Anyone else think @robertkaplinsky should write a blog post about his “Problem-Based Learning FAQ”? Drop him a line (or RT this). #CMCN14

— Michael Fenton (@mjfenton) December 7, 2014

So, here are my frequently asked questions…

How long does it take to do a problem-based lesson?

Most problem-based lessons of the type that Dan, Andrew, Graham, and I do take about one to two 50-minute periods. Clearly many of them can take as long as you want, depending on whether you want to explore all the questions students come up with. Very few will take less than the whole period; and if they do then you are probably using a lesson that is too easy for the students doing them.  Similarly there are very few that require more than two days to complete. Remember, this is problem-based and not project-based. Project-based lessons can take weeks, months, or even the entire year but these usually take a day or two.

How often do you do problem-based learning?

The Common Core State Standards (CCSS) call for educators to pursue with “equal intensity” the three aspects of the Rigor shift: procedural skill and fluency, conceptual understanding, and application. Regardless of your support for the CCSS, it seems reasonable that students should have strength in all three components. So with that in mind, consider the following scenarios.

– If you do two or more problem-based lessons per week you would be spending at least 2-4 days per week on these lessons. That would take you away from the balanced rigorous understanding we are striving for. Their procedural skill and fluency as well as conceptual understanding would suffer.

– On the other end of the spectrum, problem-based lessons are most certainly not something you do once a semester or “after standardized testing.” You definitely won’t have balance that way either.

– I believe that the sweet spot lies in doing about one to two of these lessons per unit. They are a great way to introduce a context that will provide a foundation you can use to build conceptual understanding and later procedural skill and fluency. By starting a unit with a lesson, you begin with this context you can keep referring back to (i.e., “When we did the In-N-Out burger lesson, do you remember how we kept adding $0.90?”). Having a second lesson at the end of the unit provides a nice bookend to the unit and acts as a culminating review activity. Using problem-based lessons in this manner leads well into my next frequently asked question.

Do you use a problem-based lesson to begin a unit or after you’ve already taught them everything they need to know?

First off, let me say that I believe that using a problem-based lesson at any time is better than not using it at all. That being said, I strongly believe that problem-based lessons are most effective when you start a unit with one. To make this clearer, let me give you a rough sketch of a sixth grade unit on surface area.

Let’s assume you teach five periods and have three lower classes and two higher classes.  I would begin any unit on surface area with my favorite problem-based lesson: Andrew’s file cabinet task.

<tangent>If you’ve done this task, you know how amazing it is. If you haven’t, please do it with your students. I don’t care if they are kindergarteners or college students. It is such an amazing lesson because even if you don’t know a single thing about surface area, you can do this task. It is easily scaffolded so if you are a young kid, you can count the squares on the front side . If you are an adult, you will definitely not find the problem beneath you. It also provides such abundant context for developing the idea of area as non-overlapping square units and of surface area being comprised of the net formed by the area of each of the object’s sides.</tangent>

So, you start the unit with this problem.  Not surprisingly, the two higher classes are able to finish the file cabinet problem in one period on the first day. Also, not surprisingly, the three lower classes get stuck and even if you gave them a couple more days, you are not sure they would finish it. They do understand some of the foundational ideas such as having to cover the sides with stickies and that you would add all the sides together.  Most likely they get stuck on the computations and converting between square inches and stickies as units.  What they also have, very importantly, is the realization that there is something they want to figure out but don’t know how to figure it out. Those are two realities that are frequently missing when students learn mathematics.

Accordingly, after day one you have two classes who completed the task and three who haven’t. All classes now have the file cabinet as a context.  Now continue on with however you would normally help students develop conceptual understanding and procedural skill and fluency.  No one will be saying, “When will I ever use this?” because all new learning can be actively tied back to the context you’ve already created.

When talking about nets’ connection to surface area, students will be able to connect back to the file cabinet and more solidly grasp how the knowledge is interconnected. Eventually you will be done teaching the portion of the unit on conceptual understanding and procedural skill and fluency.  To wrap up the unit, it is important to recall where the classes were at.

The two high classes have already finished the file cabinet lesson, so clearly you can’t do that again. Fortunately there are many other lessons on surface area including my aluminum foil office prank lesson. For the two high classes, have them give this a try and it will be another opportunity for them to demonstrate that they can apply their knowledge. As for the low classes, it is time for them to return to the file cabinet task.

Remember that this was something that they wanted to figure out (if you don’t believe me, look at all the positive feedback about it on Twitter) but weren’t able to. Now comes the litmus test. If they can figure it out now, it is a wonderful moment because this was a skill they needed and wanted yet didn’t have. They can now realize the growth they have made. If they cannot figure it out, then it is an honest reality check that the lessons on procedural skill and fluency were not enough and intervention needs to happen. The final result is that all students have a strong skill set that they can apply.

How is problem-based learning graded?

I have three answers to this question:

• Don’t grade it.  Do you grade all your other classwork?
• Grade it using a rubric.  I tried this and even integrated it into version 4 of my problem solving framework.  What I learned was that it was so cumbersome and difficult to use that I couldn’t even figure it out… and I made the rubric!!  Getting through five periods of students just seemed ridiculous.  Ultimately I decided this was crazy to do and haven’t used it again.
• Use a scoring scheme similar to what the Smarter Balanced Assessment Consortium (SBAC) initially published for their constructed response problems.  Essentially you get one point for a correct answer and one point for sufficient reasoning to support that answer.  You can modify this to be out of ten and be five points for a correct answer and one to five points for sufficient reasoning.

