False proxies are annoying

Posted on November 11, 2012 by Zachary_Goldman

sethgodin.typepad.com/seths_blog/2012/11/avoiding-the-false-proxy-trap.html
False proxies are results that are easy to measure and appear on the surface to be related to actual success of a person, program, or organization. The idea is that the higher these measurements seem, the greater the success at accomplishing some real mission.

The problem is that there are often ways to show growth in one of these false proxies, without actually having much success towards having a real increased impact.

Here are a few false proxies that have particularly frustrated me:

1. Volunteer hours are a false proxy for community impact.

2. Standardized test scores are a false proxy for student learning (let alone for student ability to think for herself, have innovative ideas, be ready for a career, etc.).  For that matter, in many cases, grades may also be a false proxy for all of those things.

3. Number of school events planned and dollar value of supplies collected through in-kind donations are false proxies for the likelihood of students dropping out of school. (Particularly if the opportunity cost of increasing these proxies is really a decreased focus on other things that may actually have more of an impact on dropout rates.)

In all of these situations, increasing the false proxy is not necessarily a bad thing (in fact, if done carefully, most of those things can actually have a very positive impact). However, for all of these false proxies, just making every effort to increase the proxy doesn’t automatically imply that real progress is actually being made towards achieving the real objectives. It is often possible to increase the proxy measurement without increasing real success.

What frustrates me is that people and organizations focus on increasing the proxies while sometimes forgetting about the real goals which are supposedly being measured by the proxies.

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The importance of basic infrastructure, often taken for granted….

Posted on November 11, 2012 by Zachary_Goldman

From the Chief Innovation Officer at the Acumen Fund: sashadichter.wordpress.com/2012/11/09/

His house didn’t have power for 10 days after Sandy.

“The core work of Acumen, where I work, is to support companies that provide basic goods and services – healthcare, water, housing, sanitation, education, and, yes, energy – to the half of the world’s population that hasn’t yet benefited from the global wealth creation and economic transformation that started in the 1850s.”

[…]

“Think of all the people out there not blogging, not sharing, not contributing as they could to the world because every last ounce of energy must go into just getting by.

For just a week, New York and the whole eastern seaboard got to experience how every aspect of our lives are enabled by this infrastructure.  We got to ask ourselves how resilient we would be if we lost this cushion.  A spotlight was shone on all of the invisible things that make our lives possible.”

 

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Post City Year options (your input is wanted!)

Posted on November 10, 2012 by Zachary_Goldman

My current position with City Year ends in June. I’m currently exploring post-City Year options, which City Year comically refers to as LACY (life after City Year).

I have started collecting a (so far very incomplete) list of options for myself here: post City Year options.

Update (12-3-12): I am indeed considering serving a second year with City Year as a team leader next year (but lots of other options are on the table, too)!

I need your help! Since I’m not exactly sure what I want to do next, I’m seeking input from everyone who is reading this about what to do next.

I’m looking both for suggestions of specific jobs or graduate programs you think would be a good fit for me AND also more general ideas about what you think I should pursue next both in the short term (next year) or in the longer term.

Please add your thoughts directly into the spreadsheet at the link above (google docs is magical) and/or you may contact me directly.

Please use your own knowledge of me in considering the types of things you suggest. If you think something would be a good fit for me, please suggest it!!

Here is some information about my current thinking (but if you have a good idea that doesn’t quite fit with this, still add it to the spreadsheet anyway!)…

1. I am interested in how people learn and specifically how people can learn to create innovative solutions to problems (problems of ALL kinds: math problems, structural poverty, etc.). I am interested in education research, practice, and policy in this area (and education policy more broadly). That being said, I think it is very unlikely I would ever decide to be a classroom teacher in a primary/secondary school ever again.

2. I am interested in systems, and models, and complexity (and designing systems–of all kinds–that are effective). Recently, I’ve been particularly intrigued by this course: https://www.coursera.org/course/modelthinking

3. I like economics.

4. I usually enjoy figuring out how to help people do stuff more effectively more than I enjoy actually doing stuff. (Where “doing stuff” is very broadly defined.) Update 11-20-12: Yes, operations research is an option.

5. I like math, but I have less actual expertise in math than most people who know me seem to think. I didn’t spend my time in college studying math deeply enough to set myself up to pursue a future involving lots of theoretical math. Also, I last took a stats class in high school, so I don’t currently have a particular expertise with stats or experience actually doing stats in any real context. However, that being said, I would definitely be able to pick these skills/understanding very quickly if necessary. My math background would also let me pick up other related fields very easily.

6. If I do end up doing non-profit stuff, I’d like to have some background and expertise in some particular academic area beyond “how to run non-profits.”

7. An ideal grad program will help me learn things that I am interested in, and that I wouldn’t be able to (or wouldn’t be motivated to) study otherwise. For example, as an undergrad, I minored in physics, which was interesting and which I would not have been able to learn on my own.

8. I can envision myself working in higher ed. (in a faculty or staff position), non-profits, other education policy kinds of things, etc. There are probably other things I could see myself doing, too.

9. I appreciate the ability to extend what I’m doing in a variety of different directions as I get bored with whatever my current focus is.

Please add your general thoughts or specific suggestions of grad programs or jobs to the spreadsheet!
Thanks!

