- Skills (I know how to...)
- Concepts (I understand and can explain why...)
- Connections (I see and can explain the relationship between...)
- Mathematical Habits of Mind (I can use and appreciate the process of...)
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 writeperfect squares powers of two as a sum of consecutive whole numbers.”
“Ok. We’ve been working on this staircase problem and it seems that you can’t write
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?”
“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.”
“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.”
“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.”
“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.”
“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.”
“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 100th case.”
“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 100th 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.”
“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.”
“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.”
“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.”
“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.”
“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
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
Posts
I love this! I'm going to ask my students to read this.
ReplyDeleteGreat list.
ReplyDeleteI was sent a list from the Park School by John Burke over at quantumprogress and they have:
look for patterns
tinker
guess
describe
visualize
seek proof
use plausibility
take things apart
conjecture
examine a similar problem
use inverse thinking
determine relevance
use multiple points of view
create
Shoot me an email if you want me to send the pdf I have, there are some sample problems and a description attached as well.
Oh and I sent the email to say I'd like to come to the math circle, waiting to hear back. Look forward to meeting you. And Sue if you're in the Peninsula area.
Yeah, this is hot.
ReplyDeleteFun and true list. I believe we should design similar lists in science, social science, reading, writing, and so on...(actually, many of these work for other subjects with a slight change in output and context). These habits of mind are the components that build lifelong learning and need to be better valued in education. I am so glad you are writing about this.
ReplyDeleteThis is an excellent list, but I would like to offer a minor correction. In part 1A, I think you meant to write "powers of two" instead of "perfect squares".
ReplyDeleteYour list came up in a Math 2.0 event yesterday that Sue VanHattum here lead - I think Dan Meyer brought it up. I put up a (very preliminary) draft of the Patterns chapter, which talks about some limitations of patterning, and what happens beyond: http://www.naturalmath.com/blog/patterncraft-and-beyond/
ReplyDeleteAlso, have you seen the Pirie-Kieren layers? They are relevant here. Another similar list is in "Mathematical sophistication" papers by Seaman.
Your list made me think more about how I (HS math teacher) solve problems and my students solve problems. One thing that I do and that my students have found helpful is to ask the questions: "What am I trying to find?" and "What would the correct answer look like?" This comes from the logic that in order to get somewhere you have to know where you're going. I think this might fit into either 3) Organize and Simplify or 4) Describe.
ReplyDeleteThis is a really great list and very simple for all grade levels to understand. I really like the communication section. Talking in mathematics rarely happens but it is essential to get to the depths of mathematics.
ReplyDeleteI am tasked with developing online math resources and could not agree more with the importance of getting students to ‘inquire’ and not just ‘do’. When I taught math, I would often not reveal the final solution to a problem or leave some areas of math ‘on show’ but not fully explored. For example, when looking at volume of cylinders from knowing height and cross section I might casually say “I wonder how tall a bottle of 75cl wine is? But that might be too tough for this class.” The reaction is generally ‘who you calling thick!’ Questions then begin to flow like ‘how wide is the bottle then?’ ‘do you need pi?’ ‘Can we use what we now know about volume of cylinders?’ ‘Is a wine bottle a cylinder anyway?’
ReplyDeletePolya's "How To Solve It" is probably relevant here, if not necessarily accessible for your students.
ReplyDeletehttp://en.wikipedia.org/wiki/How_to_Solve_It
I'd say this is a list of software tester's skills.
ReplyDeleteha! DragD - that's what i was thinking when reading the list. Though as Bonita and Family mentions it's applicable to all subjects... it's pretty much a (math biased) list of fundamental analysis skills.
ReplyDeleteThis is not just for maths. I can use this kind of thinking in analytics and stratgic planning at work. Great stuff
ReplyDeleteLots of great comments on here.
ReplyDelete@ Jason: Thanks for the list. I'm encouraged by the amount of overlap.
@ Dan: As hot as this?
@ Bonita: I've thought a little about what this list might look like in other subjects. I'm planning on sitting down this week with a colleague who teaches history to both get her feedback on what makes sense and what doesn't make sense and to talk about what this list might look like in history. I'll let you know how this goes.
