Statistics for Kids

Model Eliciting Activities to Investigate Concepts in Statistics (Grades 4-6)

Scott Chamberlin

• Editore: Routledge
• Anno: 2021
• ISBN:
  9781000503470
• Edizione: 1
• Argomento: Bildung
• Categoria: Bildung Allgemein

Perhaps the most useful and neglected content area of mathematics is statistics, especially for students in Grades 4-6. Couple that fact with the notion that mathematical modeling is an increasing emphasis in many standards, such as the Common Core State Standards for Mathematics and the NCTM standards, and the necessity for this topic is overdue. In this book, teachers will facilitate learning using model-eliciting activities (MEAs), problem-solving tasks created by mathematics educators to encourage students to investigate concepts in mathematics through the creation of mathematical models. Students will explore statistical concepts including trends, spread of data, standard deviation, variability, correlation, sampling, and more—all of which are designed around topics of interest to students. Grades 4-6

Chapter 1

Trends

The 10,000 Meters Problem

DOI: 10.4324/9781003238201-2

Track and field is a well-known sport in the United States. However, in the rest of the world, track and field events are referred to as Athletics, the larger category of sports to which they belong. (Henceforth, Athletics will refer to track and field and athletics to sports in general.) Athletics is considered by many to be the most international of all sports because people from every country in the world can participate in the events. Consequently, the top athletes in Athletics events come from a wide variety of countries.

For years, the three largest revenue-generating sports in the United States—baseball, football, and basketball—have been bestowed with endless statistics and statisticians to provide analysis of their games. Athletics also benefits from statistical analysis, particularly at the international level, and Athletics provides the context for this problem, designed to introduce the topic of trends. In this activity, students are asked to analyze data of 17 collegiate athletes and create a mathematical model to explain a trend in the data. Each MEA has a client (i.e., the person or group for whom the mathematical model is being produced). In this MEA, the client is Coach Mike Hall. Coach Hall needs to be more systematic in his recruiting efforts. Rather than just guessing at which athletes will develop rapidly, Coach Hall really wants to have athletes that can contribute greatly to his team and he feels that statistics may provide insight regarding which athletes to recruit most heavily. Moreover, the discovery of certain patterns (i.e., trends) might inspire Mike to change his distribution of scholarship money and award partial scholarships to more athletes rather than full scholarships to fewer athletes. For example, if athletes that have run a specified time (e.g., quicker than 32:30 for 10,000 meters) often develop as well as ath-letes who run faster in high school, then maybe Mike wants to award one full scholarship and four half scholarships to five athletes rather than award three full scholarships to three athletes. This is a very realistic scenario and one that collegiate coaches face regularly. Hence, the context for this problem is looking at performance data for past athletes in an attempt to create a mathematical model for future athletes.

Subject Focus of the Chapter

This chapter’s subject focus is on trends. Trends can be found in many places. For example, business analysts seek trends in the stock market to be able to predict what will happen in similar situations in the future. Sports and military experts seek trends to see how their competitors will react and behave given certain stimuli. Meteorologists and climatologists seek trends in an attempt to be more accurate with long-range predictions and to assist in the creation of weather/mathematical models. Retail experts seek trends to see what to order and ultimately place in stores in an attempt to have the most recent or hottest product line. It is thus not surprising that the subject of trends was selected by experts in statistics as one of the most important topics, and it is therefore the focus of the first chapter.

Links to NCTM Content Standards

This problem has many links to NCTM content standards. As a quick review, there are five content standards (NCTM, 2000). These five content areas are: algebra, data analysis and probability, geometry, measurement, and number sense and operations. It is realistic to expect that these activities will link with content standards in data analysis and probability, as per the emphasis of this book. Interestingly enough, given the open-ended nature of MEAs, many other content standards are often met, but they may not be met by all students, depending on the type of solution path selected by the group. It would be safe to assume, however, that nearly all students would meet the following NCTM content standards when completing The 10,000 Meters Problem:

• Select, create, and use appropriate graphical representations of data including histograms, box plots, and scatterplots (grade band 6–8).
• Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of best fit (grade band 6–8).
• Identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled (grade band 9–12).
• Understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference (grade band 9–12).
• Understand how basic statistical techniques are used to monitor process characteristics in the workplace (grade band 9–12).

In addition, for all MEAs in this book, the algebra standard “Model and solve contextualized problems using various representations, such as graphs, tables, and equations” is met.

Questions to Pose to Students

It is important to pose questions pertinent to the implementation of the MEA to students throughout the process. For example, it is incumbent upon the instructor to ensure that students understand what is being asked in the problem statement. The most expeditious and forthright manner in which to investigate comprehension of the problem statement is to ask groups if they understand what is being asked. However, the focus of this section is not on closed-ended questions. Instead, the focus is on questions that instructors may use to serve in the role as metacognitive coach (Papinczak, 2010). It is important that instructors do not answer any questions about solving the problem. Questions about clarifying the problem can be answered, but responses to questions that give away part or all of the answer should be avoided at all times. In this respect, instructors are serving as metacognitive coaches by using questions to guide students’ thinking (Stepien, Gallagher, & Workman, 1993). Several potential questions may arise from students. These may be referred to as “sticking points” or points in the problem when groups of students appear to reach an impasse. After repeated implementations of multiple MEAs, students rarely ask leading questions designed to retrieve a solution from an instructor because they come to the realization that instructors will not reveal answers.

Instructors will find that some general questions may be used from activity to activity, such as:

• How does that computation help you develop a mathematical model that explains the situation?
• Can you be clearer regarding the process that you used to create the model?
• How would the model change if you eliminated this procedure?

There are also questions that instructors will develop in using The 10,000 Meters Problem that are specific to this MEA. As an example, instructors may find themselves asking some of the following questions:

• Do any of the (first two) columns have anything to do with any of the other columns of data?
• Do you see any patterns in the data?
• Do any examples (i.e., individual data points, athletes) exist that might help you develop an idea about the data?
• How will your explanation help Coach Hall give out scholarships to high school athletes?
• Did you go back to read the statement to see that your final solution meets the expectations of the client (Coach Hall)?

Undoubtedly, additional questions will surface. Some may prove common to the problem, but will not appear so until the problem is implemented repeatedly.

Potential Student Responses

Given the fact that students in grades 4–6 may not completely understand how scholarships work—though some may be familiar with the process—instructors will need to check on students’ final responses to see that they have grasped the concept. It is also important to note that scholarships need not be awarded in increments of 25%. A coach reserves the right to award any percent scholarship that he or she deems necessary. For instance, a coach can split a scholarship two, three, or more ways, but when the offer becomes too minimal, athletes will start to compare the offer with offers from other academic institutions.

The focus of this problem is to look at data to identify potential trends. The value in identifying trends in this instance is to be able to make more calculated, systematic, and informed decisions for the recruitment of future athletes. In creating a mathematical model to explain the data, Coach Hall will be able to use data to draw inferences (Standard 4, Links to NCTM Content Standards) to make informed decisions. From an instructor’s perspective, the question remains: “What sort of poten...