Posted in CIty Year, Higher education, Personal Experiences, Uncategorized | Tagged , , , , , , | 1 Comment

Student data for pattern-seeking, not just assessing

Posted on September 3, 2012 by Zachary_Goldman

It is not news that, in the last decade, the use of data (often based on standardized test scores) has exploded. In addition to being used to assess how well students have learned the material for a particular class, student test scores are increasingly being used to assess teachers and schools.

Proponents of assessment vs. Opponents of hyper-testing: a false distinction

Proponents of this testing argue that students shouldn’t be able to be promoted to the next grade if they haven’t mastered the previous grade (they’re right). They argue that teachers and schools should have a way of knowing how well their methods are actually working (they’re right, too). They argue that this data can be used to pinpoint areas of student weakness, so that these can be addressed (they’re right). Some also argue that students and parents should know how well their schools are doing (they might be right). Let’s say that  people who agree with most or all of these points are called “group 1.”

Opponents of this testing argue that important things often can’t be measured by standardized tests (they’re right). They argue that testing turns schools into test-prep factories for months and even sometimes entire school years (they’re right). They argue that students start thinking of success as simply a matter of getting a high score on a test and lose sight of what actual human development and success could look like (they’re right). They argue that this will lead to schools cutting all non-tested subjects and eliminating major projects from classes and eliminating field trips and eliminating other non-test-related activities (they’re right). They argue that in the eyes of the school, students will lose all humanity and be reduced to a set of numbers that characterize how well they can repeat what their teachers have told them (they might be right about that, too). Let’s say that people who agree with most or all of these points are called “group 2.”

More and more people are starting to realize that proponents of assessment AND opponents of hyper-testing are BOTH actually making some pretty reasonable assertions. Many people are even starting to realize that group 1 and group 2 need not be mutually exclusive. That is, people might simultaneously consider themselves members of BOTH group 1 and group 2 at the same time. My guess is there are an increasing number of people who would categorize their views in that way.

This strikes me as a positive development. People who are in both groups would probably agree that feedback, assessment, and data are important parts of a student’s education, but that standardized tests don’t really provide all the information that is needed or that they might cause too much harm to other beneficial aspects of education.

How to please group 1 and group 2

Lisa Nielsen has an excellent post in which she provides some strategies for actually collecting and using data on things that actually matter: Transforming education by measuring what matters. Hint: It’s not test scores.

Here are a few things she wants schools to collect data on:

  • Which students have a plan to develop their passions?
  • Which students have strong advisers and advocates at school?
  • How satisfied are students with the support they receive at school?

She lists some more things school should be tracking and offers some models for how this could actually look when implemented in a school. I encourage you to read the full article.

At the beginning of this school year, I was thrilled to discover that the school I’m working in does actually try to track which students have strong relationships with teachers–as reported by the teachers–to try to find students who don’t (yet!) have these connections. I’m interested to see how well this system actually works this year.

Data can only be used to “measure success”–an unnecessary assumption

Those are all EXCELLENT things to be tracking and I am thrilled that there is a growing push to be focusing on these measures as opposed to just standardized test scores.

However, I’d like for people in group 1, group 2, or both to step back a bit from their assumption that the only use of data is to “measure success.”  Sure, this data that is being collected–particularly the data mentioned above–can and should be used to measure success and then to directly try to increase success. However, that is not the only use of data. Data can also be used to find patterns that wouldn’t otherwise have become clear while providing new insights into how good teaching and learning happens.

Certainly, those who use data to measure success do inherently do some pattern-seeking. When their system is working, if they see a student’s scores going down, they rightly try to intervene to get her back on track. If a certain teacher’s students show less improvement across the board than otherwise might be expected, the teacher (hopefully) gets some additional support. If a teacher sees that many of his students did uncharacteristically poorly on an assignment covering some particular topic, he knows to reteach that topic (hopefully in a different way the second time).

However, all of these examples simply take on the following format: “scores are dropping, let’s try to fix it so the scores go back up.” Besides the fact that scores don’t necessarily correlate to actual learning, there is not necessarily a problem with this model. In fact, when using some of the more meaningful types of data listed above–beyond just test scores–this is actually a pretty good way to operate a school: use changes in this data to signal that there is some problem with the school’s effectiveness and then try to explore ways of increasing that effectiveness, with the hope that these interventions will improve the data (and the hope that the improved data actually indicates improved learning and growth among the students).

Use data for pattern-seeking!

Besides using trends in the data to directly try to find ways to create better trends in the data, the collected information can also be used to search for patterns, which can provide some surprising and useful insight about the students, the teachers, and the school…..which can then be used to help teachers to teach better and students to learn better.

Here’s a hypothetical example:  Teachers certainly have access to grades and attendance information for students in their own classes, but these are rarely linked with each other, let alone with similar data from other teachers across the school. Maybe, as it turns out, students in this particular school who are absent are substantially less likely to turn in assignments that were due the day they missed. If this is the case, the school can then explore how to streamline the process of finding a missed assignment after an absence and also work with students to help them stay motivated and responsible enough to follow through with this process once they return to school.

Yeah, you could look at attendance data, and try to improve it (“95% of students were here this week…much better than 92% last week!”)….or you could look at students grades and try to improve them (possibly by encouraging them to turn in missing work), but having access to this information in a useful way–in this case, linked together–helps the school develop a deeper understanding of the problem and more insight into how it might be solved.

