The last time I wrote about my classes is over two weeks ago. I'm discouraged. (I'm up in the middle of the night because I couldn't sleep. Is my restlessness due to my concerns about my teaching?)
The most embarrassing part to report is that I still have one class that's definitely not with me. Most of the high school students are definitely not engaged. [There is a high school housed at our college. I have 13 high school students in this section. In the section that's going well, I have 5 high school students.] I've also had continued trouble with a bunch of the students who are on the women's basketball team. I can blame students who come in not caring about learning, or I can blame myself for not pulling people in. Of course blame gets one nowhere. But that class is distressing for me.
There's more. The hardest part to write about is my questions about how to handle students who continue to disrespect the needs of other students. I've mentioned my trouble to the high school principal and the basketball coach, and I've sent a few students to the dean of students. I don't like having power over students in the first place, and yet I'm using others to do what I'm not comfortable doing? Yuck.
I gave a test on graphing yesterday, and it's not good. I've looked over all the tests from my 10am section. (I marked each problem with a C or an X, but I didn't put a score at the top of most.) Only a few people did well enough for me to put a score on their paper. The rest will have "Redo" at the top. The 10am class has been delightful, people are working hard and having fun, and still, most of them did really badly. I gave them a practice test. I was clearer about which textbook sections have the problems they should use to practice. We went over the material in lots of different ways. They study together outside of class. They still bombed it. That section is a great group, and I'll be asking them what they think. But they might not know how to study more effectively. I love that group, and I feel stuck.
I feel like graphing is a great topic to do in engaging ways. I did a scaled down version of Dan's stack of cups problem with them. I want to do the egg bungee drop problem, but I haven't gotten the supplies, or tried it out. I need to find time to prepare for that. (I had time yesterday. Why didn't I pull it together?) But I feel pressure from the students (and myself?) to lecture, and to make it all organized and clear. (I cannot make it clear for them. That comes from struggling with the problems, and asking why each step of the way.) The projects feel like extras sometimes...
I'm reading blogs about Waiting for Superman and the Education Nation (the NBC series that has hardly any teachers, but purports to discuss education), and I keep running across the notion that getting rid of 'bad teachers' will improve education. (Obama really approved of the Rhode Island firing of all the teachers in a school?! That shatters my illusions. I know, that was months ago. But it's hard to digest...) I can't help feeling like a 'bad teacher' right now. I should have read more of the Doug Lemov book...
Blame my distress on insomnia perhaps, but I'm thinking my insomnia is caused by this distress... I may come back and add a bunch of links later, but I think I'll lie back down now and try again to sleep.
Thursday, September 30, 2010
Sunday, September 26, 2010
Puzzle: What Is This?
I got a very unusual gift for my birthday. I know what it is, but I'm not sure exactly how to use it yet. I'll have to experiment with it.
Can you guess what it does? (It can move at the joints.)
Saturday, September 25, 2010
September 25th is Math Storytelling Day
This holiday isn't well known yet, but over the years that will change. Maria Droujkova invented this holiday last year as a birthday present for herself. It's my birthday too (!!) and I'm publicizing our holiday as a present to myself.
Seth Godin's post, What should I do on your birthday?, inspired her. Here's some of what she wrote last year:
And now here is a story I'd like to share.
My son is in the most amazing mini-school. Right now it only has 5 kids, all about 8 years old. (They'd like to increase to about 8 kids. Contact me if interested.) Felicia runs this school out of her home. Already my son has learned to swim during school, and decided he loves science class. I love that they do sun salutations, and have a rock basket for noticing positive things that happen.
Felicia has been studying lots of different educational philosophies lately, and liked Waldorf's emphasis on story. She made up this math story...
Jolly Josh, by Felicia Jeffley
Once upon a time there was a boy named Josh. Josh was a jolly boy. He loved to laugh and play and sing and jump. He looked like any other boy, especially when he was riding his bike or swimming or drawing or reading.
But when he walked, that’s when things got a little strange. He counted. He counted by 2’s and 5’s and 10’s and 100’s. He never just walked. He always counted and walked.
When he walked to the park he counted by 2’s.
When he walked to the beach he counted by 5’s.
When he walked to the swimming pool he counted by 10’s.
When he walked to school he counted by 100’s.
“Josh, stop counting,” his sister would insist.
“Josh, I’m talking to you," his father would scold.
“Josh, are you listening?” his mother would ask.
“Josh, school’s over,” his friend would remind him.
But Josh kept counting and counting and counting and counting until ... he reached 1000.
Well, when he went to school, it was simple. It’s easy to get to 1000 by counting by a 100. It’s not too bad to get there counting by 10’s. And if he fast walked, which he often did, counting by 5’s wasn’t so bad. But counting by 2’s all the way to the park was long, long, long, long. It seemed to take forever.
None of his friends would ever walk with Josh to the park. “I’ll meet you there,” they’d say. His mother would talk on her phone the whole walk. His dad would listen to the game on the radio. His sister, well, she refused to take her little brother to the park.
One day when Josh was walking to the park with his mom, he was so busy counting he tripped over a brick. Down he fell. It hurt, but instead of crying all he kept saying was, “62, 62, 62, 62,” in a whimpery little voice.
“Josh, are you okay,” his mother said, closing her cell phone and running to him.
"62, 62, 62, 62,” he replied.
She could see that he wasn’t okay. She pulled out her phone again and called 911. While they waited, she counted softly and sweetly to him.
“2, 4, 6, 8, ….62”
That seemed to calm him down. She sang it again all the way to 62. After a 3rd time, the ambulance arrived. A nice EMT (Emergency Medical Technician) walked over to him.
“So what where you doing when you fell down?”
“Counting.”
“Counting. I used to do that too. I’d count to big numbers, really big.”
Josh’s eyes opened wide, “Really?”
“Yea, it was fun. It’s a good thing too. Now I have to use all that counting in my job. I have to find the right house and know if the house numbers are going up or down. Sometimes the house numbers skip, like by 2’s. Do you know how to count by 2’s?”
“Yes, 2, 4, 6, 8…” Josh said in a whisper.
“Yes, he can count all the way to 1000,” interrupted his mother.
“Wow, that’s impressive!” said the EMT.
“62,” said Josh.
“62?” repeated the EMT.
“Stopped at 62…”
“Are you trying to say that you had gotten to the number 62 today?”
“Yes. Don’t want to forget.” Josh added.
“Oh, that’s why you keep saying it. You don’t want to forget where you were!”
Josh nodded his head a bit.
“How about I write it down for you. Then, you can start off with 62 the next time you walk to the park?”
Josh smiled softly.
“Well, it looks like a sprain. So, Josh I don’t think you’ll be able to resume counting for a few days. We need to get some ice on that ankle."
“Well, that’s good news,” his Mom said in relief. “Could have been worse.”
“That’s true,” said the EMT. How about if I give you all a ride to your house. Let’s see. We’re in front of house number 2 and you live at 50. I’m noticing that the houses go up on this side by 2’s. Yes, that one is 4, the next 6, the next 8 and so on. So, your house is not too far from here.”
The EMT picked Josh up and put him in the ambulance. His mother hopped in and off they went.
“62, 60, 58, 56, 54, 52…” the EMT counted.
Josh’s eyes got big.
“So, Mr. Number Counter, can you do that?” asked the EMT.
Josh shook his head, no.
“It’s counting backwards by 2’s!”
The EMT counted backwards all the way to 0. Josh was impressed.
Josh was better in a few days and off he went to the park. “62, 64, 68, 70…” he began. On the way home he decided to walk backwards and count backwards. Well, that didn’t last too long. He ran into a tree. This time it was not an emergency, just funny. He laughed. His mother laughed.
“I think I should turn around,” he said. His mother agreed.
“But you can still count backwards,” she reassured him.
And that’s what he did. All the way home, from 1000 to 0. Well, with a little help from his mother.
Jolly Josh arrived home jumping for joy. “I know how to count backwards!” he exclaimed.
From that point on, ...
When he went to the park he counted by two’s on the way there and backwards by 2’s on the way back.
When he walked to the beach, he counted by 5’s on the way there and backwards by 5’s on the way back.
When he walked to the swimming pool he counted by 10’s on the way there and backwards by 10’s on the way back.
When he walked to school, he counted by 100’s on the way there and backwards by 100’s on the way back.
Eventually, people stopped asking him to stop counting. Instead they asked him questions like this:
“Josh, can you count the number of braids in my hair? They have to be even.” his sister would demand.
