
Teach 5th and 6th grade. My 13th year of teaching. Address private school.
0:15 My bias is that we tend to focus on the first and not the second. Process standards: Language used by NCTM Standards of Mathematical Practice: language used by common core Problem Solving: Language used by teachers Mathematical Habits of Mind: Language used by…me? For the most part, these all mean the same thing. I think it is worth differentiating between these processes that are taught as a means to an end with regards to content vs the processes that are inherently mathematical in and of themselves and should be taught with content as the medium, not the end goal.
#1: First year of teaching #2: Example of farmer/goat/cabbage problem Re Farmer/Goat/Cabbage Problem: Problem solving? Real world application for your students who might become goat cabbage fox farmers with rickety boats? #3: Polya is to a good scotch as these 4 problem steps is to a good scotch diluted with a gallon of water.
My current evolutionary state (in no particular order). Notice I don’t have a hovercraft here. Still a work in progress. I’m going to talk about my own version because, a, it’s less work since I’ve been working through this since before the common core standards were available, and b, I understand it better. I encourage you to make your own list. Types of patterns: numerical, geometric, etc. (why I think it’s important to know number facts…without this, you won’t see the patterns)Creating shortcuts/procedures: flipping instruction from teacher explains how to do things and students practice to students develop methods and the teacher plays a guiding role and a wrapping paper role: ie help students put their methods into a nice neat box in the end.
Guess > Evidence > Conjecture > Proof cycle with a healthy skepticism throughout Determining lower and upper bounds: “Say an answer you know is too high.” A way to scaffold and feel intermediate success.
Ttables are our best friend, but we sometime gloss over the most important aspect: what data do we collect? Re Breaking up: We have formal vocabulary like lemmas, but this idea can be powerful starting in elementary school
Show your work!
Another great opportunity for differentiation. Importance of student agency.
0:45
Why are these habits so important? Some data too shocking to not steal from David Foster’s presentation. How are mathematical habits of mind going to help me solve this problem? Not saying that students shouldn’t be able to do the second problem, but that this is a bad place to start. Habits won’t help with unfamiliar notation and vocabulary.
Problems vs Exercises Comment on minimizing vocab and notation (ideally the problem will lead to this) Comment on precision (Don’t want to get bogged down in details. Also want to let students create the precision, make assumptions, and explore the different characteristics of the problem with the different assumptions). Comment on pseudocontextual: ex of goat cabbage farmer. More subtle example of forcing students to solve a “real” problem in a way they wouldn’t solve it outside of class Scalable: Are there more problems similar enough to assess understanding but different enough that you’re not assessing parroting
10 minutes to work What mathematical questions do you have?What are the stated rules? What assumptions are you making? Familiar problem? Create an extension.
The chart on the right is not all correct!
Organized, but helpful? Chart might look differently after insight to only consider side lengths that are relatively prime.
Generalizations can be simpler or more complicated than the original problem.
Done at 1:20
What are the costs of shifting our focus?
2 pages covering all standards of mathematical practice. 76 pages covering the content standards.
Everything will be available for public ridicule shortly.
Making Standards of Mathematical Practice More than an Afterthough Avery Pickford The Nueva School Mills College Blog: Without Geometry, Life is Pointless @ withoutgeometry.com @woutgeo avery@withoutgeometry.com
Goals Share at least one intriguing (and hopefully unfamiliar) problem Share some concrete ways to explicitly teach the standards of mathematical practice Talk about the good and the bad challenging Say at least one thing you think is crazy
Process Standards Standards of Mathematical Practice Problem Solving Mathematical Habits of Mind Content Procedures/Skills Concepts Math as a noun Math as a verb
Teaching Habits of Mind: My Personal Evolution Wasn’t even thinking about this (just treading water) Justified doing problems I liked that didn’t obviously align with the curriculum as “problem solving” Started asking students to solve unfamiliar problems in class using “problem solving” but didn’t explicitly teach what I meant by this Taught “problem solving” as the 4 classic steps: 1) Read the problem. 2) Come up with a plan 3) Solve the problem 4) Reflect Read Habits of Mind: An Organizing Principal for Mathematics Curriculum by Cuoco, Goldberg, & Mark
Mathematical Habits of Mind: My version of CC Process Standards 1. Pattern Sniff On the lookout for patterns 2, 4, 6, 8,… or 4, 7, 10, 13,… or 2, 5, 11, 17, … Looking for and creating shortcuts/procedures
Mathematical Habits of Mind: My version 2. Guess and Conjecture Guess>Estimate>Use Evidence>Conjecture> Prove Determines lower and upper bounds
Mathematical Habits of Mind: My version 3. Organize and Simplify Records results in a useful and flexible way Determine whether the problem can be broken up into simpler pieces ? ?
