Gambling with Woot

MysteryBox-01

A few weeks ago, Woot.com advertised a “Mystery Box of Electronics.” One box cost $50. Woot said: “Not knowing is half the fun!” Each box would include:

  • Random consumer electronics items
  • You get 3 items per order in a brown box (nothing special about the box!)
  • Condition will be refurbished on most items. Some could be new

So, I decided, why not? And I gambled on Woot – I bought three mystery boxes of electronics! (Because not knowing really is kind of fun…)


I received my last box today! Here is what I received:

2015-09-17 13.34.58

Box 1:

I received three items:

  • A very pink “beat mixr” headset. Other listings on Amazon.com show prices for new/used (for some reason some new are cheaper than the used!), starting as low as: $110.
  • A small Acesori PowerStick. Other listings on Amazon.com show prices for new/used, starting as low as: $4.
  • An Acesori Glass Vault screen protector for an iPhone 5. Other listings on Amazon.com show prices for new/used, starting as low as:  $7.

2015-09-17 13.31.56

Box 2:

In the box, I found:

  • ifocus “Deluxe edition” educational system for kids (a set of CDs: one is an educational game and the other is a fitness program). The company that sells these says they are $199.95, but it doesn’t look like I could resell them
  • An Acesori Glass Vault screen protector for an iPhone 5. Other listings on Amazon.com show prices for new/used, starting as low as: $7.
  • Acesori Bluetooth Noise-canceling Neckband Headset with Built-in Microphone. Other listings on Amazon.com show prices for new/used, starting as low as: $30.
  • LG Electronics Gruve Bluetooth Stereo Headset. Other listings on Amazon.com show prices for new/used, starting as low as: $30.

WP_20150922_17_20_56_Pro

Box 3:

  • BlueAnt Pump – Wireless HD Sportbuds – Black. Other listings on Amazon.com show prices for new/used, starting as low as: $26.
  • Turtle Beach Ear Force Z11 Amplified Gaming Headset. Other listings on Amazon.com show prices for new/used, starting as low as: $24.
  • TruGuard Tempered Glass Screen Protector for iPhone 5/iPhone 5S & iPhone 5C. Other listings on Amazon.com show prices for new/used, starting as low as: $6.

So the big mathematics question is: Did I get my money’s worth?

In the Woot discussion (http://electronics.woot.com/forums/viewpost.aspx?postid=6481553&pageindex=60), many customers say, “No!” Others say, “Yes!” I’ll give several comments here, and let you decide!

  • The value is there for at least what everyone paid for when they ordered it. The problem is people discount the value simply because it’s something they don’t want or can’t use, or they deem to be useless. That’s on them, not Woot.
  • This sale rather was marked as getting mystery items that were either new or refurbished worth at least $200. So for $50 bucks, if thats the case, how could we go wrong. We should all consider that that this $200 value should not by any means be gauged on our perception of value, but rather fair market value, which is really the point of taking up a mystery box. It may be worth it to you, but it may not. If its not, you can go through the legwork of selling it off and making back your money.
  • (received my box 3 above) I am not satisfied because I don’t feel this box is worth the 50 bucks I paid. But still I will get use out of some of it. I understand that with these boxes you run the risk of getting items you don’t want or won’t use but you expect the value to still be there.
  • As to “value”, that is in the eye of the beholder. The great marketing dilemma. Do we want to talk about MSRP, street price at release, current street price, or the best deal available? Naturally, very few items ever sell at MSRP, but it is still used everywhere to convince everyone how much money they are saving at the current street price.
  • Overall – the value of the boxes I received is total crap – at least for me.
    IMHO no quality items. Protectors for outdated devices. MSRP that are years old and cannot really count against today’s value. Overall inflated MSRP assumptions of products that cost cents to make (like cases and protectors)… no value there.

So…. what do you think? Was it a good decision or bad?