How long does it take to create a problem-based lesson?

The short answer is don’t make problem-based lessons. Unless you are one of the crazies like me or upcoming superstars such as Graham, Dane, or Kyle you should be spending your time preparing for these lessons by implementing the five practices rather than making the lessons. There are so many resources available online or via textbook publishers that it is just a waste of time for the majority of people.

SPOILER ALERT: In addition, the honest reality is that many of your first lessons will be awful. If you don’t believe me, check out my first lessons on the very bottom of this page. They are well intentioned but really not that good.  In related news, if you are using one of my first lessons, I apologize.

If after all my discouraging, you still really do want to create these lessons, then I would say that the lessons average around 5 to 10 hours to make.  I have done a few in a couple of hours and some have taken me over 40 hours by the time I had finished filming, editing, uploading, writing up the lesson, and posting them all over social media. Seriously, did you notice how many times Andrew changed clothes during the video of him covering the of the file cabinet? I think it was at least 5 times. That is a lot of time. I am so glad that was him and not me… and that he, and many others, are gracious enough to share them for free.

Please leave me any questions I missed in the comments.

[This is one of a series of posts that explore real world examples of mathematical modeling to help educators better understand its applications. The most important post to read is this one about spies and analysts, which is a context I refer to in each of the posts.] 

I believe that there is still a lot of uncertainty about what exactly math modeling is. So, I want to share a humorous (and perhaps scary) story about how Target knew a 17-year-old girl was pregnant before the girl’s father did.

I’d read the article if you have time, but the summary is that Target (like most retailers) uses data about what you purchase to make predictions about what you’ll purchase in the future:

One Target employee I spoke to provided a hypothetical example. Take a fictional Target shopper named Jenny Ward, who is 23, lives in Atlanta and in March bought cocoa-butter lotion, a purse large enough to double as a diaper bag, zinc and magnesium supplements and a bright blue rug. There’s, say, an 87 percent chance that she’s pregnant and that her delivery date is sometime in late August.

Based on this information, Target then can send coupons to this customer to encourage her to shop at their store. It then goes on to talk about how this actually happened to someone:

“My daughter got this in the mail!” he said. “She’s still in high school, and you’re sending her coupons for baby clothes and cribs? Are you trying to encourage her to get pregnant?”

The manager didn’t have any idea what the man was talking about. He looked at the mailer. Sure enough, it was addressed to the man’s daughter and contained advertisements for maternity clothing, nursery furniture and pictures of smiling infants. The manager apologized and then called a few days later to apologize again.

On the phone, though, the father was somewhat abashed. “I had a talk with my daughter,” he said. “It turns out there’s been some activities in my house I haven’t been completely aware of. She’s due in August. I owe you an apology.”

So, let’s take a step back and think about this from the mathematical modeling perspective. There are two really important points:

• Retailers have access to rich data
• Data by itself is useless if you don’t know what to do with it

I want you to reflect on these those two points because they’re not trivial. Think about how you would even go about figuring out what data you need. Do you need to know the gender of the shopper? The time of day the purchase was made? How much money was spent? Stop and take thirty seconds to think about this. Out of all the available information, what would you want to help you make this decision? This is where the spy component comes in. What I mean is that with mathematical modeling one very important but often underappreciated part is acquiring the data. What data do you need? How are you going to get it?

Once you have all the data, now what? There are probably people who buy big purses and rugs that are not pregnant. The same thing for mineral supplements. So, how do you figure out who is and who is not pregnant? This is what mathematical modeling is all about, and I find it fascinating. This is where the analyst component comes in. Now that you have all the data you asked for, what do you do with that information once you have it?

Here’s another example for you. Think about being a Target data scientist who has access to all that data. One day you are asked to come up with a formula to predict, say, who is likely to go on a trip to warm weather location so that Target can send them coupons for bathing suits and sunscreen. What information would you look at? How would you value it to avoid false positives? Is this even possible?!

It’s the messy combination of data acquisition and formula creation that makes it so challenging. That’s mathematical modeling. If you were given the formula, then that wouldn’t be mathematical modeling at all.

While that may seem obvious, realize that this is often what happens in textbook. Instead of truly doing mathematical modeling, problem writers will focus solely on a context being real world and will sometimes even provide the formula.

So, now that you know what to look for, let’s give students as many opportunities as possible to do mathematical modeling. Here are over 70 free problems you can get started with.

Please let me know in the comments if you have any fun stories to explain what mathematical modeling is.

Sometimes we do something just because that’s the way it’s always been done. This happens in sports just like in education. For example, consider the high jump. You’re probably familiar with the basic idea of seeing how high someone can jump over a bar.

 
What you may not know though is that athletes have not always participated in the high jump in the same way. It used to be that people would jump over the bar like a hurdle by using a scissor kick method. People always did it this way and people continued to do it this way. At least it was that way until 1968 Mexico City Olympics.