@ David: Thanks! Brain fart corrected. I guess now it really is a spoiler.
@ Maria: Thanks for the references!
@ Harryks: These are good additions. My initial feeling is that they would fall into contextualize, reflect, and persevere...which brings up the question, how important is it for these processes to be categorized (or that you and I categorize them the same)? Hmmm...
@Telannia: Thanks. And I agree about the communication aspect, although this is probably an example of a habit that would (err..should) be taught and reinforced in every subject, not just math.
@Chris: Agree about the power of questioning.
@njl: "How to Solve it" was a seminal piece on habits of mind. It was also written in 1945 and (this is a good thing) I think we've come a long way in both describing these habits and thinking about how to explicitly teach these habits. I also (hope) we've come a long way in who this is written for: Polya always refers to his student as "he" and regularly talks about these strategies being used by "bright" math students. I'd like to think my list applies to all students.
@ Dragd, datadebrief, & Suhit: Shocking! You mean these habits might be useful outside the math classroom? :) One thing worth noting, I made a conscious decision to not call these "problem solving skills" because this seems to imply that the reason for learning these skills is solely to solve problems. I believe that (most of) these habits of mind can not only be used to solve math problems, but that these processes ARE math in and of themselves. Pattern sniffing might help you solve a problem, but you're doing math regardless of whether you solve a problem or not.
"brevity is the soul of wit."
ReplyDelete"B. On the lookout for shortcuts"
I love the variable approaches to turning over a problem in your mind.
ReplyDeleteThis sort of analytical checklist would be of immense value as a checklist to investors . . .
This is a wonderful tool, but I don't think you need to exclude the idea of "solving problems."
ReplyDeleteIn fact, I think it's a big piece of the toolkit that most discouraged math students bring to the table -- and I find that to be something important to emphasize in encouraging them to believe in their own ability to tackle math problems.
"Solving problems" is not just about getting an answer. Sometimes, just as in math, it's about exercising creativity, reducing suffering, increasing abundance, finding patterns, etc. And if that gets them to become curious about math and the world around them, then by gum, I'll take it!
great list indeed, thanks. readers may also find this list of mathematical habits of mind helpful. http://keepingmathsimple.wordpress.com/2010/01/27/developing-mathematical-habits-of-mind/.
ReplyDeleteThanks for this stimulating list. I'm a microbiologist and a modeler, and there is lot's here to like for anyone that needs to solve problems or analyze data. I've explicitly forwarded it to my graduate student with the implicit message they need to read it :) I've also linked to it on facebook so my science buddies see the post as well.
ReplyDeleteThis is a brilliant list with clear and flawless explanations
ReplyDeleteWhich consecutive whole numbers did you have in mind? I can't think of any whole number sequence that doesn't have intervening odd numbers, at least among the integers.
ReplyDelete@cheesemonkeysf: My intention wasn't to preclude solving problems. I didn't articulate this well. I think solving problems is very important. I guess what I meant was that I saw the term problem solving skills being used as a means to and end whereas I see habits of mind as an important part of mathematics in and of itself. Not sure if that's any clearer...
ReplyDelete@Richard: The question is which #'s can be created by the sum of at least 2 consecutive whole numbers. So 3 is good because 1+2=3. 3+4+5=12 so 12 can be made this way. One question one could ask is which numbers can and cannot be formed in this way.
Another list, broader.
ReplyDeletehttp://throughlines.blogspot.com/2007/01/in-last-three-posts-ive-been-mapping.html
Mash this up with cognitive problems and you'll have something very interesting, but probably wouldn't fit any math course. I bring up cognitive issues, because they can send us down rabbit holes. The problem can start as early as "pattern sniff".
ReplyDeleteIt is known that humans search for patterns and often find them even when they're not there.
Think statistically! Invert the problem. Remember Bayes.
Is the pattern and the following reasoning the ONLY truth? Are there other explanations that are just as rigorous? How do we know there aren't?
@drocto: I think that Avery did not intend these a curriculum for a specific math class, rather that these are habits that math students might (and should) use in any math class. I guess I'm a little confused about what you mean. How exactly are you thinking about cognitive issues in this context?