Maybe, at another school, there are two teachers: Teacher A and Teacher B. Every year, the students in these two teachers’ classes score around the school average on most assessments. In a few years, once the students graduate high school, the students have GPAs comparable to all of their classmates, have taken similar numbers of AP tests (and gotten similar scores), and gotten into colleges at the same rate as their classmates (and are going to colleges of comparable quality), on average. Maybe the school is even tracking some of the more important measures such as Lisa Nielsen’s above, and students who had Teacher A or Teacher B are still very normal compared to their peers when looked at through those lenses as well.

However, maybe, for reasons that aren’t entirely understood, it is discovered that students who had Ms. A in 7th grade and Mr. B in 8th grade do substantially better than all of their peers, including those who had only had one of those teachers. Let’s say this effect even occurs over multiple years of data over an extended period of time. This would be pretty interesting!

Why might this be? Maybe Ms. A provides students a super strong foundation in something that puts kids on track to get the most out of their year with Mr. B, who is particularly effective at building on Ms. A’s foundation. Maybe the teaching styles of Ms. A and Mr. B complement each other in a perfect way that sets students on an ideal track towards future success. Maybe Ms. A teaches something from one perspective, while Mr. B teaches it from a different perspective, and students benefit from seeing both perspectives so much, and in a such way, that they are able to excel in future classes.

The observation that students who had BOTH of those teachers are more successful in the long run–even more so than students who only had one or the other–leads to a deeper exploration of WHY that might be. The school would need to look into which of the possibilities above (or none of them) was actually taking place. Maybe whatever is discovered could then inform improvements in the practice of other teachers in the school (or even at other schools).

Noticing this pattern allows the school to figure out what can be learned from these teachers (even if they didn’t realize it themselves). It’s not a matter of saying “Mr. B., let’s figure out how to increase your scores,” but rather, “wow, this is pretty cool, lets figure out why this particular effect is happening and see if we can help other students get these same benefits.”

Finale–“Data: not just for assessment, anymore!”

Measure things that matter and then try to find deeper patterns in this information to better inform what works and what doesn’t (and even come up with some guesses as to WHY), and then try to test some of these ideas. In addition to the valid and justified desire to use data to see how you are doing and try to improve, this pattern-seeking perspective can reveal some deeper insights from your data and lead to some longer-term benefits. Collecting data is NOT just a tool for assessment!

Additional thoughts/caveats:

1. Just because some pattern is observed, does NOT necessarily tell you what is causing that pattern. For example, maybe there is an assistant principal at Ms. A and Mr. B’s school who really likes those two teachers, and takes some of the top kids every year and puts them in Ms. A’s class in 7th grade and then in Mr. B’s class in 8th grade. This may indeed be enough to explain the observed pattern of achievement for students who have had both teachers. Whenever some kind of interesting pattern emerges, more work is needed to be able to determine how interesting the root cause of that pattern might be–maybe some great new insight about how people learn, or it may be nothing of note.

2. Just like not all observed patterns are interesting, it is also not possible to observe every interesting pattern in this way–professional teachers in classrooms can pick up on things that are not represented in the data, but which have huge impacts on kids’ lives. This is meant to supplement the work of those teachers, not to replace it.

3. There is some overlap between this pattern-seeking and the concept of formative assessment. Teachers who make excellent use of formative assessment in their classrooms are probably already pretty adept at finding patterns in student information for a particular student over time or for a whole class over one lesson. The pattern-seeking mindset discussed here would seek to enhance teachers’ ability to do this well and to help them pick up on things a normal human might not naturally notice.

4. To actually start doing a better job of this pattern-seeking, school districts need better software that is designed, first of all, to have ALL of student data in one place–not spread across different software, and, secondly, to be designed to actually explore the data in this way, both by curious humans who want to play around with the data or test their guesses about patterns that may exist AND by statistical software on computers designed for this purpose.

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Must a leader be (perceived as being) full of certainty in order to be successful?

Posted on September 2, 2012 by Zachary_Goldman

I read an article by Ron Elving about Paul Ryan’s VP nomination speech at the Republican convention. The piece includes the following:

Ryan turns on conservatives with his incandescent devotion to a set of ideas. He is steeped in his world view, and the electricity he exudes comes from his deep-seated sense of certainty. [emphasis added]

Ignoring, for now, the speech itself and the specific world-view espoused by Ryan (much of which I object to in a pretty strong way), I am intrigued by the author’s claim that certainty can lead a speaker to exude political “electricity”–that appearing (and being?) absolutely sure of something can generate political support and energy.

Our society strongly values people who stick with their fundamental values even when the going gets tough (and rightly so). “Flip-floppers” who are perceived to have publicly changed their positions due to changing political conditions are vilified as indecisive and as searching only for political convenience–a situation epitomized by John Kerry.

More recently, Mitt Romney has altered his opinions on a number of issues to shift significantly to the right of the fairly centrist or even left-learning positions he held earlier in his career. Even on the short timescale between the Republican primary and the general election, a Romney adviser famously gaffed that once the primaries are over, shifting to positions that make more sense in the general election would be as easy as “shaking an etch-a-sketch” and starting over. This comment was obviously jumped upon by his primary opponents as a signal of Romney’s inauthenticity.