“Josh, how many yards are on a football field?” his father quizzed.
“Josh, can you get enough birthday cookies out, so each kid can have 2?” his mother requested.
“Josh, how much is 5+5+5+5 because I got four $5 bills for my birthday?” his friend asked.
“30, 100, 16, 20,” he answered without hesitation. No matter the question, Josh was quick with the answer. And the more they asked, the quicker he got. And the quicker he got, the more they asked.
He also listened to music and played soccer and counted stars at night and he still walked and counted. He counted forwards and backwards and backwards and forwards. Josh truly was a Jolly boy!
The end.
Seth Godin's post, What should I do on your birthday?, inspired her. Here's some of what she wrote last year:
For my birthday, I would like people to share math stories. So, for my friends and family, let it be a Math Storytelling Day. We all have some math stories to tell!
- We can use the classics, like Hilbert's Hotel Infinity.
- We can use math anecdotes and jokes.
- We can commiserate about horrible events from our childhoods that caused us bad cases of math anxiety.
- We can laugh with/at customers in search of math clues.
- We can bring in history, like the Betsy Ross star story.
And now here is a story I'd like to share.
My son is in the most amazing mini-school. Right now it only has 5 kids, all about 8 years old. (They'd like to increase to about 8 kids. Contact me if interested.) Felicia runs this school out of her home. Already my son has learned to swim during school, and decided he loves science class. I love that they do sun salutations, and have a rock basket for noticing positive things that happen.
Felicia has been studying lots of different educational philosophies lately, and liked Waldorf's emphasis on story. She made up this math story...
Jolly Josh, by Felicia Jeffley
Once upon a time there was a boy named Josh. Josh was a jolly boy. He loved to laugh and play and sing and jump. He looked like any other boy, especially when he was riding his bike or swimming or drawing or reading.
But when he walked, that’s when things got a little strange. He counted. He counted by 2’s and 5’s and 10’s and 100’s. He never just walked. He always counted and walked.
When he walked to the park he counted by 2’s.
When he walked to the beach he counted by 5’s.
When he walked to the swimming pool he counted by 10’s.
When he walked to school he counted by 100’s.
“Josh, stop counting,” his sister would insist.
“Josh, I’m talking to you," his father would scold.
“Josh, are you listening?” his mother would ask.
“Josh, school’s over,” his friend would remind him.
But Josh kept counting and counting and counting and counting until ... he reached 1000.
Well, when he went to school, it was simple. It’s easy to get to 1000 by counting by a 100. It’s not too bad to get there counting by 10’s. And if he fast walked, which he often did, counting by 5’s wasn’t so bad. But counting by 2’s all the way to the park was long, long, long, long. It seemed to take forever.
None of his friends would ever walk with Josh to the park. “I’ll meet you there,” they’d say. His mother would talk on her phone the whole walk. His dad would listen to the game on the radio. His sister, well, she refused to take her little brother to the park.
One day when Josh was walking to the park with his mom, he was so busy counting he tripped over a brick. Down he fell. It hurt, but instead of crying all he kept saying was, “62, 62, 62, 62,” in a whimpery little voice.
“Josh, are you okay,” his mother said, closing her cell phone and running to him.
"62, 62, 62, 62,” he replied.
She could see that he wasn’t okay. She pulled out her phone again and called 911. While they waited, she counted softly and sweetly to him.
“2, 4, 6, 8, ….62”
That seemed to calm him down. She sang it again all the way to 62. After a 3rd time, the ambulance arrived. A nice EMT (Emergency Medical Technician) walked over to him.
“So what where you doing when you fell down?”
“Counting.”
“Counting. I used to do that too. I’d count to big numbers, really big.”
Josh’s eyes opened wide, “Really?”
“Yea, it was fun. It’s a good thing too. Now I have to use all that counting in my job. I have to find the right house and know if the house numbers are going up or down. Sometimes the house numbers skip, like by 2’s. Do you know how to count by 2’s?”
“Yes, 2, 4, 6, 8…” Josh said in a whisper.
“Yes, he can count all the way to 1000,” interrupted his mother.
“Wow, that’s impressive!” said the EMT.
“62,” said Josh.
“62?” repeated the EMT.
“Stopped at 62…”
“Are you trying to say that you had gotten to the number 62 today?”
“Yes. Don’t want to forget.” Josh added.
“Oh, that’s why you keep saying it. You don’t want to forget where you were!”
Josh nodded his head a bit.
“How about I write it down for you. Then, you can start off with 62 the next time you walk to the park?”
Josh smiled softly.
“Well, it looks like a sprain. So, Josh I don’t think you’ll be able to resume counting for a few days. We need to get some ice on that ankle."
“Well, that’s good news,” his Mom said in relief. “Could have been worse.”
“That’s true,” said the EMT. How about if I give you all a ride to your house. Let’s see. We’re in front of house number 2 and you live at 50. I’m noticing that the houses go up on this side by 2’s. Yes, that one is 4, the next 6, the next 8 and so on. So, your house is not too far from here.”
The EMT picked Josh up and put him in the ambulance. His mother hopped in and off they went.
“62, 60, 58, 56, 54, 52…” the EMT counted.
Josh’s eyes got big.
“So, Mr. Number Counter, can you do that?” asked the EMT.
Josh shook his head, no.
“It’s counting backwards by 2’s!”
The EMT counted backwards all the way to 0. Josh was impressed.
Josh was better in a few days and off he went to the park. “62, 64, 68, 70…” he began. On the way home he decided to walk backwards and count backwards. Well, that didn’t last too long. He ran into a tree. This time it was not an emergency, just funny. He laughed. His mother laughed.
“I think I should turn around,” he said. His mother agreed.
“But you can still count backwards,” she reassured him.
And that’s what he did. All the way home, from 1000 to 0. Well, with a little help from his mother.
Jolly Josh arrived home jumping for joy. “I know how to count backwards!” he exclaimed.
From that point on, ...
When he went to the park he counted by two’s on the way there and backwards by 2’s on the way back.
When he walked to the beach, he counted by 5’s on the way there and backwards by 5’s on the way back.
When he walked to the swimming pool he counted by 10’s on the way there and backwards by 10’s on the way back.
When he walked to school, he counted by 100’s on the way there and backwards by 100’s on the way back.
Eventually, people stopped asking him to stop counting. Instead they asked him questions like this:
“Josh, can you count the number of braids in my hair? They have to be even.” his sister would demand.
“Josh, how many yards are on a football field?” his father quizzed.
“Josh, can you get enough birthday cookies out, so each kid can have 2?” his mother requested.
“Josh, how much is 5+5+5+5 because I got four $5 bills for my birthday?” his friend asked.
“30, 100, 16, 20,” he answered without hesitation. No matter the question, Josh was quick with the answer. And the more they asked, the quicker he got. And the quicker he got, the more they asked.
He also listened to music and played soccer and counted stars at night and he still walked and counted. He counted forwards and backwards and backwards and forwards. Josh truly was a Jolly boy!
The end.
Friday, September 24, 2010
Bit and Pieces
Julia asks why her proofs lesson died in the water. I wonder if anyone can help her.
I haven't seen Waiting for Superman yet. I think I'll have to, so I can argue effectively against it. Kirsten Olson, at Cooperative Catalyst, wrote a review of it that troubled me. The review I trust is by Deborah Meier at Bridging Differences. I think this movie is a powerful piece of propaganda that demonizes teachers' unions. Anyone know when it's supposed to open in the Bay Area? Anyone want to go see it with me?
Gary Davis (at his Republic of Math blog) has a long post on Richard Skemp’s Relational and Instrumental Understanding. When we talk about really understanding in math, some people think they do understand, when they've memorized something like A=LxW. [Hat tip to John Cook at the Endeavor.]
I haven't seen Waiting for Superman yet. I think I'll have to, so I can argue effectively against it. Kirsten Olson, at Cooperative Catalyst, wrote a review of it that troubled me. The review I trust is by Deborah Meier at Bridging Differences. I think this movie is a powerful piece of propaganda that demonizes teachers' unions. Anyone know when it's supposed to open in the Bay Area? Anyone want to go see it with me?
Gary Davis (at his Republic of Math blog) has a long post on Richard Skemp’s Relational and Instrumental Understanding. When we talk about really understanding in math, some people think they do understand, when they've memorized something like A=LxW. [Hat tip to John Cook at the Endeavor.]
