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Facilitators: Eric Robinson Teri Calabrese-Gray

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1 Mathematical Modeling: What it is; What it looks like in the classroom; Why it is so important
Facilitators: Eric Robinson Teri Calabrese-Gray While mathematical modeling is a well-defined (discipline) area within applied mathematics and now appears in undergraduate course offerings in many colleges and universities, it’s meaning and value is likely to be less understood in pre-college mathematics. Now, the NYSCCLS incorporates mathematical modeling as a learning goal throughout the document. Therefore, the goals of this day are to increase understanding the process of mathematical modeling and the role it should play in school mathematics—at the middle/high school levels. This [talk/workshop] is designed to focus on several issues: What is mathematical modeling? What might it look like in school mathematics? What may look like mathematical modeling but is not? Why mathematical modeling is so important for students to understand and be able to model many situations mathematically. Time: 2 minutes including speaker introductions. (introduction includes getting a sense of the audience. Teacher, mathematics curriculum specialists, administrators, etc.)

2 Presentation Overview:
What is Mathematical Modeling? Examples of mathematical modeling problems Summary: What mathematical modeling isn’t. Why mathematical modeling is so important in school mathematics. We will begin with the central issue: what is meant by mathematical modeling. We will introduce a theoretical framing of modeling at the beginning, today, While commentary in this presentation is included, the best understanding of modeling comes by working through examples of modeling problems. So, our own understanding of mathematical modeling as well as what we intend our students to learn though modeling will deepen as we work , analyze, and discuss modeling problems. We will devote the majority of the day to the examples. That said, we have taken care to choose examples so that everyone can contribute something to each example—regardless of specific background. But, this talk does include mathematical content at a variety of levels so, there is a caution that a PD session of this type needs to include an individual with mathematical expertise. Finally, we will summarize our experiences to clarify what modeling and answer two remaining issues: What modeling isn’t and Why it is educationally important. We also want you to have a chance to get a started on creating a mathematical modeling problem based on the experiences you will have. Time: 2 minutes.

3 In the NYSCCLS Mathematical Modeling is:
One of the eight Standards of Practice (that span the grades) One of the Conceptual Categories that span the high school content areas So, let’s continue to define mathematical modeling. As we do, consider the question: Why would NYSCCLS include mathematical modeling as both a practice and a content area? (Note an immediate in the Standards: Mathematical Modeling has multiple meanings!) Time: 1 minute Why is it both??

4 “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 Arguably, Henry Pollak is one of the greatest mathematical modelers of our generation and a superb educator! Here are a couple of introductory quotes. This is information. About Henry Pollak. It doesn’t have to be presented. From Wikipedia: Henry Pollak bio: Born in Vienna, Austria, he since moved to USA. He received his B.Sc. in Mathematics (1947) from Yale University. While at Yale, he participated in the William Lowell Putnam Mathematical Competition and was on the team representing Yale University (along with Murray Gell-Mann and Murray Gerstenhaber) that won the second prize in He earned an M.A. and Ph.D. (1951) degree in mathematics from Harvard University, the latter on the thesis Some Estimates for Extremal Distance advised by Lars Ahlfors. Pollak then joined Bell Labs (1951), where he in the early 1960s became director of the Mathematics and Statistics Research Center. He authored near forty papers, many of these with David Slepian and Henry Landau on analysis, function theory, probability theory, and mathematics education. He has applied mathematics to solve problems in physics and networks, communication theory, discrete systems, statistics and data analysis, and economic analysis. Pollak also holds patents in the area of signalling.[ He [currently holds a teaching position] in the mathematics department at Columbia University. (Teachers College). He is internationally renowned for his modeling ability and an exceptional educator who now spends a great deal of his time educating future teachers. The second paragraph on the slide is very significant. So, the first important characteristic to pay attention to is that: interplay between the real world and math vital. Time: 1 minute