Certainly, flip-flopping for strictly political purposes is frowned upon, and seems to be harmful politically, at least while people are still talking about it. President Obama’s “evolving” position on LGBT issues is a murkier area. To many of his supporters, in the lead-up to the 2008 election and his first several years in office, his position on gay marriage and Don’t Ask Don’t Tell seemed surprisingly far to the right and some say that his recent left-ward shift was really just an adjustment to what he really believed all along.

Situations in which a politician shifts her views in one direction (instead of going back and forth) seem somewhat more accepted, even if there is a suspicion of political strategy underlying the move.

For me to respect a politician the most (or, for that matter, any leader in any context), I must believe that this person is actually growing, learning, and exploring new perspectives deeply enough such they sometimes may change their opinion. John Maynard Keynes summed this up nicely by supposedly saying: “When my information changes, I alter my conclusions. What do you do, sir?” I expect no less of my leaders.

In general, I feel safer knowing a leader has more information than I do, more experience than I do, and more insight than I do. However, knowing the world is a complex place, and that I cannot possibly understand every important issue deeply, I also expect my leaders to realize this as well.

That being said, I also seem to have more confidence in a leader who can choose a course and stick with it–to APPEAR to have a large degree of certainty about this path, possibly based on some understanding he has beyond my limited knowledge which only leads me to my own uncertain conclusions.

Thus, I must be able to trust that a leader is continuously growing, that her views are changing on an ongoing basis as she learns, that her views are subject to change based on new information AND that she simultaneously appears certain about her chosen course of action. This is a very fine balancing act, indeed!

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My (apparent lack of) peppiness as a helpful comparison for lack of student achievement

Posted on August 28, 2012 by Zachary_Goldman

Every morning, all the City Year people at my school stand outside for half an hour as students arrive in the morning and greet them with various chants and cheers designed to get them excited about their day and to generally increase morale among the students, faculty, and staff. Basically, we want to start off their day in such a way that they feel like school is a happy/fun place where they are loved and respected.

While I have always been very quick to greet people I cross paths with in the morning, the “power greeting” we are expected to do is certainly not something I am used to, nor is it something that comes particularly naturally to me. I have been working hard to push my comfort zone and do this chanting and dancing and cheering and rhythmic clapping and stomping as well as I can. Over the last few weeks, I have definitely started feeling more comfortable doing all of these things.

Since I have been working on this and since I felt like I was improving, I was very surprised and disappointed when a colleague told me today that my peppiness is not only insufficient, but in fact my lack of peppiness actually negates the peppiness of at least two other people, so I am letting down the team. I certainly appreciate the feedback and it is very disheartening to apparently be the weak link. Later in the day, our team leader spent several minutes reminding us of the importance of this part of our work and how it is required. I have no idea what prompted this, but due to my earlier conversation, I felt like I was letting the team down somehow.

I literally have no idea how I could improve my peppiness. I felt like I was doing pretty well, and certainly better than I was a few weeks ago. I’ll certainly try to get some pointers from some other people in the group on how I could improve.

Besides making me feel like a terrible person, this experience has provided an interesting avenue for some empathy with my students. City Year focuses its efforts on students who are off-track in terms of their attendance, behavior, or course-performance. I personally have never had any struggle in any of these areas whatsoever. Academics have always been pretty easy for me, I have never really gotten in trouble for anything (once in junior high, I forgot to turn in a paper and got a lunch detention), and I’m pretty sure I have never had a single unexcused absence or tardy.

Therefore, it is often hard for me to visualize how students in these positions must feel or how my potential interventions could affect their emotions, thinking, and decisions. So, the bright side of my apparent morning cheering ineptitude is that it gives me an approximate way to take the perspective of some of my students, who may be in a similar situation related to attendance, behavior, or course-performance, as opposed to morning cheering.

More importantly, I now have some new insight into how best to structure an effective plan to improve a student’s outcomes in such a scenario (or at least how NOT to).

For me, the morning greeting is something I am very motivated to do effectively, something I have been working hard to improve on, something I feel like I HAVE been improving on, and something for which I am deeply aware of how my team depends on my personal success.

Typically, I’d consider it to be a pretty big success if I could help a previously academically unsuccessful and seemingly unmotivated student to develop one or more of those mindsets about their academic work or about school in general. If I were to observe these mindsets in a student, particularly if none were present before, I’d conclude that this student was definitely on a much clearer path towards academic success.

However, it is also the case that this student may sometimes get a test back with a grade that she is not very happy with. In this situation, reminding that student about all of the reasons (s)he was already motivated to try to get a good grade is not very helpful and could, in fact, be counterproductive. Rather, this student just needs some support and some practice with whatever it was that (s)he was unsuccessful with.

Over the next few weeks and months, when trying a new strategy to support a student, I will try to do a series of quick thought-experiments where I think about a few current mindsets the student might have (“I am good at this;” “I am bad at this;” “I really want to do a good job;” “I never get a good grade;” “my parents really want me to get a good grade;” “I wish I didn’t always get in trouble;” etc.) and think through how a student with each of those mindsets might respond to a particular thing I might say to them.

Certainly, it is helpful to try to talk to them and uncover much of their mindset about school, with takes some of the guesswork out of this process, but going through the possibilities in my head seems useful to remind myself that I really don’t always know how the student is feeling, and that I should mentally test out things I might want to say to see if they would actually be helpful in all situations.