Thursday, September 23, 2010
Optional Homework
Avery started an interesting discussion over at his blog, Without Geometry, Life is Pointless. He wants to give some challenging problems as part of the homework, and told the kids (6th grade) that they shouldn't spend more than 30 minutes on the homework - it's ok not to finish. One student complained about it, and "dislikes math for the first time". What to do...
Lots of good, interesting advice. Bowen Kerins said:
I'm teaching beginning algebra at a community college. My first two mastery tests are on pre-algebraic topics. There is one fraction story problem. On version 3 of the test, it goes like this...
I said I hate marking people wrong when they have the right answer, but that this was a fluke. Subtraction doesn't solve this sort of problem. On the spot, I made up another problem, 1/2 of 1/3. Guess what.
Both answers are the same again.
Optional homework: When does this happen? ;^)
(I give my students lots of optional homework. Most of it is: "Read this cool book and write a review.")
Lots of good, interesting advice. Bowen Kerins said:
Another way to deal with speed demons is to give several problems in a row that are related and have the same answer. Speed demons may not even notice this happening, and the result is they're more likely to "look around" a little more before and after working a problem.I liked that. And it made me think about an interesting thing that happened to me today.
I'm teaching beginning algebra at a community college. My first two mastery tests are on pre-algebraic topics. There is one fraction story problem. On version 3 of the test, it goes like this...
I have a lot of books at my house, especially after all the math books I bought during my sabbatical. Right now 3/7ths of my books are math-related; 3/10ths of those are kids’ books. What fraction of my books are kids’ math books?I had marked my student's subtraction problem wrong, and was explaining to her why it would be multiplication. As I finished up, she pointed to her answer, which was the same as mine.
I said I hate marking people wrong when they have the right answer, but that this was a fluke. Subtraction doesn't solve this sort of problem. On the spot, I made up another problem, 1/2 of 1/3. Guess what.
Both answers are the same again.
Optional homework: When does this happen? ;^)
(I give my students lots of optional homework. Most of it is: "Read this cool book and write a review.")
Wednesday, September 22, 2010
Links
Math Teachers at Play #30 is up at JD2718. Lately I've been recognizing most of the bloggers who post, but this month I discovered a new one. There aren't many elementary teachers posting about math, so I was excited to find Life Among the Elms, in which Michelle Martin writes about her 4th/5th combination class at Prairie Creek Community School, a charter school in Minnesota. So far my favorite math post at Life Among the Elms is Dinosaur Math, which was mentioned in the most recent post on the blog, along with 7 other posts about math. I also enjoyed her balanced take on standardized tests on her post Use Only a #2 Pencil, even though I personally think they're horrid.
I don't know how or when I stumbled upon Cooperative Catalyst (perhaps John Spencer mentioned it), but I recently noticed how much I enjoy reading their posts, like this one.
Steven Strogatz is back! His review of Proofiness was lots of fun to read. Can't wait to read the whole book.
Here's the ultimate in math nerdiness, lovingly portrayed at xkcd.
I don't know how or when I stumbled upon Cooperative Catalyst (perhaps John Spencer mentioned it), but I recently noticed how much I enjoy reading their posts, like this one.
Steven Strogatz is back! His review of Proofiness was lots of fun to read. Can't wait to read the whole book.
Here's the ultimate in math nerdiness, lovingly portrayed at xkcd.
Games: Turning Battleship Into Something Else
Battleship was one of my favorite games when I was young, but now I'm troubled by the military setting. I bought it for my math salon because it's a good game, and you can see me playing it with a young boy in the math salon video. But I want to turn it into something different...
I actually wrote to one of my favorite game companies a year or so ago, suggesting hide and seek. I never heard back from them. I guess most companies would be wary of a game too close to something that's already out there. They might get sued, and that's not worth it.
I'd give my idea to Milton Bradley (makers of Battleship) if they'd use it. But is there a way to make my game different enough so they wouldn't be interested in suing, if someone else were to make Hide and Seek?
Here's my idea...
Hide and SeekTM has 6 children of different sizes (3 girls and 3 boys, 2 toddlers, 2 little kids, 2 big kids), and a grid with a house in the middle and spots to place kids, in the directions North, South, East and West, 5 paces in each direction. You place the kids on your grid, and your opponent is 'it'. They 'look' by telling you things like, "3 paces East, 4 paces North". You say "No one there", or "found my head", or "found the whole little girl". After they find all your kids, you trade places. The one who finds the other team using the least moves wins.
Each player needs just one hideable grid, since you're not playing both parts at once. Playing just one part at a time also makes it easier to keep track of what you're doing.
I want to play this game! Anyone know a toy company that would be interested?
I actually wrote to one of my favorite game companies a year or so ago, suggesting hide and seek. I never heard back from them. I guess most companies would be wary of a game too close to something that's already out there. They might get sued, and that's not worth it.
I'd give my idea to Milton Bradley (makers of Battleship) if they'd use it. But is there a way to make my game different enough so they wouldn't be interested in suing, if someone else were to make Hide and Seek?
Here's my idea...
Hide and SeekTM has 6 children of different sizes (3 girls and 3 boys, 2 toddlers, 2 little kids, 2 big kids), and a grid with a house in the middle and spots to place kids, in the directions North, South, East and West, 5 paces in each direction. You place the kids on your grid, and your opponent is 'it'. They 'look' by telling you things like, "3 paces East, 4 paces North". You say "No one there", or "found my head", or "found the whole little girl". After they find all your kids, you trade places. The one who finds the other team using the least moves wins.
Each player needs just one hideable grid, since you're not playing both parts at once. Playing just one part at a time also makes it easier to keep track of what you're doing.
I want to play this game! Anyone know a toy company that would be interested?
Monday, September 20, 2010
11 Interesting Articles on Math Education
Last year I was on sabbatical, working on a book. Playing With Math: Stories from Math Circles, Homeschoolers, and the Internet is nearing completion, and will most likely be published in 2011. It's full of great stories, puzzles, and ideas from over 20 authors.
To get that sabbatical, I had to make a proposal, and I was told that just editing a book wouldn't get me a whole year sabbatical. So I added a few more bits and pieces to my proposal, and now my 'sabbatical evidence' is due in 2 days. One of the agreements I made was to read 15 books and 15 articles from a list I provided. Below are my annotations for 11 of the articles (I've left out the ones I didn't care for). I've already posted about most of the books.
These articles are all good. If you haven't read the Treisman article, and you have any interest in social justice issues in relation to math education, do read it. The Hoyles article is also a classic. It differentiates between school-math and the math people create on their own, that they don't even think of as math. You may want to skim that one, but you'll find some gems in it.
Andreescu, T., and Mertz, J, Cross-Cultural Analysis of Students with Exceptional Talent in Mathematical Problem Solving. In Notices of the AMS V55, Num 10, 2008.
Janet Mertz spoke on this at the 2009 Great Circles conference hosted at the Mathematical Sciences Research Institute, in Berkeley. This talk is available as a Quicktime movie, and includes more discussion of the data than what is in the article.
Larry Summers, while president of Harvard, made the claim that the likeliest reason for the paucity of tenured women math professors was brain differences between men and women. (He hypothesizes that there is more variability in intelligence among men than among women, so there would be more extremely smart and more extremely dumb men than women.) One woman professor walked out, and then encouraged colleagues to do some statistical analysis. The percentage of women among those who rank most highly in math competitions varies widely from one country to another, from below 5% to over 20%. This is also reflected in the percentage of women in tenured math faculty in different countries (below 5% in some western European counties and over 20% in Portugal and a few other countries). This research shows that culture is a big component in girls’ and women’s achievement in math, making clear our inability to disentangle the effects of biology and culture.
Ball, D. L. , Working on the inside: Using one's own practice as a site for studying mathematics teaching and learning. In Kelly, A. & Lesh, R. (Eds.). Handbook of research design in mathematics and science education, (pp. 365- 402). [Link is to pdf.]
Ball analyzes how 3 different teacher-researchers (herself, M. Lampert, and R. Heaton) use their own teaching as a way of researching how teachers teach and students learn. In her introduction she discussed pre-services teachers’ misconceptions about math:
Ball, D. L., & Bass, H., Interweaving Content and Pedagogy in Teaching and Learning to Teach: Knowing and Using Mathematics. In J. Boaler (Ed.), Multiple Perspectives on the Teaching and Learning of Mathematics (pp. 83-104). [Link is to pdf.]
The most valuable idea in this article for me is related to the notion of ‘compression’ – once we learn something well in math, it gets compacted, and seems simpler. To teach, we need to reverse that process:
Ball, D. L., Hill, H., and Bass, H., Knowing Mathematics for Teaching: Who Knows Mathematics Well Enough To Teach Third Grade, and How Can We Decide? In American Educator Fall 2005. [Link is to pdf.]
