Beliefs and Attitudes about Mathematics

What beliefs and attitudes about mathematics do you see in your students, in society, in the media, and elsewhere? Try and think of both positive and negative beliefs and attitudes. These can be beliefs that you agree or disagree with. I'll start with a few, but please add your own in the comments.

• If you're good at math, math problems can be solved in a relatively short amount of time.
• People do not solve math problems for fun; they do it for school, for their job, or to balance their checkbook.
• Every math problem has been solved by someone.
• Math is about numbers.
• Math is a language to describe the world.
• If you are good at math, you are smart.
• If you can do computations accurately and quickly, you are good at math.
• People who are good at math are eccentric and/or not socially adept.
• Boys are good at math.
• Asians are good at math.
• There can only be one correct answer.
• If I don't know how to solve a homework problem, I must be doing something wrong.
• Math topics/classes are sequential; I need to understand A before I can learn B.
• It is socially acceptable to say you're bad at math.
• Math is more analytical than creative.
• Using a procedure correctly to get the right answer is more important than understanding why the procedure works
• With current technology, arithmetic is not important.
• To be an engineer, you need to be computationally strong.
• Math is a gatekeeper.
• The value of math is in its connection to real world applications.
• Math teachers sleep under their desk at school. 

Ok, I guess that was more than "a few".  Hopefully I left some for everyone else.

Habits of Mind

This is still a work in progress (and feedback would be greatly appreciated), but I've decided to explicitly teach (and assess...more on that later) 4 "categories" of mathematics this year.

1. Skills (I know how to...)
2. Concepts (I understand and can explain why...)
3. Connections (I see and can explain the relationship between...)
4. Mathematical Habits of Mind (I can use and appreciate the process of...)

I've decided not to use the term "problem solving" because I believe this term is often misused to include be limited to solving problems and because the motivation for problem solving skills seems to be to solely help you get an answer.  While I believe that they can be very helpful in finding answers, I see mathematical habits of mind as also being mathematical in and of themselves.  So...while searching for patterns may help you solve a problem it is also DOING mathematics.

Here's the current version of the mathematical habits of mind I think are important.  I hope to explore (in varying depths) every one of these and have already shared the list with my 6th graders.

This is definitely a work in progress and some of these are based on work by Cuoco, Driscoll, Schoenfeld, and others.

Habits of mind

1.    Pattern Sniff

A.     On the lookout for patterns
“Ok.  We’ve been working on this staircase problem and it seems that you can’t write perfect squares powers of two as a sum of consecutive whole numbers.”

B.     On the lookout for Looking for and creating shortcuts
“It would be nice if there were a faster way to do 57x34 than adding 57 to itself 34 times. Think we can find a way?”

2.    Experiment, Guess and Conjecture

A.     Can begin to work on a problem independently
“I’m not sure how to solve this problem, but I’m confident I can make some progress.”

B.     Estimates
“Without doing any calculations, I’m guessing that it will take him 30 seconds to walk up the down escalator.”

C.     Conjectures
“Based on my work, I think the following is true.”

D.    Healthy skepticism of experimental results
“Boy, it sure seems like this 4, 2, 1 thing always repeats but we don’t have a proof yet.”

E.     Determines lower and upper bounds
“I know it will take the people at least 10 minutes to cross the bridge because the 10 minute soldier has to cross the bridge.  I also found a solution that takes 19 minutes so I know the final answer is somewhere between 10 and 19 minutes.”

F.     Looks at small or large cases to find and test conjectures
“I made a table of the first 5 cases and I think I see a pattern.  I’m going to see if this pattern holds for the 100^th case.”

G.     Is thoughtful and purposeful about which case(s) to explore

H.    Keeps all but one variable fixed

“So I’m exploring the equation y=mx+b and I’m wondering how the graph changes as m and b change.  For now, I’m going to set m to 1 and just look at how the graph changes when I change b.”

I.      Varies parameters in regular and useful ways

(Even/odd example)

J.      Works backwards (guesses at a solution and see if it makes sense)

3.    Organize and Simplify

A.     Records results in a useful way
“I’m going to make a table.”

B.     Process, solutions and answers are detailed and easy to follow

C.     Looks at information about the problem or solution in different ways

D.    Determine whether the problem can be broken up into simpler pieces
“I think I can solve this problem by solving these other 2 simpler problems.”

