Transcript Some Data on Mathematics Education
Integrating Statistics into Modeling-Based College Algebra
Sheldon P. Gordon
Florence S. Gordon
Accessing the Talk This PowerPoint presentation and the DIGMath Excel files that will be used can all be downloaded from:
farmingdale.edu/~gordonsp
College Algebra and Precalculus Each year, more than 1,000,000 students take college algebra and precalculus courses.
The focus in most of these courses is on preparing the students for calculus.
We know that only a relatively small percentage of these students ever go on to start calculus.
Some Interesting Studies
In a study at eight public and private universities in Illinois, Herriott and Dunbar found that, typically, only about 10-15% of the students enrolled in college algebra courses had any intention of majoring in a mathematically intensive field. At a large two year college, Agras found that only 15% of the students in college algebra planned to major in mathematically intensive fields.
Enrollment Flows
Based on several studies of enrollment flows from college algebra to calculus:
•
Less than 5% of the students who start college algebra courses ever start Calculus I
•
The typical DFW rate in college algebra is typically well above 50%
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Virtually none of the students who pass college algebra courses ever start Calculus III
•
Perhaps 30-40% of the students who pass precalculus courses ever start Calculus I
Some Interesting Studies
Steve Dunbar has tracked over 150,000 students taking mathematics at the University of Nebraska – Lincoln for
•
more than 15 years. He found that: only about 10% of the students who pass college
•
algebra ever go on to start Calculus I virtually none of the students who pass college algebra
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ever go on to start Calculus III. about 30% of the students who pass college algebra
•
eventually start business calculus.
about 30-40% of the students who pass precalculus ever go on to start Calculus I.
Some Interesting Studies
William Waller at the University of Houston – Downtown tracked the students from college algebra in Fall 2000. Of the 1018 students who started college
•
algebra: only 39, or 3.8%, ever went on to start Calculus I at
•
any time over the following three years. 551, or 54.1%, passed college algebra with a C or
•
better that semester of the 551 students who passed college algebra, 153 had previously failed college algebra (D/F/W) and were taking it for the second, third, fourth or more time
Some Interesting Studies
The Fall, 2001 cohort in college algebra at the University of Houston – Downtown was slightly larger. Of the 1028
•
students who started college algebra: only 2.8%, ever went on to start Calculus I at any time over the following three years.
The San Antonio Project
The mayor’s Economic Development Council of San Antonio recently identified college algebra as one of the major impediments to the city developing the kind of technologically sophisticated workforce it needs.
The mayor appointed special task force including representatives from all 11 colleges in the city plus business, industry and government to change the focus of college algebra to make the courses more responsive to the needs of the city, the students, and local industry.
Some Questions
Why do the majority of these 1,000,000+ students a year take college algebra courses?
Are these students well-served by the kind of courses typically given as “college algebra”?
If not, what kind of mathematics do these students really need?
Another Question
As calculus rapidly becomes (for better or worse) a high school subject, what can we expect of the students who take the courses before calculus in college?
Hard as it may be to believe, I expect that they will be more poorly prepared for these courses, which even more dramatically will not serve them well.
Why Do Our Students Fail?
They have seen virtually all of a standard skills-based algebra course in high school.
They do not see themselves ever using any of the myriad of techniques and tricks in the course (and they are right about that).
They equate familiarity with mastery, so they don’t apply themselves until far too late and they are well down the road to failure.
The Needs of Our Students The reality is that virtually none of the students we face in these courses today or in the future will become math majors.
They take these courses to fulfill Gen Ed requirements or requirements from other disciplines.
What do those other disciplines want their students to bring from math courses?
Mathematical Needs of Partners
•
In discussions with faculty from the lab sciences, it becomes clear that most courses for non-majors (and even those for majors in some areas) use almost no mathematics in class.
•
Mathematics arises almost exclusively in the lab when students have to analyze experimental data and then their weak math skills become dramatically evident.
Curriculum Foundations Project CRAFTY held a series of workshops with leading educators from 17 quantitative disciplines to inform the mathematics community of the current mathematical needs of each discipline.
The results are summarized in the MAA Reports volume: A Collective Vision: Voices of the Partner Disciplines, edited by Susan Ganter and Bill Barker.
What the Physicists Said
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Students need conceptual understanding first, and some comfort in using basic skills; then a deeper approach and more sophisticated skills become meaningful.
