Depends on you.
This spring a student (her real name is not Nancy) posed a problem for herself: Starting with one newborn pair of bunnies, after one month the pair matures, after the next month the pair produces a new pair, and continues doing so every subsequent month, until after six months the pair dies. Describe the number of living bunnies after n months.
If you are not sure how this is going to work, the problem is for you. See if you can figure out how many bunnies will be around for the first few months, and then see if you can describe the relationship mathematically.
If this set up is not a problem for you (if you can write the recurrence relation directly from the problem set-up), then I have a challenge for you: what interesting new problem can you create out of Nancy’s problem that would take someone who can already write the recurrence relation and make them think?
I had a student this year play with Fibonacci, then modify the problem, and give a partial solution to the modified problem. The modified problem is well-known and completely solved. You can try your own hand at it, (see next post). but here’s the student’s story:
Late in the fall of this past year (November 2014) I assigned freshmen the task of taking a problem that we had solved and discussed in class, and proposing a new problem as a modification or extension of the original. Some found it fun, and at least one remembered it later. (I’m sure it was more than this one.)
This February I started a special one-day-a-week class for freshmen (number theory and arithmetic, special topics of their choice, I did this once before).
Nancy (not her real name) worked in a team on Euclid’s algorithm. They did a very nice, very clear presentation, most of the students in the room were able to follow and perform the steps and work out a simple example. And then the team broke up.
Nancy decided to play with Fibonacci on her own. I was a little worried about real-world examples, but she stuck to the traditional “a pair of bunnies is born. In its first month it matures. In each month after that it produces a new pair. And she played it out and let the recursion and the problem statement match up fully. (My Ghost the Bunny is just word play)
And then she got bored, and played what-if. Nancy modified the problem – her bunnies would now have 6 month life spans. She carefully worked out what this would mean: 1, 1, 2, 3, 5, 8 all stay the same, but 13 – 1 = 12, and it gets interesting from there. Nancy identified the quantities that needed to be added (the two previous) and subtracted (six back) but had not written up a recursive formula when the class ended (we only met one lunch period per week, ate before we worked, and homework was not allowed).
But see why I’m excited? She played with a problem, then posed her own problem? Because she was curious. Ninth grader. Cool, huh?
I like posing and solving problems on-line. Is this reflected in my classroom? Not enough, but yes.
In the fall term of algebra classes I carve out a day here or there, or maybe a few half-periods, to work on extended problem solving. It is generally not on-topic. On-topic would allow the kids to know before they start HOW they should solve the problem, and that would spoil the joy. I usually choose problems with multiple paths to success. And I certainly do not choose problems that have an accessible formula – that would spoil the challenge.
I use an Understand / Plan / Carry out the Plan / Look Back approach with the kiddies, but too often “looking back” for them just means “check.” Over the years I have pushed “find another approach” or “find the relationship between two successful approaches” or “generalize a solution.” But this year I pushed in a new direction.
“Use your work and solution to think of a new, interesting problem.” The idea is not to simply make the problem bigger, or generalize it, but to come up with something related, but new, probably closely related and more complicated, but not necessarily so. And it was quite possible for the new problem posed to be easier than they realized or harder, to yield to a similar approach as the original problem did, or not to yield at all. After all, if they knew there would be a solution of appropriate difficulty, it would mean that there was not original problem posing going on. And after practicing generating ideas on earlier problems, we hit the checkerboard, and I assigned them to extend the problem, gave them time in class and at home, and required them to write up a problem solving “experience”:
- Understand the Checkerboard
- Devise a Plan
- Carry out the Plan
- Look Back (include posing a new problem)
- Devise a Plan
- Carry out the Plan
- Look Back (since many new problems were not solved, this included commentary on obstacles. Where problems were partially solved, we got suggestions for the next team to pick the problem up. Where problems were solved, we got ‘normal’ generalizations, but also suggestions for future work. From 9th graders. )
So, post-checkerboard, what problems got posed? Here’s a few that I recall:
- Solve for an abnormal 8 x 9 checkerboard. Generalize to squares on an m x n checkerboard.
- Solve for a checkerboard with the four corners missing. Try again with the four 2 x 2 corners missing. 3 x 3. Generalize to an n x n checkerboard with four m x m corners missing
- Variation (different group). Solve for a checkerboard with one corner missing. Then a 2 x 2 corner…. Generalize to an n x n checkerboard with a single m x m missing.
- Variation (there was a lot of removing squares going on). Solve for a checkerboard with a 2 x 2 hole in the center. 4 x 4. 6 x 6. Generalize to an n x n board with an m x m hole in the center.
- Solve for an 8 x 4 checkerboard. Account for the difference between two 8 x 4 boards and one 8 x 8 boards (the write up for this was beautiful)
- Solve for rectangles on a checkerboard.
- Leaving the board out of it, count trimonos, tetrominos, pentominos, hexominos. (I think this group got side-tracked into some fascinating but for the moment fruitless discussions of symmetry and handedness. Product? Nah. Discussion – excellent.
So, when you get an answer, are you at the end? For most of the kids the response is still “check, and that’s enough, unless the teacher makes you go on” – but for a substantial minority I think they got used to the idea that mucking around further is a good idea, and potentially fun or interesting.
Yesterday was the day after the year after sabbatical. A group of us took out a friend, first day of retirement. And I breathed, really free, for the first time in a while.
I fell down on blogging badly. Let’s see if I can’t recap the past year, over the next 20 days.
There’s my school, and teaching, and plans for next year.
There’s union stuff, a lot of it. There was some good stuff that happened last year, but lots of troubling stuff, too. And now after de Blasio / Fariña have completed 3 of 8 terms, there’s not nearly to show for it.
