Grr. Really. I have some frustration I need to get off my chest, so this will be a “constructive venting” post. Constructive, because I really do have a point, and venting because I can’t believe the things that occur in math education.
First off, I have been really interested in how the curriculum of mathematics is designed and how that curriculum actually works against the learning of math as it is presented in our current textbooks. Honestly, I am really disappointed in textbooks right now. I have spent some time examining the vocabulary in different sections of our current books, and I realized that the books set up the topics of mathematics as if each chapter, each topic, is a disjoint Venn diagram. Chapter 2 has no relation to Chapter 3, and that is how math is done.
No wonder learners struggle with math and don’t make connections. We have DESIGNED math to be taught that way. Don’t agree with me? Just look at your textbooks and tell me if the linear vocabulary is identical to the quadratic vocabulary. Then ask yourself why not? 80% of the vocab should be IDENTICAL.
Let me show you what I mean. This is the current state of affairs:
| Linear Equations | Quadratic Equations | |
| Standard Form | Ax + By = C | y = ax2 + bx + c |
| Equation forms | y – y1 = m(x – x1) | y = a(x – h)2 + k |
| y = mx + b | y = (x – a)(x – b) | |
| m = (y2 – y1)/(x2 – x1) | ||
| Vocabulary | Rate of change, y- intercept, slope, rise over run, x – intercept | x-intercepts, solutions, roots, axis of symmetry, reflection, translations, y – intercept |
Hmmm, do you see ANYTHING that overlaps there? The only thing that is the same is the fact that the x-intercept and y-intercept are both used. However, in the quadratic unit, the y-intercept is rarely used, and the x-intercepts are normally called roots or solutions. Learners = 0, Notation = 1, point goes to the book.
Seriously, even look at the idea of the “Standard Form”. Why is it “Standard”? Because that is how we would normally want to write the equation. Lines get some weird, special form that relates to nothing else in polynomials, while quadratics and all other polynomials get something that makes sense and flows from one type to the next. Learners = 0, Complicated Notation = 2; the book is clearly leading.
Oh, that Point-Slope form looks pretty promising, though. It is very similar to the Vertex form of the quadratic. Well, kind of similar. It is the only form in the family of polynomials that uses subscripts, but that is okay, subscripts are super easy to learn. Oh, they aren’t? Subscripts confuse the heck out of learners? You mean there is a difference between a number in front of a variable, a number above and to the right of a variable, and a number below and to the right of a variable? Darn. And if the Point-Slope is supposed to come from the definition of slope, where did the subscripts of 2 go? Why ? I don’t get it. Why did things change? Learners = 0; Confusing notation = 3. Clearly the learners are getting the worst end of the deal here.
How can we make this better? Well first off, we need to realize that there is only ONE type of math we are doing; Polynomials. Start there, and build a set of math from scratch.
Damn. That sucks. That is a lot of work. That is akin to what Exeter has done or what Milton Academy has done. [Terrific article on Milton’s efforts here by the way.] And there in come the rub. Here we have the idea, but it takes a team of willing people who think alike to create. It takes time, energy, money, and effort. And then, it takes only one or two teachers to block the whole thing from implementation in a school or district.
Frick.
Double Frick.
Here comes the venting. I need to move beyond this so I will get something done.
I didn’t even realize this bothered me so much until I started thinking about why I was so paralyzed for the last two weeks. Very hostile. Very frustrated and angry, and then I started asking myself why. I think I have it figured out. I was at a department lead meeting two weeks ago, and the leader of the meeting presented my idea for lines. (okay, not mine, but Exeter’s). And the department leaders from other schools at the meeting said, “Why?” “Why would we do that?” “Lines are y=mx+b”. “That is stupid, the textbooks don’t teach it this way.” etc.
I. Kept. My. Mouth. Shut. It took heroic efforts, but I did not rip into anyone. I was good. I mean, they didn’t come up with any good arguments why it doesn’t work, shouldn’t work, isn’t a good idea. They just said, “ThatisnotthewayIwastaughtthereforeitiswrong <breath> thebookteachesitonewayandwecannot/shouldnotchange.”
THEN! OMFG. The pain wasn’t over yet. I was speaking to our assessment director after the meeting, and she was just as frustrated as I was, if not more. She told me a teacher actually contacted her and told her she was doing a bad job because she left the section of the book that taught lines as y=mx+b out of the curriculum guide. Why was that bad, you ask? Because that is the ONLY way that teacher teaches lines. That teacher won’t teach it any other way because it is the easiest way.
Frick.
Really? Sigh. You know that ONE teacher will end up on the next textbook committee, and that ONE teacher will end up railroading 4 other great teachers into buying a shitty textbook simply to shut that ONE teacher up. How do we fix that? How can we?
And that is my frustration? What can I do about that? It has eaten at me these last 2 weeks, and it shouldn’t. It won’t now that I have written about it. I have gotten it off my chest, and that helps.
How do I respond?
I push forward with my Exeter project. I write a paper and submit it to Mathematics Teacher. I stay focused on what I need to do to accomplish my goals.
And remember:
“Don’t say you don’t have enough time. You have exactly the same number of hours per day that were given to Helen Keller, Pasteur, Michelangelo, Mother Teresa, Leonardo da Vinci, Thomas Jefferson, and Albert Einstein.” – Life’s Little Instruction Book, compiled by H. Jackson Brown, Jr.
I have the time. I just need to use it better.
After reading your post on vertex form of a line, i introduced it to my A2s…guess what… the use it correctly and know why it works!!! I shared it with Geometry students who were struggling with traditional eqs of lines, in less than 5 minutes, they’ve got it! So thankful to have amazing resources and teachers like you who cause me to uhhh think And questing why i do things the way i do. If it makes sense for my students, I’m willing to step out of my traditional ways… thanks for venting!
That’s so frustrating, I hate that. I mean, the form of y=mx+b has some merits (or rather, the form y=b + mx does), but refusing to even consider other ways to do something is the worst.
I have a question, though. When we’ve taught them vertex form for everything, and they get how to graph with that, what do they do when they get a quadratic in standard form, like they will on a state test? Especially since Completing the Square is Alg 2.
Pam, you are most welcome! It makes so much more sense than other forms, and it will build connections to other units as well! Now they have a real purpose to do completing the square with quadratics, because then it is in vertex form and they can graph it nice and easy.
James,
Don’t get me wrong, I am not saying vertex form is the end all be all of everything. We still have to teach standard form and factoring, we still have to teach quadratic formula, etc. All the vertex form of lines does is allow us to make connections from the linear unit to the other units. We can leverage a common vocabulary, find the y-intercepts (set x = 0), find the x-intercepts (set y = 0), rate of increase/decrease (which is constant in the first difference for lines, but in the second difference is constant for quadratics), etc.
Right now, our textbooks have no common vocabulary between the linear units and the quadratic units, and that is what I am thinking about. How do we create a common vocabulary set so the learners can learn that one vocab set and apply across multiple representations of mathematics.
Notice that regardless of the exponent, 1, 1/2, 2, etc, the vocab set should be consistent. That is how we build connections.