Teaching Teachers to Teach

The "About Me" section in this blog states that I have worked with the education departments of a couple of local colleges. That's more true than ever this semester. With the usual instructor of Vassar's Math and Science Methods course on leave this spring, I have stepped into the breach and am now trying out being "Professor Currie" (that's ADJUNCT professor Currie to you--I'm not actually sure what my official title is).

I have a lovely class of what the education people like to call "preservice teachers," and so far the class is going well, and the students seem to be enjoying the activities and the discussions BUT there's a certain disconnection I'm noticing, which is that things seem pretty theoretical without, you know, actual children in the room. And teaching is all about children, right?



So for my most recent class I brought in a few children. Three, to be exact: three students who are currently fourth and fifth graders at PDS. The kids wandered in about 3:45 (class starts at 3:10) and sat down in too-big chairs in the seminar room. They looked a tad uncomfortable at first but soon warmed to the situation.



I started by asking them some questions about their enjoyment of math (they said they liked it, and a good thing too)--what they liked best about it ("it's challenging," a couple of them said), what was not so good ("when it's boring"--but I could've told you that in advance). I asked them to talk a little about how they learned multiplication as well. (Part of the focus of the class session was on multiplication and division.) Mathups, the kids agreed, had been very helpful, and using the array model had helped them understand the concepts too.



Other questions followed, both from me and from the students. What was your most memorable math project? (The corn kernels experiment, one student said. See the blog entries from October 2009 for more details on this one.) Do you prefer mental math or pencil and paper? Why? I had asked the college students to write an explanation of why the multiplication algorithm worked; one of the students was a bit unclear about one part and asked if the kids could explain it more clearly; one was happy to rise to the challenge. What manipulatives did they like? (NOT the pattern blocks, one child explained, adding "I like numbers best.") The kids were poised and articulate and knowledgeable, which was great to see. ("My, you know a lot!" I was tempted to say at one point. "Your math teacher must be HIGHLY skilled and no doubt deserves a medal and a large cash prize!" But I resisted.)



Finally, I'd arranged for the kids to teach the students a few multiplication games they had played in the past to help them learn and practice multiplication. So we divided into small groups, each with a child in charge, and off we went! The kids really enjoyed the chance to be in charge and the experts, and took well to the role.



We could've gone all afternoon, I think, but homework and other obligations loomed for the younger ones and us older ones had more discussion to do--so I paid the kids with some Freihofer's and thanked them much and off they went.



So the kids enjoyed it, and the students enjoyed it, and I enjoyed it, and it was an excellent reminder that teaching math involves teaching actual children. Fun stuff!

Obi-Wan, Mini Hot Dogs, and the Ninety-Dollar Jar of Olives

Hungry? You’ve come to the right place. Some of our second graders have been spending a few of their math periods lately planning menus for a special dinner. If they have leftovers (and judging from the amount of food some of them are planning to buy they’ll have lots), I’m sure they’d love to have you drop by.

Each of the children was instructed first to draw up a guest list consisting of seven fictional characters they’d like to get to know better. Their lists skewed heavily toward fantasy fiction: Lord of the Rings, Harry Potter, the Star Wars crowd. Obi-Wan Kenobi, in particular, will be kept busy nearly every evening for a week with invitations to these foodfests.



Next, they sketched out seating arrangements. One child set up four tables of two people each. Another planned for two tables, each seating four. Someone else constructed an octagon. (Eight people? Yes--seven guests and a host.) Then they planned a menu. “Not just desserts,” I cautioned them when it seemed clear that this was the way the wind was blowing. “A main course, or even two, so people can have a choice. An appetizer, perhaps. Soup, salad. Fruits and vegetables. Drinks.” “Oh, okay,” they said, and proceeded to put together menus that were...well, idiosyncratic, or what else would you call it when red snapper and turkey are served side by side?

Now came the most mathematical part: shopping. We did this in the comfort of our own classroom, using grocery ads from newspaper inserts as well as the circulars that arrived in my mailbox over time. I gave the children record sheets as well. “Find the items you want to buy,” I told them. “Record what the item is and how much it costs to buy just one. Then decide how many of the item you need. What will the total cost be? Figure that out and record it here in this column; then round that to the nearest whole dollar to make estimating the grand total easier.”

The children fell eagerly to work. “Mini hot dogs, mini hot dogs,” one boy repeated over and over, scanning the ads in vain for his favorite entrée. “Pizza!” crowed a classmate, jabbing a pencil at a listing in the frozen foods section of one advertising supplement. “Pizza?” asked a classmate. “Hey, I need that too!”



