Essential Knowledge of Probability for Prospective Secondary Mathematics Teachers
Irini Papaieronymou
Michigan State University
papaiero@msu.edu
Introduction
Since the late 1950s, there have been strong calls for an increase in the inclusion of probability in
the K-12 mathematics curriculum (NCSM, 1977; NCEE, 1983; NCTM, 1989, 2000). Probability
has come to gain importance in US school curricula as a content area that students need to have
experience with in order to be well-informed citizens since its study “can raise the level of
sophistication at which a person interprets what he sees in ordinary life, in which theorems are
scarce and uncertainty is everywhere” (Cambridge Conference on School Mathematics, 1963, p.
70; as cited in Jones, 2004).
Literature Review
Research has shown that both school children and adults hold intuitions about probability that
come at odds with theory (Kahneman, Slovic & Tversky, 1982; Clifford 1989). These
problematic intuitions may explain why probability seems to be difficult to learn. Since teacher
knowledge can have an impact on student learning (Fennema & Franke, 1992), it is important
that teachers be able to tackle these student difficulties and misconceptions on probability as they
arise in mathematics classrooms. In order to be able to do so, teachers need to have sufficient
probability content knowledge as well as adequate probability knowledge for teaching. Yet, one
of the problems encountered is the inadequate preparation of teachers in probability (Penas,
1987; CBMS, 2001; Batanero et al., 2004). Many teachers have not studied probability in their
own K-12 mathematics courses and sometimes need convincing as to why they need to learn and
teach probability topics (CBMS, 2001). As a result, the teaching of probability is often difficult
for mathematics teachers.
Particular to the college level, probability content varies according to the course purpose.
Probability at this level may be taught i) as part of a statistics or mathematics course that is a
general requirement for graduation and/or ii) as part of an undergraduate or graduate degree in
the natural sciences including mathematics and statistics and/or iii) as a research tool in
undergraduate and graduate programs in mathematics, statistics as well as other disciplines.
Some university programs have recently developed a course in statistics and probability intended
for prospective mathematics teachers (e.g. Michigan State University, University of Georgia).
Recommendations for undergraduate statistics and mathematics majors and minors are described
in the ASA’s Curriculum Guidelines for Undergraduate Programs in Statistical Science (ASA,
n.d.). The guidelines suggest that the mathematical background of students should include the
study of probability with emphasis placed “on connections between concepts and their
applications to statistics” (ASA, n.d.).
While the ASA provides guidelines for statistics and mathematics majors, the Conference Board
of the Mathematical Sciences (CBMS) provides recommendations for future teachers. In 2001,
the CBMS published a report titled The Mathematical Education of Teachers which emphasized
the mathematics knowledge and mathematics knowledge for teaching that teachers should have.
Specific to the preparation of middle grades teachers, the report suggests that teachers be able to
use basic probability concepts to measure uncertainty, be able to use theory and simulation to
deal with patterns produced by probability models, know the current uses of probability in other
fields, use information about equally likely events to develop theoretical probabilities, relate
empirical probabilities to theoretical probabilities through simulations, understand and be able to
compute conditional probabilities and the probabilities of independent events as well as
understand expected value. With regards to the preparation of high school mathematics teachers,
the CBMS suggests that future teachers should be introduced to sample spaces, independence,
conditional probability, discrete and continuous probability distributions, and the Central Limit
Theorem with only little emphasis placed on combinatorial probabilities. The CBMS report also
places considerable emphasis on historical perspectives of probability, common student
misconceptions, applications, uses and misuses of technology when dealing with simulations and
inference, and connections of probability to other mathematical topics.
The aforementioned recommendations by the ASA and the CBMS provide a basis for the
knowledge that secondary mathematics teachers should have about probability. A more thorough
examination of these documents and of national and state standards, university programs and
curriculum materials to be carried out in the analysis phase of this practicum study, will reveal in
further detail the probability content knowledge and probability knowledge for teaching that
secondary teachers need to have.