 
There, American Dick Fosbury introduced the world to a new technique by going over backwards. He utilized what would eventually be called the “Fosbury flop.”

 
Now it’s important to understand how strange this first appeared. His method was laughed by the fans. They would even chant, “Ole!” as he ran towards his jump.

His fellow athletes and coaches looked down on him like he disrespected the sport. The judges were even unsure if he was breaking the rules because they had not seen it done this way before. Ultimately, they agreed that it was ok and Fosbury used the technique to win the gold medal and set a new Olympic record. Today, virtually everyone uses this technique.

I share this story as an example of something called status quo bias. This happens when we do something in a particular way, not because it’s the best way but because that’s the way it’s always been done.

Dick Fosbury challenged the status quo with his new technique. It made people uncomfortable because it violated their expectations. However, once they got past that, they realized that his method was truly better and necessitated change.

I believe the same thing is happening in education, and specifically in regards to the mathematics standards we teach. Are our mathematics standards chosen because they are still the most important ones for students to learn or because that’s what we’ve always taught in math class?

I think that we need to take a step back and realize that so much of what we teach students is instantly useless because of technology. I’ll say it again to be clear, we spend days and weeks teaching students content that they’ll likely never use again because of the pervasive existence of technology like calculators.

Obviously, I’m not the first person to think of this. I love how Conrad Wolfram articulates it in his TED Talk. Just watch the first 30 seconds of the clip below (from 4:10 to 4:40).

Here’s what he said in those 30 seconds:

Now here’s the crazy thing right now. In math education, we’re spending about perhaps 80 percent of the time teaching people to do [computation] by hand. Yet, that’s the one step computers can do better than any human after years of practice. Instead, we ought to be using computers to do step three and using the students to spend much more effort on learning how to do steps one, two and four — conceptualizing problems, applying them, getting the teacher to run them through how to do that.

 
The first time I heard that part of the clip, it blew my mind. I had never thought of it that way.

So here are my thoughts:

• When I surveyed 383 K-12 mathematics teachers in the United States, 85% of them stated that they did not have enough time to teach one year of grade level standards in one year.
• Many of the math standards we teach students are computation based and can be done with technology.
• Many of the non-computation based math standards are ones that computers (currently) cannot do such as complex mathematical modeling. For example, computers alone still can’t tell if you’re pregnant, but people with computers can.
• As of 2015, 92% of American adults own cell phones they have with them constantly and most of those phones have calculators.

Based on this information, I reach the conclusion that we are spending too much time teaching procedures that will likely be replaced by technology. For example, how often are you multiplying multi-digit numbers, dividing decimals, or figuring out sales tax by hand?

I wonder what would happen if we greatly deemphasized teaching these procedures or cut them out entirely. What could we do with that time we freed up? Like with Dick Fosbury challenging the status quo with his new technique, I’m sure the idea of changing how these standards are prioritized will certainly make people uncomfortable because it violates their expectations.

What do you think? Please let me know in the comments.

For years I’ve been trying to find a way to talk about mathematical modeling that made sense to me. I’ve finally found something I’m happy with and it has to do with the differences between spies and analysts. I’m sure this sounds strange, so let me explain.

The United States’ Central Intelligence Agency (CIA) is responsible for acquiring and using information to help American causes. There are two main groups in this agency: the people who acquire information (we’ll call them spies) and the people who break down and connect the information (we’ll call them analysts). Both are important because having more intelligence about what others are doing allows the United States to make more informed decisions. Let’s consider how this played out in both World War 2 and September 11th.
 

Now let’s consider September 11th. What if we had more spies gathering information about what was going to happen? Would that have helped? It may seem like it would have, but unfortunately it wouldn’t have been as valuable as having more analysts.

Prior to the attack on 9/11/2001, the US intelligence community had received many tips about suspicious activity. For example, they knew about suspected terrorists who had been receiving flight instruction. So, in this case, the issue wasn’t that the information wasn’t available. The issue was that there weren’t enough analysts to take the information they received and determine what was useful and what was unimportant.

So, the way I see it, there are two primary components in this process:

1. Spies figure out what information is needed and acquire it
2. Analysts take the given information and work with it

Hopefully what results is actionable information that gives the US intelligence community what they want. If it doesn’t, then they need to modify their plans and look at where the process broke down. Was it an issue of the spies not getting the necessary information? Was it instead that the information was there but the analysts didn’t put it together?

The takeaway here is that we need a combination of both spies and analysts to find success. Interestingly, I believe that this structure of spies and analysts is very useful for gaining deeper understanding of mathematical modeling. To explain what I mean, let’s consider how we predict weather and earthquakes.
 

Now let’s think about how we predict earthquakes. Right now, we’re pretty awful at figuring out when future earthquakes will take place. We know that they will happen, but we can’t tell if they will take place in a week, a month, a year, a decade, or a century. So, how can we improve? Should we be investing in better spies or analysts? In this case, the problem is with our spies, not our analysts. Earthquakes happen so rarely that it’s hard to capture data on them. We have seismographs to measure earthquakes, but is that the best we can do? What about how animals seem to go crazy before an earthquake. Could there be other available data that our spies are not aware of or are not currently able to capture? Even if you come up with a highly accurate formula for predicting earthquakes, they happen so rarely that it makes it really challenging to fine tune it.