5 “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.” -- Henry Pollak Most often, mathematical modeling starts with a question about the real world and ends with taking action or making a prediction. Modeling brings an important perspective to K – 12 mathematics education in that the real world is not just a context to highlight the value of mathematics; rather: BOTH THE REAL-WORLD…AND MATHEMATICS ARE TAKEN SERIOUSLY.” Again, this is very important—and different from what is currently practiced in most schools. In traditional mathematics education, most often a “real world” problem is a “word” problem done to show that mathematics is valuable . The focus is on the mathematics and the ability to “decode” the words in the problem. And when you find the mathematical answer you are finished. In mathematical modeling we need to reflect upon whether or not our mathematical answers really give us what we want in the real world. Indeed, “mathematics” takes a back seat to considerations about the real word in some modeling problems. That said, the modeling process has much in common with the study of pure mathematics as well. (That is another talk.) Mathematical modeling supplies additional support for the ability of students to use mathematics in their everyday and /or professional lives (including, but not limited to STEM careers.) For clarity, note that, here, we are not using the term “modeling” simply to mean modeling mathematical concepts with concrete objects. (In the latter instance, it would better to use the word “representing” in place of “modeling.”) However, we will mention a roles of concrete objects in mathematical modeling in a while. Nor are we talking about “modeling” one abstract mathematical concept with another abstract mathematical object, here. Again, a better word might be “representing” in the latter case. Time: 2 minutes

6 This entire process is what’s called mathematical modeling.”
The “practice” “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 Quote from NYSCCLS. The practice implies a perspective on a problem, mathematical habits of thinking, and an organizational scheme for thinking about and working toward a solution. Time: 1 minute

7 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 This is an example of a modeling problem from everyday real life (for many of us) to get us started. Let participants work on/think about this for 2 or 3 minutes. Ask them to think about the problem as a current student might. Ask them to think about the problem if they had to get their grandmother. Is the answer just: It will take 20 mi./40 mph = 1/2 hr.? What other considerations do you want to consider? (No, if you apply this conclusion, you probably will be late.) In mathematical modeling problems you (the student) often have to add information (in the form of factors to consider, data or assumptions) to create the actual problem you want to solve. The student must take the initiative. That is, a real world question may be incomplete as far as the information given. (In this sense, it is not like the traditional textbook application which usually contains all the information necessary.) What other factors are important in the example given here? It might be possible to include too many factors. You may have to eliminate deep consideration of some of the factors as you plan your trip. You may also introduce simplifying assumptions that can be adjusted later. Time: 7 minutes

8 (Valid) Mathematical results
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? Revise Compute process deduce Interpret Formulate Real World Mathematical Model Real World Conclusions Modeling Paradigm This is one scheme for how the Mathematical processing pieces work together. We are going to use it heavily in our work today. You have it in your packet. The process begins by identifying a question in the real world. The first “step” in this diagram is to formulate the problem (i.e. “mathematize” the question and list mathematical characteristics). This decontextualizes the problem. NOW, YOU CAN’T LIST EVERYTHING (in most real world problems). That might make everything to complicated and/or some things you list may be irrelevant. So you are going to have to make judicious choices concerning what to address in this model. The first step leads to the upper right box where the real world situation is “mathematized.” Note the inclusion of simplification as we work on problems. The second step is to solve the mathematical problem. This can include work in pure/abstract mathematics although some reference to the context might appear in explanations—foreshadowing step three below. ). Some typical signatures of this step are the construction and/or implementation of algorithms or formulas chosen/created by the student and/or deductions made from the assumptions. Multi-representations are usually considered as part of the model building. Choice of mathematical disciplines to be used appears in the model building. A traditional textbook application of mathematics GIVES the student most of the material in the upper right box and then ENDS at the lower right box. As such, a traditional application corresponds to a part of modeling—but not the whole process. Using the entire modeling process helps students build flexibility in their use of mathematics. (This does not all traditional examples need to be eliminated. However, it does suggest that these examples may be viewed as part of the modeling cycle.) The third step is to interpret the results in the real world context: Does the model work to give you an answer that makes sense. Can you test it for real, in a simulation using technology or concrete representations, or measure it against some criteria? The model is revised in a fourth step (loop) if the results do not make sense when compared to some criteria; or you want to generalize, remove simplifying assumptions, etc. Important note: When you actually work through problems, the steps are not necessarily completed in linear order. E.g. as you start to compute you might find that you need additional information so you go back to the formulate step. Or, as you compute you begin to interpret the results before you are finished to check if you should keep going. Apply the graphic to the warm up problem. Time: 8 minutes