Research Question
Consideration of the aforementioned issues gives rise to the following question:
What are the content topics of probability that secondary mathematics teachers need to be
prepared for to teach?
The identification of these content topics of probability will be achieved through an analysis of
state and national (NCTM) mathematics standards for grades 6-12, recommendations from
professional organizations about secondary mathematics teacher requirements and qualifications,
and middle and high school mathematics curricula.
Methods
Data Sources:
Mathematics State and National Standards for Grades 6-12:
An essential source in addressing these research questions is the set of mathematics content state
standards and national standards in the secondary grades. With regards to national standards,
Principles and Standards for School Mathematics (NCTM, 2000) is analyzed. With regards to
state standards on probability, a sample of 10 states was chosen and then the mathematics
standards of each state were examined.
The state documents on probability standards tend to vary in length from state to state with the
shortest set of probability standards for one state being half a page and the longest being eight
pages. Based on information regarding the length of the probability state standard documents, the
following seven states were chosen to be included in the analysis: Alabama, Arkansas, Georgia,
Indiana, Kansas, Michigan, and New Jersey. In addition, California, Florida, and Texas were
included because of their large populations. Thus, ten states and NCTM (2000) were included in
the analysis of standards for grades 6-12.
Recommendations from Professional Organizations:
In addition to state and national mathematics standards, recommendations from professional
organizations such as the MAA, the AMS and the NCTM relating to the preparation of
mathematics teachers were considered in addressing the research questions. In particular, the A
Call for Change: Recommendations for the Mathematical Preparation of Teachers of
Mathematics (1991) published by the MAA, The Mathematical Education for Teachers (CBMS,
2001) published by the AMS, and the Professional Standards for Teaching Mathematics (1991)
published by the NCTM were included in the analysis.
Middle and High School Mathematics Textbooks:
Textbooks on probability in the secondary grades are also helpful in providing information about
what mathematics school curricula embody and thus can help indicate the topics of probability
that secondary mathematics teachers need to have knowledge of and be prepared to teach. Since
mathematics textbooks are prominent elements of mathematics classrooms (Tyson-Bernstein &
Woodward, 1991; see also Jones, 2004), a sample of popular textbooks series for middle and
high school mathematics is examined in this study.
Considering the results of the 2000 report of the National Survey of Science and Mathematics
Education (Weiss et al., 2001) which indicated the most commonly used middle grades
mathematics textbooks - among them Math 76 (Saxon Publishers) and Connected Math (Dale
Seymour Publications) - and also taking into account whether a curriculum is considered reform
or traditional, four textbook series were included in this analysis of this study: Connected
Mathematics (middle school, reform), Saxon (middle and high school, traditional) and University
of Chicago School Mathematics Project (high school, reform). The teacher guides/versions of the
textbooks in these series were considered.
Data Collection and Data Recording Procedures
State standards documents were downloaded from the Department of Education’s website of
every state used in the sample. The documents were carefully read and the standards specific to
probability were recorded in an excel spreadsheet. Standards were coded by state, grade/course,
and standard code/number which was identified in the standards document itself. For example
the coding in brackets at the end of
Understand and represent probabilities as ratios, measures of relative frequency,
decimals between 0 and 1, and percentages between 0 and 100 and verify that the
probabilities computed are reasonable (IN, 6, 6.6.6)
indicates that this standard was found in the mathematics standards documents of Indiana, grade
6 and had a code of 6.6.6 in that document.
