Super-Sized, Circumcised, Circumscribed

Today was supposed to be observation day. At least once per semester since I arrived at Suffolk, I’ve been observed in the classroom by one of the department’s senior faculty. Just so that you don’t get the wrong idea, this isn’t some kinky thing where I prance in front of the examiner in a Speedo. No, outside of Europe, nobody wants to see that kind of nonsense. The faculty actually want to watch how I teach a class.

While I always try to outwardly act as if this is no big deal, the thought of being observed gets me super excited. It gives me a really good excuse to spend that extra hour preparing a special lecture. With that in mind, this observation session came at a pretty awkward time: we just finished the course material and this is review week with the final coming next Tuesday. This is usually when we kick back, tighten up the few topics that are hurting everyone and watch Raiders of the Lost Ark. Instead, though, I decided to review old material by introducing something slightly new.

Enter the Extended Law of Sines

It always surprises me how few people have heard of the extended law of sines. In short, it says that the quantity appearing in the law of sines is equal to the diameter of the circumcircle of the given triangle. If the radius of the circumcircle is , then the theorem is usually written as:

(where and represent the sides and angles of triangle, respectively, with opposite angle , etc.). Aside from giving a geometric interpretation of the terms in the law of sines, the result is also quite useful (I’ll give a fairly involved exercise at the bottom where it is possible to use it).

I wanted the students to prove the result, but I really was not confident in their elementary geometry skills, so I set things up for them. I told them to start from the triangle and draw the circumcircle. Naturally, they didn’t know what I was talking about and wondered if I’d said what they thought I’d said. Planning for this, I started explaining out loud how you physically construct the circumcircle. Eyes glaze. Planning for this, I opened up handy-dandy mathopenref.com to show them what I can’t vividly describe verbally. Next came Geogebra where we found the circumcenter of the triangle and connected it to two of the vertices of the triangle. I now referenced the Central Angle Theorem which, again, nobody knew. Thankfully, Geogebra calculated the angles for us and since these students believe the convenient falsehood that one example proves that something always works, we were golden.

After connecting a few more dots, we were ready to prove the result. Here is the worksheet I gave them to work on in groups. It is nothing fantastic, but even some of the students whose attendance has been…spotty…worked out the answers. The beauty of the last question is that they need to shift gears and use the law of cosines before they can solve the problem.

Oh, and the professor who was coming to observe me? He didn’t show up. Go figure.

 

Exercise: With the notation as above, prove that the triangle is isosceles if and only if

.

Evil SBG

It’s been a while and I really ought to do some reflection on my first true semester of SBG. But, alas, the train ride is short and I wanted to throw something out there. I just finished proctoring the Putnam exam and had a wonderful idea for how to make SBG more evil.

You see, there are times when I really want to add weights to my topics. For instance, there is no way that finding the slope, y-intercept, etc. of a line should be worth as much in a precalculus class as graphing rational functions or using the law of cosines. It isn’t that I don’t think it is as important—it is probably more important in many cases—it is just that knowledge of lines (at that level) is considered remedial material, though for some it isn’t.

So how should I weight things? Should I do it by order of (my) perceived importance? How about by difficulty level—again from my point of view? I realized that while I may think I know the correct weightings, it is unlikely that I actually do. Consequently, I came up with this evil scheme. It isn’t really something serious, but I’m not completely joking either.

At the beginning of the semester, inform the students that each topic (or standard if you prefer) will be weighted by an amount inversely proportional—you’ll have to explain what that means—to the final average class score for that topic (or standard). The topic with the lowest class average score would then be weighted the highest. Those topics which everyone figures out would be weighted the least. This would give students (especially your top tier students) an incentive to attack those topics the whole class is failing. Potentially, students could try to game the system by learning the really hard stuff so well that it doesn’t count for so much. Hah! Now wouldn’t that be something?

Integration by Parts 3

Tricks of the Trade

(with Professor Glesser)

In the first two installments of this series

Integration by Parts 1
Integration by Parts 2

we introduced integration by parts as a way to compute antiderivatives of a product of functions and we saw how certain integration by parts problems are handled more efficiently with the so-called tabular method (or, in Stand and Deliver, the “tic-tac-toe” method). In this post, we will consider the following question: As integration by parts requires the making of a choice—which is your u and which is your dv—how can we make this choice so that the resulting integral is easier to compute?