The authors looked at what special knowledge of math is needed by teachers, in order to effectively teach it. The article gives background on why they structured their research the way they did:
The article also gives some preliminary results which may be promising regarding social justice:
Benson, S. & Findell, B., A Modified Discovery Approach to Teaching and Learning Abstract Algebra. In Innovations in Teaching Abstract Algebra, MAA Notes #60, eds. Allen Hibbard & Ellen Maycock, pp. 11-17, 2002.
The author taught almost entirely through worksheets he designed to get groups of students working on the material. His ability to observe the groups allowed him to get more clarity about what they understood than when he’d lectured. When they didn’t understand the congruence relationship, he got them to discuss what they had shown so far, and would not ‘give them the answer’. In spite of letting go of a ‘calendar’ and taking the time needed for students to work out their own understandings, he found that this class actually covered more material than the usual.
Dweck, C., Caution: Praise Can Be Dangerous. In American Educator, Spring 1999.
Carol Dweck has written a book, Mindset (2006), which says about the same things this article does, at much more length. The article describes her thesis and her research much more concisely and (in my opinion) effectively. Her claim (proved by her research) is that praising a student’s intelligence makes them wary of harder tasks and of looking dumb, but praising their effort encourages them to tackle harder tasks and enjoy it. She also looks at people who think intelligence is fixed and compares them to people who think effort can change one’s intelligence. People with the second mindset are able to develop their potential much more effectively than those with a ‘fixed intelligence’ mindset.
Hoyles, C., Noss, R., & Pozzi, S., Proportional Reasoning in Nursing Practice. In Journal for Research in Mathematics Education, Vol. 32, No. 1 (Jan., 2001), pp. 4-27
Research has repeatedly shown people doing mathematics in their work situations more effectively than they can as students. Most people say they do no math in their work, because they don’t recognize that what they’re doing is mathematical.
These studies suggest that adults are adept at solving proportional problems in everyday or work situations but often employ informal strategies that are tailored to the particular situation and are not easily identified with formal school-taught methods. (From page 6 of pdf.)
The authors of this article look at nurses’ calculations of drug dosages and found these calculations to be more flexible and fluent than the 'nurses Rule’ they’d been taught.
Kato, Y., Honda, M., & Kamii, C., Kindergartners Play Lining Up the 5s: A Card Game to Encourage Logico-Mathematical Thinking. In Mathematical Behavior, 13(1), 55-80, 2006.
The authors describe their research studying video of children playing a very simple card game. They are looking for ways to describe progress in logical (‘logico-mathematical’) thinking skills. I am impressed with how much thinking kids need to develop to do things that seem utterly simple to adults.
Schoenfeld, A., A Highly Interactive Discourse Structure. In Social Constructivist Teaching, Volume 9, pages 131–169. 2002
A quarter of this article is devoted to transcripts of two very different classes, a high school physics class, and a third grade math class. The two classes turn out to share a structure in the interactions between the teacher and students. Schoenfeld has created a flowchart of this structure, but I find a summary more useful:
Teacher starts by giving context and background for topic.
• Asks class: “What (else) can you say about [this topic]?”
• Calls on a student.
• Does their response raise other issues? (If so, deal.)
• Is clarification, expansion or reframing useful? (If so, deal.)
• Would more discussion be useful? (If so, deal.)
[I have created a sheet to remind myself of this summary, to help me get out of lecture mode.] Deborah Ball taught the third grade class, which was videotaped, and is cited in many researchers' work.
Schoenfeld, A. H., What do we know about mathematics curricula? Journal of Mathematical Behavior, 13(1), 55-80, 1994.
Schoenfeld discusses what we do and don’t know about the math curricula we need. “The ‘constructivist perspective’ is better grounded in empirical and experimental evidence than the theory of evolution; we should just assume it and get on with our business (while working … hard, of course, to flesh it out and understand it more fully).” But, on the other hand, the best balance between traditional ‘content’ and the development of problem-solving skills is unclear, and “If, for example, what we now call ‘algebra’ is distributed through the curriculum in bits and pieces and learned in specific problem solving or applied contexts, how do we know when and to what degree students will have the relevant algebraic skills to deal with problems they will encounter?”
The article includes an excellent section on the value and uses of proof, which starts with:
Treisman, U., Studying Students Studying Calculus: A Look at the Lives of Minority Mathematics Students in College. In The College Mathematics Journal, V 23, #5, Nov 1992.
Uri Treisman’s work with first-year calculus students at UC Berkeley is famous. Black students were uniformly failing this course. These students were not underprepared; they were the cream of the crop in many ways. No one really understood the problem. Treisman decided to follow the students home, to get real information. He looked at the Black students and the Asian students. What were they each doing when they studied? Both the Black and Asian students started out studying and completing homework for about 8 hours a week alone. But then the Asian students got together in groups of friends and discussed the homework. If one person had a different answer, they could learn from the others. If they all had different answers, they knew they were lost on a particular problem.
Treisman knew they needed to find a way to encourage group discussions among the Black students. They put together a workshop program in which students were asked to work on especially challenging problems in groups. “Our idea was to construct an anti-remedial program for students who saw themselves as well prepared.” The students who went through this program did significantly better than the average for all students. Treisman concludes with thoughts about how to change all calculus courses to include this sort of engaging work.
To get that sabbatical, I had to make a proposal, and I was told that just editing a book wouldn't get me a whole year sabbatical. So I added a few more bits and pieces to my proposal, and now my 'sabbatical evidence' is due in 2 days. One of the agreements I made was to read 15 books and 15 articles from a list I provided. Below are my annotations for 11 of the articles (I've left out the ones I didn't care for). I've already posted about most of the books.
These articles are all good. If you haven't read the Treisman article, and you have any interest in social justice issues in relation to math education, do read it. The Hoyles article is also a classic. It differentiates between school-math and the math people create on their own, that they don't even think of as math. You may want to skim that one, but you'll find some gems in it.
Andreescu, T., and Mertz, J, Cross-Cultural Analysis of Students with Exceptional Talent in Mathematical Problem Solving. In Notices of the AMS V55, Num 10, 2008.
Janet Mertz spoke on this at the 2009 Great Circles conference hosted at the Mathematical Sciences Research Institute, in Berkeley. This talk is available as a Quicktime movie, and includes more discussion of the data than what is in the article.
Larry Summers, while president of Harvard, made the claim that the likeliest reason for the paucity of tenured women math professors was brain differences between men and women. (He hypothesizes that there is more variability in intelligence among men than among women, so there would be more extremely smart and more extremely dumb men than women.) One woman professor walked out, and then encouraged colleagues to do some statistical analysis. The percentage of women among those who rank most highly in math competitions varies widely from one country to another, from below 5% to over 20%. This is also reflected in the percentage of women in tenured math faculty in different countries (below 5% in some western European counties and over 20% in Portugal and a few other countries). This research shows that culture is a big component in girls’ and women’s achievement in math, making clear our inability to disentangle the effects of biology and culture.
Ball, D. L. , Working on the inside: Using one's own practice as a site for studying mathematics teaching and learning. In Kelly, A. & Lesh, R. (Eds.). Handbook of research design in mathematics and science education, (pp. 365- 402). [Link is to pdf.]
Ball analyzes how 3 different teacher-researchers (herself, M. Lampert, and R. Heaton) use their own teaching as a way of researching how teachers teach and students learn. In her introduction she discussed pre-services teachers’ misconceptions about math:
…what they believed was often at odds with what the teacher educators wanted them to think or know. For example, many believed that mathematical ability is innate and that many people simply cannot be good at mathematics. Most thought of mathematics as a cut-and-dried area of truths to be memorized and procedures to be practiced.She discusses the necessity of using oneself as research subject because of the rarity of teachers doing the kind of teaching one might want to analyze.
Ball, D. L., & Bass, H., Interweaving Content and Pedagogy in Teaching and Learning to Teach: Knowing and Using Mathematics. In J. Boaler (Ed.), Multiple Perspectives on the Teaching and Learning of Mathematics (pp. 83-104). [Link is to pdf.]
The most valuable idea in this article for me is related to the notion of ‘compression’ – once we learn something well in math, it gets compacted, and seems simpler. To teach, we need to reverse that process:
…Mathematics is a discipline in which compression is central. Indeed, its polished, compressed form can obscure one’s ability to discern how learners are thinking at the roots of that knowledge. … Because teachers must be able to work with content for students in its growing, not finished, state, they must be able to do something perverse: work backward from mature and compressed understanding of the content to unpack its constituent elements.The complex skills needed to teach math well are illustrated through classroom examples. [More Ball chapters and articles available online.]