E.     Considers the form of data (deciding when, for example, 1+2 is more helpful than 3)

“I’m going to leave my fraction as 6/36 because the 6 represents the number of ways you can roll a 7 with 2 standard dice and the 36 represents the total number of rolls.”

F.     Uses parity and other methods to simplify and classify cases
“Next time we play 21 Nim I’m going to keep track of whether the running sum is a multiple of 3, one more than a multiple of 3, or 2 more than a multiple of 3.”

4.    Describe

A.     Verbal/visual articulation of thoughts, results, conjectures, arguments, process, proofs, questions, opinions

B.     Written articulation of thoughts, results, conjectures, arguments, process, proofs, questions, opinions

C.     Can explain both how and why

“The algorithm for dividing fractions is simple.  Now I just need to work on making sense why this works.” 

D.    Creates precise problems

E.     Invents notation and language when helpful
“For the sugar weighing problem, I don’t want to have to write out every solution in words so I’m going to let the symbol 3w~3s stand for the act of putting the 3 pound weight on one side of the balance scale, measuring out 3 pounds of sugar on the other side of the scale, and then setting aside the sugar.”

F.     Ensures that this invented notation and language is precise
“I need to be careful that I am differentiating between sugar that I am measuring and sugar I am using as a weight.”

5.     Tinker and Invent
      A.   Creates variations

B.     Looks at simpler examples when necessary (change variables to numbers, change values, reduce or increase the number of conditions, etc)

C.     Looks at more complicated examples when necessary

D.    Creates extensions and generalizations

E.     Creates algorithms for doing things

F.     Looks at statements that are generally false to see when they are true

G.     Creates and alters rules of a game

H.    Creates axioms for a mathematical structure

I.      Invents new mathematical systems that are innovative, but not arbitrary

6.    Visualize

A.     Uses pictures to describe and solve problems

B.     Uses manipulatives to describe and solve problems

C.     Reasons about shapes

“I see how this structure is made.”

D.    Visualizes data

E.     Looks for symmetry

F.     Visualizes relationships (using tools such as Venn diagrams and graphs)

G.     Vizualizes processes (using tools such as graphic organizers)

H.    Visualizes changes

I.      Visualizes calculations (such as doing arithmetic mentally)

7.    Strategize, Reason and Prove

A.     Moves from data driven conjectures to theory based conjectures

B.     Tests conjectures using thoughtful cases

C.     Proves conjectures using reasoning

E.    Looks for mistakes or holes in proofs

F.  Uses indirect reasoning or a counter-example (Park School)

E.  Uses inductive proof

8.    Connect

A.     Articulates how different skills and concepts are related

B.     Applies old skills and concepts to new material

C.     Describes problems and solutions using multiple representations

D.    Finds and exploits similarities between problems (invariants, isomorphisms)

9.    Listen and Collaborate

A.     Respectful to others when they are talking

B.     Asks for clarification when necessary

C.     Challenges others in a respectful way when there is disagreement

D.    Participates

E.     Ensures that everyone else has the chance to participate

F.     Willing to ask questions when needed

G.     Willing to help others when needed

H.    Shares work in an equitable way

I.      Gives others the opportunity to have “aha” moments

10. Contextualize, Reflect and Persevere

A.     Determines givens

B.     Eliminates unimportant information

C.     Makes and articulates reasonable assumptions

D.    Determines if answer is reasonable by looking at units, magnitudes, shape, limiting cases, etc.

E.     Determines if there are additional or easier explanations

F.     Continuously reflects on process

G.     Works on one problem for greater and greater lengths of time

H.    Spends more and more time stuck without giving up

Top 10 Technology Advances of All Time

Kate, over at f(t) inspired (pun intended) me to write a little about technology after reviewing a TI seminar she attended.

So without further ado...

The 10 Most Important Technological Advances in Math Education (chronologically):

1.     Writing utensil (to write on cave walls)

2.     Paper (because cave walls are hard to take with you)

3.     Ruler (and, in general, the concept of measuring devices)

4.     Abacus (yeah for being able to do large arithmetic problems quickly)

5.     Printing press (makin’ a copy)

6.     Slide Rule (yeah for being able to do large arithmetic problems quicklier)

7.     The industrial revolution (education for the masses…errr…where we learn to sit still and follow directions)

8.     Sputnik (not directly used in math ed, but this played a huge role in shaping our current curriculum)

9.     Computers (beginning with its mad arithmetic skills, case checking skills and evolving into its use for modeling)

10. The internet (for resources, networking, social media learning, etc)

And as a bonus…

11. The TI n’spire

Initially, I put #11 on here in jest but I’m beginning to consider the actual importance of this technology (and not in a good way).  Specifically, I believe that the n’spire and it’s predecessors have played a significant role in shaping standardized testing and that this standardized testing has shaped recent curriculum as much, if not more, than sputnik or the industrial revolution.

