Unknown Quantity

Take The Test!

2012-01-12T11:01:00.000-05:00

What I really mean is this: the public defamation of public schools and teachers, and the concomitant policy initiatives, have been based on numerical data from tests whose contents are public, but this is the only public thing about them. Most critically, their development is opaque, the way the data is used is opaque, and the way that decisions get made about how the data is used is therefore not subject to legitimate public scrutiny, or even, in all probability, based on any real understanding of the tests. The decisionmakers don’t even know what taking the tests is like!

So, decisionmakers, take the tests! You are willing to force students to take them, to scrutinize the results, and to make important decisions about students, teachers, and schools on their basis. Finding out what you’re actually forcing on them, and opening yourself up to the same scrutiny, is the least you could do.

Amen!

New Semester, New Beginning

2012-01-11T10:30:00.000-05:00

Over break I spent a lot of time thinking about how I wanted to change my classes for second semester. I decided that Problem Solving needed to make up an even bigger portion of class than it currently did. So I decided it was going to be assessed periodically and that it would be tied into students final grades.

Phase 1:

Do lots of problem solving. More on this later, but as a teaser, I have been having students figure out as much as they can about the triangular numbers. I picked those because they are rife with patterns that are fairly easy to find and prove. I walk around and ask them about their work and periodically make announcements to the whole class about interesting ideas I see.

Phase 2:

Students write about what they have done. If they have found a pattern, solved a problem, or proven a conjecture, they let me know. If they just need some feedback, they let me know.

Phase 3:

I give the students feedback. Right now, we are pretty early into the process, and many of them are unsure how to prove something or they are trying to use an unproven conjecture to prove another unproven conjecture. So most of my feedback is in the area of trying to point out why that doesn't work.

Phase 1-3 repeat fairly often.

Phase 4:

After several cycles, students select past work (that they have turned in and I have responded to) and then they self assess according to a rubric. They can turn in more than one piece of evidence to try to hit all the areas of the rubric. I then look through their evidence and then decide if I agree with their assessment. They are then assigned a grade based on class goals (be proficient in at least two areas this time around... etc).

We will then start the whole shebang over.

So far, we have only gone through phases 1-3 once. I'll report back about how this turns out as the year progresses.

The Sorting Machine

2012-01-10T21:48:00.000-05:00

John Scammell on how high schools have turned into evil sorting machines:

We sort kids for the Universities and Colleges. We sort kids for scholarships. We sort kids for streaming into classes. When you believe you work in a sorting machine, you can justify some pretty poor assessment practices. It suddenly becomes important which student understood the material first, and which student turned in the work on time. You want to reward people who meet deadlines, and punish those that don’t.

Preaching To The Choir

2012-01-09T21:38:00.000-05:00

I think most of the readers of this blog are teachers, so I am probably saying things you all agree with. Nevertheless, I had to get this off of my chest.

I try to avoid complaining about teacher pay. I enjoy my job and I don't think it is very likely that I will get a huge raise anytime soon. However, last week, there was a "debate" in the NYT about teacher pay and there were a couple of things a pair of the debaters said that I find fault with.

The first is in the area of the "fringe benefits." According to Andrew Biggs and Jason Richwine, when you take into account the extras of our jobs, such as "generous vacation time, pensions and retiree health plans" and job security, teachers earn more than those with similar scores on aptitude tests. I'm going to completely ignore the several issues with using aptitude tests (or "intelligence") as a way of measuring how much someone should be paid. Instead, I just want to point out that these benefits are not what they used to be. Pensions are going the way of the dodo. Job security is not guaranteed in this climate of budget cuts and attacks on education. And generous vacation time? I don't get any vacation time. I don't really consider the scheduled school breaks as vacation time. They are a bit more like furlough. Sure I get paid during that time, but only because I have my pays spread out over the entire year, not because I'm on a paid vacation.

My second issue comes from the report that Biggs and Richwine wrote for the American Enterprise Institute:

Workers who switch from non-teaching jobs to teaching jobs receive a wage increase of roughly 9 percent. Teachers who change to non-teaching jobs, on the other hand, see their wages decrease by roughly 3 percent. This is the opposite of what one would expect if teachers were underpaid.

Are they adjusting the wage based on these fringe benefits? If so, then my previous argument stands and these numbers are no longer valid. However, if these are pure wage amounts without any adjustment, then I question the numbers. What qualifies as a worker switching from a non-teaching job? Does it include people who are in sub-par jobs who are working those jobs so that they can finish off their degrees and do what they really want to do? Does it include the working 18-22 year old college student? If so, then I could believe the numbers. Otherwise it just doesn't jive.

Okay, rant over.

Quick Thoughts on the Khan Academy

2011-12-11T23:05:00.000-05:00

There are already plenty of people who have written about some of the problems with the "Khan Academy as savior of math education" idea. Nonetheless, I want to share my thoughts on a troubling portion of Khan's Ted Talk. Khan was getting ready to highlight the gorgeous data mining tool that is available. Despite the prettiness of it, the thoughts here truly worried me.

