Extra topics for freshman math. Ideas? (some responses)
On Facebook I asked a question, and I got a great discussion from a bunch of thoughtful math-people. Near the bottom, Sue asks if why we aren’t doing this on jd2718, and she was right. So here’s my question, and the discussion follows:
Question for you (and others).
I am going to see a group of 9th graders one period a week, to work on non-curricular “Extra Topics” in algebra or number theory – think of this as pre-research or pre-math team. I’d like them, in 2s, 3s, or singly, to select topics to study, and to decide when to write up their work and move on to a new topic…
I think I am ok with the structure, once they’ve gotten a topic and gotten going. But I am unsure how I should help them pick their initial topics. Suggestions? I was thinking to have a number of books that they could skim/browse. There’s a high school level number theory book (which we use) which could serve that function, generating ideas. Anything for algebra? Or should I just do a full period presentation, 2 or 3 minutes per potential topic, “take notes and let me know next week what interests you”
What do you think?
The answers were helpful, and interesting:
Kate: I’d be inclined to dangle problems in front of them. Ones with low barriers to entry, but rich structures to explore. How you format the dangling doesn’t really matter, I don’t think. You could present them, or print them on cards, or whatever. My intuition is “look through these books” might be a little too open/unstructured for a 9th grader; I’d expect them to not know where to start and feel paralyzed.
Sue: So, Kate, what problems would you dangle? My Spot It problem works with highly motivated kids that age, but not so much with less motivated kids. I love exploring Pythagorean triples, but I have no idea if the problems there are useful for your kids. Jonathan, do you have the Kaplan’s book, Out of the Labyrinth: Setting Mathematics Free? It has a list of good problems in back. If you do want to point to books they might use in their exploration, I’d include the Number Devil, The Cat in Numberland, and Tanton’s Math Without Words.
Kate: Check out Paul Lockhart’s new book Measurement. Or Math Forum’s Problems of the Week? Or yes the Kaplans’ book. If you think there’s interest in grounding explorations in real-world stuff, check out the free lessons Mathalicious has available.
Me: (so I also have a small budget to replenish a classroom library – keep those suggestions rolling in!)
Andrée: I prefer none of these. Sorry. I like stuff like this book I have about NIM games and how to make all the variations. Then just play and play and toss in a question now and then (“Can you predict who will win?” “Will a table help you keep track of winners?”). Or network theory with children’s books (Virginia Lee Burton has a snow book to start) or one of the ancient math riddles (St. Ives) or Boxes with Topses (Marilyn Burns article in one of her books) which I developed into pentaminoes (we made a board and played online) which leads to triominoes, tetris. Or binary birthday cards (why do they work?). All this stuff they play with as in a real game and then can talk and think it out. It is the only math that fascinates me. There is a history of multicultural math that was fascinating: I personally worked each problem in it and learned so much. It wasn’t a text book and I can’t find it at amazon right at the moment and my copy is buried in one of my piles. But if interested, I can get title, etc. The Egyptian palimpsest is just full of problems to play with.
Andrée: you pick the first project and then see where that leads them. make a list of what questions they ask and then you can lead them to sources.
Andrée: oh and there is that game Mastermind (it has a different commercial name) athttp://nlvm.usu.edu/en/nav/vlibrary.html, which is a site FULL of ideas.
Andrée: Let them do the exploration while you rest. But you’ll get dragged into it and learn too. It’s all pretty much number theory and discrete mathematics. But please stay away from deadly textbooks. And keep a journal of your experiences.
Andrée: (Mastermind is taken down at nlvm.usu.edu for trademark stuff. You’ll have to buy the board game) (Rush Hour, online. Game analysis)
nlvm.usu.edu
Sue: Possibly The Man Who Counted. Possibly You Can Count on Monsters. Perhaps Mathematics: A Human Endeavor, though it looks like a textbook. What is your high school level Number Theory book? I’ve used and loved the one from Art of Problem Solving. (But it doesn’t look exciting.)
Sue: Andree’s suggestion of Nim games reminds me that you’ll want to check through the materials Josh. Joshua Zucker has created for the Julia Robinson Mathematics Festival. Josh, do you have any other ideas?
Sue: And if Paul Zeitz has his materials online, he offers up lots of great problems in math circles I’ve attended. Some are extensions of Nim-like problems.
