Modeling with Mathematics Training for Grades 6-8Thursday, February 4, 2016 from 4:00-5:00

Presenter: Amanda Mix

“Mathematics devoid to meaning is empty” (pg. 1).

“Problem solving, modeling, and application must be embedded throughout the process of student learning” (pg. 3).

A Shift in Thinking:There is a pressure to perform on high-stakes tests. As a result, teachers use their class time to ‘cover the standards’ and prepare students for testing. This preparation takes away a great amount of time for true problem solving, investigation, or modeling tasks (pgs. 3-4).

What is Mathematical Modeling?a. The National Council of Teachers of Mathematics (NCTM) says, “students are expected

to use mathematical models to represent and understand quantitative relationships (pre-K to 12)” (pg. 3).

b. The Common Core Standards for Mathematics (CCSSM) for K-12 students, “emphasizes a students’ ability to apply mathematical tools to solve real-life problems and to analyze his or her solution to determine whether it makes sense in context” (pg. 3).

Why Modeling? a. Mathematical modeling is a mathematical practice, not just a type of word problem (pg.

10).b. It increases student engagement, depth of understanding, and opportunities for

investigation, contribution, and the success for all learners (and helps)…develop a positive disposition toward mathematics (pg. 10).i. Engagement: (pg. 10)

Students learn to question, problem-solve, and investigate. Students need to understand that their daily life involves and requires

mathematics.ii. Deep Mathematical Understanding and Flexibility: (pg. 10-11)

Students are able to perform better on non-routine and traditional assessments.

iii. Confidence: (pg. 11-12) Mathematical modeling is a very effective for students who have a

history because a good modeling task will be based in real-life experiences.

All students have the ability to contribute due to their prior knowledge. Students, who mathematically struggle, will become empowered and

motivated to contribute and share their thinking.Mathematical Autonomy (pgs. 12-13):

a. Autonomy is key component in student motivation. When students are self-motivated, they are excited to participate in their own learning for learning’s sake.

b. A student, who is mathematically autonomous, will:i. Decide which tools/approaches are appropriate for the problem.ii. Use a variety of representations to investigate/solve a problem.iii. Decide whether an argument makes sense and is reasonable in the

context of the problem.iv. Justify the appropriateness of their solution (explain why it makes sense),

and persuade the group of their solution.v. Listen and comment on the work of others, and present their work.vi. Decide what further investigation is interesting/necessary.

c. A classroom, supporting mathematical autonomy, has (pgs. 13-14):i. A physical arrangement encouraging student collaboration.ii. Problems that promote student engagement (provides opportunities for

various approaches).iii. Appropriate manipulatives/instructional materials available for student

exploration.iv. Time to explore various of approaches/representations.v. Time for students to share work.

d. A teacher, supporting mathematical autonomy, will (pg. 14):i. Provide a classroom culture of respect for others.ii. Ask directed questions that encourage student thinking rather than

provide solutions or specific direction.iii. Spend more time listening to student questions/reasoning rather than

lecturing/directing.iv. Assure all students that their ideas are works sharing.v. Encourage students to be intrinsically motivated (not motivated by

pressure, deadlines, threats, or rewards).vi. To present tools, show when and how they are used, and then provide a

context in which students can choose the appropriate tools for a given problem.

“When a student believes he or she is truly developing as a math expert; when a teacher is willing to share decision making and provide meaning in lessons; and when a

classroom culture supports discovery, investigation, and different approaches and ideas and recognizes the contributions and approaches of all, students are more likely to dive into a problem and less likely to defer automatically to the question, ‘What do

we do now?’” (pg. 14).

e. How do we know a good (rich) modeling task from a bad one? (pgs. 16-22)i. It provides interest, motivation, and challenge for all learners.ii. It will not encourage struggling students to opt out, or proficient students

to take over.iii. It will provide challenges and extensions for more advance learners.iv. It involves real-life student experiences (authentic, “really real”) (pg. 17).

The challenge is to incorporate interesting tasks/scenarios, even if they are not “really real.”

Connections with familiar: objects, situations (patterns, poems, stories) connecting math concepts to the student’s world.