Now I come from a very data-centric reality, so we don't want that teacher to even go and intervene and have to ask the kid awkward questions: "Oh, what do you not understand?" or "What do you do understand?" and all of the rest. So our paradigm is to really arm the teachers with as much data as possible -- really data that, in almost any other field, is expected, if you're in finance or marketing or manufacturing -- and so the teachers can actually diagnose what's wrong with the students so they can make their interaction as productive as possible. -- Salman Khan

My Biggest Concern

In my classes, the most valuable interactions come when a student is stuck and then I ask a few questions, usually starting with something along the lines of "Tell me what you have done so far," or if it is clear that they have been too frightened to even start a problem, "Tell me how you think you can start."

Maybe Khan would say that he wants me to be able to skip the question and just jump straight into the task of helping students improve their understanding. Perhaps this is meant as a time saver (although it would take just as long to analyze any data that I have as it would to just talk with a kid - and the talking is part of the whole relationship core of teaching). However, it has been my experience that as the student responds to my questions he/she is often able to overcome whatever had been impeding them in the fist place. If the act of putting the trouble into words doesn't fix it, I am now armed with the knowledge of what the misunderstanding is and I can then either ask one or two more questions to get the student back on track or I can offer a quick reminder of a fact that would be helpful.

What it boils down to: education in general and mathematics education in particular should be an ongoing conversation.

Smaller Concerns

Education is not finance, marketing, or manufacturing. There is almost no way in which to compare them.
"What's wrong with the students"?? I hope he meant "what's wrong with the student's understanding." However, in mathematics, students who are struggling usually think there is something wrong with them, so it worries me when someone who is trying to help struggling students acknowledges that unspoken thought from students, even if it was a Freudian slip.
It sounds like he thinks that the measure of the productivity of an interaction is its efficiency. While efficiency is one of the key metrics in business, and while it is important for teachers to find ways to quickly assess student knowledge and then to guide them to a better understanding, the fastest way is almost never the best way in education.

Good Algebra 2 Questions?

2011-11-23T10:10:00.000-05:00

I love the work that Dan Meyer and others have done with the 3acts, #anyqs, and #wcydwt work. However, I have trouble finding similar problems that would be good jumping off points for algebra 2 curriculum. I am willing to periodically set my curriculum aside and have my students work on a problem that isn't directly tied to what we're learning in order to help them build their problem solving skills, their mathematical intuition, or their curiosity. Nevertheless, I wish I could find some problems that would be a better fit for the topics in algebra 2.

Just to be clear, my own personal taste in mathematical problems leans more towards the pure side, so I don't need applied problems necessarily. I am just looking for good problems that lack that "know-it-when-you-see-it" feeling of pseudocontext. Any suggestions out there?

Paul Cohen on teaching mathematics

2011-11-21T09:18:00.000-05:00

When teaching undergraduates, the most challenging problem thus is to give the students a feeling for the power which is in the great mathematical discoveries. For the student specialising in mathematics, the problem is to bring then as quickly as possible to the frontier. This is done more through seminars than formal classes. These informal contacts allow the professor to help the student overcome the diffidence he feels before such a highly developed discipline. The professor must reveal the essentials of mathematics and supply the personal encouragement and direction which will enable the student to make a contribution of his own. - Paul Cohen (Mathematician) (qtd from Mactutor History of Mathematics)

While Cohen was talking about teaching undergrad mathematics majors, I think there should be more seminar work in high school mathematics. I'm not saying that we chuck the whole curriculum overboard, tempting as that may be. I'm just saying that periodically we need to give students an idea that mathematics is something that we can create/discover (a whole conversation to be had around those two words).

While undergrads would benefit from seeing the frontier quickly, I think HS students would benefit from being put into a position where they can create their own methods through the explored territories. Of course we should show them the tried and true methods, but how much more valuable would those methods be if we can say, "Your method is great. Let's compare it to how the Greeks solved the problem. Their method has some merit."

Also, I think HS students would benefit from occasionally getting "letters from the frontier." They should see that mathematics is still an ongoing endeavor. There is a lot of recreational mathematics that is very accessible and yet many "real" mathematicians work on topics from this realm. Many of the ideas from number theory are also accessible and could be used to show that important work is still being done and that there are still some unanswered questions.

The professor/teacher can "reveal the essentials of mathematics and supply the personal encouragement and direction..." by creating more time in class for discovery, problem solving, exploration, and even research into the history of mathematics and what is taking place right now.

Thoughts on Pi

2011-11-16T23:45:00.001-05:00

I was driving the other day, listening to the radio (The World for those who care), when I heard the host casually say something about "... the mathematical ratio of Pi." (This was mentioned during an interview with Kate Bush, for those who want to know.)

My first thought was, "Hey wait a minute, Pi is irrational! It can't be written as the ratio of two integers."

Then I thought, "Oh, but Pi is defined to be the ratio of circumference to diameter."

This made me wonder if at least a small amount of student confusion about the irrational nature of Pi (besides the fact that Pi is a tricky number to even begin to comprehend) might be due to the usage of the word ratio in both contexts. I wonder if students hear "Pi is the ratio..." and "Pi can't be written as the ratio ..." and just tune out the other parts. That can't help but lead to some cognitive dissonance.

Of course, the above argument is supposing they even remember the definition of Pi when we start talking about properties it holds.

Regardless, next time the irrationality of Pi comes up in class, I will probably ask them about the definition of Pi in order to confront this potential confusion head on. Some obvious questions to ask them:

If the diameter of a circle is rational, what about the circumference?
If the circumference of a circle is rational, what about the diameter?
If the diameter of a circle is irrational, what about the circumference?