Paying a tithe (10% of earnings)

In Sunday School, I teach kids who are turning 7 this year. There is a lesson manual and a schedule to follow, but I like bringing in authentic tasks (especially when it is math-related!) whenever possible.

Today the lesson was “I can pay tithing.” We talked about the meaning of tithe (Merriam-Webster says: “to pay or give a tenth part of especially for the support of the church”).

I brought a sack of pennies (about one-year’s accumulation!) and gave each of the kids a pile. I asked them, “Find out how many pennies you have and then tell me how many pennies you’ll give for tithing.”

Multiple Answers

One child said, “I’ll give them all.” I responded, “Sure. You don’t have to give all of your pennies, but you can. Do you still want to?”  “Uh-huh.” “Why do you want to give all of them instead of just 10%?” “Well. I don’t need them. I could keep some.” After church was finished, he came running back to tell me “I kept some of my pennies but I put some on the ground so that someone can find it for a lucky penny!”

Another child said, “I’ll give 4 because I’ve got 43 pennies.” I responded, “Okay. How did you figure that out?” “It’s easy! Every time I count 10 pennies, I take one.”

Another child said, “Can I keep the extra pennies?” I responded, “Sure!” “Okay, I’ll give 17 because I have 39 pennies.” “Okay, how did you decide to give 17?” “Because it looks like half.” “Ok. You can decide how much you want to give. The church asks for 10% but it’s okay to give more. Do you still want to give 17?” “Uh-huh!”

I gave them each a tithing envelope, they filled out their own slip (mostly), put the pennies in the envelope, and brought them to the bishop.

Assumptions

Sometimes in textbooks a word problem is given and the question asks: “Which should you choose?” or “What’s the best choice for Rohit?” Those problems make me cringe because it is making an assumption that “optimization” means the same thing for everyone, especially when some context is given. In this example, the kids all have different reasons for wanting to keep or give away the pennies – some told me they have $20 or $30 at home to spend, so maybe they don’t value pennies much. Others were already making plans for how they would spend their pennies. Some kids are just too darn nice.

But, if I pushed them to only pay 10%, then it feels like I’m making an assumption that you should always give the minimum. At the same time, by not pushing them to pay only 10%, I feel like I’m telling them they should always pay more than the minimum. So, even for me, I’m not sure what the “right” response is in this real-world math problem.

Filling the space in length measurement

Session presented at the Michigan Council of Teachers of Mathematics annual conference in Traverse City, MI. In this session we provided a variety of tasks to enhance student learning of length measure. We examined the meaning of length as an attribute and ways of quantifying length. We explored length measurement with a variety of tasks that can push us (and our students) to think about aspects of measurement, such as filling a space without leaving gaps or overlaps.

In this session, we provided stations, each with a different measurement task and set of physical tools. Tasks were created based on tasks designed by Strengthening Tomorrow’s Education in Measurement project, based on articles from Teaching Children Mathematics, or based on observed tasks from methods courses. Participants engaged in two or more stations, tracking the tools and ideas they used, and we finished with a discussion. The hand-outs and a brief description are provided below:

  • Station01-Comparing Units – An illustration of three “wrong” strategies for measuring and comparing the lengths of paths is given. Students are asked to explain why or why not they agree with the measurement. Students could also be asked to explain the possible thinking behind each of the three strategies, and why it would or would not “work.”
  • Station02-Measuring Paths – Fourth paths are given. Each path is partitioned in different ways. Students are asked to compare lengths of paths, to explain their reasoning, and how they chose measurement units.
  • Station03-Measuring Height – Students in pairs compare their heights and indicate who is taller. They use non-standard units of measure to prove they are correct by measuring each other and comparing responses. They should each measure each other with two tools.
  • Station04-Measuring Tools – Students choose an object to measure, and use a non-standard unit of measurement to measure the object. They write or draw their strategy and then write or draw their understanding of the meaning of length.
  • Station05-Popsicle Comparison – Students measure two objects, each with a longer popsicle stick and shorter popsicle stick. The students then discuss any ideas they have or anything they notice about the measurements.
  • Station06-Strange Rulers – In order to make sense of the important features of standard rulers, students use several “strange rulers” to measure a paperclip and a piece of string and make sense of which rulers “work” and which don’t.