9 Note that the Formulate and Revise components are about “Problem Posing.”
Modeling requires both decontextualizing the problem ( moving it into the domain of mathematics) and contextualizing back to the real world. It is vital that students understand the entire cycle. They need to understand the fact that they may only be doing “pieces” of the modeling cycle in some classroom (or professional) situations. This is OK, but, they need to understand where each of the latter-type problems fits in the modeling cycle. Time: 1 minute. Real World connection “Inside Mathematics”

10 Interpret: Contextualize mathematical results and see if the model results make sense or works (e.g. if the results satisfies certain criteria). 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. Here are some definitions the words “interpret” and “revise” as they are used, here. Again, interpreting can be done using Thought experiment (minimal—but applicable to all problems) Simulation (Often requires the use of technology, but could also involve concrete objects as well.) Implementation (may not work until much later if the result was a prediction future prediction. Also note that real models may have a limited scope of influence. E.g. Population models may need to chance over time.) Revise-Validate is like self assessment: It involves both an accurate mathematical analysis and successful interpretation in the real world. Time: 1 minute. We also want to talk about the word “validate” because it is used in NYSCCLS in reference to modeling. The intent in the latter document is to make “validate” an active verb meaning consideration of the question: “are the results good enough to accept or do you need to revise?” (This subsumes a bit of what we and others have listed in the definition of “interpret” above. With either word, the intent is to BUILD IN THE NEED TO REFLECT ON THE SOLUTION in context. This is just good mathematical practice whether we are talking about a real world context or an abstract context in pure mathematics.

11 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. Modeling Practice Standard 4 as stated in NYSCCLS together with the :NYSCCLS graphic (pg. 61): The practice text clearly refers to the modeling paradigm graphic that you have. In the graphic from NYSCCLS, there are two ends: initial question and the final report (precise and often implies action or a prediction.) Both are important. The four-step cycle is in the middle. We mostly are going to concentrate on the four-step cycle. But, don’t want to forget the importance of the beginning “problem.” In the real world, the original situation could be stated vaguely as a “leading question” as in “ which route is best to take?” (Are you trying to avoid toll roads?) Indeed, it might be better to call this the modeling “question” to emphasize the fact that the student participates in the “problem formation.” (Remember the Einstein quote you saw yesterday.) Formulating the problem sometimes involves honing in on a more specific modeling question. Time: 2 minutes

12 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 This language from NYS CCLS is trying to get to the idea that modeling usually involves multiple content standards and the modeling cycle (during the “practice” of modeling”). Time: 1 minute

13 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. ★ Now, we talk about the reason some content standards are designated as “modeling standards.” (I.e. why modeling is also highlighted as a conceptual category.) Modeling-designated conceptual category content standards in NYSCCL are useful in problems that refer to any or all components of modeling. However, as the previous slide suggests, it is not sufficient for schools to implement modeling ONLY by focusing on the discrete set of content standards. Also note that some non-modeling-designated standards may fit into modeling problems. The “starred” standards signify most likely candidates. Time: 1 minute

14 Standards for Mathematical Practice
Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. You will note that in work on the modeling problems, many standards for Mathematical Practice appear frequently in modeling problems. We will be analyzing the appearance of the mathematical practices in modeling problems—using the “long form” of the practices (the statements together with the paragraphs under the statements. You should have copies of these.) Time: 1 minute

15 Storm Models
Both European and American scientists made models to predict the path of super storm Sandy. The European models came closer to the actual path of Sandy. Did the American mathematicians make wrong calculations with early models of Sandy? [No. This was not the issue.] (This is a HDTV screen shot quickly with a phone camera that shows European models and American models of Sandy as it was moving up the coast during the week before it hit. Again, the European model was more accurate! Why? Everyone had access to the same data. (Ans. modeling characteristics.) The point is that different models of real world problems are sometimes used.