Ball, D. L., Hill, H., and Bass, H., Knowing Mathematics for Teaching: Who Knows Mathematics Well Enough To Teach Third Grade, and How Can We Decide? In American Educator Fall 2005. [Link is to pdf.]
The authors looked at what special knowledge of math is needed by teachers, in order to effectively teach it. The article gives background on why they structured their research the way they did:
…there are legitimate competing definitions of mathematical knowledge for teaching....
Our aim is to identify the content knowledge needed for effective practice and to build measures of that knowledge that can be used by other researchers. The claim that we can measure knowledge that is related to high-quality teaching requires solid evidence.
The article also gives some preliminary results which may be promising regarding social justice:
… the size of the effect of teachers’ mathematical knowledge for teaching was comparable to the size of the effect of socioeconomic status on student gain scores. This … suggests that improving teacher’s knowledge may be one way to stall the widening of the achievement gap as poor children move through school.
Benson, S. & Findell, B., A Modified Discovery Approach to Teaching and Learning Abstract Algebra. In Innovations in Teaching Abstract Algebra, MAA Notes #60, eds. Allen Hibbard & Ellen Maycock, pp. 11-17, 2002.
The author taught almost entirely through worksheets he designed to get groups of students working on the material. His ability to observe the groups allowed him to get more clarity about what they understood than when he’d lectured. When they didn’t understand the congruence relationship, he got them to discuss what they had shown so far, and would not ‘give them the answer’. In spite of letting go of a ‘calendar’ and taking the time needed for students to work out their own understandings, he found that this class actually covered more material than the usual.
Dweck, C., Caution: Praise Can Be Dangerous. In American Educator, Spring 1999.
Carol Dweck has written a book, Mindset (2006), which says about the same things this article does, at much more length. The article describes her thesis and her research much more concisely and (in my opinion) effectively. Her claim (proved by her research) is that praising a student’s intelligence makes them wary of harder tasks and of looking dumb, but praising their effort encourages them to tackle harder tasks and enjoy it. She also looks at people who think intelligence is fixed and compares them to people who think effort can change one’s intelligence. People with the second mindset are able to develop their potential much more effectively than those with a ‘fixed intelligence’ mindset.
Hoyles, C., Noss, R., & Pozzi, S., Proportional Reasoning in Nursing Practice. In Journal for Research in Mathematics Education, Vol. 32, No. 1 (Jan., 2001), pp. 4-27
Research has repeatedly shown people doing mathematics in their work situations more effectively than they can as students. Most people say they do no math in their work, because they don’t recognize that what they’re doing is mathematical.
These studies suggest that adults are adept at solving proportional problems in everyday or work situations but often employ informal strategies that are tailored to the particular situation and are not easily identified with formal school-taught methods. (From page 6 of pdf.)
The authors of this article look at nurses’ calculations of drug dosages and found these calculations to be more flexible and fluent than the 'nurses Rule’ they’d been taught.
Kato, Y., Honda, M., & Kamii, C., Kindergartners Play Lining Up the 5s: A Card Game to Encourage Logico-Mathematical Thinking. In Mathematical Behavior, 13(1), 55-80, 2006.
The authors describe their research studying video of children playing a very simple card game. They are looking for ways to describe progress in logical (‘logico-mathematical’) thinking skills. I am impressed with how much thinking kids need to develop to do things that seem utterly simple to adults.
The … categories that the players created are much more abstract than those children can create in sorting activities involving squares, rectangles, “red ones,” “blue ones,” and so on. The seriation involved in “cards to be used first, second, and last” is likewise much more abstract than what can be done with Montessori sticks and cylinders. If we had to set standards for mathematics in kindergarten, we would never think of including the high-level logic that we saw in Lining Up the 5s.My belief, which continues to be affirmed by all the research I’ve done this year, is that we would do well to continue to let children learn through play, for as long as they wish. Perhaps then they’d choose to study hard later, for the sheer pleasure of the learning.
…
Play has long been valued in early childhood education, and we will do well to analyze it with depth and precision not only in card games but also in other kinds of play that naturally appeal to young children.
Schoenfeld, A., A Highly Interactive Discourse Structure. In Social Constructivist Teaching, Volume 9, pages 131–169. 2002
A quarter of this article is devoted to transcripts of two very different classes, a high school physics class, and a third grade math class. The two classes turn out to share a structure in the interactions between the teacher and students. Schoenfeld has created a flowchart of this structure, but I find a summary more useful:
Teacher starts by giving context and background for topic.
• Asks class: “What (else) can you say about [this topic]?”
• Calls on a student.
• Does their response raise other issues? (If so, deal.)
• Is clarification, expansion or reframing useful? (If so, deal.)
• Would more discussion be useful? (If so, deal.)
[I have created a sheet to remind myself of this summary, to help me get out of lecture mode.] Deborah Ball taught the third grade class, which was videotaped, and is cited in many researchers' work.
Schoenfeld, A. H., What do we know about mathematics curricula? Journal of Mathematical Behavior, 13(1), 55-80, 1994.
Schoenfeld discusses what we do and don’t know about the math curricula we need. “The ‘constructivist perspective’ is better grounded in empirical and experimental evidence than the theory of evolution; we should just assume it and get on with our business (while working … hard, of course, to flesh it out and understand it more fully).” But, on the other hand, the best balance between traditional ‘content’ and the development of problem-solving skills is unclear, and “If, for example, what we now call ‘algebra’ is distributed through the curriculum in bits and pieces and learned in specific problem solving or applied contexts, how do we know when and to what degree students will have the relevant algebraic skills to deal with problems they will encounter?”
The article includes an excellent section on the value and uses of proof, which starts with:
There are, I think, three roles of proof that need to be explored and understood: the unique character of certainty provided by air-tight mathematical arguments, which differs from that in any other discipline and is part of what makes mathematics what it is; the fact that proof need not be conceived as an arcane formal ritual, but can be seen as the mere codification of clear thinking and a way of communicating ideas with others; and the fact that for mathematicians, proving is a way of thinking, exploring, of coming to understand – and that students can and should experience mathematical proving in the same ways.
Treisman, U., Studying Students Studying Calculus: A Look at the Lives of Minority Mathematics Students in College. In The College Mathematics Journal, V 23, #5, Nov 1992.
Uri Treisman’s work with first-year calculus students at UC Berkeley is famous. Black students were uniformly failing this course. These students were not underprepared; they were the cream of the crop in many ways. No one really understood the problem. Treisman decided to follow the students home, to get real information. He looked at the Black students and the Asian students. What were they each doing when they studied? Both the Black and Asian students started out studying and completing homework for about 8 hours a week alone. But then the Asian students got together in groups of friends and discussed the homework. If one person had a different answer, they could learn from the others. If they all had different answers, they knew they were lost on a particular problem.
Treisman knew they needed to find a way to encourage group discussions among the Black students. They put together a workshop program in which students were asked to work on especially challenging problems in groups. “Our idea was to construct an anti-remedial program for students who saw themselves as well prepared.” The students who went through this program did significantly better than the average for all students. Treisman concludes with thoughts about how to change all calculus courses to include this sort of engaging work.
Sunday, September 19, 2010
Graphing: What do beginning algebra students need to know?
Here's what I said my students will need to know:
What do you all think?
- Graph a line given equation (slope-intercept form)
- Graph a line given equation (standard form)
- Find slope given two points
- Find equation of a line given two points
- Find slope given equation (any form)
- Find slope given graph
- Find y-intercept given two points
- Find y-intercept given equation
- Find equation of line perpendicular to given line and through given point
- Find equation of line parallel to given line and through given point
- Explain meaning of slope in a real problem
- Explain meaning of y-intercept in a real problem
- Create an equation based on a real problem
- Make a graph for a real problem
What do you all think?
Tuesday, September 14, 2010
Today in Class: A Good Day to Do Without the Textbook
My son had a doctor's appointment, so my morning classes didn't meet today. Maybe I came into my afternoon class with more energy, I don't know, but it felt really great. I gave a short mastery test on solving equations. Five questions; last one was a story problem. I think I'll break it up into two grades - one for solving equations, and one for the story problem. That way lots of them can know they've passed the equations portion. (Because I don't think many will have the story problem right.) After the test, I started our new unit on graphing, and it seemed to go really well. (Three of the students with difficult behavior had left after the test; that might have something to do with it.)
