Gauss… Establishing a solution method the Chinese had created over 1800 years prior…

Systems of Linear equations have always intrigued me! I am not certain of exactly why, but I found there relative simplicity in high school yet complex problem solving processes fascinating!

Graphing to solve systems of linear equations

Beginnings of Solving Systems of Linear Equations

The Babylonians around the 19th century BC were said to be the first to work at solving systems of linear equations.  The Babylonians looked at only 2 variable equations though and they became proficient at solving these 2×2 systems.  They called the two dimensions or unknowns of these systems length and breadth (Texas A&M mathematics department).  Thus, Babylonian mathematics is largely considered the beginning of looking at systems of linear equations in mathematics.

Despite the fact that the period of time the Babylonians were in predated the library of Alexandria these discoveries were made and utilized throughout Babylon.  Though the Babylonians started this field, there were no further discoveries for over 3000 years…  but not so fast, a dim view of history could see the mathematical history this way. In reality, long before Carl Friedrich Gauss came up with the modern matrix method of solving systems of linear equations, the Chinese had utilized matrices to solve 3×3 and higher systems of linear equations (Smoller, 2001).  The solution examples using matrices can be seen in the Nine Chapters of the Mathematical Art by Chiu Chang Suan Shu.  During the Han dynasty (206 BCE-220 CE ), the Chinese came up with a solution method for systems of linear equations with an unlimited number of unknowns or variables.  Later there discoveries became what we now know as Gaussian elimination.  When we teach elimination and solving systems of equations by matrices as completely separate in math class we are doing students a disservice by not connecting their learning.  Gaussian elimination is connected to solving a system of equations through the use of matrices.  I believe the reason teachers skip over this connection is because they do not believe they will have time to connect these ideas in clear detail because the connection relies on an understanding of linear algebra.

Do you have any other ideas for reasons why teachers rarely connect the elimination method and matrices method for solving systems of linear equations?  As a student in high school I remember never really understanding what the rows in a matrix stood for, so I wish my understanding of matrices could have been made more explicit somehow.

In relation to my previous post on Mathematics as a science (Gauss’s vote is that math is a science)!

Modern Day Matrices

James Joseph Sylvester is said to be the first person to name the arrangements of coefficients and constants involved in a system of linear equations as a matrix.  Sylvester has a unique story beyond that he was the chair of mathematics at Virginia University and arguably the most esteemed mathematician of his time in the United States from (1820-1840).  Prior to his naming of the matrix and finding of the discriminant of a cubic equation in London, he fled the United States because he believed that he killed a student with his sword stick when he whacked a disrespectful student in the head(more full story).  Regardless of his blunder, Sylvester went on to make some large discoveries with respect to matrices and their name came from him.

In the twentieth century, the more recent developments of matrices from a tool to a part of mathematical theory were driven by female mathematician Olga Tausky Todd(1906-1995).  The advances she made contributed to matrix theory’s creation and the measurement of vibrations on airplanes during World War II.  She was a brilliant mathematician and explorer of mathematics.

“I did not look for matrix theory. It somehow looked for me.” –Olga Taussky Todd in American Mathematical Monthly

Bringing it all together

I am really excited to incorporate all of this history into my teaching on systems of linear equations in the future to help my students to understand the value of solving systems of linear equations and its place in history!  I think I have developed a strong base to build a lesson from by exploring the interesting history behind systems of linear equations.  History matters in the learning of mathematics because it gives students a context to understand the importance of mathematical discovery!  Students often struggle to find reason to see the value of mathematics because it is taught as though it is basic and dull, but many mathematical discoveries took many years to be established and spread around the world!  As an educator I have the responsibility to build up the problem solving and history behind the mathematical operations and theory my students are learning!  As a result of this endeavor I truly believe my students can find mathematics more valuable and more beneficial when they know how something came about, and why it could be important to them today!

Olga Taussky Todd

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4 comments

  1. I think its great that you pointed out the lack of connections being explicitly made when students are taught linear algebra. Do you plan to make the connection more explicit when teaching your students? Also do you think that making the connections could possibly help students to remember what rows and columns mean in matrices? All in all an interesting post, I had no idea linear algebra has such a long history!

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  2. Systems is one of those great happenstances where we invent a notation and then there turns out to be deep reasons why it works so well. Matrix multiplication in really about linear combinations. Are vectors and matrices another kind of number? They seem to work in the same way. It’s a peek for students at how far up the ladder of generalizations goes. (All the way.) I like your historical connections here. Do you think HS students will ever be impressed that some of the stuff they’re doing as HW took supergeniuses to figure out the first time?

    5Cs +

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  3. I agree with all of those who have commented prior to me. The lack of communication of history within the classroom is hurting our students. If there were more history involved, I believe that the connections that students would make would increase significantly. When we find these connections across disciplines we are able to show a deeper understanding and comprehension of said material. Overall I thought that this was a very informative, flowing, and well rounded post. Good job Kevin.

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