16 Thanksgiving Table Example: 6th grade
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. Let’s look at a 6th grade modeling example in the classroom and analyze it using the modeling paradigm graphic. We also want to pay attention to the content and the standards for mathematical practice that appear. There are three groups of questions here: Considerations about the modeling question, how the mathematical work fits in the modeling cycle, and how the activity fits the NYSCCLS. Time: 2 minutes to introduce the questions and start the video; See discussion of questions in the Activity Notes document.

17 PISA –Like Assessment Item
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 5,000 C 20,000 D 50,000 E 100,000 This is a released PISA field test problem from several years ago. You will probably remember PISA from you last workshop. Recall that PISA is and international exam given to 15 yr. olds. PISA is an acronym for Programme for International Student Assessment Take a couple of minutes to think about this at your table. Give the answer as if YOU were asked the question. [Your TABLE can be YOU.] Also answer it as you think students might. What choice did YOU make? What choice would you think most students would make? Why? Get participant responses. Students most probably just multiplied the given dimensions together and looked at the answers—selecting B as the correct answer. The acceptable answer was C: 20,000—which makes sense if you bring in some information based in the real world: The term “packed” needs to be made more precise. It suggests that each square meter would include as many non-overlapping people as possible. Second, the size of the people has to be settled. A simplifying assumption might be that all people are the same size. For example, using “my size” of about 2 ft. wide and 1 foot thick as the size of a person, about 4 people can fit into a square meter. (Thus, a reasonable answer is 4 X 5,000 = 20,000 . You could fit in a few more people using the leftover holes in a square meter depending on the arrangement of the people—but it would not raise the total to close to 50,000 people. ) Time: 5 minutes to introduce problem and have participants think about question. 3 minutes to report out. Total: 8 minutes.

18 Activities: Fun, Fun, Fun
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? To really understand modeling, we can’t just talk about it. You have to experience it. So, most of the rest of the day will be spent working on and analyzing mathematical modeling problems. In the problems you will be working on we want you to analyze the work you do in terms of the modeling cycle using the graphic of the cycle given previously. (Refer to the graphic on slide 8.) In addition, after you have worked, we will ask you which NYSCCLS content standards were addressed in your work and which standards for mathematical practice appeared. Some of the problems will concern the entire cycle. Others were chosen to highlight specific components of the cycle. Other questions will be raised in the individual problems. There may be more than one way (or one model) to address the real world question posed. In fact, even though some of the problems are described as relating to a particular level, think about ways you might approach a similar problem at your teaching grade level. We will give wide berth to ways of tackling the modeling problems and formulating models. However, it will be important that your work and conclusions conform to the model you develop. It is recommended that, in many circumstances, classroom students should be given such wide berth in creating models. If you give such students opportunity to share their reasoning and critique the reasoning of others, another student might suggest the approach using the knowledge you wanted. Or, after students have shared their answers, you, as teacher, might ask students if there is a model that uses the knowledge you want them to use in a model. Such an approach often serves as a way of making mathematical connections between topics and/or deepens conceptual understanding. Refer to Activity Notes for discussion of activities after participants have worked on them. Time: 5 minutes

19 “Math Class Needs a Makeover”
Speaker: Dan Meyer Question to consider while watching: What is the role of mathematical modeling in the suggested “makeover?” After seeing the video, ask? The substance of this talk is broader than modeling—but modeling plays a big role in accomplishing the “makeover.” Get responses. Time: 2 minutes

20 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. (e) Solutions to modeling problems usually suggest actions or predictions. Now we summarize things that have been said (with a couple of additions) and answer the three questions in the title of the talk: What is mathematical modeling? What might try to pass for modeling, but isn’t? Why is mathematical modeling so important? d.) Different models can be created from different perspectives. E.g. use of statistics (or simulation) or use of theory only. When building functions. Either way, good reasons for assumptions should be given. Asking students why they make certain assumptions is important. Time: 3 minutes

21 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. Time: 30 seconds.