So, with mathematical modeling:

1. Spies figure out what information is needed and acquire it
2. Analysts take the given information and work with it

[This is one of a series of posts that explore real world examples of mathematical modeling to help educators better understand its applications. The most important post to read is this one about spies and analysts, which is a context I refer to in each of the posts.]
 
Do you remember reading about the man who was literally dragged off a United Airlines flight in April 2017? One of the questions people had was “Why was he the one they picked?” People speculated that it might have been because of his race or spoken language.

The reality was that, for both better and worse, that had nothing to do with why he was picked. He was picked because United Airlines used mathematical modeling to determine that he was their least profitable customer.
 

In this case, when United Airlines could not persuade someone to give up their seat, they had to decide which passenger would be forced to leave. Put another way, if you had 201 people for a 200-person flight, how would you choose the one person who would not fly?

Stop and take thirty seconds to think about this. Out of all the available information, what would you want to help you make this decision? This is where the spy component comes in. What I mean is that with mathematical modeling one very important but often underappreciated part is acquiring the data. What data do you need? How are you going to get it? When you’ve got an idea of what data you would use, read on.
 

Realize that United had to go through this same process and in their Contract of Carriage Document it states that “If there are not enough volunteers, other Passengers may be denied boarding involuntarily in accordance with UA’s boarding priority” and then “the priority of all other confirmed passengers may be determined based on a passenger’s fare class, itinerary, status of frequent flyer program membership, and the time in which the passenger presents him/herself for check-in without advanced seat assignment.”

So, out of all the data they could choose from, they picked these as the most important:

• Fare class (coach vs. business vs. first class)
• Itinerary (are there any connecting flights and will this missed flight create a chain reaction of missed flights?)
• Status of frequent flyer program membership (loyal frequent flyers generate more money)
• Check-in time (presumably those who checked in earlier should get priority)

Now with the data chosen, this is where the analysts come in. What are they going to do with all this information? How are they going to manipulate it to create a mathematical model (which could also be called a formula or algorithm) to weigh the variables and decide which customer is the least profitable for them? In other words, if being kicked off a plane made the customer so mad that they never purchased a flight from them again, who would cost United the least amount of money?

I’m definitely not stating that I like their mathematical model, that the practice of overselling flights is sound, or that they handled asking him to leave in a professional manner. I’m just trying to show you an example of how mathematical modeling is used. I want to open up some of the complexities so that we realize that if our job was creating the formula, it wouldn’t be easy.

If this was a stereotypical textbook problem, it might begin by giving the already created formula and a set of data. Then it might ask you to determine which person would not fly. That would not be mathematical modeling though. Creating the actual mathematical model, not using it, is the hardest part.

[This is one of a series of posts that explore real world examples of mathematical modeling to help educators better understand its applications. The most important post to read is this one about spies and analysts, which is a context I refer to in each of the posts.]
 
You’ve probably heard of US News and World Report’s Best Colleges issue where they rank American institutions of higher education in numerical order and sell the rankings to anyone interested. Have you ever thought about how they actually determine the rankings?

If you were tasked with doing the same, how would you even begin the process? What information would you consider? Once you had that information, what would you do with it? How would you weigh the factors you looked at? This may not seem like a big deal, but it is to US News and World Report which makes millions of dollars each year from its rankings.

The process they use is called mathematical modeling. Their goal is to use mathematics to help take a complex set of information and present it in a usable format. This process is not easy and certainly not without its many criticisms. So, my goal is to open up the complexities so that we can better appreciate how mathematical modeling is used. Think of what follows as your attempt to do a better job than what US News and World Report does.

The first part of the process requires the spies. So, I want you to stop and take thirty seconds to think about what information you would use to create a better set of rankings. If my request doesn’t feel daunting to you, it should. What would you consider? SAT scores? Total number of students? Size of campus? Location? Majors offered? The list could go on and on. Once you’ve determined what information you’d want, keep reading.
 

• Financial resources
• Student selectivity
  • SAT and ACT test scores
  • Percentage of students who graduated near the top of their high school class
  • Acceptance rate or the ratio of students admitted to applicants
• Graduation rate performance
• Alumni giving rate
• Graduation and retention rates
• Faculty resources
  • Class size
  • Faculty salary
  • Proportion of professors with the highest degree in their fields
  • Student-faculty ratio
  • Proportion of faculty who are full time
• Undergraduate academic reputation

 
Now with that information at hand, what do you even do with it all? Should each component be equally important? If not, how would you weigh each one? This is where the analysts come in. Their job is to take the data and break it down in such a way that it becomes useful. Take 30 more seconds to weigh the factors listed above and then read on to see how US News and World Report weighs them.

• Graduation and retention rates (22.5%)
• Undergraduate academic reputation (22.5%)
• Faculty resources (20%)
  • Class size (40% of faculty resources)
  • Faculty salary (35% of faculty resources)
  • Proportion of professors with the highest degree in their fields (15% of faculty resources)
  • Student-faculty ratio (5% of faculty resources)
  • Proportion of faculty who are full time (5% of faculty resources)
• Student selectivity (12.5%)
  • SAT and ACT test scores (65% of student selectivity)
  • Percentage of students who graduated near the top of their high school class (25% of student selectivity)
  • Acceptance rate or the ratio of students admitted to applicants (10% of student selectivity)
• Financial resources (10%)
• Graduation rate performance (7.5%)
• Alumni giving rate (5%)

 
What do you think about their weightings? What do you agree with? What did they get wrong? You have to realize that if you got 1000 people in a room and had them make a list of the data and corresponding weights, there would be no duplicates.
 