I started out with the function game. We had played it before, so this wasn't new. I chose a harder relationship than the ones I'd done before, and not many people were seeing it.
Here's how it goes: I have one student up front as the scribe. I ask students for a number, and then I say "Casey said 4; I say 11." The scribe writes it down in a two-column table, and I call on someone else. Each time we play I've told them: "The most important rule of the game is - don't say the rule! If you think you know what's going on, say the number I'm going to say." After a few numbers are up, some of the students will call out the number I'm supposed to say. This time only a few people were getting it.
I decided to use this function game as my entree into graphing. I had meant for it to be one example among many, but when I saw that people weren't seeing the rule in my head, I thought maybe the graphing would help some people see it. And that would help them see the power of graphing! So I told them I wanted them to plot all the numbers we had on the board so far. Their first point would have an x-coordinate of 4, and a y-coordinate of 11. I walked around as they were doing that, and helped a few people with the typical difficulties (0 to 1 is often about twice as big as 1 to 2 and all the rest, or they put tick marks between the blue lines instead of on them so it's hard to be accurate, ...). After that I got to mention axes, quadrants, and all that. We got to see the linear relationship, and we got to talk about the rule (multiply by 3 and take away one) and what it would look like as an equation, y (or output) = 3 * x (or input) - 1. This was so much more fun than doing section 3.1 in the book, where all they do is plot points, identify coordinates, and plot data for (number of years since 1970, number of Walmart stores)!
We did another function game and its graph, and then we did a worksheet from Maria Andersen's Algebra Activities workbook (from the free teasers pack pdf).
Students kept asking me how to determine the equation of a line from points. (They didn't ask it that clearly.) And I kept asking them to wait until we had built up a bit more background. I answered that question just before class ended, by doing the function game a third time, and stopping after we had just two number pairs.
I think this is the best intro to graphing I have ever done. I hope it goes as well in my morning classes!
I started out with the function game. We had played it before, so this wasn't new. I chose a harder relationship than the ones I'd done before, and not many people were seeing it.
Here's how it goes: I have one student up front as the scribe. I ask students for a number, and then I say "Casey said 4; I say 11." The scribe writes it down in a two-column table, and I call on someone else. Each time we play I've told them: "The most important rule of the game is - don't say the rule! If you think you know what's going on, say the number I'm going to say." After a few numbers are up, some of the students will call out the number I'm supposed to say. This time only a few people were getting it.I decided to use this function game as my entree into graphing. I had meant for it to be one example among many, but when I saw that people weren't seeing the rule in my head, I thought maybe the graphing would help some people see it. And that would help them see the power of graphing! So I told them I wanted them to plot all the numbers we had on the board so far. Their first point would have an x-coordinate of 4, and a y-coordinate of 11. I walked around as they were doing that, and helped a few people with the typical difficulties (0 to 1 is often about twice as big as 1 to 2 and all the rest, or they put tick marks between the blue lines instead of on them so it's hard to be accurate, ...). After that I got to mention axes, quadrants, and all that. We got to see the linear relationship, and we got to talk about the rule (multiply by 3 and take away one) and what it would look like as an equation, y (or output) = 3 * x (or input) - 1. This was so much more fun than doing section 3.1 in the book, where all they do is plot points, identify coordinates, and plot data for (number of years since 1970, number of Walmart stores)!
We did another function game and its graph, and then we did a worksheet from Maria Andersen's Algebra Activities workbook (from the free teasers pack pdf).
Students kept asking me how to determine the equation of a line from points. (They didn't ask it that clearly.) And I kept asking them to wait until we had built up a bit more background. I answered that question just before class ended, by doing the function game a third time, and stopping after we had just two number pairs.
I think this is the best intro to graphing I have ever done. I hope it goes as well in my morning classes!
Friday, September 10, 2010
Richmond Math Salon: Next Week Saturday
If you live close to Richmond, California, please join us for the next salon. On Saturday, September 18, from 2 to 5pm, we'll play with:
Games: Blokus, Connect Four, Quarto, Set, Blink!, and lots more, including my new favorite, Katamino. I’d also like to introduce two card games I’m tweaking: Make Zero and Red and Black TripleMatch.
Puzzles: Rush Hour, Tangrams, Rubik’s cube, puzzles from Math Without Words, … (Katamino is a puzzle, too!)
Polydrons are a great way to build three-dimensional shapes like pyramids, cubes, prisms, Platonic solids, … I’m also hoping my new pentomino sets will have arrived.
Books: My huge library of fun math books is available to peruse. And as folks who’ve come before know, I’ll give you my reviews of many of these books.
It's great fun, and I'd love to meet folks who've heard about it through my blog. For more information, email me at suevanhattum on hotmail.
Games: Blokus, Connect Four, Quarto, Set, Blink!, and lots more, including my new favorite, Katamino. I’d also like to introduce two card games I’m tweaking: Make Zero and Red and Black TripleMatch.
Puzzles: Rush Hour, Tangrams, Rubik’s cube, puzzles from Math Without Words, … (Katamino is a puzzle, too!)
Polydrons are a great way to build three-dimensional shapes like pyramids, cubes, prisms, Platonic solids, … I’m also hoping my new pentomino sets will have arrived.
Books: My huge library of fun math books is available to peruse. And as folks who’ve come before know, I’ll give you my reviews of many of these books.
It's great fun, and I'd love to meet folks who've heard about it through my blog. For more information, email me at suevanhattum on hotmail.
Monday, September 6, 2010
Blog Links
Maria Droujkova has started up a new version of her blog. It's now called Math Accent. She's including video every day. I've just read all the posts, and it's exciting. (I haven't watched any of the videos yet, because my son is sleeping. But I'll watch them soon.) I especially like her post on math as its own context, which relates to a discussion we're having over at Dan's blog (comments 21 to 28).
There's a new blog carnival about Standards-Based Grading. (I wish this idea/philosophy had a different name...) SBG Grading Gala #2 was posted today at Always Formative (Jason Buell). I missed contributing because it wasn't quite on my radar. If you're interested in changing the way you grade, as a way to improve your teaching, check it out.
There's a new blog carnival about Standards-Based Grading. (I wish this idea/philosophy had a different name...) SBG Grading Gala #2 was posted today at Always Formative (Jason Buell). I missed contributing because it wasn't quite on my radar. If you're interested in changing the way you grade, as a way to improve your teaching, check it out.
Sunday, September 5, 2010
Recapping the Online Math Circle
My thanks to everyone who joined me at the online math circle I hosted yesterday as part of Maria's webinar series. (Here's the full recording.) It was fun seeing those pancakes go up on the whiteboard. We spent most of our time thinking about the math problem I introduced, and didn't get much time to discuss how we would use something like this in the classroom. Some discussion of the issues has happened at Dan's blog.
I wanted to show a deeply engaging problem that doesn't take any technology to play with. But of course, we were using a high-tech way to gather together, and that changed the dynamic dramatically. Here are a few pros and cons:
Here are some of the questions and comments that came up:
Patterns - sniffing, breaking, and finding:
Sue: Avery put a list on his blog, of the habits of mind he’d like to encourage in his classes. I loved the pattern sniffing.
Sheng: Ben has that post about pattern breaking.
Maria: (a thought from a conference) Patterns do not lead to formulas at all.
Sue: With the pancake problem we're doing, we got a recursive formula, but not a functional description, of what's happening.
Online sources of math circle style problems:
Books:
For my sabbatical report, I had to write up 3 new projects I'll be doing in my classes. Here's what I wrote about this topic:
I draw a circle on the board.
“Who likes pancakes?” [I usually get enthusiastic responses when I’ve done this in math circles.]
“What type of pancakes do you like?” [Collect a variety of responses.]
“This pancake is magic. When we cut it into pieces, each person’s piece will expand to be just the right size, when they're eating it and when it's in their belly. It will also become exactly the type of pancake you love best.]
“How many pieces before we've cut it?” [one…]
Data Gathering.
“If we cut it once, how many pieces will there be?” [two]
“Twice?” [there could be 2, 3, or 4… discuss goals… we want to feed us all]
“Hmm, should we keep track of what we’re finding out? What’s a good way to do that?” [Some of them will have used what they like to call a t-chart. That’s probably what I’ll use, unless something else intrigues the group.]
“How many pieces do we get with 3 cuts?” [Discuss cutting strategies to get the most pieces.]
