A is for Vector?.

Found a copy of “One Letter Words” by Craig Conley in a cheap resale shop the day before I flew back to England, so I bought it and read it on the plane…Some interesting references to mathematical symbols and constants got me thinking…somebody should put together a mathematical book of one letter words… I offer the title above for free… it comes from the quote in the book that said , noun (Mathematics) A vector… and then justifies with a quote from Marie Vitulli’s “A Brief History of Linear Algebra and Matrix Theory”… The use of a single letter A to represent a matrix was crucial to the development of matrix algebra. Early in the development the formula det(AB) = det(A)det(B) provided a connection between matrix algebra and determinants. [Italicized sentence added by me]
Ok, did he miss the point? Or did I… I assume Vitulli meant the use of a single letter (such as A) to represent a vector was a huge leap… but; I think the give-away was calling it a noun… variables like this are more like pronouns in my mind… using x in place of the number “four” or n for “some number not named”… correct me if you disagree…
Sadly, after all that, he left out the one that is known to almost every Algebra I student, “M is for slope”. Interestingly, that one has led to more mis-history in classrooms than any other topic, with the possible exception of the life of that Great American-Indian Mathematician, Chief Soh Cah Toa.

Here is what I have found about the slope, as it appears at my MathWords page; Slope is derived from the Latin root slupan for slip. The relation seems to be to the level or ground slipping away as you go forward. The root is also the progenitor of sleeve (the arm slips into it) and, by dropping the s in front we get lubricate and lubricious (a word describing a person who is "slick", or even "slimy").
Many variations of where the idea of M for slope originated seem to be mostly myth. One of the most common is that the letter was used by Descarte because it was the first letter of some French word or another that related. In a recent post to the AP Stats discussion list, Hector Hirigoyen shared the following story:
I was told by Mary Dolciani herself, that the SMSG group "decided "to use y=mx+b because of the French (Descartes, I presume)-"montant"; I found it strange because the "logical word" would be "pente"(which is slope (and the standard term in Spanish is pendiente, which matches this). However, several years ago, while visiting a French high school, I noticed the teacher used y=sx+b. I inquired, and she said because of the "American" word "slope." I believe they are using ax+b for the most part these days.
. Here are several other clips from postings about the topic on a discussion group about math history.
In his "Earliest Uses of Symbols from Geometry" web page, ... Jeff Miller gathered the following information: Slope. The earliest known use of m for slope is an 1844 British text by Matthew O'Brien entitled _A Treatise on Plane Co-Ordinate Geometry_ [V. Frederick Rickey]. George Salmon (1819-1904), an Irish mathematician, used y = mx + b in his _A Treatise on Conic Sections_, which was published in several editions beginning in 1848. Salmon referred in several places to O'Brien's Conic Sections and it may be that he adopted O'Brien's notation.

According to Erland Gadde, in Swedish textbooks the equation is usually written as y = kx + m. He writes that the technical Swedish word for "slope" is "riktningskoefficient", which literally means "direction coefficient," and he supposes k comes from "koefficient."
According to Dick Klingens, in the Netherlands the equation is usually written as y = ax + b or px + q or mx + n. He writes that the Dutch word for "slope" is "richtingscoefficient", which also means "direction coefficient." In Austria k is used for the slope, and d for the y-intercept. In Uruguay the equation is usually written as y = ax + b or y = mx + n, and the "slope" is called "pendiente", coeficiente angular", or "parametro de direccion".
It is not known why the letter m was chosen for slope; the choice may have been arbitrary. John Conway has suggested m could stand for "modulus of slope." One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for "to climb" is monter. However, there is no evidence to make any such connection. Descartes, who was French, did not use m. In _Mathematical Circles Revisited_ (1971) mathematics historian Howard W. Eves suggests "it just happened."
Jeff Miller's web site cited above now has updated the earliest use of m for slope.
The earliest known use of m for slope appears in Vincenzo Riccati?s memoir De methodo Hermanni ad locos geometricos resolvendos, which is chapter XII of the first part of his book Vincentii Riccati Opusculorum ad res Physica, & Mathematicas pertinentium (1757):
Propositio prima. Aequationes primi gradus construere. Ut Hermanni methodo utamor, danda est aequationi hujusmodi forma y = mx + n, quod semper fieri posse certum est. (p. 151)
The reference is to the Swiss mathematician Jacob Hermann (1678-1733). This use of m was found by Dr. Sandro Caparrini of the Department of Mathematics at the University of Torino.

Testing Understanding of Slope

As part of our professional growth and school improvement process, several members of my department and I have formed a focus group to study formative assessment; testing which focuses on testing to make instructional decisions rather than for grading. One of our goals is to try and develop a deeper understanding of how to write better questions in order to measure student learning.

Recently I imposed on several members of my staff to give a brief assessment to their students that included two different forms of a question on slope. The questions, shown below, were of the lines y=3/4 x -3 and y=3/4 x - 2. I assumed their would be a slight difference in the number of correct responses since one had integer intercepts on both the x and y axis, and one did not.


What I didn't expect was the degree of difference. The students ranged from Algebra I students who have only recently been introduce to slope, through geometry students who had reviewed the topic twice this year according to their teacher, my own sections of Alg II students whom I assumed were much stronger on the topic than my results made me know, and a section of statistics students; many of whom are taking an additional pre-calc or calculus class. In all there were 47 students who took each form. Of the 47 who had the question with integers on both intercepts, a total of 26, or 55%, got the correct answer. Of the 47 who had the line translated one unit up the y-axis, only 10, or 21% gave the correct response.

Four more of the students who had the "easier" question had the expected types of mistakes; they either gave the reciprocal of the correct response, or the negative of it. I was not prepared for how strongly the students were bound to the x and y intercepts. Thirteen of the students using the second form had given slopes of .8 or 2/2.5 using the values they interpreted as the intercepts. Of the 22 students in Alg II who took the questions, 8 of 11 gave the correct answer for the "easier" problem, but only three of them were correct (that is, they answered 3/4) on the second form. Five more of them, however, gave the answer of 2/2.5 or .8 to the second form. There is no question in my mind that they understood the idea of slope. I have no quarrel with the people who would give full credit to the answer of .8 or 4/5. There is however, I think, some difference between the student who focuses solely on the intercepts and the ones who see the whole line, coordinate plane relationship. I think a question which differentiates between them tells me something about how they see and treat slope.

I guess I am more concerned than ever about learning how to write good assessment items that will help me, and my students, understand what they know, and don't know, about the mathematical ideas we are covering.