Mathematical Modeling: What it is;

What it looks like in the classroom;

Why it is so importantFacilitators: Eric Robinson

Teri Calabrese-Gray

What is Mathematical Modeling?

Examples of mathematical modeling problems

Summary: What mathematical modeling isn’t. Why mathematical modeling is so important in school mathematics.

Presentation Overview:

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• One of the eight Standards of Practice (that span the grades)

• One of the Conceptual Categories that span the high school content areas

Why is it both??

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In the NYSCCLS Mathematical Modeling is:

“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…”

--- Henry Pollak

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“When you use mathematics to understand a situation in the real world, and then perhaps use it to take action or even to predict thefuture, both the real-world situation and the ensuing mathematics are taken seriously.”

-- Henry Pollak

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“Mathematical modeling begins in the unedited real world, requires problem formulation before problem solving and once the problem is solved, moves back into the real world where the results are considered in their original context. Are the results practical, the answers reasonable, the consequences acceptable? If so, great! If not, take another look at the choices made at the beginning, and try again.

This entire process is what’s called mathematical modeling.”

-- Henry Pollak

The “practice”

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Warm Up

Your grandmother will be arriving at the airport at 6:00 pm. You live 20 miles from the airport. The speed limit is 40 miles per hour. When should you leave to get her?

-- Henry Pollak

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Modeling Paradigm

Mathematical Conclusions

Clearly identify situation

Pose (well-formed) question

List key features of situation

Include assumptions and constraints

Simplify the situation

Build math model : (strategy, concepts, data,, variables, constants, etc.)

(Valid) Mathematical results

Apply:

Do results:

make sense?

satisfy criteria?

Are results sufficient?

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Interpret

Formulate

Real World Mathematical Model

Real World Conclusions

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Real World connection“Inside Mathematics”

Note that the Formulate and Revise components are about “Problem Posing.”

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Revise: • Because the results do not seem to fit what

does/would actually happen.• You want to generalize your results thus far and

this might affect your modeling approach.• You want to remove some of the simplifying

features and/or add other features.

OR Validate: You decide this is good, accept the model results and write a report.

Interpret: Contextualize mathematical results and see if the model results make sense or works (e.g. if the results satisfies certain criteria).

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4. Model with Mathematics Mathematically 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 situations…. 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.

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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

The (High School) Conceptual Category

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Examples of Content standards and modeling:

Algebra: Seeing Structure in Expressions A-SSE

Interpret the structure of expressions1. 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. ★

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1.Make sense of problems and persevere in solving them.

2.Reason abstractly and quantitatively.

3.Construct viable arguments and critique the reasoning of others.

4.Model with mathematics.

5.Use appropriate tools strategically.

6.Attend to precision.

7.Look for and make use of structure.

8.Look for and express regularity in repeated reasoning.

Standards for Mathematical Practice

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Storm Models

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Thanksgiving Table Example: 6th gradehttps://vimeo.com/46127286

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Questions to consider as you watch the video: Is the attention to the real world realistic?Is the modeling question clearly stated? Is the modeling question phrased in a way that there value to answer in the minds of students?

How does the material in the video fit with the modeling cycle? (Refer to the modeling cycle graphic on the handout.) Specifically,• What work/information/evidence, if any, which is in the video would you put under

“formulation?” step in the cycle? Who is providing the information, the teacher or the student? (Refer to the top right box in the graphic.)

• What work/information/evidence, if any, would put under the “compute/process/deduce” step in the cycle? (Refer to the lower right box in the graphic)

• What work/information/evidence, if any, would you put in the “interpret” step of the cycle? (Refer to the lower left box in the graphic.)

• What work/information/evidence would you put in the “revise” step in the cycle?

What content standards are evident in the student activity? Give evidence.What mathematical practices are evident as students work? Give evidence.

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Rock Concert

For a rock concert a rectangular field of size 100 m by 50 m was reserved for the audience. The concert was completely sold out and the field was packed with all the fans standing.

Which one of the following is likely to be the best estimate of the total number of people attending the concert?

A 2,000 B 5,000C 20,000 D 50,000 E 100,000

PISA –Like Assessment Item

Activities: Fun, Fun, Fun

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Questions common to all Activities:

Is the attention to the real world realistic?

Is the modeling question clearly stated? Is the modeling question phrased in a way that there value to answer in the minds of students?

How does the work on the activity fit with the modeling cycle? Be specific.

What content standards are evident in the student activity? Give evidence.

What mathematical practices are evident as students work? Give evidence.

Could you use or modify this problem for the grade level at which you teach?

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“Math Class Needs a Makeover”

Speaker: Dan Meyer

http://www.youtube.com/watch?v=NWUFjb8w9Ps

Question to consider while watching:

What is the role of mathematical modeling in the suggested “makeover?”

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.

(d) Different models of some problems are viable.

(e) Solutions to modeling problems usually suggest actions or predictions.

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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.

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What Mathematical Modeling is not.

(a) 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 .

(d) 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.

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Why mathematical modeling is important.

(a) Modeling serves many everyday situations.

(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.

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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 consonant with what we know about student learning.

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Resources mentioned today:

1. Teachers College Mathematical Modeling Handbook, COMAP Inc. 2011 (www.comap.com)

2. Mathematics Modeling Our World (MMOW); COMAP Inc. 2010 - 2012 (www.comap.com)

3. NCTM Reasoning and Sense Making Task Library http://www.nctm.org/rsmtasks/

4. NCTM Focus on Reasoning and Sense Making series

5. Illustrative Mathematics Project http://www.illustrativemathematics.org/

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• Where do you put the fire station? (2)

• Why do trucks get stuck going under bridges?(4)

• Is the testing of pooled blood samples an effective technique for detecting which athletes are using drugs? (2)

• When will the Moose population in the Adirondacks reach the right size? (2)

• You just won the “Gasoline for Life” prize. Should you take the option of a lump sum of $50,000 instead? (3)

• When should you fill the bird feeder? (1)

• Can you move that huge sofa around the hallway corner? (1)

Sample modeling questions:

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Answers to: what is Mathematical Modeling?

Examples of mathematical modeling problems intended for students that illustrate all or part of the modeling cycle

What mathematical modeling isn’t. Why mathematical modeling is so important in school mathematics.

Presentation Take-aways:

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Modeling Paradigm

Mathematical Conclusions

Clearly identify situation

Pose (well-formed) question

List key features of situation

Include assumptions and constraints

Simplify the situation

Build math model : (strategy, concepts, data,, variables, constants, etc.)

(Valid) Mathematical results

Apply:

Do results:

make sense?

satisfy criteria?

Are results sufficient?

Re

vis

e

Co

mp

ute

p

roce

ss

d

ed

uc

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Interpret

Formulate

Real World Mathematical Model

Real World Conclusions

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Framing you own Modeling Cycle Problem(s): Ways to start

Create a new modeling cycle problem from an interesting real world question.

Make a list of potential contexts for modeling.

Make a list of real world questions.

Create a modeling cycle problem/activity from a favorite “application.”

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Thank You!