Mathematical Modeling. *What it is? *What does it look like in the classroom? * Why it is so important?. Welcome to the RCSD Learning Technology Grant. Year One Mathematical Modeling Using Handheld Technology-TI84 Exploration of Functions Statistical Plots Science Probes Year Two
*What it is?
*What does it look like in the classroom?
*Why it is so important?
The price of copper fluctuates. Between 2002 and 2011, there were times when its price was lower than $1.00 per pound and other times when its price was higher than $4.00 per pound. Copper pennies minted between 1962 and 1982 are 95% copper and 5% zinc by weight, and each penny weighs 3.11 grams. At what price per pound of copper does such a penny contain exactly one cent worth of copper? (There are 454 grams in one pound.)
“Math Class Needs a Makeover”
Speaker: Dan Meyer
“Mathematicians are in the habit of
dividing the universe into two parts:
mathematics, and everything else, that is,
the rest of the world, sometimes called
“the real world”.
People often tend to see the two as
independent from one another
– nothing could be further from the truth…”
“When you use mathematics to understand a situation in the real world, and then perhaps use it to take action or even to predict the
future, both the real-world situation and the ensuing mathematics are taken seriously.”
In the NYSCCLS Mathematical Modeling is:
Why is it both??
Standards for Mathematical Practice
4. Model with MathematicsMathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace…. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation…. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
List key features of situation
Include assumptions and constraints
Simplify the situation
Build math model : (strategy, concepts, data,, variables, constants, etc.)
Clearly identify situation
Pose (well-formed) question
make sense? If not REVISE
satisfy criteria? If not REVISE
Are results sufficient? If not REVISE
The (High School) Conceptual Category
Modeling Standards: Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★)
--NYSCCLS pg. 62
Examples of Content standards and modeling:
Algebra:Seeing Structure in Expressions A-SSE
Interpret the structure of expressions
1. Interpret expressions that represent a quantity in terms of its context.★
Functions Interpreting Functions F-IF
Interpret functions that arise in applications in terms of the context.
4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. ★
Thanksgiving Table Example
Problems, Problems, Problems
Fun, Fun, Fun
Nola was selling tickets at the high school dance. At the end of the evening, she picked up the cash box and noticed a dollar lying on the floor next to it. She said,
“I wonder whether the dollar belongs inside the cash box or not. “
The price of tickets for the dance was 1 ticket for $5 (for individuals) or 2 tickets for $8 (for couples). She looked inside the cash box and found $200 and ticket stubs for the 47 students in attendance. Does the dollar belong inside the cash box or not?
Summary regarding what mathematical modeling is.
(a) Problems in which both the real world and mathematics are taken seriously. With modeling problems, student need to think about both the real world and the mathematics.
(b) Modeling is about problem posing as well as problem solving.
(c) Modeling is often open-ended requiring decisions about what assumptions, information and simplifications are to be included.
Different models of some problems are viable.
Solutions to modeling problems suggest actions or predictions.
And maybe most importantly:
(f) The practice of modeling includes a multi-step process: Formulating the problem, building the mathematical model, processing the mathematics, interpreting the conclusions, and often revising the model before writing a report.
What Mathematical Modeling is not.
Just a fancy name for traditional textbook applications.
(b) An incidental context for the teaching of the decontextualized “mathematics.”
(c) Accomplished by simply “covering” the NYSCCLS content standards that are marked with a .
A learning goal that can be accomplished without student understanding of the modeling cycle.
(e) Only possible if you know a lot of complicated math.
Why mathematical modeling is important.
(a) Many everyday situations serve as modeling contexts.
(b) Some entire careers revolve around a single modeling problem.
(c) Eliminates questions regarding “what good is this stuff?”
(d) Standards from multiple mathematical domains (and multiple grade levels) can occur together in modeling problems. This serves to make connections between mathematical content.
(e) It fosters flexible (mathematical) thinking and use of concepts.
(f) Full scale modeling often engages many of the Standards for Mathematical Practice.
Additional important reasons:
(g) Modeling serves as an environment that promotes deeper understanding of concepts.
(h) Modeling problems provide context for the application of mathematics students know. In addition, such problems sometimes serve as a context to introduce new concepts in a meaningful way.
(i) It is consistent with what we know about student learning.
Resources mentioned today:
Teachers College Mathematical Modeling Handbook, COMAP Inc. 2011 (www.comap.com)
Mathematics Modeling Our World (MMOW); COMAP Inc. 2010 - 2012 (www.comap.com)
NCTM Reasoning and Sense Making Task Library http://www.nctm.org/rsmtasks/
NCTM Focus on Reasoning and Sense Making series
Illustrative Mathematics Project http://www.illustrativemathematics.org/