First day of school: let's problem solve!

2011-08-22T22:40:00.000-04:00

A quick word about classroom management at the beginning of the year

I have alternated throughout the years about how to start a class. I have heard and followed the following pieces of advice at various points in my career:

Spend the first couple of days (especially the first) setting expectations and going over the course syllabus.
Spend the first couple of days getting students comfortable in the classroom and letting them know what will be expected of them.
Spend the first day jumping straight into the mathematics to let them know that they will be expected to work and learn every day.
Spend the first day doing a fun activity so students can get to know each other and you.

I don't necessarily disagree with the ideas behind these, but I decided this year to do none of these completely and do all of these moderately. Because of an altered schedule, we only met with students for about 30 minutes. That was enough to discuss the key points from my syllabus and then to launch into the first problem solving of the year. Throughout the period, I spent time setting expectations for classroom behavior and peer interaction. However, I did it as we got to it, instead of just dropping a list of expectations on them the first day (which I have definitely done. . . I'm ashamed to say that one year I dropped a 2 page list of procedures and policies on them in addition to the syllabus (another two-three page monstrosity) on the fist day). So now onto the actual problem.

Inspiration for the problem:

I got the idea for this problem from a summer course in number theory and this excellent problem from Nrich (which, by the way, is one of the best sites ever for these types of problems). This is one of those times where I had to build a simpler version of a problem in order to eventually extend it to a more difficult one.

Also, I have been trying for most of the summer to try to think of pure math anyqs problems and this is my first foray into that field. For those of you unfamiliar with the idea, I understand it to be presenting a simple picture or video and allowing students to ask the (hopefully) obvious questions and then using that as a springboard into mathematics. Dan Meyer explains this as the first of 3 acts (like a play/movie/tv episode/etc).

I like this basic framework and one of my current obsessions is finding ways to frame pure math problems in this way.

By the way, this is not a traditional 3-act math scenario. I modified it to suite my needs for the first couple days of school.

Act 1:

I handed students the following strip of numbers and told them to stare at it for a minute.

After the minute I asked them to write down any questions they had or things they noticed. This was an easy activity for people to do. I think of 130+ students for the day, only 2 did not write something.

My hope was that students would notice a couple of things, including the boxed numbers are all perfect squares and they all seem to be in the first two columns. Not as many students noticed these facts as I could hope, but in fairness to them, I think many thought the column observation was too obvious to write down.

After discussing the observations students had, we then talked a little about patterns and conjectures and proofs. With students who haven't yet taken geometry (and even with some who have) I find it usually takes many, many discussions before I get them to understand patterns need to be verified and won't necessarily continue forever.

Sadly, this is where we had to end for the day, but I was okay with that. My main goal was to get my students looking for patterns and making conjectures.

Act 2/3:

We returned to this problem the next day and I short-circuited the problem solving process (on purpose) in order to give them an 2 examples of how to "prove" our pattern of all the perfect squares only being in the first two columns. My proofs were extremely informal, and they just had to read them and decide which one they preferred and why. One proof included some visual elements, while the other involved equations. The goal here was to get them used to reading mathematical arguments and to help them see that there are multiple ways of attacking problems.

A few days later we extended this problem more into the territory of the Nrich problem that inspired this. I will share that later.

Conclusion:

This was a relatively pain-free way to star the year. One of the goals I have for my classes is to show students the mathematical problem solving process. Not just how to solve and graph all sorts of equations, which can certainly be valuable, but I want to show them why some people dedicate their whole adult life to studying mathematics. I want my students to see the joy in searching for patterns, struggling to see why the patterns exist, and then having a glorious light-bulb moment when everything becomes clear. This is what mathematics is to me.

Stay tuned

2011-08-15T20:19:00.000-04:00

A lot of things are happening in the next couple of months that could be blog worthy.

School is starting up again.
I will be starting math club in a week and I have a lot of good stuff for them and I hope to share some of the fun.
I will have a student teacher this year and from everything I have ever heard, I will be thinking about and rethinking about my views on education so that I can share them with my student teacher. I will likely have something to say on that front.
While I gradually hand over the reigns to my student teacher, I will be using the beginning of the year to really hit problem solving and the challenging problem format that I found midway through last year. I will probably have some problems to share in the next couple of weeks.
Our math department will be using ALEKS quite heavily and I will certainly have something to say about that.
Our school will be 1-to-1 with laptops (that students will take home and eventually own after they graduate) for the first time this year. There will be lots of flops and gigantic failures, but also some awesome things happening there.
I will be starting the Couch to 5k (C25k) program tonight, so you will probably hear a lot more whining from me on Google+.
My first child is learning her numbers and alphabet and my wife is brewing up another child who will make his way into the world later this semester.

Get ready for an increase in volume from this blog and my other online outlets.

Great Quote: Piet Hein

2011-02-16T06:04:00.000-05:00

This year as I've been incorporating more problem solving activities into my classes, I found the perfect quote for my wall.

"Problems worthy of attack prove their worth by hitting back." - Piet Hein

I don't know if it is inspiring to the students or not, but it is inspiring to me. It reminds me to get "worthy" problems for my students whenever possible.