Interesting tasks promote the realization that the concepts students are learning are not completely separate from their lives outside the classroom (pg. 19).

v. It provides a variety of approaches/representations.vi. It encourages collaboration and discussion.vii. Students gain new insights/perspectives from their peersviii. Struggling students gain confidence in their ability to problem solve when

explaining their reasoning on a problem (pg. 21).ix. It sparks curiosity and promotes decision-making.x. It encourages creativity, individuality, and variety in knowledge

application.xi. It provides extended learning opportunities/challenges for advanced

learners…without the pressure to “hurry up and finish” (pg. 22). An excellent extension asks students to write their own, similar

problem and demonstrate that it works.xii. It (rich modeling) is often the missing piece in problem-solving

experiences in the classroom.

Word Problems, Problem Solving, and Modeling: (pgs. 4-5)a. Misconception: Teachers view mathematical modeling as showing students how to

approach and solve a problem. b. According to NCTM and CCSSM, the key word is “modeling WITH mathematics.”

i. The teacher is the facilitator of the process.ii. The modeling is done primarily with the students.

c. Standard word problems in school curricula do not model realistic problem situations or problem solving (pg. 5).

d. “Modeling presents students with realistic problem-solving experiences requiring strategizing, using prior knowledge, and testing and revising solutions in a real context” (pg. 5).

Cognitive Levels of Math Application (1 is the Highest, 4 is the Lowest) (pg. 5)1. Modeling2. Problem-Solving3. Word problems4. Computation problems

The Process of Modeling:1. Investigation and Problem Identification: “This first step in the modeling

process is probably the most difficult and the most important” (pg. 33-34). a. “Students consider all the information in the problem and decide what is most

important, less important, and unnecessary” (pgs. 7-8).i. Students make decisions about materials they need (tools).ii. The students should be able to (pgs. 33-34):

Restate the problem. Understand the problem.

iii. Students decide the methods (math skills) they may use to solve the problem (the approach).

iv. Students will access prior knowledge necessary for their work.

v. Students decide what they still need to investigate.b. “Modeling is challenging because we are now asking our students to take

responsibility for how they will investigate and solve the problem” (pg. 33).

Getting Started: Student AutonomyProblem with the Traditional Math Classroom (pg. 34):

a. Students are accustomed to practicing endless exercises on a given concept, and then solving a couple of word problems with the same algorithm/strategy.

b. Students, when using to step-by-step instructions and specific rules, have a difficult time stepping out on their own.

c. Teachers instructing only one right answer or method to a problem (“It’s my way or it’s wrong.”).

d. Teachers only answering the questions students ask.e. Teachers having the philosophy that “we are only trying to solve the problem.”

What is Necessary in the Math Classroom (pg. 34):a. Rich modeling problems ask:

i. Which strategy are we using here? ii. What do we do first? iii. What are we trying to find? iv. Go from learning math, to DOING math. v. Encourage students to develop mathematical autonomy. vi. It is not about one right answer and one way of doing the problems.

b. The Teacher’s Role: Exploration and Finding a Pattern (pg. 35)i. Once teachers show that there is more than one way to view a problem, and

that all ideas are respected and considered, the room will “exploded with suggestions” (pg. 35).

ii. The teacher’s role as facilitator is vital, as students begin investigating a non-routine modeling problem.

iii. The teacher must pose guiding questions. “We must remember we are not just solving this one problem-we are trying to create critical thinkers who are capable of taking the initiative when presented with real problems” (pgs. 34-35).

iv. “Teachers, when asking directed questions, should point students in the right directions without taking away the power (students) gain from decision-making and risk taking” (pg. 38).

v. When students ask, “What are we supposed to do?” the teacher should ask students questions in return: (pg. 38)

Have you read the instructions? (not in a sarcastic way) Tell me what the problem is about. What are you supposed to

find out? Can you restate the problem? Can you make a sketch of the problem? Can you represent the problem using manipulatives? Have you seen another problem like this one?

Ask students to restate the problem and describe the task after students are given time and opportunity to reconsider the problem.

vi. Teachers need to emphasize to students that any (mathematically valid) solution for which they can make a strong argument is “correct.” This flexibility in thinking, and departure from the idea of only one correct solution, encourages, and motivates students.

vii. Not all modeling problems have several solutions, but the process always has opportunities for different approaches and diversity of thinking.

