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Project M3: Mentoring Mathematical Minds; Grades 3-6

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Project M3: Mentoring Mathematical Minds is a series of curriculum units developed to motivate and challenge mathematically talented students at the elementary level.

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Model-eliciting activities (MEAs) are activities that encourage students to invent and test models. They are posed as open-ended problems that are designed to challenge students to build models in order to solve complex, real-world problems. The six essential principles of MEAs are listed at the end of this section. They may be used to engage the students in statistical reasoning and thinking and provide a means for statistics teachers and researchers to better understand students' thinking.

MEAs are created to look like authentic, real-world statistical problems and require students to work in teams of 3-4 students. The teams generate solutions to a problem via written descriptions, explanations and constructions by "repeatedly revealing, testing, and refining or extending their ways of thinking" (Lesh, Hoover, Hole, Kelly, & Post, 2000, p. 597). In this way, MEAs are thought-revealing as the students' products provide windows into the thinking processes they used to develop the solutions.

While many statistics instructors, particularly at the introductory level, use one or more traditional types of student activities in their classes, these activities are typically quite different than MEAs (see Table 1 below). Activities may also be used to engage students in statistical reasoning and thinking, but they are typically used for a different purpose than a MEA. We think that both MEAs and more traditional types of statistical activities can be used as part of a statistics class.

Some differences between MEAs and other types of typical in-class statistical activities are shown in Table 1. It is important to note that MEAs are not used instead of other types of in-class activities, but that both can be used effectively in a course, but to serve different purposes.

Table 1: Comparison of MEA to other typical in-class activities used to teach statistics

Six Principles for Model-Eliciting Activities

The six essential principles for MEAs (Lesh, et al., 2000) are described below:

1. Model Construction Principle: Problems must be designed to allow for the creation of a model dealing with:

Relationships and operations between these elements

Patterns and rules governing these relationships

In order for the MEA to be thought-revealing, the MEA can require:

An explicit construction

Description

Explanation

Justified prediction

MEAs have students:

Quantify (e.g., show how well a model works)

Coordinate information and relationships

Make predictions (apply the model to a new problem or data set)

Identify a pattern or trend.

The final product is a model that:

Has elements

Defines relationships among the elements

Defines operations for how elements interact

Identifies patterns or rules that apply to the relationships and operations.

2. The Reality Principle: Problems must be meaningful and relevant to the students and be based on real or slightly modified real data. The solution should be "real" and meaningful in the students' everyday lives. Therefore, the context of the situation should:

Identify the audience who will use or be served by the solution.

State the purpose of the solution for the audience.

State why the audience needs the solution.

State a problem that makes sense in terms of real life knowledge and experience

This can be established through a set of preparatory questions that set a (statistical) focus for reading introductory articles and other information. The preparatory questions can also form the basis for whole class or small group discussions of the articles.

3. Self-Assessment Principle: Students must be able to self-assess or measure the usefulness of their solutions. The problem statement must strongly suggest appropriate criteria for assessing the usefulness of a solution. The data must also play a part in allowing students to self-assess. These criteria should promote selection, refinement, and elaboration of models. In order to do this, the problem statement must:

State a clear purpose

Require self-assessment and a need for improvement

Clearly state when a solution is achieved (through clear criteria)

To make progress, students must be able to:

Detect deficiencies in the current conceptualization

Compare alternatives and select the most promising ones

Integrate the strengths among the alternatives, minimize weaknesses

Extend and refine promising alternatives

Assess adaptations

One mechanism for delivering these criteria and the motivation for self-assessment and refinement is to establish a "client" who is making the request and specifies the parameters.
4. Model Documentation Principle: Students must be able to reveal and document their thinking processes within their solution. Students responses must produce an audit trail by revealing the:

Givens (their assumptions)

Goals

Solution paths that are taken into account to produce a solution

The activity should foster self-reflection in order to get students to think about their thinking (metacognition). Therefore, the activity should involve group work that requires (these might be different roles for the group members.):

Planning

Monitoring

Assessing progress

The teams' products should reveal:

Statistical objects or constructs that were used

Relationships among the objects and constructs

Operations or interactions among the objects and constructs

Representational systems used

The products should allow the instructor or researcher to assess a team's statistical thinking. One method is to create a "Ways of Thinking" chart that categorizes the students' solutions.

5. Model Share-Ability and Reusability Principle: This ensures that solutions created by students are generalizable or easily adapted to other similar situations. The model should represent a general way of thinking instead of a specific solution for a specific context. This principle also ensures that students' models are communicated in a clear understandable manner that allows them to be used by others.

6. Effective Prototype Principle: This ensures that the model produced will be as simple as possible, yet still mathematically significant. The model, which should represent "big ideas" in statistical thinking, should provide a useful learning prototype or metaphor for interpreting other problems with the same underlying structure. The activity should be designed to avoid the need for numerous procedures, especially computational procedures that can circumvent conceptual understanding.

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Model-eliciting activities (MEAs) are activities that encourage students to invent and test models. They are posed as open-ended problems that are designed to challenge students to build models in order to solve complex, real-world problems. The six essential principles of MEAs are listed at the end of this section.

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Sharing some resources developed by our team when implementing Math Modeling Tasks in Elementary Grades!

