Data and Probability Strand

The Data and Probability Strand in Connected Mathematics

Connected Mathematics develops four mathematical strands: Number and Operation, Geometry and Measurement, Data Analysis and Probability and Algebra.

Within the Data strand are two distinct subjects: answering questions about situations that involve uncertainty, and analyzing data.

How Likely is It? is the first unit in the Connected Mathematics curriculum that will develop students' abilities to understand and reason about probability. Students will gain an understanding of experimental and theoretical probabilities and the relationship between them. They learn to recognize situations where they can apply rules to calculate theoretical probabilities, and when they must resort to a simulation.

The first unit on data analysis is Data About Us. The kinds of questions explored in the data strand address very practical questions about the world around us. A few years ago it was fashionable for journalists to talk and write about "innumeracy," which just means the inability of the adult population to deal with the numbers and data that they hear about in the news. In the data analysis strand CMP students have an opportunity to become comfortable with the ideas in common usage in news reports, as well as ideas and vocabulary current in physical and social science research. In units that focus on analysis of data, students learn to:

• Formulate a key question(s) to explore and decide what data to collect to address the question(s)

• Collect the data, and address issues of accuracy and efficiency.

• Organize, represent, summarize with measures of center and spread, describe the data, and look for patterns in the data, including the shape of the data, variability, and extraordinary features.

• Predict, compare, and identify relationships and use the results to make decisions about the original question(s) (From Graham, 1987)

Overall goals for CMP Data and Probability Strand

The mathematical learning goals below signify what students should be able to do in Data and Probability by the end of eighth grade.

Formulating Questions

• Formulate questions that can be answered through data collection and analysis

• Design data collection strategies to gather data to answer these questions

• Design experiments and simulations to test hypotheses about probability situations

Data Collection

• Carry out data collection strategies to answer questions

• Distinguish between samples and populations

• Characterize samples as representative or non-representative, and random or biased

• Use these characterizations to evaluate the quality of the collected data

Data Analysis

• Organize, analyze and interpret data to make predictions, construct arguments, make decisions

• Use measures of center and spread to describe and to compare data sets

• Be able to read, create and choose data representations including bar graphs, line plots, coordinate graphs, box & whisker plots, histograms, and stem & leaf plots

• Informally evaluate the significance of differences between sets of data

• Use information from samples to draw conclusions about populations

Probability

• Distinguish between theoretical and experimental probabilities and understand the relationship between them

• Use probability concepts to make decisions

• Find and interpret expected value

• Compute and compare the chances of various outcomes, including two-stage outcomes

• Use a variety of strategies to count outcomes in probability and combinatorics problems

Data and Probability Units

There are 5 Data and Probability Units in CMP, though ideas developed in these units are reviewed and practiced in relevant contexts in other units. The last unit in 8th grade is Clever Counting, a unit on combinations and permutations. While this is a number unit it is strongly related to the study of probability, though it is not on the following list.

Data Analysis and Probability

• Data About Us? (6th grade)

• How Likely is It? (6th grade)

• What do you Expect? (7th grade)

• Data Distributions (7th grade)

• Samples and Populations (8th grade)

Every important idea addressed in the CMP Data and Probability Strand is placed carefully to make initial development appropriate to student developmental level, and also to connect productively to other units already studied. For example, Data About Us, and How Likely Is It? are intended to be studied late in 6th grade so that they can take advantage of, apply, review and practice number skills about fractions, decimals and percents taught in the 6th grade Bits and Pieces units.

It is important to note that many data and probability goals are revisited in later units, in the same grade level or later, either within classroom problems or in the Connections problems in the ACE homework assignments. Units that are not focused on data are interspersed between these data and probability units, and connections and distributed practice of data ideas continue to occur. For example, in Comparing and Scaling, a 7th grade number unit, a central idea is making comparisons among measures by using ratios, percents or differences. In the ACE questions in this unit these ideas are deliberately connected to Data contexts. For example, in a Connections question about probability students compare theoretical and experimental probability, an idea from How Likely is It? and relate their understandings about probability to ratios. In an ACE question about comparing data from 3 groups a stacked bar graph is created. This graph requires students to use percents to make comparisons and extends their repertoire of data representations, begun in Data About Us. Likewise, other 7th and 8th grade non-data units deliberately review and practice data and probability ideas.

In order to have a clearer idea of the particular goals for each unit, a list of goals for each unit is available in Mathematical Help.

Relating Data and Probability in CMP to High School

It is unusual for high schools to offer a separate course in Data Analysis, except at the level of Advanced Placement Statistics. Many Algebra and Geometry textbooks have a chapter, or interspersed sections, which introduce data analysis. Teachers, facing pressure to cover a lot of algebra or geometry can be tempted to skip sections or chapters that involve data analysis. On the other hand state tests may include questions about data analysis. Students who have been successful in the data units in CMP will find many of the ideas in these high school texts very familiar. They also have a headstart in preparation for taking an Advanced Placement Statistics class later in their high school career.

Ideas about probability are often concentrated in a single chapter in high school algebra textbooks, and generally favor a theoretical approach. The approach in CMP focuses on making sense of situations involving uncertainty. CMP students have an opportunity to understand core ideas such as "equally likely events" or "independent events." Without understanding these ideas, rules for calculating probabilities can be misapplied. Many high school students find the study of probability and the calculation of probabilities to be mystifying. To complicate matters their high school teachers have often had little education themselves in probabilistic concepts, and rely heavily on textbooks. Ideas about probability are difficult, often counterintuitive, and take a long time to develop. By studying these ideas in middle school CMP students have a better chance of success than students who, in high school, come to the ideas cold.