“Now, with your partner, do 4 cuts. (We’ll go on to 5, 6 and 7 cuts…)” [As they work, they’ll be checking against neighbors. When everyone has started thinking about 5 cuts, I’ll stop them to either say how many we all got for 4, or to get consensus.]
We continue, finishing 5, and starting 6.
Searching for Pattern.
“Hmm, does anyone see a pattern?” [At this point, the conversation can go in lots of different ways.]
We’ll discuss how we know our pattern stays true, an example of inductive versus deductive reasoning.
Once they do see a pattern, it’s still described in an inconvenient way (recursively), and we’ll search for a way of describing the pattern that will allow us to predict how many pieces come from 30 cuts. We will not find this in our first hour of exploration.
Graphing.
Perhaps on this first day, perhaps later, we will graph the number of cuts on the x-axis versus the number of pieces on the y-axis.
“If we connected these points, would we get a straight line?” [no…]
“What does this shape look like?” [a parabola, but they may not see this yet…]
This will be used during the introduction to the graphing section of the course, either by reviewing what we did before, or remembering our magic pancake, and doing this step for the first time.
Quadratic Relationships.
A few months have passed since we first looked at this problem, so we’ll do it over again. [Students will direct the steps, and they’ll do it quickly. This demonstrates for them how much they’ve learned.]
We now attempt to find a non-recursive relationship between number of cuts and number of pieces. The graph now may be recognized as a parabola, which tells us we have a quadratic relationship. Unlike most parabolas, we won’t see a vertex. [Hmm, real life problems are often more complex than textbook problems.]
The final answer involves a fraction, which adds another layer of complexity to the problem.
“In the problems we’ve done so far, we’ve been given an equation and we’ve graphed it. This time we have points and we want to find an equation that fits these points. Hmm…”
We will explore adding the numbers from 1 to n. We may explore triangular numbers also.
Closure?
It’s possible the group won’t figure out the relationship entirely. I will not show it to them. If this happens, I’ll discuss real mathematical research and the fact that real problems often take years to solve. I’ll also discuss with them the benefits of leaving an open question in their minds. [It keeps them thinking.] In our last session exploring this problem, I will review the concepts they’ve used in their (perhaps partial) solution of the problem.
That's it, folks. Please bring on your questions and comments.
I wanted to show a deeply engaging problem that doesn't take any technology to play with. But of course, we were using a high-tech way to gather together, and that changed the dynamic dramatically. Here are a few pros and cons:
- (+) About 18 people from far-flung parts of the world got to work/play together, without ever leaving our homes (though one person mentioned being at the library). I was in my pajamas.
- (-) I found it very frustrating to be the only one talking audibly. (For most of the session, everyone else used the 'chat' area to ask questions and make comments.)
- (+) Many participants used the whiteboard to draw their pancake slicing attempts. Even though it was probably new to most of them, it was pretty easy to figure out.
- (-) Participants couldn't collaborate with one another easily. It's so much more fun in person, with people showing a partner their ideas. When we're really physically together, our excitement is infectious.
- (-) I wanted to mention a bunch of books. In person, I would have picked each one up at an opportune moment, shown why it's cool in less than 15 seconds, handed it to a participant, and moved on. It took longer, and disrupted the flow more when I showed a book at the webinar, so I didn't mention all the books I thought might interest people. I made a list below.
- (-) I was distracted by the new environment; that will go away over time. I forgot to give my contact information out. If you have any questions or comments, you can leave them here, or email me at suevanhattum on hotmail. I also didn't judge time well, which I'll get better at with more experience.
Here are some of the questions and comments that came up:
- John Golden: When you pose a math circle problem do you try to avoid any further scaffolding? [I said that this past summer, when one student proposed a solution the other students didn't understand, I helped get the other students up to speed. But that wasn't an optimal solution for the problem of one person way beyond the rest. I didn't have a good answer in general to John's question. Anyone experienced with math circles care to reply to this?]
- Telannia: Do you find students getting frustrated and just zoning out? [I said sure, but maybe less than in a traditional lesson. Also, the Kaplan's tell a good story of a girl who seemed for many weeks not to be getting it, and who then proceeded to propose a vital step in the solution process, blowing the minds of her fellow students.]
- Telannia: Are the math circles you have done all on your blog? [I've mainly done this pancake problem, and Pythagorean triples. I'll write a post soon about some of my favorite topics in math circles I've attended. I've put a list of some good sites for these below.]
- John Golden: Do you care if people get to the answer or are you just looking at the process? [I find getting to the answer so exciting, but if it comes from the teacher instead of the students, we haven't really gotten there. I do leave people hanging sometimes.]
- Amanda Serenevy: We have been using Fermi Estimates in some of our Math Circles this summer. [The link is to a fabulous report her students have put together.]
- Maria: I want to ask people here - what are your favorite "low-tech" problem prompts? "Stuff" that you use?
Patterns - sniffing, breaking, and finding:
Sue: Avery put a list on his blog, of the habits of mind he’d like to encourage in his classes. I loved the pattern sniffing.
Sheng: Ben has that post about pattern breaking.
Maria: (a thought from a conference) Patterns do not lead to formulas at all.
Sue: With the pancake problem we're doing, we got a recursive formula, but not a functional description, of what's happening.
Online sources of math circle style problems:
- The Math Circle (the Kaplan's site)
- Riverbend Community Math Center
- National Association of Math Circles
- Colin recommends Project Euler, but warns that it's tech oriented.
- Tom Davis' Math Circle Topics
- Josh Zucker's problem page for the Julia Robinson Mathematics Festival
Books:
- Out of the Labyrinth: Setting Mathematics Free, by Bob and Ellen Kaplan, is a great introduction to math circles.
- Circle In a Box, by Sam VanDerVelde, gives some nuts and bolts for running math circles, and includes a great selection of problems.
- Problem-Solving Strategies: Crossing the River With Dogs, by Ken Johnson and Ted Herr, has some good problems, and good ideas about how to work on problem-solving in the classroom.
- The Art of Problem-Posing, by Stephen Brown and Marion Walter, has some good problems, and a good analysis of how to get students asking the questions.
- Mathematics: A Human Endeavor, by Harold Jacobs, is an alternative textbook - it's got great problems, they're engaging and get the student really thinking.
- The Art and Craft of Problem-Solving, by Paul Zeitz, has a cool analysis of the problem-solving process, and some interesting problems.
- How To Solve It, by George Polya, is a classic, and includes some intriguing problems.
- The Cat In Numberland, by Ivar Ekeland, introduces students (of almost any age) to deep concepts related to infinity, through an engaging story. Story-telling is a great low-tech way to make problems engaging.
- John Golden recommends Symmetry, Shape, and Space: An Introduction to Mathematics Through Geometry, by Christine Kinsey and Teresa Moore.
For my sabbatical report, I had to write up 3 new projects I'll be doing in my classes. Here's what I wrote about this topic:
Magic Pancake Problem
Introduction.I draw a circle on the board.
“Who likes pancakes?” [I usually get enthusiastic responses when I’ve done this in math circles.]
“What type of pancakes do you like?” [Collect a variety of responses.]
“This pancake is magic. When we cut it into pieces, each person’s piece will expand to be just the right size, when they're eating it and when it's in their belly. It will also become exactly the type of pancake you love best.]
“How many pieces before we've cut it?” [one…]
Data Gathering.
“If we cut it once, how many pieces will there be?” [two]
“Twice?” [there could be 2, 3, or 4… discuss goals… we want to feed us all]
“Hmm, should we keep track of what we’re finding out? What’s a good way to do that?” [Some of them will have used what they like to call a t-chart. That’s probably what I’ll use, unless something else intrigues the group.]
“Now, with your partner, do 4 cuts. (We’ll go on to 5, 6 and 7 cuts…)” [As they work, they’ll be checking against neighbors. When everyone has started thinking about 5 cuts, I’ll stop them to either say how many we all got for 4, or to get consensus.]
We continue, finishing 5, and starting 6.
Searching for Pattern.
“Hmm, does anyone see a pattern?” [At this point, the conversation can go in lots of different ways.]
We’ll discuss how we know our pattern stays true, an example of inductive versus deductive reasoning.
Once they do see a pattern, it’s still described in an inconvenient way (recursively), and we’ll search for a way of describing the pattern that will allow us to predict how many pieces come from 30 cuts. We will not find this in our first hour of exploration.
Graphing.
Perhaps on this first day, perhaps later, we will graph the number of cuts on the x-axis versus the number of pieces on the y-axis.