Stehr, E.M., Gönülateş, F., & Nimtz, J.L. (2013, August) Filling the space in length measurement. Paper presented at the Michigan Council of Teachers of Mathematics Conference, Traverse City, MI.

Questioning in College Algebra

Questioning in College Algebra

College Algebra courses serve as gatekeepers to higher mathematics courses in universities, as well as to many technical and science-intensive college majors (Small, 2002).  More students enroll in College Algebra and similar courses than in all Calculus courses (Lutzer, Maxwell, & Rodi, 2007). Despite high enrollments, Small quotes a Dean of Science and Mathematics as stating, “Traditional College Algebra is a boring, archaic, torturous course that … turns off students and discourages them from seeking more mathematics learning” (Small, 2002). Small’s recommendations include a more student-centered College Algebra.

One way for teaching to be more student-centered is through focusing on types of questioning to become more intentional in using questions to support student learning. Driscoll (1999) presented a framework for questioning fostering algebraic thinking.

Theoretical Framework

Driscoll (1999) recommended teachers use variety in questioning to support development of algebraic habits of mind. Driscoll organized five categories of teacher intentions: (a) Managing, (b) Clarifying, (c) Orienting, (d) Prompting Mathematical Reflection, and (e) Eliciting Algebraic Thinking, as described in Table 1 below (examples are from my observations). Note that I am not including the first category (Managing) since the course I observed is lecture-based.

My research question is: How does questioning by college algebra instructors to questioning intentions recommended by Driscoll (1999) for fostering algebraic thinking in grades 6-10?

Method

I observed three 80-minutes class sessions for a small-lecture College Algebra section at Michigan State University that meets on Tuesday and Thursday evenings. The department provides class notes to instructors and this instructor emailed them to his students. Notes include definitions and examples with space provided to write additional notes and solutions. The instructor is in his final semester of a mathematics master’s program and intends to not continue teaching mathematics.

I wrote field notes focused on instructor questions and their context. I spent an hour after each session writing my impressions. I coded instructor questions were according to Driscoll’s (1999) four categories of teacher intentions, described in Table 1 with examples of questions from my observations.

Questioning-01

Results

I recorded 182 instructor questions across the three observed sessions. Table 2 below shows the results of coding each session according to four categories of Driscoll’s (1999) framework for intentions.

Questioning-02

For the most part, each of the questions asked by the instructor did fall into one of the four categories. Most of the questions that did not fall into one of these categories were of the type: “Any questions?”

Discussion

The instructor asked a variety of questions, and most questions fit into one of Driscoll’s (1999) five categories. The majority of questions fell into the Orienting category.  Almost half of the Orienting questions were “Doing” questions, that is, they ask about how to do the next step of the problem. For example, “So what’s first?” or “Okay, and we have one step left, now what do we want to do?” These questions fit in the first half of Driscoll’s description: “Intended to get students started, or to keep them thinking about the particular problem they are solving…” (Driscoll, 1999, p. 6). The remainder of the questions coded as Orienting fall into the second half of Driscoll’s description (see Table 1).  For example, [finding a radius] “Okay, and what does the square root of 40 simplify to?” or “I want you to write this formula in standard form. So what does it look like? Start with x – so what goes right there?” I felt in my coding that these types of questions were different enough that the Orienting category might be more usefully broken into smaller categories such as “Doing” as opposed to “Leading to Correct Answer.”