From the Mailbag

Über-reader CalcDave wrote in the comments to the last post in this series that,

I usually make a show of how sometimes the order does matter…That is, I’ll let and the first time and then go through it and say something like, “Well, that didn’t get us much of anywhere. What if we switch up our u and dv this time? Let’s let and .” Then when you work it through, everything cancels out and we’re back to the original problem.

Indeed, Dave. Let’s take a look at what happens if we switch it up.

Egad, Dave is right. Since the product of the terms in the last line of the table is what we will need to integrate, doing it this way just makes things worse.  Ah, but what if we start with the cosine on the left and then switch it up? Oh, yeah, we’ll just get back what we started with. This suggests that we should always put a polynomials on the left so that it doesn’t go up in degree. It turns out that there are several examples where this is precisely the wrong thing to do. We implicitly saw this in the first post, but let me give you a couple of more explicit examples.

If we split this up using our ‘rule’ to always put the polynomial on the left, then we are forced to integrate . Let’s say you just happen to know the antiderivative of is (I didn’t, although I can use integration by parts to figure it out!). You would now get:
and be forced to integrate the monstrosity on the right. Not for me thank you. However, if you put the on the left, we get:
and at the very least we have gotten rid of the . In fact we have done more, but we’ll have to wait until the next post to resolve this.

Another example is . Although we did integrate in our first post, it gave an answer of and we don’t want to integrate that since it we don’t know how to integrate (in a future post, we will resolve this last problem directly). On the other hand, if we put the on the left, the derivative will return and the natural logarithm is gone. So when does it pay to put the polynomial on the right? Whenever the derivative of the other function changes it into an algebraic function, it will be right to integrate the polynomial. Otherwise, you should differentiate the polynomial. If we also include trigonometric functions and exponential functions, the rule of thumb is:
Logarithms Inverse-Trig Algebraic Trig Exponential

This list represents a good order in which to choose your u in the following sense: if you have two functions, whichever comes first in the above list should be your u. Some people enjoy a good mnemonic to memorize the order. I’ve heard the following:

LIATE rule (or alL I ATE rule)

Lions In Africa Tackle Elephants

Liberals In America Typify Elitists

Little Indians Are Tiny Engines

Lets Integrate All The Equations

This says, for example, that when confronted with , differentiate the and integrate .

Next Time

In our next segment, we will introduce the box method for handling several of the integrals left unsolved in this post.

You don’t know it until you know it

During my finite mathematics exam today, I had a little bit of time to ponder and I came up with two ideas worth exploring. Most likely, I shouldn’t implement them this semester, but I haven’t ruled it out. For posterity and potential discussion, I offer the following.

Filling the Empty Bucket

As a first approximation to the process of learning course content, imagine an empty box. This does not represent the knowledge of most of our students. Everyone has a few items in their box: some old postcards, two dead AA batteries, a dirty single sock, a couple of credit card applications, etc. The goal throughout the course is to fill that box. On occasion, a student might lose some things in their box after partying with Jack from next door and his cousin Daniels. However, with any reasonable effort, a student can keep the flow going into the box rather than going out. At the end of the course, their grade might simply be the percentage of the box that gets filled.

From a grading point of view, this suggests that my usual way of posting grades has a significant flaw. At any time, a student can see what their grade is, computed as a percentage of the possible points so far. Ah, there is the rub. A student might do well on the first quiz, regress a bit on the second, look at their grade and see 78%. Not great, but not too shabby considering they only need a C to graduate and they didn’t expect to be so close to a B-. Somehow, I’ve already lost. What if, instead of telling them how they’re doing relative to the available points, I gave them their grade in terms of the whole box? If my philosophical position is that their knowledge at the end of the course rather than their intermediate knowledge should determine their grade, then I should be telling them things like, “You have now mastered 21% of the material.”

There is an obvious problem with the original box metaphor: boxes are generally of constant size. It can’t all be about knowledge, though. I want their box to grow. This is not about content standards, but about them learning to think: the application of knowledge in logical and creative ways. How should I reflect that in the gradebook? Or is a one semester course insufficient to judge such growth? These aren’t new questions. David Cox explains things quite well here. I don’t have any new answer yet…but I’m still thinking.

