Math for Love and Love for Math

Dan Finkel loves math. His blog, Math for Love, is one of my favorites. Here's a short video of him talking about the perennial question, "Why are we doing this?"




I've slept better the past two nights, but I was still awake for about an hour in the middle of the night last night, thinking about my difficult class. I might get some videos of people in the fun class, talking about what's starting to change for them, and then tell my difficult class that they can become a community of learners, too, if they choose it.

Here's are some snapshots in words...

A has taken this course 4 times before this, and has always dropped when they got into graphing, because she just wasn't getting it. She's starting to get it now. I asked her to help B, who has been an amazing community organizer, emailing other students, suggesting times to study together, encouraging them to stick with it. A promised to do that.

B came by my house this afternoon for some tutoring. When I was teaching at a community college in Michigan, I had long lines of students waiting for their turn with me during my office hours. Here in California, they don't come for help as much (at first). Perhaps I've changed, but I think it's a difference in the students here, they don't expect as much help from their Professors, and think they have to go to tutors for help. (Is this really true?)

C walked out once, angry with me. She gets pretty angry at herself too. Each test that comes back with a 'Redo' at the top makes her mad. But then she comes in, and aces her retake. She's doing great in this class, but she would have gotten C's if I used a 'normal' grading system. [I do not call what I do Standards Based Grading, even though I was inspired by those folks to expand my Mastery Tests to be most of the grade.  Mastery Tests seems like a simpler name for it. They get to retake until they've got a score of 85% or better.] Won't it be great when grading systems like this are the new normal?!

D complained at the beginning of the term about all the pictures I use (trying to get them to really understand fractions, instead of memorizing steps). She also complained about me asking them to do things before I explained how. "Most teachers explain first, and then let us work on it." She struggles with math, and got 100% on her retake yesterday. I asked if she liked my method better now. She hesitated, and I think she's not really believing in the pictures and the trying things out before getting fed the steps. But she's happy about her progress, and that's what counts.

This class starts at 10am two days a week, and at 9:40 the other two days. Many of the students are coming in around 8:15 to study together. We are very lucky that our classroom is empty before our class meets - that's rare. They're making great use of this lucky break. One of them, E, often goes up to the board during this time to explain things to the others. He got a chance to do that in class on Monday or Tuesday, and he did a great job. (He made a mistake at one point, and I waited, on edge, hoping the others would correct him. It got fixed, but never got pointed to directly.) They were all shouting things out, as they moved together through the problem. I told E after class that I knew he'd make a great teacher some day, and his eyes lit up. Watching this beautiful scene during class, I thought this group would do great on the test. When they didn't, I was really discouraged. (What I said to them in class the next day is that I know they're working hard, and working together, which can make a big difference; we need to figure out together what they need to do to work more effectively.) But they are keeping their spirits up.

Most of the students I've described are older students, and have made a commitment to themselves that they're going to do it - they're going to pass this class. My commitment to them is to push them to do more than pass. I love their dedication.

Middle of the Night, Stuck in a Bad Place

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.

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.)

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:

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.

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.]

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:

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.")

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.

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 Seek^TM 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?

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:

…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.
…
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.

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.



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.

Graphing: What do beginning algebra students need to know?

Here's what I said my students will need to know:

• 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

Am I missing anything? I'm thinking of adding 'convert any equation in two variables to slope-intercept form'. And I'm thinking of taking out 'graph a line given the equation in standard form'.

What do you all think?