Helping Students Get Started:1. The teacher will provide wait time for student thinking (pg. 45):

a. Allow students time to investigate the problem.b. Allow students time to discuss strategies for solving the problems.c. Allow students time to plan the actual problem solving.

2. The teacher will base his or her own questions on the student’s own ideas. 3. The teacher will identifying more problem examples:

a. Old MacDonald’s Farm (pgs. 40-41)b. 1,000 Paper Cranes (pgs. 41-44)

4. “The teacher will allow group discussions and directed questioning to encourage mathematical autonomy. Students’ ideas are used to clarify and refine the monitoring and reflecting process” (pg. 45).

5. “It is important for the students and the teacher to get the first steps of mathematical modeling problems right. When students, with the teacher’s guidance, understand what the problem is asking-what information is important and what is not-and can formulate a plan to solve the problem, their path to completion of the task is smoother and more direct” (pg. 45).

2. Mathematical Foundation of the Model and Data Collection: This is where many teachers hit the proverbial wall” (pg. 46):

a. Students begin to collect data and formulate the mathematical model.b. Teachers must start slowly when introducing modeling with plenty of support and

supervision (pg. 47): i. Teachers need to provide hints and guided questions. ii. Teachers need to remind students of the different tools they have in their math

toolkits (various approaches), and provide opportunities for multiple representations.

iii. Start students on tasks with require fewer tools/concepts that are comfortable and familiar.

iv. Let students experience success and gain confidence on these problems. Then, gradually, require more of your students and provide less specific information and fewer suggestions.

v. “If teachers progress slowly and patiently through this ‘letting go’ process, you will produce autonomous leaners who are capable of formulating a model, making decisions regarding problem solving, and incorporating a variety of techniques and representations into their models” (pg. 47).

c. Formulating a Model:i. “If we ask students to formulate a model for a given task, we are guiding them to

use mathematics to represent a problem in the real world” (pg. 47).ii. Ladder of Abstraction (Concrete-Representational-Abstract Model) (pgs. 48-

49): “The objective in formulating a model is to find the mathematical representation that provides the most useful solution” (pg. 49).

d. Using Multiple Representations:i. Multiple representations: A variety of methods to describe what is happening

in a math problem (sketch, graphs, pictures, diagrams, tables, equations, symbols, concrete models, and verbal descriptions).

The Barbie Bungee Problem (pgs. 61-63) The 1,000 Paper Cranes Problem (pg. 63-70)

ii. Benefits of Multiple Representations: Multiple representations help students understand the characteristics of

a problem. Students gain deeper, more flexible knowledge about the problem. “When students presented their results in a variety of ways, they found

that they were not only providing a clear explanation of their reasoning and solution, but had a better understanding of the connection between the equation, the picture, and the actual problem they were solving (table and guests)” (pg. 103).

iii. Earliest Stages of Multiple Representations: “Early in the process students may not consider the various ways in

which they might represent data or solutions, so it is important to remind them of the representations available to them” (pg. 101).

“When presented with rich modeling tasks, students may become overwhelmed with information and decisions. When student are communicating and implementing the solution to a modeling task, they begin to understand that different representations convey ideas in different ways and appeal to different learning styles” (pg.101).

When students ask, “What do we do?” encourage them to use whichever representation makes sense to them, and see if they can make progress.

“Abstract is not the final goal. By moving back down the ladder of abstraction, moving from the highest abstract to the concrete, students gain a deeper understanding of the process of abstraction and the power of generalization and algebraic thinking” (pg. 55).

“In the beginning, most basic version of mathematical modeling, we might be very specific about requiring students to represent a problem in several ways” (pg. 50).

“As students gain confidence… (in)using a variety of representations, they will become more autonomous and better able to decide which representations would work best for a particular problem” (pg. 56).

“As students gain more experience with independent work and develop a stronger sense of mathematical autonomy, you can provide students with less direction/specificity” (pg. 57).

“When the representations are completed, students can analyze the various data in order to make decisions about solving the problem. They

may discuss the problems and possibilities of each representation, and decide which they prefer and why” (pg. 57).