Problem Solving: Pirates

2011-02-06T08:11:00.000-05:00

Like most of my problem solving puzzles, this one is not my own. I found this in information on The Math Circle website, and there it is credited as a classic puzzle by David C.P. LaFrance-Linden.

Hook

Pirates! Monkeys! Coconuts!

Problem

Three pirates are on a desert island and they gather up some coconuts. It is late by the time they finish, so they decide to wait until morning to divide up the pile. Secretly, the first pirate gets up divides the pile into three equal piles, but there is one coconut left over which he gives to a monkey that is watching from a nearby tree. The first pirate takes one of the small piles and hides it, then recombines the two piles and goes back to bed. Later, the second pirate gets up, and does the same thing: after dividing the pile into three equal parts, he has one coconut left over, which he gives to the monkey. The pirate hides one pile for himself then recombines to other two piles and then returns to bed. After this happens, the third pirate gets up and does the same thing. Once again there is a coconut left over for the monkey. Finally morning rolls around and the pirates get up and divide the pile up between themselves equally. Once again, there is a coconut left over for the monkey.

What is the smallest number of coconuts that could have been in the pile at the beginning of the evening?

What I like about this puzzle is that it lends itself to working forwards or backwards. Most students will solve this by guessing and checking (though it will take them awhile to do so). A few of them will attempt to put together an equation. The equation isn't terrible, but it is tricky for students to generate. In addition, there are really two unknown quantities, so even with the help of an equation it still boils down to guess and check (although the equation helps reduce the number of possibilities for guessing and checking).

Extensions

What other numbers of coconuts could have been in the pile at the beginning of the evening? (As there are an infinite number of possibilities, it would be nice if we could categorize all of them somehow).
Add monkeys or pirates into the mix. What is the answer to the above question if there were 4 pirates and 1 monkey? If there were 4 pirates and 3 monkeys? 9 pirates and 12 monkeys?
The generalization of this problem to handle any number of pirates and monkeys: p pirates and m monkeys?

Index of problems

Focus

2011-01-02T08:26:00.000-05:00

A couple nights ago we were eating with my parents and after dinner my dad says something like, "Since we all give up on our New Year's resolutions pretty quickly, I don't want to know about your resolution, but what one word would you hope describes the new year for you?"

I originally said that I was hoping for calm or peace or smoothness (I think I said three things, not being able to hone in on the word I was hoping for in a short amount of time), but as I was reading a few things later that night it occured to me that I would like to change my answer. If I could choose one word for the next year it would be "Focus."

I'm going to break the rules of my dad's question and I'm going to turn my hope into my goal for the year. I'm talking about focus in two senses of the word. The first is that I want to be focusing on what is really important instead of allowing the small distractions to gnaw away at my life. The second is that I want to be present in the moment instead of always thinking about what's next.

These thoughts are the culmination of two things I have read recently and which I would encourage you to read as well.

Focus (PDF link) - This is a short (about 120 pages) book about improving your focus and limiting (but not eliminating) your distractions. One of his main points is that we have become addicted to the connection the internet offers and we should try to break free.
"Is Google Making Us Stupid?" - This is an article in The Atlantic magazine from a couple years ago. It is a long article, but it completely worth the read.

The Arby's Problem (or how I learned something new recently)

2010-12-30T21:14:00.001-05:00

One of the funny things about mathematics is the way that we continue to find better and better ways of doing things as we go along. Several years ago, I stumbled upon something that made me wonder and that kept me up late a few nights in a row. I call it the Arby's problem and my wife still uses it as an example of my nerdiness. Anyway, I was at Arby's one day and I noticed they were running their 5 for $5.95 deal. I could pick any five items off of a menu of 8 items to qualify for the deal. I was also allowed to double up on items if I wanted.

For example, I could have ordered 3 Arby's melts, 1 soft drink, and 1 shake for $5.95 or I could have ordered 5 shakes, or I could have ordered five different items. All would have qualified. I wasn't hungry enough for the deal, but the advertisement on my tray mat claimed there were over 790 possible combinations. Now so my interest was piqued. I couldn't help but wondering, were they right? How precise was that statement? Were there really a couple million combinations and they were selling themselves short? I quickly thought about permutations and combinations but permutations would give way more than 790 ways and combinations would give way less than 790.

So I set to work thinking about this. I originally answered my question the hard way. Yesterday I found there was a better (faster) way to accomplish this. I'm going to show both, so that you can get a sense of how I wanted to smack my forehead after I learned the new way.

This problem is part of the mathematical world of combinatorics. As such, I will talk be using the multiplication principle and combinations. The ideas are not too tough, and if you are unfamiliar with them, feel free to take a few minutes to look at the links I've provided before diving on. I will be using the nCk notation for combinations, meaning choose k items out of a set of n objects. This corresponds with the binomial coefficient notation that Wikipedia uses, which is
.