Check out these math modeling lesson Infographics:

Middle grades extension https://docs.google.com/document/d/1YwtSmcWEnpf-bPJ0qRESvn6Xcx1oN5uFPQ2x0V_ATbA/edit?usp=sharing https://drive.google.com/drive/folders/1Oajd5FYYHWmnfAynPNHCKEb7zZCQTPmT?usp=sharing
Graphic Organizer
http://completemath.onmason.com/math-modeling/
Related readings

Websites with Project Tools

Contact: Dr. Jennifer Suh George Mason University

Our IMMERSION Year 2 groups are off implementing Math Modeling Tasks in Elementary Grades!

Mathematical modeling is a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena. Most short definitions we find emphasize this most important aspect, namely the relation between modeling and the world around us. - Using the language of mathematics to quantify real-world phenomena and analyze behaviors. - Using math to explore and develop our understanding of real world problems. - An iterative problem solving process in which mathematics is used to investigate and develop deeper understanding. (Gaimme report, p.8)

Math Modeling tasks use math to make decisions! Some of the types of modeling problems that we have had success in elementary grades include:
* Descriptive models: Using real world data to describe/represent/ analyze a phenomenon
For emergent math modelers, this would engage them in math discourse that might start with-
I can use math modeling to describe __________.
I can use math modeling to describe how many buses Ms. Green will need to take us on the field trip.
This may lead to a math model like : (# of students + # of teachers + number of parent chaperone)/number of passengers allowed on each bus= number of buses to order for the field trip.
This model may need revising if the modeler found out that the bus allows for 3 students to each seat and 2 adults per seat. (# of students/3)+ (# of adults/2)/#of seats in each bus= number of buses ordered for the trip.

Predictive models: Using trends and data analysis to predict an outcome
For emergent math modelers, this would engage them in math discourse that might start with-
I can use math modeling to predict ______________.
I can use math modeling to predict how many pencils we will sell at our School Store based on our data.
With our Buy one get 2 free pencils sale, if we make at least 5 sales, that would mean
number of pencils sold=5 +2(5)=15. Some students might be able to say it in words or with a number sentence, while students ready for algebraic reasoning might be able to use variables to represent the model.
P=5+2(5)
P=n+2n

Optimizing models: Using data to find the "best" by optimizing or in some cases minimizing some situation.
For emergent math modelers, this would engage them in math discourse that might start with-
I can use math modeling to find the best ______________.
I can use math modeling to find the "best" way to design an edible garden.
For optimizing models in earlier grade, it provides a great opportunity for students to engage in mathematics argumentation because the criteria for "best" can be determined by the assumptions and constraints they put around the real world problem. For example, Ms. Farmer wants to maximize area with a plot of land and has 24 feet of fencing or Ms. Farmer has a narrow plot of land that is only 4 feet wide but had lots of room for a long garden or Ms. Farmer wants to use the wall of her house as one side of the garden. These different assumptions and constraints provide lots of different solutions that all could be mathematically viable.

Rating and ranking: Using criteria and mathematical measures as a way to rate and rank options to make decision.
For emergent math modelers, this would engage them in math discourse that might start with-
I can use math modeling to rate and rank to make decisions about ______________.
I can use math modeling to rate and rank to make decisions about the best college basketball player/team. I can use math modeling to rate and rank to make decisions about the best vacation spot for our family.

Rating and ranking provides a great way for students to quantify the world around them. This modeling activity is primed for using collected data to make decision. Data scientists use strategies like rating and raking to make important decisions in our daily lives. Sometimes even exciting decisions like which team to rout for during the March Madnesss game 🙂

Resources

Math Modeling Talking Points

What do you wonder about?

What do you know or notice?

What information do you need?

What assumptions can you make?

How will you solve the problem?
Does the solution/model make sense when you go back to your problem? What might you revise and refine in your model to better solve your problem?

Table 1.

Emergent Mathematical Modeling situations and questions posed and curricular alignment

Grade Level and Topics

Math Modeling Questions & Type

Curricular Alignment to Math Standards

Students were able to count cubes for each item and them put them all together like counting collections, draw pictures to represent numbers, add up to 20, and a few subtracted down.

While making Thanksgiving baskets, students worked to determine the total cost of each of the baskets. To find the best deal for an item led students to learn how to calculate unit rate.

What is the best field trip? (5th grade)

Family day trip (3rd graders)

Optimizing time and minimizing cost/Ranking

Solving single-step and multistep practical problems, using proportional reasoning and planning for the trip using budgetary constraints and elapsed time.

Too much Texting (5th-7th) Predictive

Describing and comparing data, using ratios, and solve single-step and multistep practical problems, using proportional reasoning.

Solving single-step and multistep practical problem using four operations. Analyzing patterns and learning about linear functions.

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Sharing some resources developed by our team when implementing Math Modeling Tasks in Elementary Grades! Math Modeling Taxonomy of Tasks Math Modeling Lesson through Math Happenings (Google folder) Check out these math modeling lesson Infographics: Field Trip Dilemma Too muc ...

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Statistics for Kids by Scott Chamberlin, Ph.D.

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Statistics for Kids by Scott Chamberlin, Ph.D.

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Statistics for Kids: Model Eliciting Activities to Investigate Concepts in Statistics (Grades 4-6) (Statistics for Kids, Grades 4-6)

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Amazon.com: Statistics for Kids: Model Eliciting Activities to Investigate Concepts in Statistics (Grades 4-6) (Statistics for Kids, Grades 4-6): 9781618210227: Chamberlin, Scott: Books

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Spaghetti and Meatballs for All- Student Sheet 

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Spaghetti and Meatballs for All- Student Sheet 

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Spaghetti and Meatballs for All Student Sheet

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100 Questions That Promote Mathematical Discourse 

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100 Questions That Promote Mathematical Discourse 