I believe that this is the future of mathematics for our students. We can’t say that we’re creating college and career ready students when we spend years of their K-12 experience teaching them to do things that calculators can already do and are never to be used again. We need to find a way to spend more time teaching them how to do mathematical modeling, which will be a very useful skill.

[This is one of a series of posts that explore real world examples of mathematical modeling to help educators better understand its applications. The most important post to read is this one about spies and analysts, which is a context I refer to in each of the posts.]
 
What if I told you that you could make billions of dollars by using mathematical modeling? Well, that’s exactly what Amazon has been using when they recommend products for you to buy. These product recommendations frequently come in the form of “People who bought __________ often buy __________, __________, or __________” and result in additional purchases that generate significant revenue.

So, what if you worked for Amazon and they tasked you with creating a formula to make these recommendations? Where would you begin? What information would you want to know? What would you do with that data once you had access to it? These are the topics I’m exploring in my spies and analysts post. I want to walk you through the process so that you can better appreciate the complexities of mathematical modeling.

The first part of the process requires the spies. So, I want you to stop and take thirty seconds to think about what information you would use to make your product recommendations. What would you consider? Where the customer lives? The time of day? Upcoming holidays? The customer’s gender? The list could go on and on. Once you’ve determined what information you’d want, keep reading.
 

• Previously purchased items may suggest interests
• Items added to carts but not purchased
• Pricing experiments online where they offer the same products at different prices and see the results
• Experiments where they offer products in different “bundles”
• Your wishlists
• Other sites you’ve visited may provide information about your interests
• How long you look at an item before moving on
• Product ratings you or people in your social network have left
• Demographic information such as your shipping address to know what people in your general area like
• Clicking on a link in an email
• Items you viewed on their site
• Number of times you viewed an item before final purchase
• Purchase history from other partner stores

 
If you’re not feeling overwhelmed by the sheer magnitude of that level of data, you should be. It’s a blessing and a curse because while the information is very helpful Amazon likely has more information in those categories than we can possibly imagine. Now what do you even do with it all? Should each piece of information be equally important? If not, how would you combine and weigh each one? This is where the analysts come in. Their job is to take the data and break it down in such a way that it becomes useful. Take 30 more seconds to think about which of those factors you would prioritize.

At this point, there are no computers or calculators that can figure this out on their own. This is where the jobs are at. If we truly want to focus our time and energy in a skill that will really help our students become college and career ready, mathematical modeling is where we need to be.

I want to begin this blog post with a question: why do we have word problems?

I don’t mean why do we need or use word problems but rather why do they even exist in the first place? This may seem unimportant or trivial, but I think that understanding the answer to that question will be useful in understanding their role in math education today.

Take a look at the image below. It is from the book Milne’s Inductive Algebra which is an Algebra textbook published in 1881 by Eclectic Press. When I first received this book as a gift about ten years ago, I was very curious as to what would be inside. How had Algebra in the United States changed over the last 125 years?

What I came to realize was that not much has changed. There were definitely many word problems, but I realized that when this book was published there probably weren’t great options for incorporating context into problem solving.

Let’s compare that to present day instruction. Today we have many alternatives for providing authentic and engaging contexts (such as my problem-based lessons). So, why do we still use so many word problems?

Is it because word problems exist in real life? Not that I can see.

Is it because using word problems are the best way to instruct students? Almost certainly no.

Or, is it because of status quo bias where we continue to use them because that’s the way we’ve always taught math? This is what I think is happening.

This is not a trivial point to me. While word problems can be useful, losing focus of what’s important (making sense of mathematics) leads to teaching students strategies like the ones below.

There are three very important takeaways here:

• Teaching strategies like CUBES is not teaching mathematics. It’s teaching how to decode a kind of problem writing style that has no application in real life. At best, it teaches students how to be a math robot like I was. I could give you the correct answer but I had no idea what I was actually doing. It’s very similar to the Chinese Room thought experiment and leads to creating students like the last student in this short and terrifying video.
• The main reason educators teach strategies like CUBES is because they may help students solve the kinds of problems they will likely see on standardized assessments. No one is teaching these because this is something that is used in real life.
• Finally, word problems are virtually non-existent in real life. There are certainly plenty of real life contexts, but very few of them are set up as neatly written word problems. In real life you have to create your own problem by figuring out what information is important and using it to solve the problem.

I think it’s important for us as educators to take a step back and think about what tools we use and why we use them. To be clear, I’m not saying that teachers should never use word problems, but it depends on both why they’re used and how they’re used.

For example, word problems are commonly used in Cognitively Guided Instruction (CGI). In those situations, it’s never about mindlessly using a technique to find the answer but rather about providing a context where conversations about sense making can come out. You’re never going to see a CGI-trained educator teaching CUBES.

So, what do you agree with and why? Or, where am I missing the point and need to think more deeply? Please let me know in the comments.