Telling me I’m letting down my team is not very helpful when trying to improve my morning greeting cheering ability. Giving me concrete advice is. For someone else in a different situation, it might have been just the opposite.

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Lesson idea for geometric series: Loan Amortization!

Posted on August 19, 2012 by Zachary_Goldman

Here’s a non-trivial application of the concept of geometric series with some actual context: calculating monthly payments on a mortgage.

The question:

You take out a loan for $50000 at an annual interest rate of 12%. If you want to pay off the loan in exactly 30 years, how much would you have to pay each month?

The details:

Interest is compounded monthly, so at the end of each month, the TOTAL amount that is owed is the remaining principal plus 1% interest. The monthly payment is then subtracted from this and the result becomes the starting amount for the next month, and the process is repeated each month. (The only reason I picked 12% as the interest rate is to make the monthly rate a nice round number. Real-life interest rates are–hopefully–much lower. At the end, students can compare results with different interest rates!)

An interesting concept:

Loan amortization leads to the surprising result that even if you pay the same amount each month, the principal decreases very slowly at first, and much more quickly towards the end of the loan period.

Here’s why: the monthly payment will pay off the interest from that month plus some amount of the principal. Early on in the loan period, while there is still a large amount of principal remaining to be paid off, the dollar value of the interest is higher than it will be later on (since the interest is being calculated from a higher principal). So, early on, each monthly payment will cover that month’s interest with only a small amount left to start paying down the principal. Over time, the principal gets smaller, so the interest each month on that principal gets smaller. Thus, if the payment is the same each month,  less and less of that payment will simply be covering that month’s interest and more and more will be going towards reducing the principal. The further into the repayment period, the FASTER the principal will be reduced. This is pretty surprising.

The problem:

At the beginning of the 30 years, how can I determine a fixed monthly payment that will actually result in exactly zero remaining balance at the end of 30 years (360 monthly payments in the future)?

A slightly easier question:

If I pay a fixed amount (say $700) each month, with the same principal ($50000) and same interest rate (12%), will I ever pay off the entire loan and, if so, how many months will this take?

One way to solve the “slightly easier question:”

Calculate month-by-month:

total owed at the beginning of next month = total owed at the beginning of this month + this month’s interest – this month’s payment.

Total owed at the end of month 1:

= $50000 + $50000*(1%) – $700

= $50000 + $500 – $700

= $49800    (the payment of $700 went to cover $500 of interest and $200 of principal).

Total owed at the end of month 2:

= $49800 + $49800*(1%) –$700

= $49800 + $498 –$700

= $49598 (the payment of $700 went to cover $498 of interest and $202 of principal)

This $49598 is the starting amount for the third month, and so on. Each month, as the principal slowly declines, less and less interest is charged each month, so more and more of the monthly payment goes towards paying off the principal.

Paying $700 a month, you would have to repeat this process 126 times to eventually reach a balance of zero. Thus, it will take 126 months (which is 10.5 years) to pay off this loan.

Doing this process repeatedly until getting a balance of zero (which would take 126 steps in this example) is a good exercise for any budding computer programmers in your class, or a good opportunity to have the whole class practice using Excel!

Now a solution to the real problem:

Recall that we wish to calculate the monthly payment that will allow the loan to be paid off is exactly 30 years. One way to do this would be to do exactly what we did above, pick a payment, and figure out how many months it would take to pay off the loan. If it will take less than 360 months, try a lower payment next time; if it will take more than 360 months, try a larger payment next time. Repeat this until you narrow in on a monthly payment that takes exactly 360 months to reach a balance of zero. This method will actually get you pretty close very quickly (the magic of a binary search)!

However, there is a more elegant mathematical solution involving geometric series:

A geometric series is a list of terms added together where each term is some constant multiple of the previous term. For example:   1 + 3  + 9 + 27 + 81 is a geometric series with each term being exactly 3 times the previous term. The series 4 + 2 + 1 + 1/2 + 1/4 + 1/8 is a geometric series with each term being exactly 1/2 of the previous term.

So how does this relate to the loan payments? First, some setup: let’s call the amount owed at the end of the first month, T1, the total at the end of the second month T2, and so on. Also, let p be the monthly payment (which we are trying to calculate).

Using the same interest rate as above (12% yearly), recall that total owed at the beginning of next month = total owed at the beginning of this month + this month’s interest – this month’s payment, so the total at the end of the first month could be written like this: T1 = 50000 + 50000*(1%) – p.

For simplicity, we can combine the first two terms and write 50000*1.01. This just represents the principal and that month’s interest combined (this is the total amount you owe). Thus, T1=50000*1.01 – p.

The total at end of the second month is T1 plus interest minus the monthly payment: T2= T1*1.01 – p.

Recalling that we already have an expression for T1, we can plug that in the expression above to calculate T2 = T1*1.01 – p = (50000*1.01 – p)*1.01 – p. Simplifying a bit gives: T2 = 50000*(1.01)2-(1.01)p – p.

Now, T3=T2*1.01 – p = (50000*(1.01)2-(1.01)p – p) *1.01 – p = 50000*(1.01)3 -(1.01)2p-(1.01)p – p.