The kids needed help deciphering some of the prices: supermarkets, we learned, don’t often write prices the way teachers say you should. Instead of $1.59, for instance, they often write a big 1 followed by a smaller 59 and no decimal point or dollar sign at all: “Looks like a hundred and fifty-nine dollars,” one child said disapprovingly. “5 for $2” was confusing enough, but “4/3” to mean “you can get four for three dollars” was enough to make strong second graders cry. (Well, not literally. They were good sports about it and chalked it up as yet another example of the weirdness of adults.)

Determining how many of each item they needed was again idiosyncratic. One child decided that each person would probably eat 5 cookies, no, make that 6, so he settled on 48 as the number to buy. Somebody else, less generous or perhaps expecting that everyone would be full when dessert came around, decided to purchase just one small apple pie for the table. Did you need one carton of orange juice, or were you better off with 10? Different kids answered similar questions in wildly different ways.



Determining the cost of n cartons of orange juice, when n was more than 1, was often tricky as well. Some of these hosts have a beginning knowledge of multiplication, which they used to good effect. “Each cookie costs ten cents,” mused our 48-cookie-buying friend from before, “so 48 of them would cost, um, $4.80.” One child bought two jugs of milk at $2.99 apiece, set up the costs in columns, and laboriously regrouped to arrive at the total, $5.98; another, trying to determine the cost of two $4.98 items, rounded up to $5 for each, added the estimates to get to $10, and finally subtracted the extra four cents for a total of $9.96. We shared strategies and tried out newer, more efficient ones.




What about items that appeared on menus but were nowhere to be found in the ads? I hear you ask. Surely there were some of those. There were indeed, the aforementioned red snapper and mini hot dogs were among them. “Not a problem,” I said. “You have an idea of what similar things cost. Estimate the price. Just put a circle around the price on your record sheet to indicate that it’s an estimate.” Some estimates were IMHO pretty good. Others were...well, one cent for salad dressing was perhaps overly optimistic, and over $100 for a few other items seemed a tad excessive, but hey, who am I to complain?



Once the estimates were done, it was time to find a grand total. Making an estimate (using the rounded-to-the-nearest-dollar figures on the shopping list) was a job for mental arithmetic. Finding the actual total was a job for a calculator. Children compared the answers to make sure their totals were reasonable, then looked again at the total bill.

At first each was flabbergasted by the high cost of groceries (especially given the $90 jar of olives one of them purchased--an estimate, natch), but after a moment flabbergast vanished to be replaced by pride. Yes, indeed, they seemed to be saying, a meal that costs over $500 must certainly be something special. Now if only they’d found more expensive broccoli and purchased twelve loaves of bread instead of just two--

So the menus are done, the shopping lists are ready to go, and all that remains is to check on the availability of the guests. Anyone know if Dumbledore is free the evening of the 23rd? The invitation says six pm sharp, but he can arrive at seven if he doesn’t mind missing the goldfish and gummiworms being served as appetizers...

The Pirates of Poughkeepsie

It was Pirate Day in the 1-2s a little while back (arr!), and the buccaneers were out in force. Their captains, um, teachers, had a number of activities planned for them. One of these was locating a treasure chest that pirates of a previous era had buried somewhere in the Chapman Room. Luckily, these modern pirates had an appropriately antique map and faded directions.



“What’s it say next?” one pirate asked.

“Turn left,” another replied, squinting at the directions, and as one they all turned right.

“No, no, the other left,” their teacher explained helpfully.

They rotated 180° to the proper orientation. “Next,” said someone, “it says to take ten steps forward.” and they all took ten steps forward except for those who took twelve or thirteen because if ten steps are good then twelve or thirteen are even better, besides which if you’re the pirate in the lead you might get a bigger share of the treasure if and when it appears.



“Ten steps,” suggested the teacher....

So there they were, these young terrors-of-the-seas, thinking they were engaged in a search for hidden booty. But how wrong they were! Us Math Guy types knew the truth: They were actually engaged in Spatial Reasoning. Left, right, forward, backward, mapping, following directions, the whole nine yards. Proving that even pirates are mathematicians at heart--scuse me, at h(arr!)t.