[This is one of a series of posts that explore real world examples of mathematical modeling to help educators better understand its applications. To learn about Spies and Analysts, I recommend watching this webinar (with elementary, middle, and high school versions) or reading this blog post.]
 
If you’ve ever used Pandora, then you know that it has an amazing ability to recommend music to you that you may have never heard but really enjoy. They play a song for you, and you can give it a thumbs up if you really like it, thumbs down if you really don’t like it, or neither if it’s just ok. After a little while, it seems to somehow know your taste in music better than you know it! This leads you to stay on their site longer, and as a result, they can play more ads or get you to pay for them to go away.

So, what if you worked for Pandora and they asked you to create a formula to predict which songs people have never heard but will probably like? Where would you begin? What information would you want to know? What would you do with that data once you had access to it? These are the topics I’m exploring in my spies and analysts post. I want to walk you through the process so that you can better appreciate the complexities of mathematical modeling.

The first part of the process requires the spies. So, I want you to stop and take thirty seconds to think about what information you would use to recommend songs. Would you look at which songs are being played the most on radio stations or selling the most records? Maybe where a person lives affects the kind of music they listen to? Does gender or age matter? The list of questions could go on and on. So, think about what information you’d pick if this was your job. Once you’ve determined what information you’d want, keep reading.
 

A given song is represented by a vector containing values for approximately 450 “genes” (analogous to trait-determining genes for organisms in the field of genetics). Each gene corresponds to a characteristic of the music, for example, gender of lead vocalist, prevalent use of groove, level of distortion on the electric guitar, type of background vocals, etc. Rock and pop songs have 150 genes, rap songs have 350, and jazz songs have approximately 400. Other genres of music, such as world and classical music, have 300–450 genes.

 
Apparently it takes 20 to 30 minutes to categorize each song. Can you imagine the amount of work it would take to do this for every single song in existence!? Crazy enough, this is just part of how Pandora works. Specifically, even if you had all that information about the songs, how do you write a formula to figure out which song to play? Is “gender of lead vocalist” more important than “prevalent use of groove”? Remember, if your formula isn’t good, customers won’t stick around and you’ll be out of business.

This is where the analysts come in. Their job is to take the data, figure out what parts are more or less important, and break it down in such a way that it becomes useful. Take 30 more seconds to think about how you might even begin to work with the data.

The system depends on a sufficient number of genes to render useful results. Each gene is assigned a number between 0 and 5, in half-integer increments.The Music Genome Project’s database is built using a methodology that includes the use of precisely defined terminology, a consistent frame of reference, redundant analysis, and ongoing quality control to ensure that data integrity remains reliably high.

For the record, I don’t completely understand what that just said either! The reality is that they created a formula to take all of that information, determine what was most important, and make it into a product that earns them significant revenue.

At this point, there are no computers or calculators that can figure this out on their own. This is where the jobs are at. If we truly want to focus our time and energy in a skill that will really help our students become college and career ready, mathematical modeling is where we need to be.

[This is one of a series of posts that explore real world examples of mathematical modeling to help educators better understand its applications. To learn about Spies and Analysts, I recommend watching this webinar (with elementary, middle, and high school versions) or reading this blog post.]
 
Perhaps you’ve heard of Zillow, which is a real estate website that can give you a rough estimate for how much a real estate property is worth. While the property values they share are not as researched or accurate as what you would get from a professional appraisal, the fact that they instantly provide this data for free for over 100 million homes is pretty amazing.

So, what if you worked for Zillow and they asked you to create a formula to more accurately predict how much a home is worth? Where would you begin? What information would you want to know? What would you do with that data once you had access to it? These are the topics I’m exploring in my spies and analysts post. I want to walk you through the process so that you can better appreciate the complexities of mathematical modeling.

The first part of the process requires the spies. So, I want you to stop and take thirty seconds to think about what information you would use to accurately predict a home’s value. Would you look at what the home previously sold for? Would you look at how close it is to freeways and schools? Would you look at the crime rate? Would you look to see what other nearby homes have sold for? The list of questions could go on and on. So, think about what information you’d pick if this was your job. Once you’ve determined what information you’d want, keep reading.
 

Physical attributes: Location, lot size, square footage, number of bedrooms and bathrooms and many other details.

Tax assessments: Property tax information, actual property taxes paid, exceptions to tax assessments and other information provided in the tax assessors’ records.

Prior and current transactions: Actual sale prices over time of the home itself and comparable recent sales of nearby homes

So now that we know some of the data they use, how would we go about turning that into a price for the home? Is “lot size” more important than “number of bedrooms and bathrooms”? Remember, if your formula isn’t good, customers won’t spend their time on your site and you’ll be out of business.

This is where the analysts come in. Their job is to take the data, figure out what parts are more or less important, and break it down in such a way that it becomes useful. Take 30 more seconds to think about how you might even begin to work with the data.

We use proprietary automated valuation models that apply advanced algorithms to analyze our data to identify relationships within a specific geographic area, between this home-related data and actual sales prices. Home characteristics, such as square footage, location or the number of bathrooms, are given different weights according to their influence on home sale prices in each specific geography over a specific period of time, resulting in a set of valuation rules, or models that are applied to generate each home’s Zestimate.