To calculate T4, again take T3 and multiply by 1.01 and subtract. After simplifying, T4=50000*(1.01)4 -(1.01)3p – (1.01)2p -(1.01)p – p.

Recall that 1=1.011, so the very last term becomes (1.01)0p instead of p (you’ll see why this matters in a moment), and we have T4=50000*(1.01)4 – (1.01)3p – (1.01)2p – (1.01)1p – (1.01)0p.

Let’s look a little more closely and try to break down what this tells us. The first term, 50000*(1.01)4 can be thought of as the total amount we would owe on the loan after 4 months is we didn’t pay at all—this is the just the principal (50000) plus the interest (1%) added each month for 4 months.

The other terms just tell you how much you have saved by making a payment of p each month since the loan began. The last one, (1.01)0p, which is really just p since (1.01)0=1, is just the payment made this month (which is subtracted from the total you will owe next month). The next to last term (1.01)1p is LAST month’s payment (plus an extra month’s interest that would have been paid on that amount, if you hadn’t already paid it last month).

The next term, (1.01)2p, is the payment you made TWO months ago (plus the TWO extra months of interest that would have been paid on this amount had you not paid it two months ago). Likewise, the (1.01)3p term is the payment you made THREE months ago (plus the extra THREE extra months of interests that would have been paid on this amount had you not paid it three months ago).

Ignoring, for a moment, the leftmost term, we have the following series:  -(1.01)3p – (1.01)2p – (1.01)1p – (1.01)0p. Each term is exactly 1.01 times the term to its right, so this is a geometric series! There is a term in this series for every month you make a payment. We’re assuming this is a 30 year mortgage, so there will be 360 months and 360 payments, so by the time you have paid off the loan, this series looks like: – (1.01)359p – (1.01)358p – (1.01)357p – … – (1.01)2p – (1.01)1p – (1.01)0p, where the ellipsis indicates lots more terms in the middle.

For simplicity, I’ll multiply through by -1 and call the resulting series S. Thus, S =  (1.01)359p + (1.01)358p + (1.01)357p + … + (1.01)2p + (1.01)1p + (1.01)0p.

Now to calculate T360 (the total amount of money still owed after 360 months) , we have to put the “principal” term (that we had temporarily been ignoring) back in, so T360 = 50000*1.01360 – (1.01)359p – (1.01)358p – (1.01)357p – … – (1.01)2p – (1.01)1p – (1.01)0p = 50000*1.01360 – S.

However, we already know what T360 should be! After 360 months (30 years), our loan should be paid off, so the remaining balance should be zero. Thus, T360 = $0.

Since T360 = 50000*1.01360 – S and T360=0, we now know that 0 = 50000*1.01360 – S. Adding S to both sides gives S = 50000*1.01360, evaluating the expression the right gives S = 1797482.

Now, solve for p, the monthly payment:  S= 1797482 and S = (1.01)359p + (1.01)358p + (1.01)357p + … + (1.01)2p + (1.01)1p + (1.01)0p, so we have 1797482 = (1.01)359p + (1.01)358p + (1.01)357p + … + (1.01)2p + (1.01)1p + (1.01)0p. Call that equation 1.

Luckily, this is a geometric series, so we can use a cool algebra trick to solve for p. First, create equation 2 by multiplying both sides of equation 1 by an extra factor of 1.01. So, equation 2: 1797482*1.01 = (1.01)360p + (1.01)359p + (1.01)358p + … + (1.01)3p + (1.01)2p + (1.01)1p. Note that this just has the effect of increasing the exponents on each term on the right by 1.

Now, subtract equation 1 from equation 2:

On the left side, we get 1797482*1.01 – 1797482 = 1797482*.01=17974.82.

On the right side something interesting happens. For most of the terms on the right side of equation 2, the exact same term also appears in equation 1, so when we subtract, most of the terms fall away. We are left with (1.01)360p – (1.01)0p. Evaluating the exponents and simplifying gives us 34.95p. (This step exactly models the steps to derive the “formula” for summing a geometric series. In fact, that’s what we just did!)

Setting the left side equal to the right side, we have 17974.82 = 34.95p. Divide both sides by 34.95, and we get that p = $514.31. If we pay $514.31 each month, we’ll pay off the loan in exactly 30 years! We just solved our equation to calculate the monthly payment by simplifying a geometric series!

It is interesting to think about what that geometric series actually represents. It does NOT represent the payment each month, the interest each month, the balance at the end of each month, etc. It DOES represent the amount of money you have paid, plus the future interest that was never charged on that amount since it had already been paid.

In other words, S,  the sum of our geometric series, above, represents the total amount you would have paid by waiting until the very end of the loan and paying it all off as one lump sum (although your house would certainly be foreclosed long before that!). In this case, S=$1797482, which is almost 1.8 million dollars! Paying our calculated monthly payment of $514.31,  you will pay 514.31*360=$185151.60 over the life of the loan–a much better deal!

Conclusion:

Call a bank! Find out their rates and loan details and have the students calculate a monthly payment. Ask the bank to see if their numbers match!

There are also lots of free amortization calculators available online. Have you students do the calculations and then plug their principal, interest rate, and number of months into one of these websites to see if their results match the software’s results!

Possible extensions:

Students can explore the implications of changing the principal, down-payment, interest rate, number of months, etc.

What happens if the amount someone pays in a particular month is less than the interest charged that month?