Mathematical Habits of Mind: My version 4. Describe Conversational, verbal, and written articulation of thoughts, results, conjectures, arguments, process, proofs, questions, opinions Can explain both how and why
Mathematical Habits of Mind: My version 5. Experiment and Invent Original problem: Variations: Changing the case (eg the number) Goal: Find an algorithm for finding fractions between two consecutive unit fractions. Generalizations: Solving sets of cases or all cases Extensions: Changing the “rules” (axioms)
Mathematical Habits of Mind: My version Visualize Uses pictures to describe and solve problems Uses manipulatives to describe and solve problems
Mathematical Habits of Mind: My version Strategize, Reason and Prove Moves from data driven conjectures to theory based conjectures Strategizes about games such as “looking ahead”
Mathematical Habits of Mind: My version Connect Describes problems and solutions using multiple representations Finds and exploits similarities within and between problems
Mathematical Habits of Mind: My version Collaborate and Listen Asks for clarification when necessary Gives others the opportunity to have “aha” moments
Mathematical Habits of Mind: My version Persevere and Reflect Embraces productive failure Can reduce or eliminate "solution path tunnel vision”
All fine and dandy, but… For us: Determine the subgroup lattice of GL(2; 2). For our students:
A Good Problem is… Accessible Minimizes vocabulary and notation Is only as precise as necessary Has multiple entry points Includes ways to collect data Has multiple solution methods Deep Naturally leads to variations, generalizations, and extensions Leads to and connects different aspects of mathematics Motivates developing procedures, vocabulary, notation, and mathematical concepts Can be worked on for as long as you want Captivating Consists of benchmarks along the way where one is reenergized by the feeling of success Could be real world, but doesn’t have to be (and certainly shouldn’t be pseudocontextual) May lead to a surprising result May feel like a puzzle waiting to be solved May be necessary to solve a different, interesting problem (which is not the same as “you’ll need to know this next year”). May be posed by students. Sideways scalable. Assessable in meaningful ways Can be scaffolded 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.
But enough about me… Time to do some math Draw a rectangle on a square grid. An example 9 by 3 rectangle is drawn for you below. Draw one diagonal. How many squares does the diagonal pass through? Develop a rule to determine the number of squares a diagonal passes through for any rectangle of any size. Tasks *Do some math *Think about where students might get stuck & helpful habits that could get them unstuck
Organize & Simplify: Student work
Organize & Simplify (less successful)
Organize and Simplify 12
Generalizing 3D Version
Another good problem What mathematical questions do you have? What are the stated rules? What assumptions are you making?
Comedy comes in threes… along with good math presentations What mathematical questions do you have? What are the stated rules? What assumptions are you making?
The Onion, 11/17/11 What are the costs?
Challenges of Common Core Standards of Practice They are broad and not broken up by grade. Few available resources Easier to implement with good problems. Good problems are hard to find/create. Not how most of us learned the subject Not sure they are valued. Really not sure they’re valued beyond their ability to help students access content. Can take longer to see success/need to redefine success Will we care if they’re not explicitly assessed? If so, how do we assess?
Avery Pickford The Nueva School Mills College Blog: Without Geometry, Life is Pointless @ www.withoutgeometry.com @woutgeo avery@withoutgeometry.com
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