Just when you thought it was hard enough to figure out what information you needed, you start to realize that even doing something with that information is challening too! The reality is that they created a formula to take all of that information, determine what was most important, and make it into a website that earns them significant revenue.

At this point, there are no computers or calculators that can figure this out on their own. This is where the jobs are at. If we truly want to focus our time and energy in a skill that will really help our students become college and career ready, mathematical modeling is where we need to be.

[This is one of a series of posts that explore real world examples of mathematical modeling to help educators better understand its applications. To learn about Spies and Analysts, I recommend watching this webinar (with elementary, middle, and high school versions) or reading this blog post.]
 
You’ve probably heard of dating websites (like eHarmony, OkCupid, or Match.com) that help you find someone you might be interested in. You enter your information, take a survey, and then hope you’re connected with the right people. That’s what the websites hope too, because they make their money through people signing up for memberships. The reality for these websites is that people have to see others they are interested. If they do, they will stick around, refer their friends, and the company will make money. If not, their customers will leave and they will go out of business.

So, what if you worked for eHarmony and they asked you to create a formula to decide which people to pair up? Where would you begin? What information would you want to know? What would you do with that data once you had access to it? These are the topics I’m exploring in my spies and analysts post. I want to walk you through the process so that you can better appreciate the complexities of mathematical modeling.

The first part of the process requires the spies. So, I want you to stop and take thirty seconds to think about what information you would use to match potential customers. Would you look at where they lived? How old they were? How much education they have? The kinds of foods they like to eat? Whether they want to get married? The list of questions could go on and on. So, think about what information you’d pick if this was your job. Once you’ve determined what information you’d want, keep reading.
 

As for eHarmony, they are so proud of what they ask you about that they advertise their “29 dimensions of compatibility” as a selling point for their website. Those dimensions are:
 

Character & Constitution:

• Good Character
• Dominance vs. Submissiveness
• Curiosity
• Industry
• Vitality & Security
• Intellect
• Appearance
• Sexual Passion
• Artistic Passion
• Adaptability

Emotional Makeup & Skills:

• Emotional Health
• Anger Management
• Quality of Self Conception
• Mood Management
• Communication
• Conflict Resolution
• Kindness
• Autonomy vs. Closeness

Personality:

• Obstreperousness
• Sense of Humor
• Sociability
• Energy
• Ambition

Family & Values:

• Feelings about Children
• Family Background
• Education
• Spirituality
• Traditionalism
• Values Orientation

 
In reading this list, many definitely make sense. Some, like “Obstreperousness” I had never even heard of. So, let’s imagine that you made a survey that measured all of these factors. Now what? How do you turn them into a person to show? Is “Education” more important than “Ambition”? What about the reality that what people say they want often differs from the people the date? This is where the analysts come in. Their job is to take the data, figure out what parts are more or less important, and break it down in such a way that it becomes useful. Take 30 more seconds to think about how you might even begin to work with the data.

The Chief Scientist at eHarmony has revealed that although singles are asked to choose likes and dislikes on a sliding scale, unless they pick the extreme ends their answers will be largely ignored. Dr Steve Carter said it stopped daters ending up ‘in a universe of one.’

 
Think about that! Sometimes to make customers happy, they have to be saved from themselves by ignoring what they say!

At this point, there are no computers or calculators that can figure this out on their own. This is where the jobs are at. If we truly want to focus our time and energy in a skill that will really help our students become college and career ready, mathematical modeling is where we need to be.

[This is one of a series of posts that explore real world examples of mathematical modeling to help educators better understand its applications. To learn about Spies and Analysts, I recommend watching this webinar (with elementary, middle, and high school versions) or reading this blog post.]
 
We now live in a world where ride sharing services like Uber and Lyft are considered commonplace. You just open the app on your mobile device, request a vehicle to pick you up, get in, and you’re on your way. If you’ve used such a service a few times, you’ve also probably noticed that the same trip might cost significantly different prices, depending on when you use it. So, how does a ride sharing service like Uber know how much to charge you? This really matters because if they charge too much, you might use a competitor, a taxi, or public transportation. If they charge too little, they may not make enough money to stay in business.

So, what if you worked for Uber and they asked you to create a formula to competitively price their trips? Where would you begin? What information would you want to know? What would you do with that data once you had access to it? These are the topics I’m exploring in my spies and analysts post. I want to walk you through the process so that you can better appreciate the complexities of mathematical modeling.

The first part of the process requires the spies. So, I want you to stop and take thirty seconds to think about what information you would use to competitively price customers’ trips. Would you look at how far the driver has to go? Does traffic matter? Does the driver’s experience level matter? Should fancier cars cost more? Should it cost more if you have more passengers? Should it cost less when demand is low and more when demand is high? Does the city you’re in matter? As I hope you realize, the list of questions could go on and on. So, think about what information you’d pick if this was your job. Once you’ve determined what information you’d want, keep reading.
 

• Time the trip takes
• Distance the trip takes
• Tolls and fees
• Whether there are enough drivers in the area to meet demand
• Whether you are sharing a ride with other passengers you don’t know
• The vehicle’s luxury level

So, much of what I imagined them using was included, but the reality is that we are still far from a dollar amount to charge. How do you take all those factors and turn them into a formula that pumps out a competitive price?
This is where the analysts come in. Their job is to take the data, figure out what parts are more or less important, and break it down in such a way that it becomes useful. Take 30 more seconds to think about how you might even begin to work with the data.