The recommendations from the professional organizations were also collected and the
expectations specific to probability were recorded. A separate excel spreadsheet was created for
the recommendations from each organization. Specific to the various textbook series used, the
specific textbooks that included chapters/sections specific to probability were identified and the
goals from each textbook specific to probability content were recorded in excel spreadsheets, one
spreadsheet per textbook. Overall, the number of probability standards/expectations/goals in each
data source, prior to any type of data analysis, was as follows:
Source
Number of Probability Standards/Expectations/Benchmarks/Goals
State and National
Standards
CMP
252
UCSMP
136
Saxon
109
MAA
16
AMS
25
NCTM
6
Total
661
117
Table 2. Number of probability standards/expectations/benchmarks/goals in each data source
Data Analysis and Results
Research Question 1: What are the content topics of probability that secondary mathematics
teachers need to be prepared for to teach (as suggested by state and national standards,
secondary mathematics curriculum materials, and recommendations from professional
organizations)?
The work involved in analyzing the data to address this research question was completed in the
following stages: i)identifying major topics within the data; ii) Coding of standards/expectations
within an identified topic; iii) identifying concepts within the coded standards/expectations/
goals; iv) Counting the occurrences of these concepts; v) summarizing the results
The following table below summarizes the major topics identified in the data along with the data
sources in which this major topic appeared:
Major Topic
Sources
History of Probability
CBMS-AMS, Saxon, UCSMP
Terminology
CMP, UCSMP, Saxon, MI
Misconceptions of Probability and
other instructional issues
CBMS-AMS, MAA, NCTM (1991), UCSMP, Saxon, MI
Uses/Misuses/Applications
of Probability
CBMS-AMS, MAA, NCTM (2000), CMP, UCSMP,
Saxon, AR, FL, KS, MI, NJ
Representations of Probability
UCSMP, Saxon, GA, IN, AL, MI, CA, KS, NJ
Odds
NCTM (1991), UCSMP, Saxon, FL, KS
Sample spaces and
Counting Principles
MAA, CBMS-AMS, NCTM (1991, 2000), CMP,UCSMP,
Saxon, AL, AR, CA, FL, GA, MI, IN, KS, NJ, TX
Representations of Outcomes
CMP, UCSMP, Saxon, AL, CA, FL, GA, IN, NJ, TX
Theoretical versus Experimental
Probability including simulations
MAA, CBMS-AMS, NCTM (1991, 2000), CMP, Saxon,
UCSMP, AL, AR, CA, GA, KS, MI, IN, NJ, TX
Probabilistic Events
CBMS-AMS, NCTM (1991, 2000), UCSMP, Saxon, AL,
AR, CA, FL, GA, KS, MI, IN, NJ, TX
Probability Distributions
MAA, NCTM, CMP, UCSMP, AL, CA, MI, IN, NJ,
AR,GA
Hypothesis Testing
MAA, CBMS-AMS, UCSMP, AL, AR, CA, GA, MI, IN
Expected Value
CBMS-AMS, NCTM (1991), CMP, AR, CA, NJ
Central Limit Theorem
CBMS-AMS, UCSMP, AR, CA, GA, IN
Markov Chains
UCSMP, IN
Modeling
CBMS-AMS, NCTM (1991, 2000), CMP, UCSMP,
Saxon, AL, KS, NJ, TX
Table 3. Major Topics identified in the data sources
Summary
In recent decades, probability has come to gain importance as one of the content areas of school
curricula in the United States since its study “can raise the level of sophistication at which a
person interprets what he sees in ordinary life, in which theorems are scarce and uncertainty is
everywhere” (Cambridge Conference on School Mathematics, 1963, p. 70; as cited in Jones,
2004). However, research on teacher knowledge in this content area is scarce.
Preliminary results of this study (as indicated in Table 3) indicate the important probability
content topics that secondary mathematics teachers should know and be able to teach. The
identification of these topics is an essential tool that can be used in future research in this area.
In addition, this study can contribute to research efforts on teacher preparation and statistics
education by developing a systematic way of identifying the important aspects of content
knowledge for the teaching of probability in the secondary grades. Further refinements of the
major themes identified in Table 3 are currently under way in order to identify the probability
concepts and teacher knowledge associated with these themes that secondary mathematics
teachers should acquire and use in their mathematics classrooms.
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