Minimally Defined Problems, Chapter 2

There is understandable skepticism out there (and some not so far away in the comments of previous posts) about students' ability to engage and come up with rich mathematical problems starting with what I am calling minimally defined problems. First, an anecdote of success with this approach that happened yesterday with some real, live 2nd, 3rd, and 4th graders at Math for Love.

These kids were given a chessboard and asked to come up with some math questions. Link on over if you want to see the great things the kids came up with.

Now don't get me wrong, I'm sure I could find an anecdote showing how a bear dancing on the back of a goat can lead to better proficiency with multiplication tables. One anecdote does not make a theory and should not send everyone running to rewrite the curriculum.

On the other hand, this doesn't mean that this anecdote should be ignored...

Anyway, I first want to address some of the issues/concerns about minimally defined problems and student posed problems that were brought up in earlier posts (comments that actually hit on many of the concerns that I am addressing in my proposal). Concerns/questions addressed here can be loosely separated into the following five questions:

1. Can this approach (or any approach for that matter) increase intrinsic motivation?
Ms. V: “I was a good math student, and I like puzzles, but I have very low "task-commitment"...Certain types of problems in certain contexts will totally get me working doggedly long past when anyone else cares. But other types of puzzles bore me (once I realize there's a trick/strategy/etc)....when I hit a wall and have run through everything I can think of...[So] how do you build stamina for such things in your students? Or is it just a mismatch between problem and personality? Or are these actually not such great problems, and the really great ones DO promote commitment? Or am I just lazy?”

2. How do you change the dominant culture of explanation-example-exercise-answer-satisfaction and should this culture be changed?
O: “I think I would be initially frustrated because the question is so open ended and there doesn't appear to be a right answer." 
Perdita: “If I'd had the gumption I have now I'd have written "What comes after 5? Answer: 6". Not sure what I'd have done then, but you'd have lost me for ever.”
Perdita: " you were apparently targeting it at middle schoolers, but my 6yo answered it instantly, and I'd certainly expect any 11yo with reasonable number sense to do so, if not distracted by wondering what teacher wanted today...”

3. Are students (specifically, sixth grade students) capable of engaging in this kind of work?
Max: “It's very hard for me to see how a student without any background or context could approach that assignment in a productive way.”
Perdita: "the space of mathematical problems that are solvable with effort by a given person is very, very narrow by comparison with the space of all problems that that person can formulate. Preduction: if you ask children to make up their own problems, almost all of them will be either trivial, or insoluble."

4. What does this look like on a day to day basis?
Ms V: “What you do when they arrive in class could drastically change the results on future similar assignments, though. What *would* you do?”
O: "I think it can be frustrating to do work which you are uncertain if it is in the right direction because you can feel the work is pointless.”

5. What should we value in a mathematics education?
Perdita: "[This site] seems to be an exemplar of a trend to favour trivial creativity over solving hard problems, and it seems likely to encourage teacher-pleasing rather than mathematical work. "

Answers Responses forthcoming...

Paper Puzzle

I've been off the grid for about ten days, but still have things to share from my math teacher circle workshop from before I left.  Here's a puzzle for you.  Make the following with one piece of normal paper and a pair of scissors.  Nothing else.  No tape, glue, or photoshop magic.

A coin flipping game...

I'm attending a math teacher circle workshop this week.  Yesterday afternoon we had a great speaker who talked about an old favorite of mine.

You and a friend play a game where each of you pick a sequence of three heads or tails.  Let's say your friend chooses THH and you choose HHH.  You then repeatedly flip a coin until one of the sequences shows up.  So if you flipped THTHH, your friend would win.  Is this a fair game?

Oh, if you have a solution feel free to write a comment, but give people a spoiler alert.

Math Circles Workshop

Today was the first day of a week long workshop put on by a local chapter of math teacher circles.  What a fun day!  We spent the morning working on problems and talking about problem solving.  An amazing group of high school students from Los Angeles presented a workshop this afternoon about their work in The Algebra Project and the Young People Project.

This was a big take away for me from the morning session.  Any guesses on what it is?

Meet Deep Blue's offspring, Watson

I just read a fascinating story in the New York Times about Watson, a supercomputer that plays Jeopardy. I'm sure I enjoyed the article in part because I was a computer science major in college. I am intrigued by tasks that are easy for computers and hard for humans versus tasks that are easy for humans and hard for computers (the most famous probably being the Turing Test). I gained a real appreciation for this difference when I had a project in college to write a computer program to count the number of people in a lecture hall. Wow. Hard. It was around this time that I thought about this exact problem--getting a computer to play Jeopardy.

I didn't find myself reading the article through the lens of a person interested in computer science, though. I was thinking as a teacher. Assuming this technology advances, and assuming it becomes readily available (it's current incarnation will cost companies a million bucks just for the hardware), what are the ramifications for education?