There’s education stuff outside of my union. Common Core and Opt Out, and vicious battles in other states and cities.
There’s politics, education and otherwise. In that respect, the year ends on a high note, with marriage equality the law of the land.
And there’s always math. State exams, national exams, new courses, old courses. And puzzles. I’ll start tomorrow with puzzles.
They have “wrap-around” services – which can mean a lot. They have drop-out prevention programs. They have medical and dental care. They have mental health services. They have expanded guidance services. And each school is supposed to develop further services to meet the needs of its community.
But for high schools, there is no community.
It is a farce to call them community schools.
Mayor de Blasio says: “Every Community School is different and reflects the strengths and needs of its students, families, and local community. ”
The report the UFT posts says “Community in this model is defined in the broadest sense possible, including not only non-profits, but also private-sector businesses, hospitals, universities and communities of faith. ”
But under Bloomberg’s DoE, most communities do not have their own high schools. Students are assigned to a school by OSEPO, after going through an insane process of listing 12 choices, and sometimes getting none of those. Students are often drawn from across their borough, or beyond. There is no neighborhood. And how do we call it a community without a neighborhood?
When a kid does not apply to attend a special school, or a school with a special program, where is the neighborhood school that is his or her default? Under Bloomberg, such defaults were eliminated or destroyed.
Carmen Fariña and Bill de Blasio have been running the show for a year and a half now. Every neighborhood or group of nieghborhoods should have their own neighborhood high school. They should be good choices, with good programs, good extracurriculars, good course and elective options. And yeah, it would be very cool if they could provide the wrap-around services in the Community School model.
But until then, please don’t tell me about community schools that have no community.
How can New York State test kids in math when it can no longer consistently write appropriate questions? This gaffe is almost two years old, but it looks like no one noticed the problem, until it showed up on the Association of Mathematics Teachers of New York State (AMTNYS) listserve this week.
On the August 2013 geometry regents, students were asked to find the slant height of a cone, given the lateral area.
It’s easier than it sounds. There is a formula sheet in the back that gives
L = πrl,
where L = lateral area and l = slant height and r = radius.
Heres’s the question:
14. The lateral area of a right circular cone is equal to 120π cm. If the base of the cone has a diameter of 24 cm, what is the length of the slant height, in centimeters? (1) 2.5 (3) 10 (2) 5 (4) 15.7
Since radius is half the diameter, r = 12 and plugging in: 120π = π(12)l, or l = 10, choice 3.
But wait. The height of the cone (like a flagpole from the base to the highest point), the radius (like a stripe from the base of the flagpole to the edge of the cone), and the “slant height” form a right triangle, with the slant height being the hypotenuse. So how is the hypotenuse (10) shorter than the base (12)? Can’t happen in the real world… but in New York State?
Here is an insightful comment from the listserve:
This is another example of the type of error that has been occurring on Regents exams since the early 1990s when the math bureau of NYSED was downsized from 7 very experienced and talented people (a bureau chief + 6 math specialists) to an inexperienced few. It is also a product of contracting out the writing of exams to rich companies that had no experience in this area.The errors often occur from the creation of questions that require substitution into formulas without looking at a drawing to see if the numbers are possible.
Probably not. But we should not gloss over the formal shift.
In this week’s NY Teacher there is a brief report on the April 15 UFT Delegate Assembly:
The first resolution, introduced by Vice President for Academic High Schools Janella Hinds, articulated the union’s view on the role of testing in public education. The resolution voiced the UFT’s support for the right of parents to opt their child out of state tests, called on the state to break Pearson’s monopoly on testing and condemned Gov. Andrew Cuomo’s proposal to increase the weight of standardized tests in teacher evaluation, among other things. (Resolutions: Delegates approve resolution on proper use of assessments)
But there’s something missing.
As originally written, the resolution said that tests were good for rating teachers and schools.
RESOLVED, that the UFT affirms its support of standards and its support of multiple measures to assess student progress, evaluate teachers and gauge the success of schools;
The resolution was introduced at the March 7 UFT Executive Board. New Action moved to strike “evaluate teachers” and Unity sent up several speakers to argue against the amendment, and defeated it. (It would be easy to add “overwhelmingly” since Unity controls all but ten of the 102 votes, but there were clearly some who quietly chose not to vote).
But at the Delegate Assembly a few weeks later, I moved to strike “evaluate teachers and gauge the success of schools.” There was another proposed amendment (changing “standardized assessments” to “state-mandated assessments,” shifting both meaning and tone). I saw LeRoy Barr move from the podium to the floor of the meeting hall – usually a sign that he would take the mike to speak against. A Unity loyalist whispered to me that it was a good amendment, but that it would be voted down. But after some discussion, I saw that LeRoy returned to the podium. They changed their minds? Sure enough, the amendment went through, either unanimously or close to it.
What happened? Mulgrew was touting the new Federal Legislation that would stop mandating sanctions for schools with low test scores, and stop forcing states to rate teachers based on student test scores. It would be hard to favor such changes, if the union maintained its traditional stance that said that test scores should be part of rating teachers and schools. In fact, a delegate asked if the amendment was just echoing the federal change. Also, Unity has been taking heat for helping bring in the lousy evaluation system. And the DA was during state testing, and Unity was taking heat for opposing MORE’s opt out resolution. They stepped aside.
It’s not a small thing. The largest teachers’ local in the country dropped its official support for using tests to rate teachers. The Delegate Assembly said that we do not support using “multiple measures” to”evaluate teachers and gauge the success of schools.”
The DA coverage in the New York Teacher omits that the Delegates don’t want multiple measures used to rate teachers and schools. It is easier (not easy) to delete language from a resolution. It’s another thing to change policy, on the ground. We have not seen any change in practice.