Without much thinking, my first criticism of this list is that it is pretty 20^th century-centric.  Anyway, something to ponder.  Anything you would add? 

My Blog Post about Math For Love's Blog Post about Math Mamma's Blog Post about...

Math for Love just wrote a post that I really enjoyed about play.  Go there and check it out and then come back here for the post-game analysis.  I was originally just going to comment over there, but this got too long.  Anyway, I found that it was a really well articulated process of how I want students to work with math problems.  Since you're good at following directions you've already read this but let's recap.

1. Play (Expansion)
2. Ask questions (Reflection)
3. Choose a question that you really want to explore (Contraction)
4. Try to answer the question with what you already know
5. If that fails (i.e., if your question is deep enough), start exploring other ways to approach the question, with your own ideas and by looking at what others have done. A teacher or expert can be very useful here (“Oh, you want to figure out how far apart those two dots are? Have you ever heard of the Pythagorean Theorem? It works like this…”)
6. Refine your question
7. Repeat Steps 4-6 until you solve your question
8. Explain your work to people
9. Realize that you can’t really explain it, and refine your solution until you can actually write down an explanation.
10. If in a class or group, share your work with the group, or have the group come to an agreement on how things really are.
11. Satisfaction.

I would stop at #11 (and although this is hard, I would also aim to have students feel #11 after every step).

12. Tweak the original "rules" in some way and return to #1.

In terms of addressing the floundering that can occur in progressive models, I think the key (not that this is easy) is teaching kids how to do #1 through #11 (well, imho the cycle of  #1 through #12).  Even "playing" can be taught. Consider the development kids go through when playing boardgames.  They start with games like Chutes & Ladders which has absolutely no strategy and children simply enjoy the unknown of what you're going to roll next and where you'll end up.  This eventually evolves into greater sophisticated game play where students have to make decisions (whether or not to buy that property in Monopoly, for example).  This evolves into more sophisticated strategies such as thinking ahead (Chess), thinking about your opponent's thinking (Settlers of Catan), making inferences based on past play (Poker), etc.

I believe the game metaphor can be extended to math problems, even traditional ones.  Just as an example, consider how the "rules" of subtraction evolve throughout elementary school:
1. Single digit whole #'s with the first being larger than the second
2. Multiple digit whole #'s with the first being larger than the second
3. Multiple digit whole #'s with the first being larger than the second, but requiring borrowing
4. Whole #'s with the second being larger than the first
5. Fractions
6. Subtraction of integers
7. Subtraction of rationals
etc.

We change (usually relaxing) the rules as we progress through the curriculum.  It's nice that this also mirrors what mathematicians often do, relaxing or restricting conditions and exploring the ramifications.

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...

What Makes a Problem Great

I've put together an ever evolving list of characteristics of rich math problems. Anything you'd add, subtract, or edit? These are in no particular order...

1. The problem should be accessible. It should minimize vocabulary and notation, have multiple entry points, and include ways to collect data of some sort. It should have multiple methods that promote different learning styles and celebrate different ways of being smart.

2. The problem should be deep. It should be rich enough to spend hours, days, weeks, months, or years working on variations, generalizations, and extensions. It should lead to and connect as many different aspects of mathematics as possible, as this can then be the motivation for developing procedures, vocabulary, notation, and mathematical concepts.

3. The problem should be able to scale sideways so that students can explore related ideas in different contexts to reinforce concepts.

4. The problem should be captivating. This does not mean that it has to be a “real world” problem and it really shouldn't be a contrived real world problem (please explain to me how in the world I would know the number of total animals and feet I had on my farm, but not the number of cows and chickens). It might mean that it leads to a surprising result. It might mean that it feels like a puzzle waiting to be solved.

5. The problem should be mathematical. Progress should be able to be made by using problem solving heuristics and resources. The language of mathematics should benefit the student in solving the problem.