Questioning frameworks are useful focusing devices to help instructors reflect on their teaching. The questioning framework developed by Driscoll (1999) is intended to help secondary mathematics teachers focus on developing students’ algebraic habits of mind. I would argue that this framework would also be useful for College Algebra instructors and that the instructor I observed uses questions from this framework (whether consciously or not) to support students’ development of algebraic habits of mind. Using this framework to track questions types over time could help a College Algebra instructor become more planful about question use in support of students’ learning.

References

Driscoll (1999). Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10. Westport, CT: Heinemann.

Lutzer, D. J., Maxwell, J.W., and Rodi, S. B. (2007) Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States: Fall 2005 CBMS Survey. American Mathematical Society: Providence, RI. Retrieved from http://www.ams.org/cbms/cbms2005.html

Small, D. (2002) An urgent call to improve traditional college algebra programs. Focus: The newsletter of the mathematical Association of America, 12-13.

What is Algebra? (Historically)

At the beginning of my second year, I thought it would be interesting to explore definitions of Algebra in historical texts: both how the author described Algebra and also what topics were included in Algebra. Unfortunately, I have not had time to do a more comprehensive search, but I found five British and American texts that were interesting and spanned about a century. (I chose to focus on these countries because the texts are readily available in digital form and also because I am limited in my language).

Please see below for additional details, but I list their definitions here:

  • Hutton (1831) wrote, “Algebra is the science of investigation by means of symbols. It is sometimes also called Analysis; and is a general kind of arithmetic, or universal way of computation.”
  • Smith (1848) defined Algebra, writing “By Algebra, which is a branch of mathematics, in which the quantities considered are represented by letters.” He continued, “Algebra dates a more recent origin than either Arithmetic or Geometry. It was not until Arithmetic had advanced to a considerable degree of perfection, and mathematicians had commenced to feel the necessity for abridging, as well as generalizing, its operations, that Algebra was introduced. In the early part of the 13th century, Leonardo, a merchant of Pisa, having made repeated visits to Arabia, returned to Italy with a knowledge of Algebra. A manuscript of his is quoted as far back as 1202.”
  • Davies (1861) wrote that “Algebra is that branch of mathematics in which the quantities considered are represented by letters, and the operations to be performed upon them are indicated by signs. These letters and signs are called symbols.”
  • Taylor (1889) wrote that “Algebra is the science of algebraic number and the equation. It differs from arithmetic (i.) in its number. The number of algebra has quality as well as arithmetical value. The double series of numbers in algebra gives a wider range to operations. Thus, in algebra to subtract a great quantity from a less is as natural as the reverse, while in arithmetic it is impossible. (ii.) In its symbols of number. Arithmetical symbols of number represent particular values; while in algebra any value may in general be attributed to the letters employed. Thus, arithmetic is confined to operations upon particular numbers, while algebra is adapted to the investigation of general principles. Again, in arithmetic all the different numbers which enter a problem are blended. (iii.) In its method of solving problems. In arithmetic we use identities, but very seldom equations; and a problem is solved by analyzing it. In algebra the characteristic instrument is the equation. To solve a problem, we translate its conditions into equations and solve those equations. The algebraic method renders easy the solution of many problems of which the arithmetical solution would be very difficult, or impossible.”
  • Wentworth (1902) wrote that “In elementary Algebra we consider all quantities as expressed numerically in terms of some unit, and the symbols represent only the purely numerical parts of such quantities. In other words, the symbols denote what are called in Arithmetic abstract numbers.”

Charles Hutton

Charles Hutton had been a professor of Mathematics for the Royal Military Academy. You can see additional details about his life here: https://en.wikipedia.org/wiki/Charles_Hutton

Slide2

In his “Course of Mathematics,” he included many Algebra topics, starting from basic operations, moving through roots and powers, to arithmetic and geometric progression, to some equations, and finishing with interest formulas.

Slide3

Because his definition of Algebra depended on his definition of Mathematics, I include both here. He wrote, “Algebra is the science of investigation by means of symbols. It is sometimes also called Analysis; and is a general kind of arithmetic, or universal way of computation.”

Slide4

Frances H. Smith, A.M.

Again, find additional details at https://en.wikipedia.org/wiki/Francis_Henney_Smith

Slide5

Smith (1848) defined Algebra, writing “By Algebra, which is a branch of mathematics, in which the quantities considered are represented by letters.” He continued, “Algebra dates a more recent origin than either Arithmetic or Geometry. It was not until Arithmetic had advanced to a considerable degree of perfection, and mathematicians had commenced to feel the necessity for abridging, as well as generalizing, its operations, that Algebra was introduced. In the early part of the 13th century, Leonardo, a merchant of Pisa, having made repeated visits to Arabia, returned to Italy with a knowledge of Algebra. A manuscript of his is quoted as far back as 1202.”

Slide6

Charles Davies, LL.D.

Find additional information about Charles Davies at the MAA website.

Slide7

Davies (1861) wrote that “Algebra is that branch of mathematics in which the quantities considered are represented by letters, and the operations to be performed upon them are indicated by signs. These letters and signs are called symbols.”

Slide8

J. M. Taylor, A.M.

James Morford Taylor served 57 years at Colgate University as a professor of Mathematics.

Slide9

Taylor (1889) wrote that “Algebra is the science of algebraic number and the equation. It differs from arithmetic (i.) in its number. The number of algebra has quality as well as arithmetical value. The double series of numbers in algebra gives a wider range to operations. Thus, in algebra to subtract a great quantity from a less is as natural as the reverse, while in arithmetic it is impossible. (ii.) In its symbols of number. Arithmetical symbols of number represent particular values; while in algebra any value may in general be attributed to the letters employed. Thus, arithmetic is confined to operations upon particular numbers, while algebra is adapted to the investigation of general principles. Again, in arithmetic all the different numbers which enter a problem are blended. (iii.) In its method of solving problems. In arithmetic we use identities, but very seldom equations; and a problem is solved by analyzing it. In algebra the characteristic instrument is the equation. To solve a problem, we translate its conditions into equations and solve those equations. The algebraic method renders easy the solution of many problems of which the arithmetical solution would be very difficult, or impossible.”

Slide10

George A. Wentworth

Slide11Wentworth (1902) wrote that “In elementary Algebra we consider all quantities as expressed numerically in terms of some unit, and the symbols represent only the purely numerical parts of such quantities. In other words, the symbols denote what are called in Arithmetic abstract numbers.”Slide12

Why study the history of Mathematics Education? The case of multiplication.

I had the privilege to take a “History of the K-16 Mathematics Course” from Dr. Sharon Senk at Michigan State University. I loved the opportunities I had to dig into old mathematics textbooks – it’s amazing how many are available digitally through Google Scholar or other resources.

I argue that looking back at old textbooks helps me in thinking about mathematics education by broadening my perspective on what “traditional” means, how often the goals of textbook authors have changed, and giving me some additional support when I talk to parents and preservice teachers about why they need to learn “new-fangled” methods for understanding basic arithmetic operations.

I like to pull out some images I found in Swetz (1995), showing that multiple methods of multiplication (yes, even the “new-fangled” methods like the lattice method) were included in textbooks as early as the 13th century:

Multiplication

I like looking in old textbooks because they also help by allowing me to “rediscover” old methods or strategies that fell into disuse, and yet could be used to help students choose from additional methods, find methods that make more sense to them, and accept that mathematics isn’t a set of rigid procedures invented by a group of old math teachers somewhere, but that creativity can enter the mix – students can come up with their own strategies!

After looking at these methods, I also found this site that gives many other strategies for mulitplying:

https://threesixty360.wordpress.com/25-ways-to-multiply/

Reference

Swetz, F. (1995). To know and to teach: mathematical pedagogy from a historical context. Educational Studies in Mathematics, 29(1), 73-88.

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

­

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.