“If we connected these points, would we get a straight line?” [no…]
“What does this shape look like?” [a parabola, but they may not see this yet…]
This will be used during the introduction to the graphing section of the course, either by reviewing what we did before, or remembering our magic pancake, and doing this step for the first time.
Quadratic Relationships.
A few months have passed since we first looked at this problem, so we’ll do it over again. [Students will direct the steps, and they’ll do it quickly. This demonstrates for them how much they’ve learned.]
We now attempt to find a non-recursive relationship between number of cuts and number of pieces. The graph now may be recognized as a parabola, which tells us we have a quadratic relationship. Unlike most parabolas, we won’t see a vertex. [Hmm, real life problems are often more complex than textbook problems.]
The final answer involves a fraction, which adds another layer of complexity to the problem.
“In the problems we’ve done so far, we’ve been given an equation and we’ve graphed it. This time we have points and we want to find an equation that fits these points. Hmm…”
We will explore adding the numbers from 1 to n. We may explore triangular numbers also.
Closure?
It’s possible the group won’t figure out the relationship entirely. I will not show it to them. If this happens, I’ll discuss real mathematical research and the fact that real problems often take years to solve. I’ll also discuss with them the benefits of leaving an open question in their minds. [It keeps them thinking.] In our last session exploring this problem, I will review the concepts they’ve used in their (perhaps partial) solution of the problem.
That's it, folks. Please bring on your questions and comments.
Saturday, September 4, 2010
Later Today: Math Circle Online - Please join us!
Please join me for Math Circles: Low Tech, High Engagement - Good for Classroom Use? at 2pm Eastern time / 11am Pacific time, today.
I'll conduct a math circle for the first part of the session. (If you haven't seen the problem before, you get to participate. If you have seen it before, you get to watch a math circle in action.) Then I'll host a discussion about how it went, how it might go if done in a class, and what topics in the curriculum might lend themselves to math circle format. We may also want to discuss what other low-tech ideas folks have, and how we can deeply engage students through both low and high tech methods.
Hope to see you there!
I'll conduct a math circle for the first part of the session. (If you haven't seen the problem before, you get to participate. If you have seen it before, you get to watch a math circle in action.) Then I'll host a discussion about how it went, how it might go if done in a class, and what topics in the curriculum might lend themselves to math circle format. We may also want to discuss what other low-tech ideas folks have, and how we can deeply engage students through both low and high tech methods.
To join:
|
Hope to see you there!
Friday, September 3, 2010
This week was rough...
Sam Shah wrote about some of his struggles this past week. His openness helps us all, and I will try to be as open about what I'm struggling with...
It's too late now. I should have asked last week for interesting ways to approach the basics of solving equations. I did a lot of lecture, and some small group practice.
I have a few projects I'm planning to do later, but I'm doing a lot of lecturing now. Some students say they like that best, actually. (But I know it's not what helps them learn the most.) In one section, I asked what percentage of the time we should have: projects, lecture, and small group practice. Students called out all sorts of percentages for each, and I had them do averages. We came up with 1/4 projects, 1/3 lecture, and 2/5 small group practice. (No, that's not quite 100%.) I like that as a goal. But I'm not there yet. This week, I probably gave them about 10% projects (we did the function machine game), almost 20% small group practice (sometimes alone or in pairs, sometimes groups of 4), and over 70% lecture. Yuck!
I asked that question in the one section that's going really well. In another section, I've told 2 students they're on probation (for repeated disrespectful behavior), and had a similar behavior talk with one other student. I am not teaching high school; I am teaching at a community college. Yes, I often have discipline problems. That's hard to admit; it makes me feel inadequate. I often think that if I were interesting enough, students would cooperate perfectly.
I find myself lecturing more in the classes where students aren't into it. Getting group work going well is pretty hard when there's resistance. It's hard to write this. I wish I could sail into the classroom with interesting activities every day. I am not there yet. (After 20 years of teaching, Sue? When will you be there?...)
No one (3 sections, about 140 students total) passed the mastery test I gave on pre-algebra topics. (FIDO = fractions, integers, distributive property, and order of operations). They'll all be doing retakes. Yikes! Everyone has this homework assignment: for each problem on the test, do it correctly, explain the steps in words, and make up a new similar problem.
I think it will get better. A lot of students struggle with graphing, and there are so many cool projects to do for that.
Today I have no regular classes, but I come in for two hours of "hour by arrangement". Many of my students come in, and they can get help, play mathy games, or just do a worksheet that's been provided. The best thing about the hour by arrangement, in my opinion, is that the students start to bond - it helps them form a community.
It's too late now. I should have asked last week for interesting ways to approach the basics of solving equations. I did a lot of lecture, and some small group practice.
I have a few projects I'm planning to do later, but I'm doing a lot of lecturing now. Some students say they like that best, actually. (But I know it's not what helps them learn the most.) In one section, I asked what percentage of the time we should have: projects, lecture, and small group practice. Students called out all sorts of percentages for each, and I had them do averages. We came up with 1/4 projects, 1/3 lecture, and 2/5 small group practice. (No, that's not quite 100%.) I like that as a goal. But I'm not there yet. This week, I probably gave them about 10% projects (we did the function machine game), almost 20% small group practice (sometimes alone or in pairs, sometimes groups of 4), and over 70% lecture. Yuck!
I asked that question in the one section that's going really well. In another section, I've told 2 students they're on probation (for repeated disrespectful behavior), and had a similar behavior talk with one other student. I am not teaching high school; I am teaching at a community college. Yes, I often have discipline problems. That's hard to admit; it makes me feel inadequate. I often think that if I were interesting enough, students would cooperate perfectly.
I find myself lecturing more in the classes where students aren't into it. Getting group work going well is pretty hard when there's resistance. It's hard to write this. I wish I could sail into the classroom with interesting activities every day. I am not there yet. (After 20 years of teaching, Sue? When will you be there?...)
No one (3 sections, about 140 students total) passed the mastery test I gave on pre-algebra topics. (FIDO = fractions, integers, distributive property, and order of operations). They'll all be doing retakes. Yikes! Everyone has this homework assignment: for each problem on the test, do it correctly, explain the steps in words, and make up a new similar problem.
I think it will get better. A lot of students struggle with graphing, and there are so many cool projects to do for that.
Today I have no regular classes, but I come in for two hours of "hour by arrangement". Many of my students come in, and they can get help, play mathy games, or just do a worksheet that's been provided. The best thing about the hour by arrangement, in my opinion, is that the students start to bond - it helps them form a community.
Wednesday, September 1, 2010
Free Math Book Online: Shadows of the Truth
Alexandre Borovik describes his book, Shadows of the Truth: Metamathematics of Elementary Mathematics, as collaborative. He's asked mathematicians to describe their memories of their childhood understandings and confusions of math, and has woven their stories into a rich tapestry in which he analyzes the learning of mathematics.
I've only read the first chapter so far, and am looking forward to enjoying the rest. I loved his examples of how language affects our mathematical understandings (page 29). The words 'and' and 'or' have particular meanings in mathematics. These meanings can be quite different from the meanings given in a person's natural language:
This confusion seems to have led him to some of the insights he shares in this book. Enjoy!
I've only read the first chapter so far, and am looking forward to enjoying the rest. I loved his examples of how language affects our mathematical understandings (page 29). The words 'and' and 'or' have particular meanings in mathematics. These meanings can be quite different from the meanings given in a person's natural language:
... what matters in the context of the this book are invisible differences in the logical structure of my students’ mother tongues which may have huge impact on their perception of mathematics. For example, the connective “or” is strictly exclusive in Chinese: “one or another but not both”, while in English “or” is mostly inclusive: “one or another or perhaps both”. Meanwhile, in mathematics “or” is always inclusive and corresponds to the expression “and/or” of bureaucratic slang. In Croatian, there are two connectives “and”: one parallel, to link verbs for actions executed simultaneously, and another consecutive .I also loved his confusion as a child about units (page 17):
When, as a child, I was told by my teacher that I had to be careful with “named” numbers and not to add apples and people, I remember asking her why in that case we can divide apples by people: 10 apples : 5 people = 2 apples. Even worse: when we distribute 10 apples giving 2 apples to a person, we have 10 apples : 2 apples = 5 people.
Where do “people” on the right hand side of the equation come from? Why do “people” appear and not, say, “kids”? There were no “people” on the left hand side of the operation! How do numbers on the left hand side know the name of the number on the right hand side?
This confusion seems to have led him to some of the insights he shares in this book. Enjoy!
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