“Implementing rich modeling tasks and multiple representations presents a challenge to teachers, who must themselves develop a deep understanding of the different methods and the layers of the problem, but it also opens an exciting new way of

teaching and learning. When you begin to incorporate these tasks into your classroom…teachers are rewarded by the engagement, motivation, and depth of

understanding they reveal” (pg. 71).

3. Obtaining a Mathematical Solution to the Model: The calculation stage; includes the solution. a. If the problem needs to be identified, the data collected, and the math methods to

solve the problem chosen before this stage can occur.

4. Interpreting the Solution and Comparing to Reality: (pgs. 72-88)a. While most traditional word problems stop a Stage 3, the solution, a vital part of

modeling is the fourth stage: interpreting the solution and comparing it to reality. i. “We must interpret the solution in light of the problem, and we must

compare the solution with reality-ensuring that it makes sense in the context of the problem” (pg. 73).

ii. If this does not take place, then students enter a state Boaler, a mathematician, calls “Math Land” (pg. 74).

According to, “Boaler’s Mathland,” students put aside their knowledge of how to world works (common sense) to solve math problems.

Pseudo contextual problems do not interest or engage students. Students are not asked to make connections between the real world and mathematics, making the problems less relevant. They are not required to check the reasonableness of their solutions since the solutions, themselves, are not reasonable (pg. 74).

b. Interpreting the Solution (pg. 75):i. Students need to know how to determine what the solution means (pg. 75).ii. Students need to ask themselves, “What did I just figure out?” (pg. 75)

c. Explaining/Justifying Your Solutions (To Ensure Meaning) Examples(pg. 75):i. Using Incorrect Answers: It is a powerful learning experience to ask

students to find the error on a problem or compare the work of two imaginary students (pgs. 76-77).

“Through the process of finding others’ errors in thinking, students begin to examine the reasonableness of their own solutions rather than simply giving an answer and moving on” (pg. 78).

ii. Non-Routine Problems: Sharing non-routine problems before students engage in their own modeling of non-routine tasks encourages them to seek to avoid the same sort of mistake” (pg. 78).

iii. When Pure Calculation Doesn’t Make Sense (Consider the Context): “Discuss a variety of problems where pure calculations do not produce a solution that makes sense in context” (pg. 78).

iv. Listing Assumptions: “Have students list some assumptions they are making when providing a solution that is a result of pure calculations” (pg. 79).

v. Encountering Problems with Counterintuitive Answers: If we can provide students problems with counterintuitive solutions, we spark interest, curiosity, and thought regarding the solutions (pg. 79).

A great deal of thinking is done after the calculations. Students will express surprise at unexpected answers, have excitement for the problems, and figure out why the answer is not expected (pg. 80).

5. Communicating and Implementing the Solution (By being active.):a. The final stage may take many forms (a poster, a digital presentation or video,

pamphlet, instructional guide, description, written explanation. Etc.).b. For students to truly understand mathematics, “they must do more than follow

rules and procedures to arrive at an answer: they must be able to explain what their answer means” (pg. 89).i. Students can only construct meaning “by exploring, justifying, representing,

discussing, using, describing, investigating, predicting” (pg. 89). ii. Good communication is vital to a successful classroom.iii. Good teachers understand:

The importance of effective communication. Getting information and content across to students. The importance of students communicating with others. “To teach students how to communicate their ideas, reflect on their

strategies, and justify their reasoning in the math classroom…they may clarify understanding for themselves” (pgs. 89-90).

c. NCTM emphasizes, “students who have opportunities, encouragement, and support for speaking, writing, reading, and listening in mathematics classes reap dual benefits: they communicate to learn mathematics, and they learn to communicate as well” (pg. 90).

d. Communicating the Solution: (pgs. 90-93)i. “Communicating the solution in the math classroom can take many forms

(writing a sentence defining the solution, recording units and brief descriptions of work as students proceed, drawing a picture or sketch, making a table or writing an explanation” (pg. 90).

ii. The communication should convey (pg. 90): The solution to the problem Some reasoning about the answer The solution into the context of the original problem. “Putting the solution back into context, using very few words, is

often the nudge a student needs to recognize their mistake” (pg. 91).iii. “On more complicated modeling tasks, we might need to remind students to

use written or numerical notation or units on each part of the problem as they proceed” (pg. 92).

e. Communicating the Solution to Peers: (pgs. 93-94)

i. “Too often, both students and teachers see the final product (or) ultimate goal (is working toward) getting a good grade, missing the point that the work itself can enhance everyone’s understanding” (pg. 93).