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Conceptual understanding is more important than computational skill.
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Computational skill without theoretical understanding is shallow.
What the Physicists Said
•
The learning of physics depends less directly than one might think on previous learning in mathematics. We just want students who can think. The ability to actively think is the most important thing students need to get from mathematics education.
What the Biologists Said
• • •
New areas of biological investigation have resulted in an increase in quantification of biological theories and models.
The collection and analysis of data that is central to biology inevitably leads to the use of mathematics.
Mathematics provides a language for the development and expression of biological concepts and theories. It allows biologists to summarize data, to describe it in logical terms, to draw inferences, and to make predictions.
What the Biologists Said
• • •
Statistics, modeling and graphical representation should take priority over calculus.
The teaching of mathematics and statistics should use motivating examples that draw on problems or data taken from biology.
Creating and analyzing computer simulations of biological systems provides a link between biological understanding and mathematical theory.
What the Biologists Said
• • • •
The quantitative skills needed for biology: The meaning and use of variables, parameters, functions, and relations.
To formulate linear, exponential, and logarithmic functions from data or from general principles.
To understand the periodic nature of the sine and cosine functions.
The graphical representation of data in a variety of formats – histograms, scatterplots, log-log graphs (for power functions), and semi-log graphs (for exponential and log functions).
What the Biologists Said
• • •
Other quantitative skills: Some calculus for calculating areas and average values, rates of change, optimization, and gradients for understanding contour maps.
Statistics – descriptive statistics, regression analysis, multivariate analysis, probability distributions, simulations, significance and error analysis.
Discrete Mathematics and Matrix Algebra – graphs (trees, networks, flowcharts, digraphs), matrices, and difference equations.
What the Biologists Said
•
The sciences are increasingly seeing students who are
•
quantitatively ill-prepared.
The biological sciences represent the largest science
•
client of mathematics education.
The current mathematics curriculum for biology majors does not provide biology students with
•
appropriate quantitative skills. The biologists suggested the creation of mathematics
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courses designed specifically for biology majors.
This would serve as a catalyst for needed changes in
•
the undergraduate biology curriculum.
We also have to provide opportunities for the biology faculty to increase their own facility with mathematics.
What Business Faculty Said
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Courses should stress problem solving, with the incumbent recognition of
•
ambiguities.
Courses should stress conceptual
• •
understanding (motivating the math with the “why’s” – not just the “how’s”).
Courses should stress critical thinking.
An important student outcome is their ability to develop appropriate models to solve defined problems.
What Business Faculty Said Mathematics is an integral component of the business school curriculum. Mathematics Departments can help by stressing conceptual understanding of quantitative reasoning and enhancing critical thinking skills. Business students must be able not only to apply appropriate abstract models to specific problems, but also to become familiar and comfortable with the language of and the application of mathematical reasoning. Business students need to understand that many quantitative problems are more likely to deal with ambiguities than with certainty. In the spirit that less is more, coverage is less critical than comprehension and application.
What Business Faculty Said
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Courses should use industry standard technology (spreadsheets).
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An important student outcome is their ability to become conversant with mathematics as a language. Business faculty would like its students to be comfortable taking a problem and casting it in mathematical terms.
The Common Threads
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Conceptual Understanding, not rote manipulation
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Realistic applications via mathematical modeling that reflect the way mathematics is used in other disciplines and on the job
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Statistical reasoning is primary mathematical topic in all other disciplines.
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Fitting functions to data/ data analysis
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The use of technology (though typically Excel, not graphing calculators).
Implications for College Algebra Students don’t need a skills-oriented course.
They need a modeling-based course that:
•
emphasizes realistic applications that mirror what they will see and do in other courses;
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emphasizes conceptual understanding;
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emphasizes data and its uses, including both fitting functions to data and statistical methods and reasoning;
•
better motivates them to succeed.
Further Implications
If we focus only on developing manipulative skills without developing conceptual understanding, we produce nothing more than students who are only Imperfect Organic Clones of a TI-89
Another Question
As calculus rapidly becomes (for better or worse) a high school subject, what can we expect of the students who take the courses before calculus in college?
Hard as it may be to believe, I expect that they will be more poorly prepared for these courses, which even more dramatically will not serve them well.
Should x Mark the Spot?
All other disciplines focus globally on the entire universe of a through z, with the occasional contribution of
through
.