What happens if the amount someone pays in a particular month is exactly equal to the amount of interest charged that month?

Students could also do some research on loans such as adjustable rate mortgages.

HT: Gautam Kaul’s Introduction to Finance course on Coursera.
P.S. I wish wordpress had a LaTex editor!
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Teacher fired from Art Institute (of California–OC) for not requiring an e-textbook…

Posted on August 18, 2012 by Zachary_Goldman

http://www.huffingtonpost.com/2012/08/14/mike-tracy-art-institute-textbook_n_1776544.html

http://www.takepart.com/article/2012/08/18/teacher-fired-refusing-make-students-buy-pricey-textbooks

According to the articles above, at this school, owned by a for-profit company, the policy is that each teacher is supposed to assign students to “rent” an e-textbook for each course. Students are required to pay a specific e-book distributor (with ties to the school) in order to access their e-books.

The teacher apparently hadn’t required any textbook when he had previously taught the course (neither a physical book nor an e-book). He decided the e-book wasn’t necessary, and continued to not require a book for the course since he didn’t think a book was necessary and to save his students some money.

The president of the school threatened to fire him for insubordination if he didn’t require his students to rent an e-textbook with the particular company. The teacher still refused, and was forced to resign.

I am definitely curious to hear the school’s side of the story. It is hard to believe that this really happened as it was described in these articles. However, assuming we do have all of the facts, this is a pretty obnoxious example of a for-profit college making decisions that are not in the best interest of student learning.

I don’t have any inherent objection to a for-profit company owning/running a college . I (maybe naively) assume that in most cases, companies with a profit motive still have an incentive to do what is best for their students since this should directly translate into a greater number of happy students and faculty, which seems like it would lead to financial benefit for the company in the long run.

It is possible that this is one (egregious) counter-example to that principle. There are probably other counter-examples that I’m not aware of.

The other possibility is that the principle (making decisions to promote student learning has a long-term positive effect on profits) actually still applies, but the school just did a poor job of recognizing that this particular e-book policy raises profits in the short term (a favored corporate partner sells more e-books), but only at the expense of a long-term loss in profits due to bad PR and pissed off students.

Either way, props to this teacher for standing his ground. I hope it will be pretty easy for him to find a new job…

Posted in Higher education | Tagged , , , | 2 Comments

Effects of repeating content in future courses

Posted on August 16, 2012 by Zachary_Goldman

I continue to be impressed with Coursera, a provider of free online courses from an ever expanding number of big-name universities. I am lightly participating in several courses (and I am signed up for dozens more over the next several months). By “lightly participating,” I mean that I am watching the lectures and occasionally participating in discussions with other students around the world on the forums (but not doing–or turning in–most of the official coursework).

As I pointed out before, while it is debatable as to in what ways a Coursera course might be pedagogically worse or better than an in-person course on the same subject, it is NOT debatable that a Coursera course is pedagogically superior to not studying a particular topic at all.

As more and more courses are offered on Coursera (and other similar sites) and as more people participate in these courses, there will be more and more rising freshman at colleges across the world who will enter with a history of having taken one or more such courses as a high school student and in some cases even as a middle-schooler.

How should colleges deal with these students? As of right now, most of the classes offered on Coursera are for intro courses in a particular subject. My guess is that most colleges, at least in the near future, will still require students to actually take the in-person version of these courses in order to count the credit and in order to register for the next higher level courses in that subject area.

Here’s my question: is this a good thing pedagogically?

For students, like me, who don’t tend to do much of the actual work for an online course, it is hard to deny that the additional practice gained by doing homework and studying for tests is extremely valuable on top of just keeping up with lectures and discussing with other students online. Students in that situation, and even students who actually do all of the practicing associated with an online class would most likely develop a deeper understanding by seeing the material again.

As I entered college, I had never done any online courses, but I had taken a large number of AP courses which were supposedly comparable to intro courses in college. In a few cases, based on departmental requirements, I actually ended up taking intro classes that overlapped heavily with what I had studied in prior AP courses. Seeing similar material a year or two later, and sometimes with a slightly different emphasis or context, definitely helped me to develop a deeper and wider understanding of the material.

However, while seeing similar material presented again in an intro class did indeed help me strengthen my understanding, this did not help me strengthen my understanding nearly as much as did taking higher level courses which expanded upon the earlier material, applied the earlier material in new ways, or provided a wider context in which the earlier material formed a small piece of a larger conceptual understanding. Taking the intro classes certainly strengthened my understanding, but at the cost of having the time for one fewer higher level class (that might have increased my understanding of the subject, and even the specifics of the intro course, more thoroughly than just doing the intro course).

When I was teaching math in Georgia last year, the curriculum (which has since been transitioned to the Common Core) seemed to take into account that cycling through material again–in a deeper or wider way, the second time–seems to help students understand the material a little more clearly. However, the problem with that math curriculum was that in order to do this cycling through topics, some concepts and techniques ended up seemingly randomly in strange grades levels, with no context for their relevance or importance. For example, in 7th grade, students were expected to learn how to use a compass and straightedge to do some basic constructions (copy an angle, bisect an angle, etc.).  Students are asked multiple choice questions like “when copying an angle, after you draw an arc through the angle, what is the next step?” The students have no idea how this relates to the future concept of proofs they’ll start seeing in geometry class in high school. Many of the TEACHERS have no idea how this relates to the future concept of proofs which the students will start seeing in high school. The intent seemed to have been to touch on topics early and then cycle back in later grades, but teaching random topics out of context creates much more harm than any benefits that might exist by studying a topic twice.