The hard way:

Lets call the menu items A,B,C,D,E,F,G,H for simplicity. I broke the problem down into several smaller problems, that are similar to poker hands:

Getting five of a kind (e.g. AAAAA) . . . there are 8 ways to do this (since there are 8 menu items).
Choosing 4 of a kind (e.g. AAAAB) . . . . there are 56 ways (8 ways to choose the item being repeated and 7 ways to pick the other item, so 8 * 7 = 56).
Choosing 3 of a kind (e.g. AAABC) . . . 168 (8 ways to choose the repeated item and 7C2 ways of choosing the other two items, so 8 * 21 = 168).
Choosing one pair (e.g. AABCD) . . . 280 (8 ways for the repeat and 7C3 ways of choosing the other three items; 8 * 35 = 280).
Choosing a "Full House" -- three of one item and two of another (e.g. AAABB) . . . 56 (8 ways to choose the three of a kind and 7 options for the two of a kind; 8 * 7 = 56).
2 Pairs (e.g. AABBC) . . . . 168 (8C2 ways to choose the paired items and 6 ways to choose the lone item 28 * 6 = 168).
Nothing in common (e.g. ABCDE) . . . 56 (8C5).

Add up all the results (8 + 56 + 168 + 280 +56 + 168 + 56) and you get 792. So Arby's was correct and they weren't lowballing it at all.

The easy way:

So a couple of days ago I watched the following video by James Tanton.

I had never heard of "multi-choosing" or "k-multicombinations" as Wikipedia describes it. So it turns out that my problem really boiled down to 8 multi-choose 5. In terms of the circles from the video, there would be 12 circles and I would fill in 7 of them, leaving 5 blank ones that would signify a particular combination. For those who didn't watch the video or who just want another example,

would be equivalent to CCCDF in my letters above. Each black dot represents the end of a choice, so the two black dots at the beginning signify that there are no A's or B'as (since there are no white dots immediately before them). The three white dots followed by a black dot indicate 3 C's. The pattern continues in this fashion.

As Tanton points out in the video, this reduces my problem to 12C5 (or equivalently, 12C7), since we have to decide which of the 12 dots to leave blank (or fill in if you go with 12C7, which is why the two options are equivalent). The calculation is 792.

The moral:

One of the features of mathematics is the way that we continue to find new and better ways to do things as we go. This often happens because we grow tired of doing things the "hard" way and so we start to look for a better way.

Some fun tools:

Just for the fun of it, I created a few widgets on Wolfram|Alpha, and I put them here so you can check my calculations or try a few of your own. On a side note, the widgets on W|A are unbelievably easy to create. If you have the inclination, you should try it out and feel free to share about it in the comments.

(edit: I noticed that the widgets aren't showing up in Google Reader, so you may need to click through to see them.)

Songs I Like: "You've Been on My Mind" by John Heart Jackie

2010-12-28T20:05:00.000-05:00

NPR's Song of the Day had another great song a couple of weeks ago. Because of the busyness of this time of year I was only able to listen to it recently, but I've been playing it repeatedly for the last several days. The song is "You've Been on My Mind" and the duo is John Heart Jackie.

This mellow reminiscence is the perfect thing to be listening to as you consider the past year and the one that has just begun. Put on your comfy clothes, brew some tea, and then head over to the song link above to listen the the song from NPR's site.

Challenge Days

2010-10-26T17:30:00.001-04:00

The Basics

Besides problem solving days, I have also started Challenge Days this year. After we have reached the end of unit I throw a challenge day. This is a group work day. Sometimes I assign the groups and sometimes I let them choose. I assign 10 very challenging problems. Each problem is worth 3 "pride and glory points" I hand each group three hint tokens, each of which is worth 1 "Pride and Glory" point. Students then spend the rest of class working these challenging problems. It is a group competition against the other groups, but most of the students very quickly get in the mindset that it is also a competition against the problem set. If they aren't sure where to start or if they get stuck, they can turn in a token to me in order to receive a hint on any problem. After class is over, I grade the problems and the team with the most points wins pride and glory: they get their pictures on my wall of fame.

Questions

The questions I ask aren't ridiculously difficult; they are just at the edge of what a student knows. Some of my questions have been about the next topic that we are about to talk about (or a topic we won't talk about for awhile). For example, in calculus we had been discussing differentiation and then a couple of the challenge problems I gave were anti-differentiation problems (one with initial conditions and one without). When I give students problems like this later in the year the goal is for everyone to be successful on them. But now, it was okay that not everyone got the answer right. They were challenging. We had never seen anything like that before.

Another example from algebra 2 this time was I had students discover the meaning of the cube root of the square root of a number by using rational exponents. My students had seen rational exponents. My students had seen roots of many different indices, but they hadn't seen roots of roots. It was a new idea for them.

Results

Scores range depending on how difficult the problem set is. I try to have at least one problem that is just nasty calculations so that anyone who understands the current topic can answer the problem if they work carefully. I also like to have a few pieces of "low hanging fruit" for the low-to-average students to be able to answer. But the biggest part is the 4-5 problems that are a challenge for my brightest students and 1 problem that may be out reach. I do this not to be cruel, but to help my bright kids see that there is plenty more ground to cover and a lot more that we can learn about.

I have found that these days are very good for my advanced students. I feel like I spend so much of my time working specifically with the low-performing students or the average students that often the high-ability students will slip right by me. These days are dedicated to those students. I have really enjoyed writing these problem sets and my students seem to enjoy working them.

