Really clear explanation of how Warren Buffett's investment strategy creates high returns year after year.
Really clear explanation of how Warren Buffett's investment strategy creates high returns year after year.
If you live on the East Coast, chances are you’ve heard about this summer’s emergence of Brood II cicadas. Cicadas are winged insects that live underground most of their lives, sucking on the xylem (sap) from roots of trees. When the ground temperature 8″ below the surface reaches 64 degrees Fahrenheit, they will emerge and transform into their final adult form. They ascend into nearby trees, shed their skin one last time, grow wings, and set off to mate. Some species are annual cicadas and emerge every year. However, Brood II cicadas only emerge every 17 years and do so by the millions.
I teach at an Expeditionary Learning school, and teachers are encouraged to design curriculum through expeditions. Expeditions are rich units that engage students by investigating some compelling question(s). When I first read about the 17-year cicadas that would swarm North Carolina to Connecticut, it seemed like the perfect topic for my sixth grade math students.
We began the unit by Building Background Knowledge. First we listened to audio of the male cicada mating call. Students wondered what it could be, and then we showed them the associated video. We watched another video and discussed some cicada facts. Then students created a KWL chart of what they Knew and Wondered. (Later, we would add the other things the class Learned). Our guiding question was, “Where can we find mathematics in nature?”
The guiding question led us to explore why periodical cicadas only emerge every 13 or 17 years. We looked at the Sieve of Erastosthenes, prime numbers, factors and multiples. Then we practiced converting between fractions, decimals, and percents by creating surveys and representing the results in various ways. Students wanted to know if they could outrun a swarm of cicadas flying towards them, so we went outside and measured how fast different sixth graders could run. We explored measures of central tendency to see how fast an “average” sixth grader was. Finally, we looked at a cicada cookbook and practiced scaling recipes. Here is a link to many of the resources I used for this unit.
Students were disgusted and enthralled by the idea of eating cicadas! It was unfortunate that they made such a minimal appearance in New York City, especially after all of my build up. Fortunately, they did emerge in large numbers in Staten Island, so I took a trip to harvest enough Brood II cicadas to sate my students and colleagues! Here are some pictures of the preparation.
Anyone brave enough to eat one earned a button that became a badge of honor. (Thanks to Maurice Principe for designing them!)
As a strong student with a good deal of intrinsic motivation, I sometimes find it hard to understand my own students’ academic dispositions. What makes them choose to put forth effort, or not? What do students find motivating? Since many studies suggest that time spent practicing is highly correlated with growth, as a teacher I want to know how to structure class to increase time on task.
Which brings me to the panel discussion we had with high school juniors and seniors about their experiences in math class. None of the students who participated love math or see themselves as strong math students, which is much more typical of the students I teach in 6th grade. Here are some highlights from our discussion:
What makes you participate during group work?
What advice do you have for middle school students / teachers?
At one point, a student was talking about logarithms, but couldn’t think of the name. All he had was a vague sense that they were the “opposite of exponential functions.” The video below gives students a bit of math history and motivation for learning about logarithms. Maybe there’s even an interesting discussion in here somewhere 🙂
One of my favorite math bloggers, Sarah, had a great post recently about pencils. It was serendipitous, since I’d just had a conversation with a first year teacher about school supplies. Her lesson came to a halt because students didn’t have loose leaf paper!
I, too, am always baffled when students don’t come to school with what I consider basic supplies. For me, the battle this year has been with pencils. At the beginning of the year I had a bin of maybe 50 golf pencils. They’re distinctive, don’t have erasers, and students really don’t like them. My supply lasted several months, disappearing little by little, until one day I was out for Professional Development. Upon my return, the whole supply was gone, along with my stock of erasers.
I had a discussion with each class, reminding them that I had paid for those supplies so that students could borrow them during class. I wasn’t going to go out and pick up more, so now everyone would need to be prepared for class. The unfortunate consequence was students going around trying to borrow a writing implement from classmates 10 or 15 minutes into the period!
So I tried collecting the pencils students left behind: a sort of “take-one, leave-one” system. The obvious flaw here is that a student who doesn’t have anything to write with in math is going to have the same problem next class as well. It ended up being more of a “take-one” system. The whole thing would’ve collapsed within days if not for one valiant student who brought me several pens and pencils a day! Alas, I needed a better solution.