Me: Which of you taught me “puppies and kittens”? I used it last week – a kid said it reminded her of (and then she described Nim, without using its name)
Me: OK, so it is not high school level. But the chapters are very short, and it is kind of readable. This course, or potential course, grew out of me asking some advanced freshmen to skim the book, and see if there was anything that grabbed there interest – and there was. The book is A Friendly Introduction to Number Theory by Joseph Joseph Silverman
Andrée: I just read 1 review of that one at goodreads and it’s on my “to read” list now. The one I was talking about is http://www.amazon.com/Multicultural-Mathematics-David-Nelson/dp/0192822411
Josh (who I think I interacted with years ago, but not in a while, and not on FB ever): Silverman’s book is pretty good. http://jrmathfestival.org has links to a bunch of the activities Sue was talking about. There may be some good stuff at http://mathteacherscircle.org — it’s aimed at teachers, so a lot of stuff there is left more open-ended than we might usually do for kids, plus some of you might enjoy the session notes as well. There’s also lots of good game theory (like the Divisor Game linked at the JR festival site for instance, as well as Nim and the like). The recent Mathematical Magic book might lead to some good ideas too and it’s definitely a good one for your shelf.
Sue: That one comes from Paul Zeitz. (I don’t remember discussing it with you or blogging about it, so I don’t think I showed you.)
Me: It could have come from a discussion about four years ago. I dragged it out of an old lesson. But it definitely came from one of the math blogging people. Dave Marain ? Anyhow I changed kitten to an archaic form, catling… And I think I remembered the rules correctly…
Me: Take as many puppies as you want. Or as many catlings as you want. Or an equal number of catlings and puppies. But if you take the last animal, you lose.
Ben: I second Kate’s general recommendation of dangling problems. If a subject has pretty pictures (e.g. fractal geometry or low-dimensional topology) then those can also be dangled, although I think the problems support initial exploration by a student better. For sources, here’s some quick thoughts: LOGIC: Raymond Smullyan’s puzzle books especially his classic _What Is the Name of This Book?_. ALGEBRA: the book Visual Group Theory by Nathan Carter is a good place to go for inspiration although it’s not designed as a compendium of problems. Also I think the Mattress Problem, and the problem of counting the number of distinct ways to, say, 2-color the vertices of an octagon (or a dodecahedron, etc) are great for motivating group theory. COMBINATORICS: I like the book Aspects of Combinatorics by Bryant, and it’s pretty accessible, though there are lots of good choices here. Okay, now I’m blanking. Books with a motivated history of math might also be useful for dangling: coming to mind are The Calculus Gallery by William Dunham and Gamma by Julian Havil.
Jackie: Thinking along the lines of “math team” type problems — you could get old AMC 10 (or even old AMC 8 contests) I have bunches of both as .pdf’s if you’d like). You could also use AoPS’s ALCUMUS site to set up a “classroom” and pick the problem types on which you’d like them to work — or just pick all that are appropriate and let them work on them until they find an area in which they’d like to delve deeper.http://www.artofproblemsolving.com/Alcumus/Introduction.php The upside of that would be that they could continue to work on their own if they’re interested. The downside would be not working together during your time with them.
Jackie: If you’re looking for something more “group like” you could work on the Problems of the Week from the IMP program. (I don’t have those electronically but I could scan a few of my favorites and send them).
Rishana: For student inspiration, you could show some of Vi Hart’s videos.
Rishana: One of my students also introduced me to Numberphile on YouTube. Some really cool topics are explored in those videos,too.
Sheik: Let them start with basic sequence of numbers, find the patterns and built on that and come up with their own function .
Sue: There are two regions in a circle problems, one easier, one harder. I love them both. I call the first the magic pancake problem (magic because a small piece is just as delightful as a big one): Cut a pancake n times, count the maximum number of regions. The second involves putting (n) points on a circle, connecting them with straight lines, and counting the (maximum) number of regions formed.
Sue: Shouldn’t we be doing this at your blog, where the results will be google-able later? This is becoming a great list.
Andrée: of course there is a math ed (research) community at google+ which is just boring links to journal articles. yawn. you could get the starch out of their shorts . . .
https://plus.google.com/u/0/communities/104846032595782747037
Mathematics Education Research – Google+
Jack: yea, The Math Teacher Circle problems is a really good place to start. Also, there’s always Martin Gardner’s Mathematical Games series.
JD2718 
I like having all our suggestions up in the main post. Nice collection.
Art of Mathematics is a series of books (in draft form?) that could provide a lot of trigger material for further exploration. With some light supervision, like you provided in the extra hour sessions, these books could be great for a really wide range of students. Also, they are free online.