“Students writing just to get a good grade write on what they think the teacher wants, limit their own justification /explanation” (pg. 93).

ii. Students who must explain their solutions and justify their reasoning…quickly become aware of the need to be clear and specific…and when students see a problem approached from different perspectives, using different tools and methods, and hear a problem explained in different ways, everyone gain access to new ideas…and gain a deeper understanding of the problem” (pg. 94).

f. Implementing the Solution (pg. 105):i. “Whenever possible, students should physically implement the solution to

rich modeling tasks in a real, concrete way” (pg. 105). “If some tasks have time or material constraints, it may be possible to

replicate the problem on a smaller scale” (pg. 105) “Students should implement the solution through a written

recommendation, proposal, or explanation” (pg. 105). ii. “Whenever students can actually implement their solutions, they continue to

lean and actively engage in the mathematics even after the problem is “solved” (pg. 105).

g. Writing in Mathematics: (pgs. 94-100)i. The NCTM emphasizes the importance of writing in mathematics in its

Principles and Standards stating, “Reflection and communication are intertwined processes in mathematics learning…writing in mathematics can also help students consolidate their thinking because it require them to reflect on their work and clarify their thoughts about the ideas” (pg. 94).

ii. “Writing about mathematics can enable students to analyze, compare and contrast, and synthesize information and data. It requires students to think about their own thinking and reasoning, focus on what is most important, and internalize an construct meaning out of their work” (pg. 94).

iii. Writing to Help Teachers Assess Understanding: Reading students’ writing about math can be informative and

interesting (pg. 94). “When students discuss and apply their solutions, they continue to

engage the problem, make connections to other mathematics and real life, and gain a depth of understanding they did not have when they initially reached the solution.”

Writing in mathematics helps students consolidate and reflect on their work, thoughts, and ideas according to the NCTM (pg. 95).

iv. Teacher’s questions on student understanding when students are writing about mathematics (pgs. 96-97):

Do students use math to make sense of complex situations? Can they formulate hypotheses? Can they organize information? Are they able to explain concepts? Can they use computation skills in context?

Do they use mathematical language appropriately? Are they confident about using mathematical procedures?

v. Prompts for Writing Tasks (Could be pasted inside the cover of students’ math journals) (pg. 98):

What do you notice? What do you find interesting? What patterns do you see? What surprises you? What do you predict? Why? What do you findings make you wonder? What does this remind you of?

vi. “As (students) progress through the writing process, and…summarizing and communicating solutions, briefer prompts may be used to inspire student reflection. The following words, along with a brief question or two, might prompt students to better focus their written responses” (pg. 98).

Analyze Describe Evaluate Justify Reflect/Question Summarize Synthesize

“It is not necessary to use all prompts on every problem, but it may be helpful to use one of these prompts occasionally to get students comfortable with their meanings…(students will) begin to produce more elaborate written summaries and implement

their solutions.” (pg. 100).

Finding Rich Mathematical Modeling Tasks (pgs. 24-26):1. MARS Tasks (Mathematics Assessment Resource Service): “Offers students grade-

appropriate standards-based problems that spark interest and present challenge to students of varying abilities, and are good resources for teachers” (pg. 24). They provide a rich mathematical experience for students, and are easily accessible to teachers: http://map.mathshell.org/materials/tasks.php

2. Three-Act Math:a. Act 1: a visual (often video) introduction of the challengeb. Act 2: Students are given some of the information they might need to solve the

problem. It is the job of the student (with the teacher’s help) to dig into a toolkit to figure out what tools and skills might be needed to solve the problem.

c. Act 3: provides the resolution to the conflict. The 3-Act Math tasks vary by grade level and topic, and are easily accessible online. For a spreadsheet of all the tasks, see https://docs.google.com/spreadsheet/ccc?key=0AjIqyKM9d7ZYdEhtR3BJMmdBWnM2YWxWYVM1UWowTEE#gid=0

3. NCTM Illuminations: Includes rich modeling opportunities at http://illuminations.nctm.org/

As you, “develop your own rich mathematical experiences for your students, and you will begin to see the return of your investment of time and energy in the

enthusiasm, motivation, and depth of understanding your students” (pg. 32).