The Exam Enigma

I mentioned in my last post how frustrated I am by the students adherence to the typical college way of studying. I had considered changing the names of exams to something less obvious; perhaps it might fool them. But who I am kidding. If I put it on the schedule, the students will figure it out. Maybe the whole problem is that I separate things in the first place. What if instead of treating these three exams as tests of retention, I simply clumped them in with the assessable standards portion of the grade? Right now, if a student does poorly on the standards, but studies hard and pulls off a good score on the exam, they get two grades: one says they don’t know what they’re doing, the other says they do. That makes no sense to me. I suggest (to myself) that I don’t differentiate quizzes and exams in the gradebook. The 75 minute exam should count no more than a daily quiz, except that the student will see more standards than usual on the test. This remands retention to the final exam.

Does it give the student an incentive to study throughout the term? Maybe it has the opposite effect. Maybe students will think nothing of the 75 minute exam and save up their study power for the final exam. It would be suicide, of course. They certainly won’t be able to cram the material in such a short time. Moreover, under this system, the final exam would be 20% and the standards + exams longer quizzes testing standards would be 70%. Some would still put all of their eggs in the final exam basket (another basket metaphor?), but that person won’t be helped by any system of assessment I try.

A Final Thought (read “Vent”)

I was told today that one of my students was complaining about homework not being assessed. This student is having trouble because he or she isn’t doing any of the homework, nor asking about homework problems, nor writing me emails, nor attending office hours, etc. The student freely admits all of this, but, for not requiring homework, blames me. What is going on when people want to be treated like little children? Nothing is stopping the student from doing the homework problems but himself or herself. “But if I do the homework, how will I know if I got it right?” Aside from asking me, they could just look up the answers in the back of the book. I put up videos on youtube of me working the review problems. I put up links to videos of people explaining each standard. I beg for students to communicate with me outside of class. But it is my fault this student is struggling and because I don’t require him or her to do the homework? Well, sign up one more for the nanny state. Perhaps a class discussion is in order; let the kids doing the work hear from the kids not doing the work and see what they say to them.

An exam by any other name

As I’ve mentioned in the past, I don’t have full control of my finite mathematics course. I asked for certain leeway in my grade breakdown, but had to settle on leaving the four class tests at 60% of the overall grade (in contrast to my precalculus where there are two tests making up 30% of the grade). Consequently, the students still treat my class as if nothing counts until exam week. Sure standards comprehension make up 30% of the overall grade, but they aren’t EXAMS!!!! How do I convince these kids that the typical college attitude will not cut it? They have an opportunity to spend four years living the life of the mind. Instead, they spend four years taking their mind off life. The night before a quiz, I stayed up late making screencasts of linear programming problems. One student told me how much he appreciated it, only he hadn’t had time to watch it because, “you know, the Yankees were on.” Guess how he did on the quiz?

Maybe I have to follow the rules and set my percentages like everyone else, but is there a chance that I could change the name (retention assesments?) and de-emphasize the importance of exams so that the day-to-day stuff gained relevance? Or are students socially constructed to work this way: slack, slack, slack, cram, rinse, repeat?

Integration by Parts 2

Last time on

Tricks of the Trade

(with Professor Glesser)

we introduced integration by parts as an analogue to the product rule. We start this post with an example to show why the method can become tedious.

Consider

As there is a product of functions, this seems ideal for integration by parts. A question we will take up in our next post is which term we should look to differentiate (i.e., be our ) and which we should antidifferentiate (i.e., be our ). For now, I will give you that a sound choice is

With this, we get

.

Using the integration by parts fomula:

we get

Using linearity, we reduce the question to solving .

Hold on, now. Is that really an improvement?

Yes, because the power of is smaller. But, I’ll grant you that life doesn’t seem much better. Essentially, we need to do integration by parts again. So, we rename things:

and we get

and after using linearity, we only need to compute .

Check please!

Before you get up and leave, notice that the power of is one less again.

Whoo-hoo. Yay, capitalism!

Seriously, each time we do this process, the exponent will decrease by one (since we are differentiating). So we “only” need to do it two more times.

You suck, Professor Glesser

Agreed. This is why it is nice to automate the process. I first learned this by watching Stand and Deliver over and over while in high school. I am not much of a fan of Battlestar Galactica (nerd cred…plummeting) and the few times I watched, I thought Edward James Olmos’ portrayal of William Adama was really flat; I thought Olmos was mailing in the performance. The most likely reason for my feelings? If you’ve never seen it, watch Stand and Deliver and Olmos’ portrayal of math teacher Jaime Escalante. Now that was a performance. Anyhow, here is the clip I watched incessantly.

