue) also ran a smart race, keeping a steady pace the entire time. Fake John Lee (in green), however, started off on pace with Priscah Jeptoo, but slowed down more and more as the race went on. He went out of the gates too fast, and as a result, his graph is concave down, meaning it curves downwards.

We can see this even more clearly if look at the speeds of the three runners over the course of the race. Speed is equal to distance over time, so a runner's speed between two checkpoints is the slope of the line between those points. Here's the graph of the three runners' speeds:

At certain points along the race (like around mile 13), all three runners slow down. These are probably the uphill parts of the course! And it looks like there's a nice downhill stretch around mile 22. But you can also see that Priscah Jeptoo ran a smart race, with a pretty steady speed. The Real John Lee also maintained a steady speed throughout the race. But Fake John Lee kept slowing down throughout the marathon.

So by looking at the slopes at different locations of a runner's graph of distance vs. time, we can see how quickly that runner is moving. And in this case, we can see who ran a smart, steady race, and who hit the wall.

Why airlines don't allow cell phones

The Federal Aviation Administration (FAA) just announced that most electronic devices can now be used on airplanes, all the way from takeoff to landing. But you still can't make a call on your cell phone during a flight, and all devices must be set to "flight mode." Why can't you make calls on flights? To get a better understanding, let's use some trigonometry.

Airplanes communicate and navigate using a band of radio waves called the airband, which uses frequencies between 108 and 137 MHz (or megahertz). Radio waves are electromagnetic waves, or waves of light, which travel through space like the sine function. Waves with higher frequencies move up and down, or "oscillate," faster, while lower-frequency waves oscillate more slowly. Here are examples of waves that have different frequencies:

How do the frequencies of cell phones compare to those of the airband? Well, it depends on the carrier (Verizon, AT&T, etc.), but for the most part they're between 500 and 2500 MHz. So cell phones send and receive radio waves that are close to the Airband, but are slightly higher in frequency.

If you were to make a call on a plane, then the cell phone's radio waves and the airplane's radio waves would add together. The airplane's waves are probably a lot stronger than the waves coming out of your phone. So what happens if we add a very weak cell phone signal (say, at 700 MHz) to a strong airplane signal at 120 MHz?

The sum of the two waves looks pretty close to the airplane's signal. But what happens if the cell phone signal were a lot stronger?

Suddenly, the sum looks quite different from the airplane's signal, and that's what worries the FAA. While there are mathematical tools that tease apart signals with different frequencies that have been added together, pilots and officials are concerned that cell phone radio waves could still interfere with the communication and navigation of airplanes.

If cell phones instead used frequencies that were really far away from the airband, then the risk of interference would be even smaller (even if the phones emitted really strong signals). Why do you think that is?