Develop Your Own Modeling Tasks:1. You can begin with” textbook word problems, writing tasks based on students’

interests, and connecting mathematics to students’ current areas of study in other classes” (pg. 27).

2. “Two of the most important attributes of real problems are that they should be meaningful and interesting” (pg. 28).

3. “You will never find a topic every student is excited about, but if the students can make real-world connections with the mathematics, they will certainly be more engaged and motivated to succeed” (pg. 30).

4. “It’s easy to transform students’ many real life interests into rich modeling tasks” (pg. 30).

5. “Another way to creating engaging, rich modeling tasks is to incorporate students’ areas of student in other classes” (pg. 31)

In Terms of Assessments: “It is important to remember that assessment is far more than scoring an exam or giving a grade. Assessment of modeling tasks takes place not only at the end, but also during the course of the work” (pg. 128).

1. “Assessment is not a dirty word” (pg. 109). 2. Good Assessments Can:

a. Place a focus on learning goals for teachers and students.b. Inform students/teachers about what students know/need to know.c. Give students feedback about performance and understanding. d. Actively involve students in the learning process.e. Inform teachers about the direction of further instruction (pgs. 109-110).

3. “Assessment is not synonymous with testing” (pg. 110). a. Assessments happen in the classroom daily. “Informal assessment involves the

continuous task of listening questioning, observing, and interacting with students” (pg. 110).

4. NCTM Principles for School Mathematics: (pg. 110).a. Assessment should support the learning of important mathematicsb. Furnish useful information to both teachers and students. c. Inform teachers at the conclusion of instruction what students know and are able to

do.d. Be an integral part of daily practice in the classroom. e. Assessments, used in this way, enhance student learning and students demonstrate

significant gains (pg. 110).5. Informal Assessments:

a. ABWA! Assess by walking around (pg.110): This informal assessment is crucial when students are engaged in modeling tasks.

b. Fairly Common Misconception (pg. 111):i. Many teachers believe that modeling involves presenting students with a

task and then leaving them to their own devices to tackle this task.

ii. The truth is, the teacher’s role is vital when students are working on a rich modeling task.

iii. Teachers must engage in ongoing instruction.iv. Formative assessments evaluate student understanding, progress,

misconceptions, and needs-and it occurs while we, the teachers, are waking around, listening, and talking with students (pg. 111).

v. “By assessing the students’ understanding and misconceptions, the teacher was able to redirect…and keep (students) from moving too far down the wrong path” (pg. 114).

c. “When we engage and direct students during the modeling process, we reduce student frustration and help students get “unstuck” by starting with what they already understand and guiding them as they proceed from there“ (pg. 115).

d. Other forms of Informal Assessment: Summaries, reflections and exit slips (pg. 115):i. Many teacher use exit slips that students hand in…at the end of the period”

(pg. 115).ii. The teachers ask students to assess their understanding by answering

the following questions: (pg. 115). What is one thing you learned today? What is one thing you are still confused about? What did you accomplish today? What did you learn today? What do you need to do tomorrow? What questions do you have?

6. Formal Assessments on a Modeling Task (pg. 116):a. Formal assessment can be a formative and a summative assessment.b. They can assist students in self-and peer-assessment on a task.c. They can “serve as an instrument for effective and efficient final assessment at the

conclusion of the task” (pg. 116).7. Effective Rubrics: “A rubric is a ‘coherent set of criteria for students’ work that

includes descriptions of levels of performance quality on the criteria” (pg. 117).a. “Good teachers clearly identify and quantify what they want their students to know

and be able to do before they begin teaching the material. They plan their instruction around learning goals. They also plan their assessments around these goals. A really good teacher (helps)… students understand before, during, and at the conclusion of a lesson what it is they are expected to know and be able to do” (pgs. 116-117).