I decided on a different notational scheme, but the method is the same. We make the following observation: when doing integration by parts repeatedly, the term that we differentiate will usually be differentiated again. That is, (abusing notation) the becomes our new . If you like, the formula for integration by parts has us multiply diagonally left to right () and then subtract the integral of the product left to right along the bottom ():

The next iteration of integration by parts gives:

Essentially, this creates an alternating sum. In practice, it means we can set up the following chart where, going down, we differentiate on the left until we get and antidifferentiate on the right as many times as we differentiated.

Notice here that we are condensing quite a bit of notation with this method since we are no longer using the u, v, du, and dv notation. But, we are getting out precisely the same information. We draw diagonal left-to-right arrows to indicate which terms multiply and we superscript the arrows with alternating pluses and minuses to give the appropriate sign.

We don’t need to draw a horizontal arrow on the bottom since that would simply give us the antiderivative of . Following the arrows and taking account of signs, our antiderivative is

Could you do that again?

Let’s try a different example, a little more complicated. Say we want to compute . We simply set up the chart where, going down, we differentiate on the left and antidifferentiate on the right:

and follows the arrows to get

as the antiderivative for .

I think I need a break

Indeed. Next time we’ll take this a step further and show how to handle some situations where neither function is a polynomial. This will also bring up the question, again, about how to choose which function to differentiate and which to integrate.

Integration By Parts 1

This is the first in a series of posts on one of my favorite methods of antidifferentiation: integration by parts. I didn’t love it at first, but a little practice and a few tricks made me appreciate it. Teaching it, well, that is where the love affair begins.

Tricks of the Trade

(with Professor Glesser)

What are you talking about?

Let me assume that the reader is familiar with basic differentiation (including the product rule) and antidifferentiation of some basic elementary functions, i.e., the reader knows such facts as the power rule and how to antidifferentiate exponential functions as well as sine and cosine.

Integration by parts is an analogue to the product rule for derivatives (which tells you how to differentiate a product of functions). In the language of differentials, we have

for functions and of some common variable, say . Integrating both sides, we get

.

The usual form of the integration by parts formula is now obtained by subtracting a term:

.

Uh…What?

An example may be helpful. A canonical first example is . The typical calculus student, fooled by the simplicity of the sum rule and not having the product rule in mind, will incorrectly assert . Of course, differentiating shows that this answer is wrong. Why? Well, because antidifferentiation is additive but isn’t multiplicative.

So let’s try the integration by parts formula. We start by noting that is the derivative of , i.e., . Consequently, we could write

.

We may now apply the integration by parts formula where and . This gives

.

Could I see one more?

Sure, here is a less obvious example. Consider .

Wait, there is no product of functions.

There is a product; it is just a bit silly. You see . Yes, it is one of those kinds of tricks. Now, I know that is the derivative of and so I can use the integration by parts formula with and . This gives:

.

How do I keep everything straight?

A very common bookkeeping measure is to make a little table including and . For our first example, you would start with:

You then compute and to complete the table:

You can then simply plug everything into the integration by parts formula.

This isn’t so bad. Why do you need multiple posts?

For those who don’t know the punchline, I won’t spoil it here. It suffices to say that there are some harder problems out there and there are some really efficient ways of handling these difficulties. Stay tuned!

Collecting Unlike Terms

The school is year is in full swing and I’ve been so busy grading that I’ve been neglecting to record my ultra-profound thoughts. As I’ve forgotten those, you’ll have to settle for my ultra-not-so-profound thoughts.

My flavor of SBG is working

One of the downsides of this new approach for me is the amount of grading I’ve thrust upon myself. Between two classes of twenty-three students taking two quizzes a week, I feel like the grading is non-stop. I apologize to all the high school teachers reading this, as they probably consider my grading load a dream; there is an acclimation process, I assume, and I’m still at the front end of it. However, I’m giving feedback and learning a heck of a lot about what the students aren’t doing well. The grade breakdowns (see here, for example) help me identify weak areas (either in my explanations or the background of the students) and I can spend extra time on things. For instance, we are currently covering circles and ellipses (this is my way of introducing function transformations) and one of the things I have the students do is convert the standard form into the center-vertex form. This uses completing the square, a skill I incorrectly assumed they had. I suppose I knew they wouldn’t have that skill (most of my math majors struggle with it), but I guess I just wanted to believe. In this case, I didn’t even have to give a quiz to know that the students needed to spend some quality time working on this skill. I broke the students into groups, threw some problems on the board and walked around giving guidance, but mostly letting them self-teach. This is quite a departure for me as I feel most comfortable lecturing and working examples.