Finally I bought a hundred pack of Ticonderoga pencils from Amazon. The next time a student needed a pencil, I told him that I’d sell him one for 25 cents. At first, a few students were outraged that I was selling them pencils. So I reminded them that I’d already bought a class set of pencils and erasers that had all gone missing. I wanted to make sure they were prepared for class, but I can’t afford to buy school supplies for over one hundred students.
It’s been almost two months now, and students have requested that I add erasers and sharpeners to my store! I sell them basically at cost, which doesn’t seem to trouble kids who already spend several dollars a day on candy and bottles of Arizona. Last week I had to make another order to restock my store, but I no longer feel angry or frustrated when students don’t have pencils–I just sell them one.
Alex* did both worksheets today, however his sense on how you’ve taught him to convert whole numbers into fractions appears to be different than what I know (keeping whole number as numerator and making the denominator 1). I think he must have been confusing whole number conversions with something else you were teaching but he was obstinate. Thus I think he had trouble with the generic rectangle calculations and I could not help him and we lost patience with each other.
Perhaps you can revisit whole number conversions with him as he would not believe me. I’ll have him do the three problems once he gets the concept correctly — or that you confirm that in this case I am smarter than a 6th grader….and if I am not, I am so sorry. It is probably better I write for a living.
*Not his real name
Don’t worry, you’re certainly smarter than a 6th grader. Certainly more wise, too, because 6th graders have a problem not realizing what they don’t know.
I haven’t taught students that they can write any whole number as a fraction over 1 (e.g., 4 = 4/1). As you realized, they sometimes don’t “get” why it works and end up getting more confused. Currently we’ve just been reasoning through multiplicatively via repeated addition: for example, if you had 4 times 2/3, that just means 2/3 + 2/3 + 2/3 + 2/3. To go further, drawing a picture of 2/3 four times and then counting the pieces to recognize you’d have 8/3. A lot of students are still struggling with the idea, and want to say the answer is 8/12 (because they multiply top and bottom by 4). But then you can present them with the question: Aren’t 2/3 and 8/12 equivalent fractions? This creates the dissonance for them to see that the answer should be more than what they started with, not the same. Put another way, multiplying by 4 changes the number of pieces you have (the numerator), but not the size of the pieces (the denominator).
He’s okay right where he is–a lot of students are still there, since we only just learned how to multiply fractions. We’ll be doing more work multiplying fractions by both fractions and whole numbers via the mixed number generic rectangle problems. I think with concepts like this, it’s important to foster both a continual emphasis on why the answer makes sense in addition to repeated drill, which gives students several times to make mistakes / correct them and notice the patterns themselves.
If he wants, we could do some problems together at lunch. I often have a few kids up for extra practice / more individual instruction.
But knowing how important the start of a class is and making it priority number one at the beginning of a year. It’s the first thing you do with students every period. If you set the tone well at the beginning, they fill in the blanks and end where they expect themselves to be. People are predictors, constantly imagining how things should be. As mentioned in Steve Pavlina’s recent post, the essence of frustration is when our predictions don’t match reality. So teaching kids how things should begin is half the battle.
I’ve recognized a few things that work really well with my 6th graders:
So that’s the start of my class. Then we’re off into the day’s lesson. And each day keeps getting better!
Next post, I’ll talk about the management strategies I’ve found successful during class.
Second, I wonder what sort of accountability Pearson has for their testing materials. Considering the company just won a $32.1 million contract to provide testing materials for the next several years, it seems that they should provide quality assurance measures. Perhaps Pearson should pay a $100,000 penalty for each question that needs to be thrown out. There were several in this year’s math & ELA exams. (For example, this 5th grade math question was thrown out after teachers realized it was impossible for students to solve.)
Finally, how can we make standardized tests more useful for students and teachers? As a student, I just see a summarized score of 1 through 4. For example, what would a “3” really tell me? Research tells us feedback must be timely to be useful. Currently the standardized test results come back too late to be useful for students or teachers. The full breakdown of student results by standard don’t arrive until well after the end of the school year. And even then, teachers don’t have access to the original questions. I understand that norm-referenced tests need large samples of data and are often field-tested for validity. But more transparency would be wholly appropriate and ease concerns of all parties involved. (Except perhaps the testing company.)