Will we be having the good ol' calculator debate in history class? Will educators fight over whether or not students should be allowed to use their Watson on their science test? In some ways, we're already close to being there with google, although I think there is a fundamental difference between the current technology that necessitates some analysis and the Utopian potential of Watson. On the math end, could/would/should a tool like this change instruction? Would it maybe help resolve the long going debate on what tools are appropriate when? (On somewhat of a side note, I read an article today by R. James Milgram (pdf) who argued that we should continue to teach the standard algorithm for long division to elementary school kids even though calculators (and phone with calculators) are ubiquitous because it introduces them to the important task of estimating and revising)

I'm also intrigued by the different algorithms and checks that this computer does to test the plausibility of its answers. The article only mentions a few, but I wonder if humans do these same (or similar) checks or if these checks could/should be taught.

It made me think...which is almost always a good thing.

A Call for Problems

This post is solely a request and includes absolutely no insightful, bland, or even obtuse (no pun intended...ok, pun intended) repartee.

With that disclaimer out of the way, what is/are your favorite math problem(s)? I'm specifically looking for questions that are be accessible for middle schoolers (doesn't mean they have to be easy for middle schoolers).

Non-math-teacher-types, sorry if you're feeling left out but still feel free to contribute.

Thanks!

Oh, and just so you don't think I'm slacking, I'll contribute one that I like.

---------------------------------

I'm not sure why, but I know that 1+2+3+4+5+6+7+8+9+10=55.

Find 2+4+6+8+10+12+14+16+18+20.

EDIT: Ok, I just decided that I was getting too into the middle school spirit by asking for the most bestest favoritest problem of, like, all time. Share a problem you like. The one I shared isn't my favorite problem; I'm not even sure I could come up with a true favorite.

The Common Core Standards

The final draft (for now) of the Common Core Standards for English and Math were released today. I flipped a coin, so I guess I'll talk about the math section. I'd read parts of earlier drafts, but spent much of today reading through the final version. Here's a summary and my first take:

We start with a two-page introduction, with lots of quotes from people who helped write these standards, and then a paragraph on mathematical understanding.

Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from.

Me like...the hard part now will be writing valid and reliable assessment questions.

A page describing how to read the standards follows, and explains that grade level standards will be broken up into three hierarchies: standards, clusters, and domains.

"Standards define what students should understand and be able to do. Clusters are groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject. Domains are larger groups of related standards. Standards from different domains may sometimes be closely related."

"grade placements for specific topics have been made on the basis of state and international comparisons and the collective experience and collective professional judgment of educators, researchers and mathematicians"

This concerns me a tad. We're making decisions based on what's been done in the past (isn't the whole point that what we've done in the past hasn't worked particularly well...I really hope this is more than just trying to get all 50 states to do the same mediocre thing) and what's being done in places like Singapore (where reportedly everyone is good at math). Now while Singapore Math and I are not friends on facebook, there are aspects of this curriculum that I think are fantastic. Leinwand and Ginsburg wrote a good article about what they believe make it successful. I wonder if the core standards goes far enough to link problem solving with concepts and skills, especially since the authorsexplicitly claim that these standards "do not dictate curriculum or teaching methods." I feel that this was a political decision, and question whether this was a really bad idea. If we are trusting "math experts" to determine the content that is taught in schools to students who will be entering the workforce in 2030, why not entrust experts to dictate best teaching practices? I've heard there's lots of research on the subject.

Three pages are then devoted to defining and describing "standards for mathematical practice."
They are:

problem solving, reasoning and proof, communication, representation, connections... adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

These practices are organized into eight categories that the authors elaborate on.
1 Make sense of problems and persevere in solving them (very Polya-esque).
2 Reason abstractly and quantitatively (ability to decontextualize and contextualize).
3 Construct viable arguments and critique the reasoning of others (conjecture, proof, and critique).
4 Model with mathematics (connect math to "the real world"...not sure how this is different from #2 and I definitely worry about how this will be done in lame, inauthentic ways).
5 Use appropriate tools strategically (can you in good faith make this 1 of the 8 principals of doing mathematics and not allow students to use these tools on high stakes tests?).
6 Attend to precision (accuracy, on the other hand, is overrated...I jest, precision is a practice that will be beneficial to students in every realm of their life).
7 Look for and make use of structure (understanding symbols/structures and characteristics of these symbols/structures).
8 Look for and express regularity in repeated reasoning (looking for shortcuts and generalizations).

It's unclear whether the authors believe that these practices will lead to a better understanding of the content (which will be assessed) or whether they believe these practices to be an integral part of doing mathematics (which would then imply that these practices should be assessed independently of content). Boy, I could probably be convinced to do some unsavory things to see the latter. The authors leave this decision largely to textbook writers: "Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction." which, in my experience, are beholden to test writers. Unfortunately, my guess is that these will be treated much like the current "mathematical reasoning" standards that are"inherently embedded in each of the other strands", in my humble opinion a specious claim confirmed by the low cognitive levels involved in some of the standards and most of the test questions.

Also...where's pattern sniffing? Did I miss it? That was a surprise.
What about problem posing? Ok...I'm less surprised that this wasn't in there. At least I still have a purpose in life.

And I still can't help think that this isn't as good as Cuoco's Mathematical Habits of Mind, but I digress.

Anyway, that ends the first section. The remainder of the document is devoted to grade level standards. This post is already way too long, so I'm going to stop here and do a separate write-up on the grade specific standards.