Your Uber fare is first calculated on 4 main criteria:

• Base fare – A flat fee charged at the beginning of every ride
• Cost per minute – How much you are charged for each minute you are inside the ride
• Cost per mile – How much you are charged for each mile of the ride
• Booking Fee – A flat fee to cover Uber’s operating costs

 
Here’s how Uber uses the 4 main criteria above to calculate your fare:

Base Fare + (Cost per minute * time in ride) + (Cost per mile * ride distance) + Booking Fee = Your Fare

From there it gets a little more complicated. First, comes the issue of driver supply and demand. If there are more people requesting rides than there are drivers to supply them, Uber uses something called “Surge Pricing” which will multiply the cost of your trip by a multiplier. For example, when a sporting event ends and everyone wants a ride at the same time, your fare might be multiplied by 2 or 3. While this might seem like price gouging, it is also a huge incentive for drivers from distant areas to come to your area and pick you up.

Then come many other factors that get added in like tolls and fees, how luxurious the vehicle you request is, whether you’re sharing a ride with strangers, and the city you’re traveling in.

At this point, there are no computers or calculators that can figure this out on their own. This is where the jobs are at. If we truly want to focus our time and energy in a skill that will really help our students become college and career ready, mathematical modeling is where we need to be.

[This is one of a series of posts that explore real world examples of mathematical modeling to help educators better understand its applications. To learn about Spies and Analysts, I recommend watching this webinar (with elementary, middle, and high school versions) or reading this blog post.]
 
In March 2001, Russia’s Mir space station was falling out of orbit and was going to crash into the Pacific Ocean. Taco Bell used this as an bizarre marketing opportunity and placed a 40′ x 40′ sign in the middle of the ocean with a target on it that said “Free Taco Here!”. If the Mir hit the sign on the way down, everyone in the United States would get free tacos.

Here’s a short video to refresh your memory:

What Taco Bell ultimately decided to do was take out an insurance policy from SCA Promotions, a company that guarantees prizes. What this means is that SCA Promotions charged Taco Bell a fee, and if the Mir hit the target, Taco Bell would not have to pay anything more. In fact, Taco Bell was probably hoping and praying that it hit the target as it would have boosted sales and cost them nothing.

Consider though what SCA Promotions had to do to figure out how much to charge Taco Bell. If they charged too much, Taco Bell wouldn’t feel like it was worthwhile. If they charged too little, it wouldn’t be worth the risk to make the deal. If the Mir missed the target, they’d get all the money. If the Mir hit the target, their company could go bankrupt!

So, what if you worked for SCA Promotions and you were asked you to figure out how much to charge Taco Bell for this insurance policy. Where would you begin? What information would you want to know? What would you do with that data once you had access to it? These are the topics I’m exploring in my spies and analysts post. I want to walk you through the process so that you can better appreciate the complexities of mathematical modeling.

The first part of the process requires the spies. So, I want you to stop and take thirty seconds to think about what information you would use to figure out the cost of the insurance policy. Would you look at how how big the sign is? The speed of the Mir? The number of people in the United States? The cost of a taco? The number of people who like tacos? The time of day? As I hope you realize, the list of questions could go on and on. So, think about what information you’d pick if this was your job. Once you’ve determined what information you’d want, keep reading.
 

• The sign was 40′ by 40′.
• In 2001, there were an estimated 281 million people living in the United States and the cheapest Taco Bell taco cost 60 cents. Taco Bell estimated the cost of the free tacos at $10,000,000. Accordingly, that seems to assume that about 6% of the population would actually take advantage of a free taco.
• According to the video, the area that the Mir was expected to crash into was approximately 3600 miles (5794 km) x 120 miles (193 km) by of the that the Mir

What’s important to realize is that the only solid fact is the size of the sign. The cost of the tacos and the size of the area where the Mir might crash are estimates. That being said, based on the video, it sounds many of the factors I thought of were included. So, how do you take all those factors and turn them into an amount of money to charge Taco Bell?

This is where the analysts come in. Their job is to take the data, figure out what parts are more or less important, and break it down in such a way that it becomes useful. Take 30 more seconds to think about how you might even begin to work with the data.

If that’s the case, then the area of the sign is 1600 square feet while the area of the crash zone is 432,000 square miles. Converting square miles to square feet gives us an area of 12,043,468,800,000 square feet for the crash zone. This results in a ~0.0000000133% (or 1 in 7,500,000,000) chance of hitting the target. So now what? How much do you charge Taco Bell?

The way I see it, a ~0.0000000133% chance of having a $10,000,000 payout leads to an expected value of having to pay $0.001 on average. So, I guess that charging anything over a penny should earn you money over the long run. I don’t know what Taco Bell was actually charged, but with odds like this, I’m surprised that they didn’t just cover the costs themselves.

At this point, there are no computers or calculators that can figure this out on their own. This is where the jobs are at. If we truly want to focus our time and energy in a skill that will really help our students become college and career ready, mathematical modeling is where we need to be.