Problem Solving 1

2010-10-25T22:39:00.000-04:00

Okay, time for specifics. This was the first problem solving puzzle I used this year. I'm really glad I started with this one; it set the stage perfectly for the type of work I expect from my kids during this process. My students were also pleasantly surprised with how much they enjoyed thinking about this. In the longstanding teacher tradition, I stole this from another teacher. Well, I stole and modified; I added the spiderwebs and made the grasshopper (which one of my students thought was a cow) a little smaller.

Hook

I handed my students a paper with just this graphic printed on it. After allowing them to stare at it for a few moments I asked "What question comes to your mind when you look at this?" Most of the classes started a little goofy at first, but they very quickly came around to the big question.

Question

"Does the grasshopper ever land on a spiderweb?"

A few of the classes had to be informed that the numbers and the patterns they saw continued off of the page, at which point, they were able to ask the important question.

There is a lot of fun math here, and I just let my students mess with it for awhile. After a bit, some of the class thinks it never does (at which point I ask for a reason that moves beyond, "well I've tried it up to . . . .") and some of the class has a number they think it lands on. I post the "lands on" numbers on the board and allow the class to confirm or refute as they see fit.

Extensions

Can the grasshopper start on any particular number (not counting shaded boxes) and be safe by taking hops of 4?
What if it took hops of 5? 6? 7? You get the idea. . . .
What if the spiderwebs were on perfect cubes? What if they were on triangular numbers? Fibonacci numbers? etc.

My index of problem solving.

Problem Solving Days

2010-10-24T14:00:00.009-04:00

The Setup And A Warning

This year I have been incorporating "Problem Solving Days" into class on an almost weekly basis. On these days, I walk in with some wonderful problem, usually of the pure math variety because I love the simplicity of the problem and the beautiful solutions they provide.

[Warning]: I will sometime dress the problem up in a story or a picture. I do this not to convince the students that the problem we are working is "real life" or any such garbage. I hate pseudo-context as much as the next guy. I dress the problem up to provide a hook for the students. Not a hook that pulls them in, but a hook for them to hang their ideas on as they develop. It helps them to relate to the problem a little bit more and it also provides us with some funny vocabulary to use together. A quick example: I recently told a story about a man with a crazy mustache who had a crazy coffee drinking schedule. Each class started to call the days on which the man had coffee "mustache man numbers." However, the students weren't fooled into thinking this was real or that anyone would have the problem Mustache Man had, and I wasn't trying to fool them.

The Method

On the best of these days I can get the students to ask the question before I do. If I have to, I provide the question. I then let the students fall to the work of problem solving for awhile. I meander among them, making comments about their approach and point out how they are using one of the problem solving approaches that I have posted in my room to reinforce good ways to tackle a problem.

Eventually, several students think they have found the answer. When a student calls me over to tell me they've solved it, I usually ask them to explain why they think they have solved it and to explain the method they use. If they are obviously right, I usually press them with a couple of extension questions and allow them to choose which one to pursue. Occasionally I have to offer a few students 2-3 extension problems because they solve each of them quickly. Obviously, the best problems are the ones that offer many avenues to explore after the main problem is solved.

There have also been a few times that I have given a problem that has no solution or that the answer to the question "when does [fill in the blank] happen?" is "never." Of course I never let on that the problem has no solution. The students work for awhile and eventually a few of them become convinced that it won't work. The best reasons they give have to do with a pattern they see. The worst reasons usually sound like this: "well, it didn't work for the [pick random number from 2 to 100] different things I tried." (No kidding, some kids threshold for trying things is as low as two, but we're building that up slowly.) I usually then press them to find a way to explain that for sure it won't work...in other words, can they find a pattern (if they haven't already) and will the pattern continue forever? How do they know?

Closing

With about 10-15 minutes left I pull the class back together and we discuss what we have found so far. By this time, at least 75% of the class has a solution. I point out the "brute-force" solution or the guess-and-check solution and then I allow students to offer other solution methods. I mention 1-2 extension questions to the students and then class ends. I'm under no illustions that anyone tries these extenstions outside of class, but I'm just trying to promote the idea of mathematical thinking (and curiousity in general); answering a question opens up many more avenues of exploration.

The days have been wildly successful and a lot of fun. Most of the students enjoy the challenge they offer and all my students enjoy the break from the routine of class. So far I have had only one student who has been tough to get involved. He just sat there the first time. I asked what he was doing and he replied "Waiting for you to tell us the answer." I hassled him a little bit, but then let him be. The next time he at least joined a group of people who were working, but he didn't really participate. Slowly but surely the problems have been drawing him in. I think he is the prime example of our education system killing the curiosity of our students.

Specifics

I have done several of these Problem Solving days, and I will be sure to post specifics soon.

A new experiment: No Backspace

2010-10-23T13:00:00.000-04:00

I was reading Leo Babuta's free e-book Focus (ht: Lifehacker) and I was reminded of an application I had previously read about called Typewriter. It is a full screen text-editor (which have been hot for the past several years), but it also prevents you from editing. The basic idea being to get a first draft completed then you can use another program to actually edit the draft into something useable.

I was intrigued by the idea, so I decided to try it. Typewriter is a program that runs on Java, so theoretically it should work on any system. However, my ubuntu system gets goofy when it comes to separate java programs and this was no exception. The program worked, but when I made it full page it would freeze up. So I decided to try something different. I just used a text editor I liked that had full-screen capabilities (most do) and then turned off the backspace key.