ReplyDeleteGreat list and comments! I so liked it that I linked to this blog entry in Facebook. When I did so, the text beneath my "What's on your mind?" field looked like this:
ReplyDelete"Without Geometry, Life is Pointless: Habits of Mind
mathteacherorstudent.blogspot.com"
followed by the first comment, stripped of end-of-line characters:
"Great list. I was sent a list from the Park School by John Burke over at quantumprogress and they have:look for patternstinkerguessdescribevisualizeseek proofuse plausibilitytake things apartconjectureexamine a similar problemuse inverse thinkingdetermine relevanceuse multiple points of viewcreateSh"
Not a pretty sight.
@betweenthenumbers: I just meant that some warning is due regarding being trapped by models. For example, after I happened to read the Congressional testimony of an economist regarding the failure of economists to predict the recent financial and economic meltdown. Of course, the explanation involved incentives.
ReplyDeleteThe key thing was that the profession has strong incentives to work with extremely detailed quantitative models, while ignoring the messy stuff that isn't currently modeled.
For example, he states the profession "over-researched a particular version of the dynamic stochastic general equilibrium (DSGE) model that happened to have a tractable formal solution, whereas more realistic models that incorporated purposeful forward looking agents were formally unsolvable".
In other words, the profession becomes ensnared in the "beauty" if you will of the DSGE - the mathematics are compelling. But "DSGE models were of only limited direct policy relevance, since by changing the assumptions of the model slightly, one would change the policy recommendation".
Yet I'm sure all of the mathematical habits of the mind have been at work perfecting the DSGE.
So the point is to remind people that the mathematical habits of the mind are very powerful and very useful in nearly all aspects of life. However, there is no single framework for conceptualizing things or "solving problems". One must always remember that all frameworks are limited and be willing to jump out of a model that is elegant and useful, but inappropriate to situations outside of its proven domain.
@drocto I agree that we must be careful about not getting trapped in models, but I think you then go on to conflate using models to make things more explicit (habits of mind) and using models to predict (the financial crisis). I don't see this list as a model in the latter sense (at least in how I conceptualize the word model). I see this list as no different from listing the skills a student will learn in a particular class (adding fractions with different denominators, converting between fraction representations and repeated decimal representation, etc).
ReplyDeleteFurthermore, I think that mathematical habits of mind were NOT used effectively in "perfecting the DSGE" since, as you said yourself, slight changes to the model would change the policy recommendation. That might fit into the category of:
"Be skeptical of experimental results." or "Continuously reflect on process."
<3 Schoenfeld
ReplyDeleteSeriously, send me articles.
Can you *please* change #9 to "Collaborate and Listen"? (:
ReplyDeleteRegarding 4E (inventing notation), you might be interested in a blog post by Mark Jason Dominus about the article or book (I'm not sure which) in which Robert Recorde invented the equals sign (scan included). It can be insightful to see how that sign that everyone takes for granted was once just shorthand notation for "is equalle to".
ReplyDeleteThe sum of consecutive integers problem becomes more interesting if you look at the sum of k consecutive integers (n, n+1, ..., n+k-1) modulo k. For odd k, you can pair up all the numbers to see that the result is congruent to 0 mod k. For even k, there is one unpairable number, giving you a sum congruent to k/2 mod k. Any odd number can be done as the sum of 2 consecutive numbers. Any multiple of an odd number k can be done as the sum of k consecutive numbers. That covers everything but the powers of 2.
ReplyDeleteHabits of Mind
ReplyDeletethanks for sharing
Wow, very impressive and comprehensive list/outline. I remember this term, "habits of mind" from my grad classes a few years back....
ReplyDeleteI just started up my own math blog a couple of weeks ago: Rick's Algeblog at www.algeblog.com. I hope some of my own insights and experiences taken from my 25+ years in the classroom will be insightful to others like you in our math-blogging world!
Thank you for sharing this with the math community. I am impressed by how much more, regarding conversations of math, are out there now. This blog really parallels what math is about. Not just DOING math!
ReplyDeleteTad