b. Rubrics Resources: (pg. 119).i. Examplars Standard Based Rubric: www.examplars.comii. The Mathematics Assessment Resource Service (MARS) tasks:

www.insidemathematics.orgiii. Rubistar: http://rubistar.4teachers.org/index.phpiv. teAchnology: www.teachnology.com/v. iRubric: www.rcampus.com/indexrubric.cfmvi. For the larger modeling tasks, focus on the big ideas and goals. Teachers

should write rubrics in student-friendly language while still identifying specific criteria for each category (pg. 119).

vii. “The goal is to clarify for your students what you expect from them to make student assessments more meaningful and less time-consuming for you” (pg. 120).

viii. “Even though modeling tasks often involve more in-depth investigations than typical math classwork, and they require some time for students to develop, revise, and test their models, the assessment of these tasks is relatively quick and easy with the rubric” (pg. 123).

ix. The students should be given the rubric as a guide to let them assess their progress (pg. 127).

Final Thoughts Regarding Modeling with Mathematics:1. “You can teach a student a lesson for a day; but if you can teach him to learn by creating

curiosity, he will continue the learning process as long as he lives. Clay p. Bedford” (pg. 129).

2. “Mathematical modeling is a powerful practice that can engage our students and increase their understanding of mathematics…students are preparing for a lifetime of learning and applying…mathematics in real life” (pg. 130).

3. “You are not going to use modeling tasks every day in your class…However it is well worth the time investment to trade, perhaps, a day of “Do problems 2-34 even” for a modeling task now and then. You will still be able to address all of the standards, but you will produce mathematically autonomous thinkers in the process. Try one task each month or each grading period at first” (pgs. 130-131).

4. “Remember not all modeling tasks have to be major, time-consuming endeavors. You can ease your class into modeling tasks with briefer, yet still rich tasks, and you will be rewarded by your students’ engagement and depth of thinking” (pg. 131).

Frequently Asked Questions Regarding Modeling with Mathematics:Q: How can I possibly grade open-ended problems? (pg. 131)A: “You can use teacher notes to plan in advance for man of the students’ responses and questions. Also, once you begin using the modeling tasks, it will get easier ever time…Often it requires a back-and-forth discussion with the students, asking them to explain their work and reasoning (a good thing), and then testing their solution to be sure it works (also a good thing). “The students’ presentations will often demonstrate their understanding of the mathematics and whether their answer is correct. Finally, if you are assessing work with a good rubric, you will see that it is possible to understand variations on open-ended problems without doing infinite calculations of every possible answer” (pg. 131).

Q: “I’m a math teacher. How am I supposed to grade writing?” (pg. 132)A: “When we ask students to write in the math classroom, we are not grading the writing. We are using that writing to assess understanding and inform our future instruction. We are also using that writing to help students organize and articulate their reasoning and explain their thinking to others.” Also, remember that you are reading brief explanations, questions, and reflections. You do not have to correct grammar and spelling if you are uncomfortable doing that, but at the very least, read for content. I can read through a class period’s slips in less time than it takes to grade a page of computation problems” (pg. 132).

Q: “I am afraid of losing control of my class.” (pgs. 132-133)

A: “If you crave order and quiet, this may no be an easy transition for you, but I encourage you to try. The key to successful classroom management with modeling tasks is to set clear expectations and norms for cooperative groups. Some norms include:

Respect each other’s ideas. Only one person speaks at a time-don’t interrupt. Everyone must participate and contribute. Listen to one another. Each person has a specific job. Stay with your own group.

It is important to set norms early in the school year. If you do not have norms for group work, it might be advisable to first do one group project or activity” (pg. 133).

Q: “This sounds great for your students, but it will never work with mine.” (pg. 134-135)”A: “You students can do mathematical modeling tasks!...From elementary school through graduate school…the most struggling learners…the most advanced learners…students who have a long history of failure in mathematics, and who believe, ‘I can’t do math’” (pg. 135).

More Hints (pgs. 133-135) : 1. Change groups frequently.2. On a multiday task, set a clear goal about what should be accomplished by the end of

each class period.3. Ensure that each group member has at least one specific “job” and that all members are

contributing to a task. When presenting, each group member must present a part of the work, and it must be evident that each member understand the work.

4. Try to promote an all for one and one for all mentality.5. Wrap up each class period with a quiet reflective time.

“Give modeling a try. Have confidence in your students and in yourself. With rich tasks and direction, they (your students) will achieve great things” (pg. 135).