Solving Systems of Linear Equations: Changing Methods from 1820 – 2000

Looking through mathematics curriculum with a historical perspective has been interesting.  I am reminded that mathematics truly is evolving – the mathematical landscape is constantly changing.  When considering the topics that should be taught in algebra courses, or other mathematics courses as well, we should not throw topics away without careful consideration but we also should not keep topics only through loyalty to the past.  Unfortunately, it is difficult to understand fully the interdependence of topics and thus difficult to choose which topics are needed and which can be disposed of.

I began with a contemporary view of possible solution methods – graphical, tabular, matrix and elimination – and assumed that elimination methods have remained more or less stable (forever) and that matrix solutions (determinants) would also have been included even in early books.  I guessed that tabular and graphical methods would be used but not until mid-1900s.   Originally, I had especially been interested in the use of matrices in general in these textbooks, but it turns out that very few high school textbooks include use of matrices in general and then only as solution to systems of equations. I was not surprised to find that there are some consistent, basic elimination methods that have been taught for the past 200 years.  I was surprised to find that there had been additional methods that seem to have disappeared more recently.

For this project, I selected 82 algebra textbooks, intended for all levels of students, which were published in the United States between 1800 and 2000.  I attempted to create a list that was representative of typical algebra classroom texts. For textbooks published between 1800 and 1900, I based my selections on the catalog of algebra textbooks held by the Educational Research Library that was published in Svobodny (1985).  In the catalog were listed algebra textbooks including multiple reprints of some, as well as answer keys, exercise books and textbooks we would today consider as calculus. First, I went through the list to find textbooks that were intended as a treatment of algebra, including many that were ungraded (especially in the early 1800s) and many that included arithmetic, trigonometry or calculus as well as algebra.  I was able to find a digital copy of every title listed, even if occasionally the copy was an earlier printing or a reprint of the text.  I attempted to categorize each textbook according to the level of the intended audience (i.e., university, secondary or grammar) since the twentieth-century textbooks were all intended for high school students, however not every textbook was graded and not every author stated the intended audience.  Some textbooks were intended for any level from grammar-school to university students, but gave instructions that the younger student should begin by omitting certain sections or not attempting to understand the deeper mathematics on the first reading. In general, because systems of linear equations seemed to be acceptable for any level and because it was so difficult to separate the earlier textbooks into grade-level categories, I did not exclude the lower and higher levels. I feel this was worthwhile since I was then able to make comparisons between topics taught at different grade levels.

For textbooks published between 1900 and 2000, I used the list of “typical and other” algebra textbooks created and discussed in Donoghue (2003).  I was able to obtain a copy of each title on her list except John Swenson’s Integrated Mathematics with Special Applications to Algebra. Donoghue (2003) created this list based on a survey of algebra textbooks published in twentieth-century United States.  She included 2 or 3 textbooks for each twenty-year period with at least one textbook being an example of a “typical” high school algebra textbook and one being an example of an integrated or otherwise atypical algebra text.  In this case, Swenson’s text was considered atypical, so I felt that replacing it with a textbook published in that same timeframe would provide useful data, even if it was chosen for convenience (from Michigan State University Mathematics Library) rather than as being representative of a typical textbook.  I am worried that this made my sample less representative, but I felt it was already problematic. Because of this, I am not able to make strong claims about American algebra textbooks in general, but only the group of textbooks that I analyzed.  If I had the time to develop a better and more representative list of textbooks, I am still unsure whether that would be possible.

Once I had created this list of textbooks, and obtained digital or physical copies of each text on the list, I conducted a superficial survey of each.  In this superficial survey I recorded general data about the book, the author, the types of solution methods presented for systems of linear equations, and the types of systems of linear equations presented (number of unknowns, literal, numerical, application, etc.).  Regarding the author, I recorded the listed credentials (degree, position, previous textbooks published) as well as the author’s stated purpose for writing the textbook.

I did not analyze the credentials or stated purposes of the authors.  Without the numbers, I cannot state a certainty, but in general there seemed to be some differences between authors’ intents from the 1800s to the 1900s.  In the 1800s, the author generally referred to writing the textbook to follow a specific method: French (rigor or deeper mathematics); German (begins with the more advanced topic and then unpacks it); English (practical mathematics); subdivisions (breaking each topic into the smallest, most easily digested “chunks”); analytic or synthetic (rule-based); or inductive (“proceeds from the particular to the general” (Van Velzer & Slichter, 1890)) methods seemed to be the most common.  Even particular authors showed signs of changing. For example, Thomson and Quimby (1887) stated that they included many examples to “stimulate thought on the part of the student” but a few years later Thomson (1896) stated that he would not include so many examples that merely were “puzzles, calculated to waste the time and energy of the pupil.”

In contrast, in the 1900s, it seemed that authors often referred to the new educational theories, recommendations or standards, if they stated a purpose at all.   Many authors, whether in the 1800s or the 1900s, stated that students would learn to think by learning mathematics and others stated that in order to learn mathematics, students needed to learn to think. Authors in the 1800s seemed to be at all levels of education (A.B. to A.M. to Ph.D.) and placement (instructors at common schools, academies, normal schools and universities). If I had more time, I would analyze this in more detail to see if there are any trends over time related to these intents.

I saw a trend from the 1800s to 2000 of authors moving from presentation of algebraic methods (comparison, substitution, addition-subtraction, Bezout’s, division, etc.) to multiple representations (graphical, tabular, determinant, etc.). Of the methods employed, there were three or four methods that were taught for solving standard systems of equations before 1900: elimination through comparison, substitution, addition/subtraction and Bezout’s method (also called “method of undetermined multipliers” – rare in secondary algebra but relatively popular in college algebra).  Additional methods include: dividing one equation by the other, multiplying each side of an equation to the corresponding side of the other equation, and creating simpler equations using linear combinations of the equations in the set.

Systems-01

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In Figure 1, the percentage of textbooks with a specific solution method for each twenty-year period is shown.  I provide brief descriptions ­­­­­of the less well-known methods below. It can be seen that the addition-subtraction and substitution methods seem to be stable since 1820.  On the other hand, the comparison method all but disappears in the mid-1900s, while the graphing and tabular methods gain popularity.  In the textbooks I looked at, determinants showed up briefly in the mid-1900s but then disappeared.  The increasing popularity of alternate representations, such as graphing, tables and determinants, is consistent with policy documents  throughout the twentieth century which called for multiple representations (MAA, 1923; NEA, 1894).

The “other methods” category includes a variety of solution methods: Bezout’s method (or undetermined multipliers), division methods, multiplication methods, methods using a general formula and methods using linear combination simplifications of the equations. These other methods, as well as the comparison, substitution and addition-subtraction methods are all algebraic in nature.  In the 1800s, almost every textbook included at least the comparison, substitution and addition-subtraction methods, usually starting with comparison.  In later books, the variety of methods used for elimination dwindled to substitution and addition/subtraction.  In the chart above, it can be seen that certain newer methods replaced the others – solutions using determinants, as well as graphical and numerical methods became standard. Only two books intended for secondary schools included determinant methods – one in 1897 and one in 1912.  Only one book intended for colleges (1898).  Four books intended for secondary schools included graphing methods: 1906, 1912 and 1920. No books intended for universities included graphing methods.

Comparison

Systems-02

The most popular method in the 1800s was solution by comparison, as illustrated by this image from Loomis (1870).  The method is to solve each equation for the same variable and then set them equal to each other and solve for the remaining variable.  I assume that the substitution method must generally result in fewer steps which is why this method fell out of favor in the popular algebra textbooks.

Linear Combinations

When given a name, some authors refer to this method as the method of “derived equations” (Collins, 1893):

Systems-03

This example is from Van Velzer and Slichter (1890).  The method is to add and subtract the given equations (or multiples thereof) in order to derive two new independent equations that are easier to deal with.  Once the new equations have been created, one of the more common methods of elimination is used to find the solution.

Bezout’s Method (Undetermined Multipliers)

Systems-04

The method of undetermined multipliers shows up relatively often in Algebra textbooks intended for college or secondary students in the 1800s. It is always referred to as the method of undetermined multipliers, but many authors also reference it as “Bezout’s method” or the “French Method.” This example from Collins (1893) shows the basic process.  First, the student would multiply one equation by an unknown value (represented by m). Next, add the transformed equation to the other equation.  Finally, choose a value for m so that one of the variables will be eliminated.

Division

This is a cute little method. First, manipulate the equations so all terms are on one side of the equals sign.  Then (to avoid fractions) multiply one equation so the coefficients are multiples of the initial coefficient of the other.  Divide the equation with the larger coefficient by the other. The result will be an expression with only one unknown. Set this equal to zero and solve for the unknown. Finally, use substitution to find the other unknown.

Systems-05

This example of the method of elimination by division is from Thomson and Quimby (1887).  This method is similar to the method of undetermined multipliers. In essence, we are subtracting a multiple of one equation from the other in a way that eliminates one variable, but the determination of which multiple to use is embedded in the standard division algorithm.

Multiplication

Only one textbook used the multiplication method, but I include it because it serves as an interesting example of the algebraic depth of some earlier textbooks. This example is from Van Velzer and Slichter (1888). I find this solution method interesting since it seems that to be only a mathematical exercise for the authors in equivalent systems of equations rather than an effective solution method. The authors point out that it will introduce additional solutions in certain cases (when each equation is equal to something other than a known quantity) so that the resulting system of equations is not necessarily equivalent. My interpretation of this is that it is intended to show students why certain methods are valid and why other methods are not, similar to teaching elementary students today about alternate base systems when learning base 10. This might help students think more deeply about the meaning of equivalent systems, which might help them later when solving systems of higher-degree equations.

Systems-06

This example shows the solution of a system of equations with two unknowns by multiplying each left-hand side by the other left-hand side and each right-hand side by the other right-hand side.  Because the author solved the first equation for x (instead of leaving both unknowns on the left-hand side), the solution introduced an extra solution as well.  This method could only be for curiosity’s sake since it does not always result in an equivalent set of equations and the process is very similar, but more complicated, than solving using the substitution method.

References

Collins, J. V. (1893). Text-book of algebra: Through quadratic equations. Chicago Albert, Scott & Company.

Dolciani, M. P., Berman, S. L., & Freilich, J. (1965). Algebra : Structure and Method: Houghton Mifflin.

Donoghue, E. F. (2003). Algebra and Geometry Textbooks in Twentieth-Century America. In G. M. A. Stanic & J. Kilpatrick (Eds.), A History of School Mathematics (Vol. 2, pp. 1629 – 1700). Reston, VA: National Council of Teachers of Mathematics.

Giffin, W. M. (1895). Grammar-school algebra: seventy-five suggestive lessons for beginners: Werner School Book Co.

Hutton, C., Adrain, R., & Gregory, O. (1831). A Course of Mathematics: For the Use of Academies as Well as Private Tuition : in Two Volumes: W.E. Dean.

Loomis, E. (1870). Elements of algebra: Harper & Bros.

MAA. (1923). The reorganization of mathematics in secondary education: a report of the National Committee on Mathematical Requirements under the auspices of the Mathematical Association of America, Inc.: The Mathematical Association of America, Inc.

Milne, W. J. (1920). High school algebra: American Book Co.

NEA. (1894). Report of the Committee of Ten on Secondary School Studies: With the Reports of the Conferences Arranged by the Committee: Published for the National Education Association, by the American Book Company.

Svobodny, D. (1985). Early American Textbooks, 1775-1900. A Catalog of the Titles Held by the Educational Research Library. Washington, D.C.: U.S. Department of Education.

Thomson, J. B. (1896). New practical algebra: adapted to the improved methods of instruction in schools, academies, and colleges with an appendix: Maynard.

Thomson, J. B., & Quimby, E. T. (1887). The collegiate algebra: adapted to colleges and universities. New York: Clark & Maynard, Publishers.

Van Velzer, C. A., & Slichter, C. S. (1888). A Course in Algebra: Being Course One in Mathematics in the University of Wisconsin. Madison, Wisconsin: Capital City Pub. Co., Printers.