If you are running linux, here's how you can achieve a similar set up. At a command prompt run the following:

xmodmap -e 'keycode 22='

22 is the keycode associated with the backspace key. (If you want to learn other keycodes in linux—I'm thinking of some funny april fool's jokes that could be played—you can run 'xev' and then press keys. In the terminal window look for the keycode #.)

Next fire up your favorite text-editor and bask in the glory of edit free writing. When you are ready to call it quits you can run the following at the command prompt to bring back the BackSpace functionality.

xmodmap -e 'keycode 22=BackSpace'

Of course, this is just a gimmick. You can still edit by using the arrow keys and the delete key, but I find it is enough to keep me focused on just writing without worrying about perfection. If you were really trying to stop yourself from editing, you could use the xmodmap command to turn off any other editing key you would like.

A side benefit of this experiment is seeing how bad of a typist I really am. I thought I was pretty good and fast. Turns out I was just fast. I bet I make 2-3 errors every 10 seconds or so. Pathetic.

Homework update

2010-10-22T15:42:00.000-04:00

Quick Review:

I am not grading homework this school year in any of my classes.
A student's grade is based on what level of understanding they have shown on a test.

So far the experiment is going pretty well. Now that we are settled into the year more, I haven't been as vigilant about making sure they have worked on their homework. This isn't laziness on my part, I have just decided they should have the freedom to choose the amount of practice they need for their success.

On the teacher side of things, it is nice to have substantially less work. In the past I have either graded for accuracy or I have just checked for completion. The first method is a time-suck that I cannot afford. The second way of handling homework is still an administrative hassle that I have decided I can live without.

As for the students, reactions are surprisingly mixed. I recently asked for feedback on how the course is going (more on that at a later date) and I received some wildly different thoughts concerning the policy. There was a strong contingent that either made no comment about the homework or said that they liked that I didn't grade it. And then of course there was the obligatory "you assign too much homework" response. I ignore this though, because my homework isn't required and it should only take them about 15-30 minutes max. . . usually more like 5-10 minutes. There are some who wish that they would be given credit for the work they have done. Several mentioned that they wanted the cushion for their grades that homework usually provides. (Successful outcome #1 - grades aren't tied to how much effort a student gives, but how much they know.) Other students wished I would give a grade so that they would have the motivation to do their homework. (Successful outcome #2 - students are made more responsible for their own success.)

While I try not to be a difficult teacher just to be difficult, I sometimes feel that we are too easy on our students. We "force" them to be successful. I don't want any of my students to fail, but if I don't give them the freedom to fail, have they really succeeded? There is a reason why the first step in recovery programs is to admit you have a problem; without this admission, there can be no recovery. If my students don't realize they have an problem understanding a particular concept, I can't really help them. I am trying to let students call for help before I rush to their aid.

Nonetheless, after reading the evaluations, I am planning on making a few changes.

I have been providing a key for each homework set they work on (most of which I create) so that they can check their answers. I now plan on giving them worked out solution keys so that they can also see how to correctly work the problem.
I will make it explicitly clear to students that if they want feedback on any problem from an assignment they can turn it in and I will review it and give feedback.

I think these two changes will make my expectations for what they should be able to do for a test even more clear than they currently are.

An update from my classroom

2010-09-25T22:25:00.000-04:00

So far my experiment of not grading the homework is working fairly well. As class starts each day I circulate around the room while the students check their work from a key. What I've noticed is that homework completion rates are pretty much the same as what they were when I was picking up the homework and checking it for a grade. Sadly, the kids who need the most practice are the ones who are most likely to skip the work. I've seen this happen in the past with the ability to retake quizzes.

Now on to my response plan for the students who skip an assignment. If I notice a student doesn't have their homework, I make a note of it on my roster and then make a point of following up with them the next several days to see if the behavior is chronic or acute. Depending on the student and my relationship with them, I will either apply a little guilt over skipping an assignment, or I will ask if they thought they didn't need the practice, or I will just let it slide. Knowing teenagers, I am confident that many times skipping the homework is a poor choice, but I figure I would rather have them make the poor choice and then help them to see what the consequences are once they arrive.

There are a few special cases. I have a few students (fewer than normal actually for this point in the semester) who already have D's or F's. I have to take a harder line with these students, because they often have so little expected of them from home. So I have let them know that I will be riding them especially hard until they have at least a C+ or B- (which would be the best grade they have ever had in a math class). I force them to see me outside of class (usually during our homeroom period each day).

Overall, I would say the new homework policy is a success. It is a success for me because I'm grading much less and the administrative side of my work has decreased substantially. I think it will be a success for the student as it shifts even more responsibility to their shoulders.

Teaching Plans 2010-2011

2010-08-25T22:36:00.002-04:00

So now that the school year is upon us, let me take a few minutes and detail what my classes will be like this year. My class schedule hasn't changed at all in the last several years. I teach algebra 2, precalculus, and calculus.

Grading
I mentioned earlier that I was moving to a method of grading that was going to award no points for homework. That is still in effect. I assign homework nightly. I have the students check their work with a key at the beginning of class the next day. We discuss misconceptions and any problems that they can't fix after looking at the key. I make a quick pass around the room noting how they did on the homework and answering individual questions. It is at this time that I make a note of who didn't do their homework. First time I notice it, I say something to the student about how I expect them to keep up with the work. Second time I ask them to spend some time with me in homeroom or before/after school.