Only mathematics focuses on a single spot, called x. Newton’s Second Law of Motion: y = mx, Einstein’s formula relating energy and mass: y = c 2 x, The Ideal Gas Law: yz = nRx. Students who see only x’s and y’s do not make the connections and cannot apply the techniques learned in math classes when other letters arise in other disciplines.
Should x Mark the Spot?
Kepler’s third law expresses the relationship between the average distance of a planet from the sun and the length of its year. If it is written as y 2 = 0.1664x 3 , there is no suggestion of which variable represents which quantity.
If it is written as t 2 = 0.1664D 3 , a huge conceptual hurdle for the students is eliminated.
A Modeling-Based Course
1. Introduction to data and statistical measures.
2. Behavior of functions as data and as graphs, including increasing/decreasing, turning points, concave up/down, inflection points (including normal distribution function).
A Modeling-Based Course
3. Linear functions, with emphasis on the meaning of the parameters and fitting linear functions to data, including the linear correlation coefficient to measure how well the regression line fits the data.
A Modeling-Based Course
• • • •
4. Nonlinear families of functions: exponential growth and decay, applications such as population growth and decay of a drug in the body; doubling time and half-life; power functions; logarithmic functions; Fitting each family of functions to data based on the behavioral characteristics of the functions and deciding on how good the fit is.
A Modeling-Based Course
5. Modeling with Polynomial Functions: Emphasis on the behavior of polynomials and modeling, primarily by fitting polynomials to data
A Modeling-Based Course
• •
6. Extending the basic families of functions using shifting, stretching, and shrinking, including: applying ideas on shifting and stretching to fitting extended families of functions to sets of data statistical ideas such as the distribution of sample means, the Central Limit Theorem, and confidence intervals.
A Modeling-Based Course
6a. Functions of several variables using tables, contour plots, and formulas with multiple variables.
A Modeling-Based Course
7. Sinusoidal Functions and Periodic Phenomena: using the sine and cosine as models for periodic phenomena such as the number of hours of daylight, heights of tides, average temperatures over the year, etc.
Some Illustrative Examples and Problems
The following table shows world-wide average Year temperatures in various years.
1880 1900 1920 1940 1960 Temp 13.80
13.95
13.90
14.15
14.00
1980 14.20
1990 14.40
2000 14.50
(a) Decide which is the independent variable and which is the dependent variable.
(b) Decide on appropriate scales for the two variables for a scatterplot.
(c) State precisely which letters you will use for the two variables and state what each variable you use stands for.
(d) Draw the associated scatterplot. (e) Raise some predictive questions in this context that could be answered when we have a formula relating the two variables.
The following table shows world-wide wind power generating capacity, in megawatts, in various years.
Year Wind power 1980 1985 1990 1995 1997 2000 2002 2004 10 1020 1930 4820 7640 13840 32040 47910
50000 40000 30000 20000 10000 0 1980 1985 1990 1995 2000 2005
(a) Which variable is the independent variable and which is the dependent variable?
(b) Explain why an exponential function is the best model to use for this data.
(c) Find an exponential function that models the relationship between power P generated by wind and the year t. (d) What are some reasonable values that you can use for the domain and range of this function?
(e) What is the practical significance of the base ( 1.1373
) in the exponential function you created in part (c)?
(f) What is the doubling time for this function? Explain what it means. Solve: 52.497(1.1373)
t
= 2 × 52.497
.
(g) According to your model, what do you predict for the total wind power generating capacity in 2010?
A Temperature Experiment
An experiment is conducted to study the rate at which temperature changes. A temperature probe is first heated in a cup of hot water and then pulled out and placed into a cup of cold water. The temperature of the probe, in ̊C, is measured every second for 36 seconds and recorded in the following table. Time 1 42.3
2 3 4 5 6 7 8 36.03 30.85 26.77 23.58 20.93 18.79 17.08
31 8.78
32 8.78
33 8.78
34 8.78
35 8.66 Find a function that fits this data.
36 8.66
A Temperature Experiment
The data suggest an exponential decay function, but the points don’t decay to 0.
To find a function, one first has to shift the data values down to get a transformed set of data that decay to 0.
45 40 35 30 25 20 15 10 5 time (1 - 36 seconds)
Then one has to fit an exponential function to the transformed data. Finally, one has to undo the transformation by shifting the resulting exponential function. T = 8.6 + 35.439(0.848)
t .