If I didn’t need the quiz, why do I say SBG is working for me? Well, as Cornally might say, SBG ain’t all about SBG. Giving my students feedback has already helped train me. I know better what questions to ask of my students (during lecture) to gauge their comprehension. I don’t have to assess them to find out what they know. This isn’t new for me, but the approach has heightened my sensitivity. Students still aren’t sure of the system and not one has come to remediate during office hours, but I think this is because I promised that each standard will show up on at least two quizzes. However, no one is complaining to the chair or dean, so I’m good for now.

Oral Evaluation is having its ups and downs

If the few students who have commented on my system of lecturing to only three students are an indication, the idea  is a success. I’ve had at least three students volunteer to do it again and one begged me to make an exception to my rule that you can only do it once per third of the course (this rule is a practical one, not an attempt at cruelty). On the other hand, I find two things difficult with the system. First, it slows me down (probably a good thing). Second, I need a lot more practice in asking questions. Too often I ask either the wrong questions or answer the right questions myself. I may be trying to channel Michel Thomas in the classroom, but Michel Thomas I am not. Third, I haven’t found a good rubric for grading it. I take off points for tardiness and being unprepared, but I don’t really have a good feeling for how to translate their performance into a grade. In the end, I’m giving obscenely high grades and think that it is going to act as unintentional grade inflation. So, that’s not so cool.

Teaching to the test is wrong…now let’s see what is on the test

My precalculus course is basically mine to do with as I please, so I don’t have to teach to the test. I can test to the teach…er test to what I taught. My class on finite mathematics, though, is one of twenty or so sections and I don’t get to write the final exam. As such, I feel an obligation to get them prepared to ace that exam, especially since I have a darned good idea what will be on it. So, I throw out meaningful topics to focus on material I know will be on the test. Am I selling out? No. Well, yes. You see, I’m not. But…I am. I don’t like it and I’m having trouble looking at myself in the mirror these days (of course, being bitten by that vampire last week may also be contributing to my mirror problems).

Words of wisdom from a biologist

My buddy and pedagogical sound board from the biology department had this little nugget for me last week:

Sometimes the hardest thing is convincing the students that I can see them.

He explained to me that students are so used to watching TV, which can’t see them back, that they forget a professor can see what they’re doing. This certainly explains why they are so quick to text message, fall asleep, pick their nose, or try to cheat while I’m standing six feet away. A girl in my class was visibly trying to read the whiteboard residue during a quiz while I stood close enough to squeeze out the holy ghost. C’mon. He suggested that I appeal to their experience in band or choir. Those who’ve been involved in such activities will surely understand looking out at the audience and seeing someone paying absolutely no attention the performance.

What’s coming

I’ve decided that I’ve put it off long enough. I’ve held back my favorite tricks of the trade post, well posts, as I didn’t want to it being read by so many people. Now that school has started and readership is down, I think it is safe to release it into the wild. One of my all-time favorite tricks involves integration by parts. I have a pretty long post on the subject that I’ll break up into three or four posts. The end result is pretty cool and as close to original as I get. I also still plan on putting up my multiplication songs, I just need to figure out when to record them (the baby now sleeps in my office and that might be problematic).

Take care all.

The Circle Is Now Complete

Tricks of the Trade

(with Professor Glesser)

I watched a very nice video on remembering the values of sine and cosine at the various standard angles (or, in degrees, ) and their second through fourth quadrant analogues. My method is only slightly different and is likely well-known, but as most of my students have never seen it, I figure there is at least one person out there I can help. This also gives me an opportunity to throw out a useful mnemonic that naturally attaches itself to the unit circle.

Get your students fingering with each other!

The first step is for your students to be absolutely, 100% comfortable with the standard angles mentioned above. I will assume that this is done. Hold up your hand (either will work) with your palms facing either away or toward you, and starting from the leftmost finger, count or .

You don’t need to remember which value goes with which finger; you only need to be able to recite the list and stop at the correct finger. When you get to the desired angle, put the corresponding finger down. Now flip your hand over. We will use the number of fingers on the left of the bent finger to determine the cosine of the angle and the number of fingers on the right to compute the sine of the angle. The formula for the value is “ROOT FINGERS OVER 2″

For example, to compute the cosine and sine of , starting with my palm facing in, I put down the second finger from the left and flip over my hand:

The number of fingers on the left of the bent finger gives you the cosine, the fingers on the right give you the sine. Just remember ROOT FINGERS OVER 2. Why the hand flip?