Van Velzer, C. A., & Slichter, C. S. (1890). School algebra. Madison, Wisconsin: Tracy, Gibbs & Co.

Welchons, A. M., Krickenberger, W. R., & Pearson, H. R. (1976). Algebra, Book 1: Elementary Course. Lexington, MA: Ginn and Company.

Wells, W., & Hart, W. W. (1912). New High School Algebra. Boston: D. C. Heath & Co., Publishers.

 

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Algebra and Graphing Calculators

Calculators are widely used in classrooms today to support student learning of mathematics.  Ten years ago, Dion et al. (2001) conducted a wide-spread survey of school calculator use.  Of the 4,568 schools that responded, 95% answered that they either require (46%) or at least allow (49%) the use of graphing or scientific calculators.

Both the Common Core State Standards in Mathematics (CCSS-M) and the National Council of Teachers of Mathematics (NCTM) Standards make strong recommendations for the use of technology in the classroom. Technology should be a tool “in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning” (NCTM, 2000, p. 24). The Common Core State Standards (2010) Standards for Mathematical Practice states, as an example of student tool use, that “mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator.  They detect possible errors by strategically using estimation and other mathematical knowledge” (p. 7).

The widespread use of calculators in the classroom is not necessarily equivalent to widespread effective use.  Milou (1999) surveyed 146 teachers and concluded that “successful integration of the graphing calculator into the mathematics classroom cannot take place without the aid of enthusiastic teachers.”

Both the NCTM and CCSS-M standards previously quoted encourage the use of technology as tools, when appropriate.  Hiebert et al. (1997)7) stated: ‘Students must construct meaning for all tools…. As you use a tool, you get to know the tool better and you use the tool more effectively to help you know about other things.’ (p. 54). This suggests that teachers and students both must learn to use technology such as graphing calculators by co-constructing that knowledge.

Calculators as Tools not “Dispensers of Answers”

In 1981 a study was conducted to study the process of student thinking in estimation tasks.  As one part of the study, the researchers asked 33 participants of various ages from child to adult, who had been identified as the top 10% of estimators in previous interviews, to estimate a calculation and then to use a calculator to check the answer.  They used a specially programmed HP-65 calculator that would make increasingly large errors in calculation.  In the study, 36% of females and 77% of males said something about the calculator being in error before they reached the last problem. One conclusion drawn by the researchers was that: “[t]he unwillingness of these good estimators to reject unreasonable answers suggests that a challenging task lies ahead in preparing students to be alert to unreasonable answers” (Reys, 1980, p. 231).

This study was redone by Glasgow and Reys (1998) almost 20 years later with 25 undergraduate participants each of whom had been identified as a top student in his or mathematics course.  In this second experiment, only 28% of the participants questioned the calculator before the last question.  Several of the participants in this second study were pre-service teachers.  One statement by the authors in the conclusion mentioned the importance of convincing these pre-service teachers “to question or reflect on their own mathematical thinking”.  The authors also argued that students needed to see calculators “as tools, not ‘dispensers of answers’” (p. 388).

In order to be able to use the calculator effectively in the classroom to support student learning of effective use of this tool rather than dependence on it, teachers must first construct their own knowledge of the calculator as a tool rather than as a sort of magic or “black box” that is unconnected to reality and is unpredictable.

Knowledge for Teachers

Chamblee, Slough, and Wunsch (2010) conducted a study to measure the concerns of in-service teachers before and after a year-long professional development program.  This program intended to support mathematics and science teachers in their use of graphing calculators, probes and sensors in the classroom.  90 teachers received 105 days of professional development throughout the school year and following summer.  The teachers also received a classroom set of graphing calculators, probes and sensors. The researchers tested 22 teachers before and after 105 days of professional development.  One of their findings was teachers needed professional development activities that allow them to learn how to use the graphing calculator in their own day-to-day classroom, instead of watching the professional development facilitator “show-and-tell” possible uses and activities.

This seems to suggest that seeing the calculator and its functionality is not enough to incorporate it into classroom life. The teacher needs opportunities to meaningfully construct her own knowledge – she needs to be involved in her own learning process in order to construct the knowledge necessary to strategically use the calculator in the classroom.