The Species-Area Model Biologists have long observed that the larger the area of a region, the more species live there. The relationship is best modeled by a power function. Puerto Rico has 40 species of amphibians and reptiles on 3459 square miles and Hispaniola (Haiti and the Dominican Republic) has 84 species on 29,418 square miles. (a) Determine a power function that relates the number of species of reptiles and amphibians on a Caribbean island to its area. (b) Use the relationship to predict the number of species of reptiles and amphibians on Cuba, which measures 44218 square miles.
The accompanying table and associated scatterplot give some data on the area (in square miles) of various Caribbean islands and estimates on the number of species of amphibians and reptiles living on each.
Island Redonda Saba Montserrat Puerto Rico Jamaica Hispaniola Cuba Area 1 4 40 3459 4411 29418 44218
N
3 5 9 40 39 84 76
100 80 60 40 20 0 0 15000 30000
Area (square miles)
45000
A Tale of Two Students
The Next Challenge: Statistics Based on the Curriculum Foundations reports and from discussions with faculty in the lab sciences (and most other areas), the most critical mathematical need of the partner disciplines is for students to know statistics. How can we integrate statistical ideas and methods into math courses at all levels?
The Curriculum Problems We Face
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Students don’t see traditional precalculus or college algebra courses as providing any useful skills for their other courses.
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Typically, college algebra is the prerequisite for
•
introductory statistics.
Introductory statistics is already overly crammed with far too much information.
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Most students put off taking the math as long as possible. So most don’t know any of the statistics when they take the courses in bio or other fields.
Integrating Statistics into Mathematics
•
Students see the equation of a line in pre algebra, in elementary algebra, in intermediate algebra, in college algebra, and in precalculus.
•
Yet many still have trouble with it in calculus.
They see statistics ONCE in an introductory statistics course. But statistics is far more complex, far more varied, and often highly counter-intuitive, yet they are then expected to use a wide variety of the statistical ideas and methods in their lab science courses.
Integrating Statistics in College Algebra
Data is Everywhere! We should capitalize on it.
1. A frequency distribution is a function – it can be an effective way to introduce and develop the concept of function.
2. Data analysis – the idea of fitting linear, exponential, power, polynomial, sinusoidal and other functions to data – is already becoming a major theme in some college algebra courses. It can be the unifying theme that links functions, the real world, and the other disciplines.
Integrating Statistics in College Algebra
But, there are some important statistical issues that need to be addressed. For instance: 1. Most sets of data, especially in the sciences, only represent a single sample. How does the regression line based on one sample compare to the lines based on other possible samples? 2. The correlation coefficient only applies to a linear fit. What significance does it have when you are fitting a nonlinear function to data?
Integrating Statistics in College Algebra
3. The z-value associated with a measurement x is a nice application of a linear function of x:
z
x
It can provide the source of many algebra problems that have a simple underlying context.
Integrating Statistics in College Algebra
4. The normal distribution function is
1
e x
2 2
It makes for an excellent example involving both stretching and shifting functions and a function of a function.
Match each of the four normal distributions (a) (d) with one of the corresponding sets of values for the parameters μ and σ. Explain your reasoning.
(i) μ = 85 , σ = 1 (ii) μ = 100, σ = 12 (iii) μ = 115 , σ = 12 (iv) μ = 115 , σ = 8 (v) μ = 100 , σ = 6 (vi) μ = 85 , σ = 7
0.1
0 50 0.1
0 50
(a) (c)
100 100 0.1
150 0 50 0.1
150 0 50
(b)
100 100
(d)
150 150
Integrating Statistics in College Algebra
5. The Central Limit Theorem is another example of stretching and shifting functions -- the mean of the distribution of sample means is a shift and its standard deviation
x
n
produces a stretch or a squeeze, depending on the sample size n.
Some Conclusions
Few, if any, math departments can exist based solely on offerings for math and related majors. Whether we like it or not, mathematics is a service department at almost all institutions.
And college algebra and related courses exist almost exclusively to serve the needs of other disciplines.
Some Conclusions
If we fail to offer courses that meet the needs of the students in the other disciplines, those departments will increasingly drop their requirements for math courses. This is already starting to happen in engineering.
Math departments may well end up offering little beyond developmental algebra courses that serve little purpose.
Accessing the Talk This PowerPoint presentation and the DIGMath Excel files that will be used can all be downloaded from: