Teaching Problem Solving, Part 3: Mathematical Habits of Mind

After working on some problems that I think are good, we'll discuss the two questions I posed in my earlier post.


1.  Why am I working on this problem instead of one of the other six?  Why am I working on this problem instead of watching tv, organizing my sportscaster bobblehead collection, or eating that chocolate santa that's still sitting on my kitchen counter?

2. What problem solving skills/mathematical habits of mind helped you make progress on a particular problem?


Unless there is outright mutiny, we may avoid sharing solutions all together.  Instead, I hope to really explore how we--experts--use mathematical habits of mind when solving unfamiliar problems and how this can be taught to students.  The plan is to explore the framework I am developing (here's where it all began) for mathematical habits of mind. The ones in bold are habits I believe could be helpful in solving one or more of the problems.

Mathematical Habits of Mind

1.  Pattern Sniff

A.          On the lookout for patterns

B.          Looking for and creating shortcuts

2.  Experiment, Guess and Conjecture

A.          Can begin to work on a problem independently

B.          Estimates

C.          Conjectures

D.          Healthy skepticism of experimental results

E.          Determines lower and upper bounds

F.           Looks at small or large cases to find and test conjectures

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

H.          Keeps all but one variable fixed

I.            Varies parameters in regular and useful ways

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

3.  Organize and Simplify

A.          Records results in a useful way

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

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

F.           Uses parity and other methods to simplify and classify cases

G.          Uses units of measurement to develop and check formulas

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

D.          Creates precise problems

E.          Invents notation and language when helpful

F.           Ensures that this invented notation and language is precise

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

D.          Visualizes data

E.          Looks for symmetry

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

G.          Visualizes 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

D.          Looks for mistakes or holes in proofs

E.          Uses indirect reasoning or a counter-example

F.            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.   Collaborate and Listen

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

Teaching Problem Solving, Part 2: Some Problems...

Here are seven problems that we'll be using in tomorrow's PD on problem solving that, as I mentioned in my previous post, make for half-decent problems.  I'd love for you to try your hand at one or seven of them.  What I'd really love is for you to set aside your focus on answering the problem and instead reflect on the following questions:

1.  Why am I working on this problem instead of one of the other six?  Why am I working on this problem instead of watching tv, organizing my sportscaster bobblehead collection, or eating that chocolate santa that's still sitting on my kitchen counter?

2. What problem solving skills/mathematical habits of mind helped you make progress on a particular problem?

That's right, solutions pollutions. I want to hear about your process. Anyway, without further ado:

Some Problems to Work On…

Judah Schwartz: http://gseweb.harvard.edu/~faculty/schwartz/formulating%20measures1-v2.pdf

“Dividing Triangles”

1. Construct the point in a triangle at which segments from the three vertices meet at 120 degree angles.  

2. Prove that it truly is the 120 degree point. 

3. Prove that the sum of those three segments is minimized at this point.  

4. Find the triangle with minimum perimeter which is inscribed in an acute triangle.

Contributed by Peter

“Finding Staircase Numbers”

12=3+4+5.  15=7+8.

Turns out 15 can also be written as 1+2+3+4+5 and 4+5+6. 

In other words, the numbers 12 and 15 can be written as the sum of consecutive whole numbers.  Let’s call these numbers staircase numbers.

2 cannot be written as a sum of consecutive whole numbers. 

1. What numbers can and cannot be written as staircase numbers (sums of consecutive whole numbers)?

2. Why do you think I call these staircase numbers?
3. What other mathematical questions could we ask related to staircase numbers?
4. Work on trying to answer one or more of your own questions.

Adapted from Marilyn Burns’ Math for Smarty Pants by Avery

“Drawing Paths”

www.jamestanton.com

“Weighing Sugar”

You have a balance scale, a 2-pound weight, a 3-pound weight, and a 100-pound bag of sugar. 

You can only use the balance scale three times per “weighing”.

1. Describe how to measure 7 pounds of sugar.
2. Describe how to measure other amounts of sugar (remember you can only use the scale 3 times for each weighing).

Contributed by Avery

“Breaking a stick”

If you break a stick in two random places, what is the probability that it can be made into a triangle?

Contributed by Kyle

“Watching Chameleons”

A new species of chameleons has recently been discovered on an uninhabited island. They have been named the Patriot Chameleon because some are red, some are white, and some are blue.  These chameleons are special in that if two chameleons of different colors meet they will both change to the third color. This continues in other meetings. Is it possible that sometime in the future all the chameleons will be red?

Contributed by Kyle. Adapted from www.sosmath.com

Teaching Problem Solving, Part 1: Starting with a Good Problem

I'm helping lead a PD on problem solving tomorrow.  We'll start by working on some problems that I think do an a'right job of passing my "Characteristics of a Good Problem" test.  I blogged about this a while ago, but here's a shiny new version:

Characteristics of a Good Math Problem

1.     The problem is accessible. It minimizes vocabulary and notation (and vocabulary and notation that does exist should simplify, not complicate).  It should only be as precise as necessary.  The problem should 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.  It may have multiple valid solutions.

2.     The problem is deep. It is rich enough to spend hours, days, weeks, months, or years working on variations, generalizations, and extensions. It leads to and connects different aspects of mathematics. The problem motivates developing procedures, vocabulary, notation, and mathematical concepts.

3.     The problem is captivating. This does not mean that it has to be a “real world” problem and pseudocontext (from Jo Boaler's What's Math Got To Do With It?) should be avoided. A captivating problem may lead to a surprising result. It may feel like a puzzle waiting to be solved. It may be necessary to solve a different, interesting problem (which is not the same as “you’ll need to know this next year”). The problem may be posed by students. The problem consists of benchmarks along the way where one is re-energized by the feeling of success.

4.     The problem scales sideways. This characteristic may be more applicable to school mathematics, but sideways scaling allows for practice and quality assessment (beyond just solving exercises measuring the ability to follow procedures).

5.     The problem is mathematical. Problem solving skills and/or the language of mathematics help make progress in defining, simplifying, quantifying, dividing and/or solving the problem. Exploring the problem promotes mathematical habits of mind. 

Poor Teaching

Test prep.


I am currently teaching four sections of AP Calculus.  My students took the exam this morning.  I am thankful for this, not because it means that we can stop doing math (sorry kiddos, tomorrow we begin with some graph theory), not because this marks the finishing line of an intense race (sorry kiddos, I really don't think you're going to talk me into watching Stand and Deliver), and not even because this means that summer is right around the corner (although I can't say I'm disappointed by this). The reason that I am happy that the AP is over is because my teaching has really gone downhill in the past month.

My suckage.


It's been about a month since we finished the AP curriculum, a little longer for my AB classes and a little less for my BC classes (those AP Calc teachers who are now wondering how in the world I managed to do this, my students began Calculus midway through the year of their junior year). So part of my decent into mediocrity is due to lack of freshness.  A month is a long time to review and I didn't develop ways to approach old material in new ways or synthesize material in ways that aren't possible earlier in the course.  Why?  Time constraints?  Lack of will, creativity, or resources? My physical stature?  Who knows.

I fell for the song of the Direct Instruction Sirens with refrains of "teacher assigns problems/students do problems/teacher does problems students can't do."

I was less patient.  I was less willing to let kids struggle and internally frustrated (and probably externally frustrated) when kids did or said things that exhibited misconceptions "they shouldn't have."

I said yes and no much more frequently, interrupted kids when they took false paths, and even found myself saying "Don't worry about that. It's not going to show up on the AP."

Why?

Here's the thing I can't explain. Why did this happen?  I don't think I succumbed to the pressure.  I really don't care how students do on the AP (at least I don't think I care).  I see myself first as a math teacher, second as a Calculus teacher, and third as an AP teacher.  I care most about sending kids off to college excited about math (with some success), second about exploring the beautiful tool that is the calculus (with some success), and third about the inter-workings of the debt ceiling. The AP is way down on the list (or at least I think it is). It's a 3 hour snapshot with lots of uncontrollable external variables that have little or nothing to do with calculus (in both directions...if everyone got a 5 it would not change my view that all of my students are still at different stages of learning calculus).

It is the end of the year. I am leaving for a new school. All of my students are leaving for college. Maybe it's just laziness. All I know is that this thinking is making me tired...time to go take a nap.