Support for Student Learning

In Doerr and Zangor (2000) this co-construction of the graphing calculator as a tool was studied in two pre-calculus classes all taught by the same teacher. The students used graphing calculators in class, along with probes and sensors for data collection.  The researchers suggested that both teachers and students can use a graphing calculator in different ways: as a computational tool, transformational tool, data collection and analysis tool, visualizing tool, or checking tool (p. 151).  As a computational tool, it is merely used to evaluate numerical expressions.  The danger is that students will see it only as a computational tool (Herman, 2007; Ruthven, 1995).  As a transformation tool, the calculator transforms “tedious computational tasks… into interpretative tasks” (p. 153).  In support of this role, the teacher must be often re-focusing the students’ attention away from the computation and to interpretation.  As a checking tool, the students might use the calculator to verify results acquired through different methods.  One danger of allowing students to use the calculator too often as a checking tool might be that it would lead to the students abdicating responsibility to their calculators, as discussed above in Reys (1980) and Glasgow and Reys (1998).

Pugalee (2001) also suggested that graphing calculators are not a magic device that result in student learning.  He argued that technology must be used with intentional discourse in order to be effective.  He performed a study using constructivism and graphing calculators in an algebra course for “at-risk” students.  The study covered only one section of 16 students, but the results were favorable. The use of graphing calculators, along with questioning, supported the students’ construction of mathematical knowledge. These results support Doerr and Zangor (2000) above since they also argued that the teacher needs to focus the students’ attention on interpreting calculator results instead of only focusing on the calculation in order to increase their mathematical knowledge.

References

CCSS-I. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Chamblee, G. E., Slough, S. W., & Wunsch, G. (2010). Measuring High School Mathematics Teachers’ Concerns About Graphing Calculators and Change: A Year Long Study. Journal of Computers in Mathematics and Science Teaching, 27(2), 183.

Dion, G., Harvey, A., Jackson, C., Klag, P., Liu, J., & Wright, C. (2001). A survey of calculator usage in high schools. School Science and Mathematics, 101(8), 427-438.

Doerr, H. M., & Zangor, R. (2000). Creating meaning for and with the graphing calculator. Educational Studies in Mathematics, 41(2), 143-163.

Glasgow, B., & Reys, B. J. (1998). The authority of the calculator in the minds of college students. School Science and Mathematics, 98(7), 383-388.

Harvey, J. G., Waits, B. K., & Demana, F. D. (1995). The influence of technology on the teaching and learning of algebra. The Journal of Mathematical Behavior, 14(1), 75-109.

Herman, M. (2007). What Students Choose to Do and Have to Say About Use of Multiple Representations in College Algebra. The Journal of Computers in Mathematics and Science Teaching, 26(1), 27-54.

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Wearne, D., Murray, H., . . . Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.

Hollar, J. C., & Norwood, K. (1999). The effects of a graphing-approach intermediate algebra curriculum on students’ understanding of function. Journal for Research in Mathematics Education, 30(2), 220-220.

Mayes, R. L. (1995). The application of a computer algebra system as a tool in college algebra. School Science and Mathematics, 95(2), 61-68.

Milou, E. (1999). The Graphing Calculator: A Survey of Classroom Usage. School Science and Mathematics, 99(3).

NCTM. (2000). Principles and Standards for School Mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc.

O’Callaghan, B. R. (1998). Computer-intensive algebra and students’ conceptual knowledge of functions. Journal for Research in Mathematics Education, 21-40.

Pugalee, D. K. (2001). Algebra for all: The role of technology and constructivism in an algebra course for at-risk students. Preventing School Failure, 45(4), 171-176.

Pullano, F. (2005). Using Probeware to Improve Students’ Graph Interpretation Abilities. School Science and Mathematics, 105(7), 373-376.

Reys, R. E. (1980). Identification and Characterization of Computational Estimation Processes Used by Inschool Pupils and Out-of-School Adults. Final Report.

Ruthven, K. (1995). Pupil’s View of Number Work and Calculators. Educational Research, 37, 229-237.