22 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. Pre high-school modeling is important. First, mathematical modeling is a practice that extends throughout the grades. Second modeling needs to be phased in: Consider the scaffolding in the first video. Skill needs to build in each step of the modeling cycle as more problems are encountered. The goal in later grades would be to (sometimes) simply state the modeling question and let the students have at it. We want to transition from giving everything needed to getting students to formulate the problem, process the mathematics, and consider the results as they are (or might be) applied. We want students to understand how mathematics models real world situations (E.g. Models don’t include “everything” about the real situation and often use simplifying assumptions.) We want to support student initiative and provide more ways student can participate in the “creation” of problems—not just canned processing of problems. See the Dan Meyer’s video. Time: 1 minute

23 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. Mathematical modeling supports literacy and usefulness of mathematics as well as content and mathematical reasoning. E.G issues sustainability or eradication of invasive plants. Time: 1 minute

24 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. (g) You saw this in the first video looking at the relationship (or non-relationship) between perimeter and area Student learning is deepened if students can attach new learning to something they are familiar with. Further note that there are many careers in modeling. Time: 1 minute

25 Resources mentioned today:
Teachers College Mathematical Modeling Handbook, COMAP Inc ( Mathematics Modeling Our World (MMOW); COMAP Inc ( NCTM Reasoning and Sense Making Task Library NCTM Focus on Reasoning and Sense Making series Illustrative Mathematics Project These resources are mentioned in the modeling Activities participants do. The quotes from Henry Pollak are from the first reference. Time: 1 minute

26 Sample modeling questions:
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) Numbers refer to which references on the previous slide include reference to such problems. Note one more thing. U.S. population growth was nicely modeled by an exponential function from 1790 to about 1850, but then the model failed. (Why?) Here is an example where you do the revisiting much later. For the “truck problem, see NCTM’s Focus on High School Mathematics: Reasoning and Sense Making as well as Focus on High School Mathematics: Reasoning and Sense Making in Geometry. The latter volume goes through the “loop” and revises the problem by removing one of the simplifications. Time: 1 minute

27 Presentation Take-aways:
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. The major goals if this day were to gain a deep understanding of the modeling cycle both theoretically and in term of what we are asking students to do. This is all new to most school teachers. This is about changing culture with regard to what students are being asked to do. They are asked to bring their initiative to formulating problems. In the best modeling problems they are the ones to decide upon in terms of the mathematical assumptions and strategies they will use to solve the problems. That is what happens in real life. They will be the ones to determine if they did a good job when they interpret their results in the real world. They will become independent problem solvers. You will be purveyors of this understanding to colleagues. Talk is cheap. The best understanding of modeling comes by working through examples of modeling problems. Finally, with regard to changing culture in the classroom, it is important to be able convey how modeling differs from what may be done currently i.e. What modeling isn’t. It is imperative to understand why modeling is educationally important. We want to do the right thing as we prepare students to be literate citizens and successful in college and in professional lives. I hope this day has been beneficial. It is up to you to carry the message. Time: 1 minute

28 (Valid) Mathematical results
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? Revise Compute process deduce Interpret Formulate Real World Mathematical Model Real World Conclusions Modeling Paradigm A take-away: This is one scheme for thinking about components of the practice on mathematical modeling. This helps in both the creation of modeling problems and the analysis of such problems. Time: 30 seconds

29 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.” If there is time.

30 Thank You!

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