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!)
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.)
Literacy is much more than the ability to read and write. When I first started teaching I believed being literate just meant having decoding and comprehension skills. Students at a low level would be stuck decoding words. Students at a higher level would be able to read and understand, provided they weren’t hindered with unfamiliar vocabulary. Thus my view was a blend of emergent and functional literacy, a murky picture at best.
These days I have a much richer understanding. For one, literacy is impossibly entangled with the specific subject being considered. One can enjoy Harry Potter or The Hunger Games with a shallow level of functional literacy, but reading the New York Times or The Economist demands a degree of cultural literacy. Likewise, someone who has access to a range of American cultural allusions may find himself lost while reading a major Spanish newspaper, even if he is fluent in Spanish. So literacy is not a unitary skill that is acquired and transferred effortlessly across disciplines.
Literacy is the ability to read and understand what others have written, along with the ability to write as a means of recording information and for communicating with others. Using this definition with regard to a specific discipline yields the idea of content literacy. As a teacher of mathematics, I am interested in developing students who have content literacy in mathematics. This ability is predicated on 1) general literacy skills, 2) prior mathematical knowledge, and 3) math-specific literacy (such as familiarity with the notation used in Algebra).
I began to realize more and more the truth that content literacy is not content knowledge during my student teaching this semester. In eighth grade the most advanced math students are typically those with strong numerical fluency, which allows them to learn procedures and notice patterns while their less savvy classmates are bogged down by the computations. However, if I were to ask my advanced students to justify their procedures or describe a pattern, they were often unable to do so. I don’t mean that they lacked the vocabulary, like numerator or quotient or distributive property—use of correct terminology is icing on the cake. They could not articulate why two different procedures yield the same result. For example, calculating 7% tax and adding it to the bill versus just multiplying the bill by 107%. And so it was that I set out to encourage meaning making in math class.
My first step was to create a need in students by showing them there were things they could not do and then tell them that I would teach them what they needed to know. I designed a pre-assessment that included basic mathematical tasks as well as asking them to write how they would explain an idea to a sixth grader. When we anonymously reviewed the class’s responses, I pointed out that even students who were able to perform the skill were unable to explain it to someone else. I told them that I would teach them how and help them get better at communicating.
Rather than give the standard lecture (this is how you do X, now let’s all do X, now you do X), I had students read a short bit of text I’d created explaining the idea. Beforehand, I introduced the concept of marginalia in order to get them engaged in self-cognition as they read. (Many students read the page without making any marks, which I can only assume meant the words washed over them with little understanding.) Then I asked them to write how they would do X. I was surprised by the difficulty students had with what I thought was essentially an exercise in copying. Just rewrite my explanation with different numbers, right? Wrong. A majority of the class was unable to answer the prompt. So what that told me was that students needed more support in constructing, verifying, and extending meaning as they read.
We did several of this type of lesson throughout the unit, and students gradually improved. I experimented with prompts; here are a few:
I expected students to balk at this sort of difficult metacognitive, especially when many have conceptualized math as memorizing what to do. But an anonymous survey a couple weeks in found the majority of students really enjoyed the reading and writing. I guess it levels the playing field for those who don’t have strong numerical fluency, because we are discussing ideas and not just blindly computing.
Since writing is a time-consuming activity, I also wanted to introduce mathematical discussions for shortest bursts of reasoning. Much of the time in math classes I’ve observed (and my own as well), the teacher asks a question with one correct answer. If I were to teach content literacy, students should be able to explain their reasoning, not just produce the correct answer. So I turned to Zweirs’ Building Academic Language to build a framework for classroom discussions. It declares that classroom talk is a tool for working with information such that it becomes knowledge and understanding.
The first thing to consider was the balance between closed (or display) questions and open-ended questions. Display questions, while suitable for activating prior knowledge or displaying what students know, rarely lead to deep discussions. Zwiers provides four main categories of open-ended question:
Despite all this, he suggests questions are overused in classroom discussions and often still lead to a teacher-centered pseudo-discussion.
For that reason, I designed a series of four posters with sentence stems for agreeing, disagreeing, observing, and questioning. Then I created a homework assignment that asked students to solve a problem and write up notes that would help them explain what they did to a classmate the next day. Before the discussion I had students turn and talk with partners for 2 minutes sharing their answer and how they got it. Then I said that I noticed many different answers as I walked around checking homework and wanted to have a class discussion about the problem. I introduced the sentence stems I expected students to use and modeled along with some examples of how not to participate. And then students discussed, with me merely moderating who was talking. Occasionally I had to ask students to write down their ideas so that we could come back to them in order to allow each thought to reach its conclusion. Within 15 minutes we had come to consensus and I sensed there was a high level of understanding from the engagement and what students were saying. Towards the end I began cold calling students to explain whether they agreed or disagreed with what someone had said, which aided me in checking for understanding and holding everyone accountable for participating.
Even more so than with reading and writing about mathematics, students universally enjoyed our mathematical discussions, and I have been infusing them into class whenever possible. I have been quite pleased myself with the development of students’ reasoning skills and ability to justify their answers.
In the future I will focus more on the challenges I faced incorporating literacy into a mathematics curriculum. At times it felt like I was devoting a lot of the class period to teach through literacy-based instruction what could be direct taught in a fraction of the time. Asking middle school students to struggle with text and then write their understanding appeared to bear very little fruit at first. In the future I will slowly introduce the idea of gaining knowledge through text by having students first read about something they are already familiar with. After they have been successful with that several times, they can begin to use text to study new mathematical concepts.
I also struggle to find a way to consistently include literacy-based components, especially with topics like simplifying algebraic expressions. There are more mathematical discussions happening, but I find it difficult to find time writing about a skill that they still haven’t mastered. I think literacy has worked best when the writing was assigned towards the end of the unit. Discussion may help sort out what to do as students are learning a new skill, but writing in particular forces us to (re-)organize our thoughts in a way that they become crystallized. I think this may be one of the keys to long-term knowledge retention. This is perhaps the biggest bane of math teachers, finding students who learned something last year, but have no recollection of what to do now. Sometimes it doesn’t even take a year—they can’t remember the skills they learned last month. Taking the time to incorporate literacy strategies to solidify students’ knowledge would be time well spent.