Truthfully, I never used it. However, I added it because I wanted to reinforce the fact that when you define cosine and sine via the unit circle (as I do), then the coordinates on the unit circle are of the form .
To make this consistent with the hand trick, you need to flip the hand.

What was this mnemonic you mentioned?

It is actually pretty silly. The following (ridiculous) diagram should help a student remember where sine and cosine are positive and negative. They need only remember where the S goes and where the C goes. “S stands for summit, so it goes on the top” has worked for my students.

Furthermore, this diagram, when read clockwise, gives you derivatives of sine and cosine (as well as their negatives). That is not too shabby.

Cleaning Out the Gutters

School is starting this week. Hooray!!!

No more summer
Here comes class
Time to get up
Off my chair

The syllabi are written. The schedules are finalized. The students are being informed of their impending doom. Mmmm, it’s good to be alive.

Let’s see, what can I offer you today? First, in case you aren’t checking out the awesome SBGBeginners Wikispace, here is a link to my (always in flux) standards list for precalculus. I would post my standards list for Topics in Finite Mathematics (and maybe I will still), but  I doubt there are too many people teaching a course covering precisely what this course covers. Nonetheless, I should tell you a bit about my experience with producing the grade distribution for that course.

First, my precalculus course is controlled by me. I get to decide what, where, when and how much. So when I decided to make standards comprehension 70% of the grade and to reduce the midterm and final to 10% and 20%, respectively, that was all good. Unfortunately, the other course is not under my control. I had to do some serious negotiating with the course coordinator who is worried that my not grading homework policy will be a disaster. There are four exams in that class and, in the end, they will make up 60% of the grade. I’m making 30% of the grade standards comprehension. Those of you adept at addition will notice the missing 10% and that is the cool/scary/”oh gawd, what am I doing” part. This last bit is called Oral Evaluation. Way back in May, I mentioned my desire to teach only three students, instead of twenty-five. Well, I’m giving it a shot in one of my courses. Every student will sign up three times during the semester to be one of my three interloculars for the day. Their grade will be determined by their preparedness and their performance. More on this story as it develops.

In an effort to waste time, I collected links to videos targeting the standards for my classes (see here for precalculus and here for finite mathematics). Originally, I was going to record the videos myself, but even I can’t justify spending that much time right now on duplicating previous efforts. In the process of creating these lists, I started hanging around on the Art of Problem Solving website. This reminded me of all the time I spent taking the California Math League contests in high school. Man, those were fun. And…

Homeschooling Connection

As I’ve mentioned before, my wife and I are starting homeschooling with our five-year old this fall. Actually, since I happen to believe that

1. Summer vacation should be a time to do things you enjoy
2. School is something you should enjoy

we started a proper curriculum in April and went all summer, forgetting to tell him that he shouldn’t want to learn over the summer. (Mwoo hah hah!!! Kids are so gullible.) So, we’ve covered addition, subtraction and multiplication and are just starting with division. Incidentally, division is the first operation that I think my son finds practical. At the store today, he was constantly telling me how much each member in the family would get of the things we were buying. Anyhow, I was thinking to myself that knowing the four basic operations at his age is pretty good and just think how far I can take him over the next few years. But then, I thought to myself how my biggest complaint with my students is how they are so weak at the basics. Even those who can perform calculations efficiently, can rarely apply that math to solve real-world problems or interesting abstract problems. What if I got my son up to the level of understanding the basic vocabulary and then, instead of striving for breadth, went for depth? What if I started feeding the kid competition style problems and applied problems, not with the goal of getting him into the Math Olympiad, but rather aiming towards mastery of the foundations of arithmetic and mathematical thinking? The idea gives me goosebumps…oh, wait, the air conditioning was set to 63.

One last bit: I made up a few songs to help my son learn the multiplication tables and I think I’ll record them and post them on the blog for posterity. I searched hard for such songs and hated just about everything I found. Mine aren’t really any better, but they don’t annoy me as much and my son sings them all the time.

More Bad Poetry

There once was a teacher who dared
To teach as if all students cared
But his methods all stunk
‘Til he read Think Thank Thunk
Now his class is no longer impaired

Happy New School Year!!!