I've been doing concept based testing for the past several years. Each year I hone down my list more and more. After discussing all the concepts in a unit (bound together by some characteristic) I will give a test made up of separate sections for each concept. Each concept will be graded individually and will be assigned a letter grade based on the mastery of the concept. After returning tests, if students don't like what they have earned, they can come in and show me that they have learned the material. This will result in improved grades.

Each concept will get a separate spot in my gradebook. For those who have been following the concept based grading (or standards based grading) conversations around the internet that have been going on for awhile now, this will be no real surprise.

Finally, I will give a midterm and a final. My goal here is to get the students to put the material together and to show me they have retained all the things I consider essential.

In Class
I started this year off shaking every student's hand and greeting them as they walked in my room. It felt goofy at first. Actually, it still feels a little goofy. However, I'm keeping it up for awhile for several reasons. It forces me to interact with each individual student. Since I have about 120 students, this is about the only way that I can have a positive moment with each student. In addition, it is a thing unique to my class. I think it will probably help build an interesting class culture.

I also made a point from the first day of calling on whomever I felt like while discussing material. (In the parlance of Teach Like a Champion this is called "cold calling.") We are one week in, and I am surprised at how much it has opened my classes up. They are participating more than normal, especially for this early in the year.

Looking ahead
This year has started off strong for me. I hope things continue to go smoothly. There are several challenges facing our district, so outside of my classroom things are going to be stressful. I'm hoping that I can distract myself with success in the classroom.

One thing I will be doing in the near future that I have never done before is a data collection project in my Algebra 2 class. More on that in the future.

End of the Summer

2010-08-23T22:37:00.000-04:00

Last week was my first week back at school, and I thought I would take some time and reflect on the past summer. It was a busy one with a lot of new experiences. One of the big things I did this summer was to work for the tech department at school. I worked about 20 hours a week and I mostly moved computers around and cleaned computers out. It was a busy time at school. They had ordered a ton of new computers and were implementing a new phone system and installing new printers/copiers throughout the buildings. I had a lot of fun working there, plus it earned me a little money and a few new things for my classroom (bookshelves, desk, and chair . . . all from being at the right place at the right time).

I also was able to spend a ton of time with family. Keltron worked some but I was able to spend a lot of time with my infant daughter, which was quite a lot of fun and a growing experience for both of us. As a family we tried to do something exciting once a week, and many of the weeks we succeeded. There were several trips to the farmer's market, a trip to a county fair, a family reunion, and a family sickness. The last one wasn't much fun, but we all survived.

I always enjoy summer. It is a time for me to recharge and a chance for my wife and I to reconnect after the marathon that is the school year. I spent quite a bit of time planning things for this new school year, so I am hoping that will save some time that I can spend carrying the relaxation and peace from the summer throughout the upcoming months.

Goodbye summer; I will think back on you with fondness and will look forward to our next meeting.

Music I Like: Till The Sun Turns Black (by Ray LaMontagne)

2010-08-21T23:28:00.000-04:00

Image via www.raylamontagne.com

My awareness of Ray LaMontagne crept up slowly. I had been hearing a few of his songs ("Three More Days" and "You Are the Best Thing") on the radio station. I liked them, but I never really tied the two songs together or to a particular performer. Later, our friends used "You Are the Best Thing" at their wedding reception. Ends up, that is their song. So they are big LaMontagne fans. They would mention stuff about him periodically. I began to notice his music on the radio more often, and I began liking them even more.

Recently, I was able to check out Till The Sun Turns Black. This album is exceptional. The album is mellow (much like many of Duncan Sheiks albums). It has a wonderful blend of piano, guitar, and string arrangements. Ray LaMontagne's voice is wonderful and low.

The CD is full of great songs; it is hard to pick a favorite from this album, but one of my favorites is "Barfly." It has LaMontagne singing slowly, a nice acoustic guitar driving the song, a B3 organ that provides a smooth background, and an electric guitar riff that ties it all together. Some of the other highlights from the album (but you really should check out the whole thing) are the songs "Be Here Now," and "Three More Days."

Music I like: Battle Studies (by John Mayer)

2010-08-14T15:00:00.000-04:00

via: wikipedia.com

I've enjoyed John Mayer's albums for a long time and I finally got my hands on his most recent cd, Battle Studies. I've been enjoying listening to it for the past couple of weeks. By this time, I'm sure many of you have heard several of the singles on the radio already. Of those, "Heartbreak Warfare" is my favorite. But as usual, there are several gems on the cd that aren't singles.

My two favorites from the album are "Assassin" and "Edge of Desire." "Assassin" starts low and mellow. The song is driven more by the bass than any other instrument. It feels different from many of Mayer's other songs, but I love it even more because of that. On the other hand, "Edge of Desire" is a throw back to his older albums. The guitar lick sounds almost as if it could have been floating around inside his head since Room for Squares or maybe even Inside Wants Out. Lyrically, the song is much closer to his current songs or something from Continuum. Specifically, the song is one of longing and because of that, I am reminded of his earlier song "Dreaming with a Broken Heart" off of Continuum.

Overall, it is a great album. I have yet to be disappointed in one of his cd's.