Math Relearning/Progressions/High School Modeling
These are my comments on the Common Core Math Progressions document Modeling: High School (PDF).
Modeling in K-12
The Modeling Process
"Complex models are often built hierarchically" - I've wondered. I thought the designers of those were just geniuses.
"nominal dollars ... [vs] constant dollars" - You really do have to understand the context to choose the right model. I don't understand these money concepts, and I'm sure I'd get the model wrong.
Modeling in High School
The Modeling Cycle
"Judgment, approximation, and critical thinking" - This will be a leap in difficulty, so it's important to prepare students for it and guide them through it. I think modeling will help some of them understand that math is useful. Some will still wonder why *they* have to learn math, since they won't be using it for any of these situations. They're content to let experts do the math for them.
Units and Modeling
Modeling and the Standards for Mathematical Practice
"mathematics and statistics" - I wonder why people separate these.
"a capstone experience" - That's my impression. This progression hasn't taught anything new mathematically, just tied together what was taught in the other areas.
"looking for entry points" - Yes, this is how I was thinking of the process.
"reason inductively about data" - This is all the kind of reasoning people have to do in general, not just about math. It's probably worth mentioning this to students. Mathematical modeling is good practice, though I can see some students protesting that they could practice on the non-mathematical problems they actually care about. Math provides some rigor, though, that keeps a learner from fudging. But it's worth thinking through the parallels between mathematical modeling and reasoning in other contexts.
"educed from some context" - I thought this was a typo. I learned a new word.
"technology can enable" - I'm glad the standards don't ignore technology and insist that students do all the math with the power of their minds. Technology exists, and students will use it both as students and in their careers. Plus mathematicians use it.
"the issue of uncertainty" - I'm glad the standards list all these aspects of modeling. It's a complex enough activity that a list will help me feel less overwhelmed.
Modeling and Reasonableness of Answers
"Stat-Spotting" - Sounds like a helpful book.
Statistics and Probability
Developing High School Modeling
"situations that can become more complex" - You know, the Standards might move more slowly in the upper grades and not cover as many concepts as our curriculum did, but modeling sure sounds demanding, and I don't think we did much of it.
Linear and Exponential Models
"a distance d in t hours" - This reminds me that I'd like to learn physics alongside math, partly as a source of math applications to help me think about how math relates to the world. I don't know if I'll want to take the time for all that though. I might put physics off till later.
"comparing quantities and making decisions" - Good point, and it's hard to argue with the fuel-efficient car example. People probably aren't going to bring their car buying decision to their local math expert. Or they should feel bad if they do, if they really could figure it out themselves.
"horizontal intercept ... is the break-even point" - It's interesting to see this everyday example being expressed in mathematical terms and to know they're relevant to the problem. Being technical has a point. I wonder if students would be more interested in math if more of the examples came from everyday parts of their lives like video games. You could model Pokemon and learn interesting things about it that could help your gameplay.
"question the assumptions" - Math gives you a more disciplined way to do this. Many of the assumptions have to be articulated as parts of the equations, so they're easier to notice and vary later.
"learn to question why" - This is a good discussion of the example. It would be good to observe what kinds of questions come up in these discussions. What does math make it easier to ask?
Counting, Probability, Odds and Modeling
"reconcile accounts of probability" - Very interesting and important. Another set of examples it's hard to argue with.
"everyday language and feelings" - Also interesting and important.
"the birthday problem provides rich learning experiences" - Indeed! They get a lot of mileage of it.
Key Features to Model
Formulas as Models
"Formulas are mathematical models" - I've been using formulas as a general term for expressions, equations, and whatever else, so I'll need to pick a different term.
"mulch" - Another good example. You can't always just look mathematical things up or plug them into an app, although you might in this case. But sometimes you have to solve things dynamically, in uncommon situations that arise as life happens and aren't anticipated by software developers. Or instead of searching for an app, you could solve it yourself and feel more in control of the situation. I think those subjective benefits should be emphasized. There's a sense of satisfaction and power in doing math. It might also help to present examples humorously in terms of regret--"if only he had paid more attention in math class."
"used in forensic science" - Interesting. An application of an application of math.
Where the Modeling Progression might lead
"extend simpler models" - Is this circular motion example supposed to be one that can be understood simply at first and then be examined more in depth as new concepts are introduced?