Cycling back through material may sometimes be beneficial, but certainly not always. This makes me even more curious to see how colleges will respond to the increasing number of cases in which students will be put in a position to possibly retake (in in-person form) a course they have already (partially?) completed online.

This becomes even more important as students in the next few years start entering college not just with experience in intro courses, but also some (or even nearly all) of the topics explored in higher level courses at that college. What will colleges do when a student comes in claiming to have taken classes equivalent to all of the required courses in a major? More importantly, I wonder how colleges can reassess their assumptions about how education is supposed to happen (a major, some number of general education classes, and maybe a minor or three) in the new situation in which people have the opportunity to explore some of these things–with varied and uncertain levels of sophistication and depth–outside of a “normal” path through college…

Posted in Math education, Online education, Personal Experiences | Tagged , , , | Leave a comment

Is Algebra Necessary?

Posted on August 7, 2012 by Zachary_Goldman

Andrew Hacker recently had an op-ed in the New York Times entitled Is Algebra Necessary?   

His answer is “no” based on an argument that algebra is hard and that people who do poorly in algebra tend to drop out and that most people don’t ever use algebra in their jobs.

(Some of) his points actually do make sense, but I don’t agree that with his conclusion that algebra should be dumped from the curriculum. First, his suggestion that the history and philosophy of mathematics should also be a focus of math classes and his desire to more explicitly teach quantitative reasoning are exactly right and there should absolutely be more emphasis on exploring math from these perspectives in class. This helps students realize that math is a field that is actually real and has a basis in something, and is not just some random set of tasks your teacher tells you to do.

He is also right that the total quantity of mathematical topics that are taught could be reduced. For me, however, this would be for the sake of actually having an opportunity to explore topics more deeply and then apply topics more deeply (as opposed to his apparent desire just to “waste” less time on math). As more and more math standards get added to the curriculum each year, students actually understand each one less and less (don’t even get me started on the harm caused by how disconnected some of these standards are from each other).

He is also right that courses in “citizen statistics” should be offered to everyone. However, I think his description of such a course is not nearly rigorous enough. For example, learning how to calculate the CPI and other similar tasks, as he suggests, fail to explore the most important concept embedded in any stats course: correlation does NOT imply causation (his conclusion that algebra should not be taught because people who do poorly in algebra tend to drop out would, itself, have benefited from a deeper focus on that particular principle, for example).

Even if it is true that algebra is a major factor causing people to drop out, if it is eliminated from the curriculum, as he suggests, this makes a high school diploma pretty watered-down in the eyes of most people, and may actually further reduce the quality of jobs available to someone with only a high school diploma. Using his employment perspective, graduating from high school without having taken algebra seems fairly similar to just dropping out in terms of the real or perceived depth of knowledge of potential employees.

Of course, the argument that employers will think that passing algebra means you have at least a basic understanding of reality is actually one of the smaller reasons why algebra is important. Here are some grander reasons:

1. Algebra is all about finding patterns in the world and then using them to your advantage to help understand a situation and make predictions about the future behavior of a system. Algebra is about developing a deep understanding of something and then using that understanding to create tools for yourself. This is a critical skill in any field of work (and life).

2. Algebra is beautiful. For example, the obvious statement “if two things are equal, they will still be equal if you do the same thing to both of them,” sounds pretty boring and obvious. It turns out to be a surprisingly powerful insight to help create tools to solve problems such as, “If I start with a number and then add 7 to it, and then multiply it by 9, and then divide it by 3, and then square it and the result is 64….what was my original number?” You could certainly do a bit of (tedious) trail and error, or you could turn it into an equation and use that seemingly trivial but surprisingly potent observation above to solve it! Cool!

3. The concept of a variable is fundamental to my understanding of the world. The idea of a box with some meaning which contains a (possibly changing) quantity is a super-powerful way of conceptualizing quantities in the real world.

4. The concept of a function is ALSO fundamental to my understanding of the world. The idea that the quantity which lives in one of those meaningful boxes described above might depend on a quantity that lives in another (or possibly many other) meaningful boxes is an even more super-powerful way of conceptualizing reality! (Next month, the number in the box called “selling price of my house” probably depends on what is contained in the boxes that describe what improvements I have made, where it is located, the value of my neighbors’ houses, and even *gasp* the current number that is in that same “selling price of my house” box.)

5. Variables and functions form the basis of all computer programming and computer science, which is currently a pretty important avenue of human progress (let alone a pretty important factor in lots of people having jobs).

6. There are probably more good reasons. Feel free to add your own in the comments below.

P.S. Let’s consider a sentence near the end of Hacker’s piece: “Yes, young people should learn to read and write and do long division, whether they want to or not. But there is no reason to force them to grasp vectorial angles and discontinuous functions.”   Why on Earth would he advocate that students SHOULD learn long-division, but SHOULD NOT learn algebra? I seriously cannot come up with a coherent theory which could explain this. Also, using a bunch of complicated-sounding terms to make math sound complicated and irrelevant is an unnecessary cheap shot.

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