Camera Shy

For those of you loyal readers who don't know what I look like, well, wonder no more! Though photographs of me are hard to come by on the web, a student kindly sketched my picture on the whiteboard that lives in the Chapman Room. Thanks to the miracles of modern technology, I can get you that image faster than you can say "Jackie Robinson." Here you go:



Thank you, Joanna in the 3-4 for the extremely accurate likeness (well, not quite; I am not nearly that handsome, but never mind). In any case, if you ever run into someone who looks almost exactly like that picture, you will know it must be me.

Cheerful Charlie and the Cupcakes

My good friend Cheerful Charlie, I told the third and fourth graders, was having a party, at which he planned to serve cupcakes. One hundred cupcakes, to be precise. I'm not sure who-all is on his guest list (other than me, of course), but he's either inviting a lot of people, or a few big eaters, or else he just wants a lot of leftovers.

Cheerful had (wisely) decided not to make these cupcakes from scratch (his measuring skills aren't what you would call real accurate), but was having some trouble determining which store he should go to. He had three choices, I explained, and he wanted to spend as little money as he could and get as good a deal as possible, and if the students could advise him that'd be great.

So, there was Cupcakes R Us, I told the kids, which sold baskets of 50 cupcakes at a shot, and each basket cost $40 but he also had to pay a fee of $5 to park.

And there was Cupcake Depot, where cupcakes were $9 per bag for a bag of 10, plus which Cheerful had a coupon for $5 off his total purchase.

And there was Cupcake's Discount Warehouse...

Look at all the information, I told them, and do some calculations if you need to, and decide if there are other considerations Cheerful should be thinking about, and then write Cheerful a letter suggesting what he ought to do. The students could turn in a handwritten letter which we'd forward on to Mr. Charlie, I explained, or they could email him directly at cheerful.charlie@yahoo.com, an account which he checks but not as regularly as he should because he often forgets or mistypes the password.

For some students, the assignment was just a relatively straightforward problem in arithmetic. You determine how many bags, boxes, or baskets of cupcakes Cheerful needs to buy at each store so he has 100 cupcakes in all; you multiply that number by the cost per box/bag/bucket/basket; you add the parking fee or membership fee, you subtract the coupon...and if you've done it right, you inform Cheerful that the cost at all three stores is the same, eighty-five bucks, and he can go wherever he likes and it doesn't make any difference. And this is a fine way of looking at it.

But what makes the problem interesting is the real-world nature of it. Sure, price is important. But is the lowest price always the best deal? I remember when my wife and I discovered WHY generic spaghetti sauce was so cheap (hint: it contained mostly water)... So the question becomes, what other things should Cheerful be taking into account?

Well, a lot of kids came up with lots of ideas.

"You should go to Cupcake's Discount Warehouse," one student wrote. The membership fee of $5 was annoying, she pointed out, "but if you have to go back again you will have your membership card so it'll be cheaper."

"Maybe you can walk to Cupcakes R Us," said somebody else. "Or ride a bike. Then you might not have to pay to park." (Note the "might." Hedging your bets, we call it.)

"It depends," wrote another student. "How far away are the stores?" A good question. If the nearest Cupcakes R Us outlet is in Albany, Scranton, or Paramus, it isn't worth the time and the gas to get there.

"You should ask them each for a free sample," suggested one optimist. "If you say you'll buy a lot they'll probably let you have one. Buy the one that tastes best." Not much point in saving five dollars if the cheaper cupcakes tasted like sawdust or carbon paper. --Or were made primarily of water, like the generic spaghetti sauce referred to above.

"Buy them in bags of 10," someone else advised. "The prices are all the same but if you need more then you only have to buy 10 more, not 20 or 25 or 50, and that will save you money."

And another student went right to the heart of the matter: Measure the cupcakes. "Buy the ones that are biggest," he suggested.

Unfortunately, Cheerful (who likes things simple) is still uncertain what to do: he's definitely bummed at the prospect of needing to find more information. We'll keep you posted. In the meantime, it was nice to see how many students recognized that there might be other considerations besides the cost of the cupcakes. We like to remind kids that math is about real-world situations, and in particular we like to point out that answers may not be as cut and dried as the textbooks sometimes suggest they are. This was a good example of both--and a good example of how a little knowledge can be a dangerous thing.

(Though regarding "other considerations," you CAN have too much of a good thing: see one of my favorite cartoons of all time, http://xkcd.com/309/. --Cut and paste the URL into your browser window if the link doesn't work for you. The two folks on the extreme right? That would be me and my wife...)