Complexity-Breakout-Group
From Kaput Center Wiki
Thu Nov 20, 1 - 4 pm
Attending:
- Dick Lesh
- Annie Selden
- John Selden
- Fred Martin
- Deborah Tatar
- Jim Middleton
- (scribe) Sara Dalton
Data Modeling and Complex Systems
- you would get at underlying ideas
- great way of bringing in new kids (this is the avenue to bring new kids in)
- viz. and looking at the data.
We started with these observations:
- Curriculum change – what do we want without worrying how to get there
- Consult people other than math educators, teachers e.g. engineering, comp. sci., science
- How to connect with outside of classroom
Then we defined complexity as:
- situations of more than one function
- that involve feedback
- and many have emergent properties
With this approach, you can
- reorder topics
- put proof back on the table
- visualize the analyses
- move back and forth between visualizations and the underlying data.
Regardless of complexity, *data modeling* is an area that begs to be organized for a number of reasons:
- it provides a different avenue of access for students into deep levels of mathematics, including
- bringing proof back on the table
- attention to underlying structure
- when done in a manner than emphasizes modeling problems of importance to key use communities of mathematics,
- like engineering, mathematics, sustainability
- It provides a useful set of 21st century skills that are directly applicable to fields that use mathematics.
- Get kids to think more cleverly about scientific, engineering design, other concepts
Not just about access, also it's likely that children/students of any age that engage in data modeling, will see themselves differently in relation to the mathematics, there is some modicum of utility that they can grasp, can tie to their personal interests, provides a richer variety of solution methods and ways of undrstanding mathematics that links to their prior knowledge and predilections toward problem solving
take each of these things, create a section that deals with given this huge potential, how do we think about curriculum, viz tools, economic power, rigor, application, motivation
- Everything we said had a heavy dose of technology
- We can work with teachers too
Lesh: Purdue had tried to get middle school-ers involved in engineering
Common traits amongst those who are successful in technical/ complex fields – developed “kid versions” of real problems - People who can work in teams - Almost always have radically changing tools that are evolving, describe things so tools can be used - Almost always describing a (complex) system Operationalize a construct like drag or floating paper airplane
Most systems involve more than 1 agent – in math books there is 1 agent. Can be solved with 1 function; not the case in the real world.
Minimalization/ maximization of systems Stabilization of systems
Getting into complexity theory
Not thinking about second or third order effects
Lesh: What math do people need to be developing in order to be successful after school?
Lesh: GIS systems – Interactions amongst various systems Can not be characterized by 1 function Lesh: want to see kids working with more than 1 function Interesting topics intersect more than 1 field – cross topic
What is Complexity?
*Systems *“NP-complete” *Theorem proving *Hyphenated departments *Interactions *Information *Feedback
Jim: organized versus disorganized complexity (Weaver, 1948)
Disorganized: complicated-ness
Organized: what we would call complexity now in complexity-theory sense; get at boundaries of system; could determine some sort of probability; emergent property not entirely predicted by initial variables
Jim: What mathematics do you need for technological fields?
John: take something from real world, abstract it, apply math to it, move it back to real world to see if it holds; modeling is based on
If you do not have abstract def., they will change over time; theorems will change over time – have to stay proved Moving back and forth between real and symbolic worlds
Deborah: what is left out of the model? Industry wants students’ to model. Some things are model-able but not included in the model; increasingly think about the extension of our models; so what is left out of the model (e.g. quality of life)
Uri Wilenski simulations
All lump into one corner of the model matrix – talk to D. Lesh about this matrix
Manipulating variables and what they have to do with the model
1-rule describable
Systemic problems
Feedback & Emergence
STELLA & Swarm vs. NetLOGO & Vensim
John: “non-routine problem” – are these model-eliciting activities?
- Routine problems: like problems at the end of the chapter - Non-routine problems: like you’ve never done before
Domain of complex systems
Lesh: order of modeling versus given a representation
Had a set of ideas to think about and create a modeling V. having a whole and decomposing it
Lesh: Do you first learn something then apply it? Not always.
Interpretation system versus a rule (involve different interpretation systems)
John & Annie: for models – need to see patterns. Can be blind to a pattern if you haven’t gotten the concept
Mathematics: vast realm of concepts and patterns that can be used to see the world
Lesh’s matrix: Systems: discontinuity & irreversibility (catastrophe theory) and resonance (hard for people to understand -> e.g. gulf stream, mental health)
DT: don’t model things might not have tools for modeling or thinking about some things
Repeating patterns versus time scale
Jim:
- Exponential growth (explosive growth)
- Asymptotic decay (somewhat explosive)
- Logistics curve (S-curve; positive followed by negative)
- Cyclic curve
All are non-linear in global geometry; might be linear up close; need some mathematics in your pocket to understand what is going on. Have to bring it to the table; cannot come in without some initial understanding
What math do you need in your pocket? For what you are trying to do?
Lesh: Proportional reasoning -> develops over years If there is a need for it; they will develop it because they partly had it already
Jim: S-curve example. Fluxtion in Newton’s Principia Look at what is happening when sweep out equal area of time as you travel left to right
Lesh: Different math between describing a system that’s “out there” versus systems you are trying to build.
Jim: what is our definition of math?
Deborah: “growth is good” “even numbers are nice”
Lesh: Rates of change – In the past this has meant calculus. It does not anymore. Many computation based approaches have replaced calculus based approaches.
Least squares versus least distance lines to model a set of data. Often very different lines
John: Need to do deductive reasoning, specifically proofs, is not going to go away. Much of the proof is psychological in nature, take specific actions; brainstorm ideas, which may or may not be important for the task at hand.
Lesh: Proof is back on the table; not out of bounds
John: think of it more generally; interested in actions students’ take with proofs -> look where they are going; keep end in mind. Reasoning in the context of doing proofs. People can look at a proof and tell it is right; versus modeling data which is not so easy to tell it is “right”. Not enough math educators focusing on proof.
Jim: students capable of justifying every step in an algebra solution process We know young students can solve these and make sense of them; there are aspects in solving algebraic problems in looking at standards of evidence in data analysis Certain deductive reasoning patterns in developing subsets of the data to analyze the data and look at different representations to understand the underlying picture of the data. Mapping of that back onto the world.
Have symbol systems that we work with and they become ritual – have power to us. Have to be careful about data analysis not to change the data to fit the model.
John: proof as exercise in careful thinking
Jim Middleton: Paradigms defined by methods and analytical techniques Build studies based on an existing model before they’ve got data
Lesh: if we start looking at data analysis and modeling -> proof back on table Sequential order of topics breaks down – data modeling had been out of bounds because of complexity
Jim: Exploratory data analysis Need to know what measurement is – develop into law of large numbers, central limit theorem etc. Lots of ways to organize a stats class – so analysis of variance is not the end goal. It is a small part of understanding rate of expected value to error; test-statistic
What do you expect and how sure are you – the ratio of those two things
Deborah:
10% of stats class is hypothesis testing
90% on complex to accomplish statistical tests; visualization
Should we spend more time on the things we want students to be doing?
Lesh: Visualization is important & we can now do this!
Data Modeling & new mathematical tools now available (back on the table!) Visualization Radically re-order topic Deductive Reasoning/ Proof
Jim: philosophical change; ascribe structure to something; use a model to think about a relationship between two variables.
