Quadratic Formula....Exponential Functions...yikes! Hi there, I am preparing to take the GACE for Middle Grades Mathematics, and while studying I have come across some material that is difficult and am wondering at what grade level it is covered. It seems advanced. The topic is the Quadratic Formula, more specifically narrowing into the discriminant and what kinds of roots come out of if the discriminant is negative, zero, positive, etc...It even discusses "i". Furthermore it discusses exponential functions of base "a", and that they take the form f(x)=a^x, where a>0 and not equal to 1. Wow. This is all out of a study guide for teacher certification, but even I can't figure some of this stuff out! Can anyone with experience teaching at the middle school level advise me as to whether or not this is expected of kids to learn at the middle school level? I am assuming this is considered Algebra, and I remember taking Algebra in high school before I took Trig. Thanks!!

Expected of the kids? No. Expected of the teacher? Yes, and more. It is not good enough to know the topics one is teaching. One must know the material far beyond the level of what is being taught. It is our responsibility to teach in a manner that not only leads to competence in the given topics, but also lays the foundation for what is to come. If we do not know, and fully understand what is to come, then we simply cannot be effective teachers. To be anything less is a disservice to our students.

I agree with mm - and the same is true in any other domain one can name. But is this the study guide that's on the GACE Web site, or what?

I agree. They probably want to assure real competance in math, above and beyond what's taught to the kids. For what it's worth, I DO mention imaginary numbers when I systems of numbers to middle schoolers; it seems silly to give them most of the list and leave something out. I let them know that they'll learn about imaginary numbers in a year or two. Inevitably some kid googles the term and comes back asking about it. I'm real big on the "whole truth" to my kids. So I admit that I CAN factor the sum of perfect squares (and numbers that aren't perfect squares) and that I CAN take the square root of a negative and so on, even if they don't yet have the skills necessary to do so. When they ask if ANY letter can be used as a variable, I explain how we don't use "i" or "e" and why. When we do simple interest, I explain how it's not the system actually used by banks, but that they won't be able to do complex interest until they learn about "e" in 11th grade. (Plus, many (in my opinon, waayy tooooo many) middle schools now teach Algebra I to 8th graders. So knowledge of the basics of Algebra is now even more necessary for anyone teaching middle school math.

I am using the XAMOnline study guide for Middle Grades Math. I totally understand the concept of having to know far more than the kids, of course, and it is in my nature to dig deeper and know the root of everything so that I can answer the "whys?". Maybe that's where my Accounting background comes in, I like to know where things come from and where they go Everything has a place/reason, etc...So I really want to be prepared for these kids. The GACE website study guide seems less involved, which for testing purposes is great, but I want to be sure I know MORE! So I am starting with this guide and will end with the GACE study guide. I really appreciate all of your responses.

These items were all covered in my precalc class. I believe that class was my combined geometry and trig class. Have you taken any math classes at the college or was this years ago? Are you able to take any math classes to freshen up? When I took my multiple subject CSET years ago, the math portion covered pretty much all higher level math topics, even though this test was for teachers obtaining their elementary teaching credential.

I did take some math classes in college, I think Statistics, and Calculus. The rest was high school. Most of the study guide comes back to me once I dust the cobwebs out but there's just been maybe a couple things that are going to require me to dig further and study more on that particular topic in order for me to feel as though I could "explain" or "teach" it. And the studying continues.....

Can you, though, explain to that child who asks about taking the square root of a negative number, how while he doesn't have the tools for it quite yet, it's not impossible. Can you teach beginning equations and graphing in such a way that it lays the foundation for functions later on, and refrain from teaching in such a way that a later teacher has to "unteach". Do you know WHY the standard algorithms work...from a theoretical point of view (can you explain to me in every day English the fundamental theorem of arithmetic and the euclidean algorithm, and then tell me why I just asked you that). Do you understand, from an abstract algebra point of view why operations, inverses, and identities work the way they do, and why the order of operations is what it is? Most middle school teachers in this country couldn't answer the last few of my questions, but they should be able to....all that and more. In my own teaching, most of my "eureka" moments came because of my knowledge of Abstract Algebra and Real Analysis, not because of my knowledge of 7th grade math.

Hi! I'm new to this and I'm trying to add my own post. I am having difficulty navigating this. Can anyone lead me in the right direction? I'd really appreciate it.

Welcome to A to Z, ibateacher. Which "this" are you having difficulty navigating? The forum, or the math?

Goodness! Do you have a degree in Mathematics? How many years did you or have you taught? I don't even know where to start to reply to your post, it is actually very discouraging. Yes, I can explain the Fundamental Theorem of Arithmetic in everyday English. It's actually very simple and just b/c it is worded in such an overpowering way doesn't mean it is tough. The other questions you asked, well, I am not Euclid, and am not striving to be. While I understand the importance of knowing the material you are teaching, I also think there are parameters to what you are expected to know.

Well, from what I hear from SEVERAL of my colleagues...math is so EASY to teach...much EASIER than something like reading...it's just black and white, cut and dry, in and out, here's the method, now take the test... Yeah, it's sad...

Middle schoolers have a knack for asking questions that are at the edge of the teacher's knowledge, or of what they think the teacher's knowledge is - in some cases because they're being little snots, but in other cases because They Just Gotta Know, whether or not the state standards for that age cover that topic. In any case, at least two states that I know of cover the topics you mentioned in their middle-school math exams - so someone else out there believes that it's appropriate for middle-school math teachers to know this stuff. If NCTM doesn't concur, it would be surprising.

I completely agree with you about sharing that there is more to math than what they are currently learning. I also stress that learning is a journey and that every single lesson I teach builds in some way on top of a lesson before so it is all critical. They seem to appreciate knowing that we are on a journey and that each concept is just a step on the path.

Math is a beautiful and wondrous subject that has sadly stricken many non-math teachers with fear and loathing. Hope is not lost though because math isn't impossible to understand (I didn't study math in college), and once you do, you'll come to appreciate the beauty of its logic and order. Middle school math covers bridge topics. This means that you need to understand what came before (K-5) and what comes later (high school). With a good understanding, you can emphasize how current topics will reappear in high school (and beyond). It is absolutely critical to understand the connections if you want to prepare the kids effectively for math after they leave your classroom. For example, I really appreciate Alice's comment elsewhere about the problems with applying the lattice method of division to algebraic division of polynomials. Without understanding long division (the old fashioned way circa 1980s), kids will probably have a tougher time understanding this topic in high school. As for test preparation, the best suggestion is to review key topics in algebra, trig, pre-calculus, basic probability and basic statistics from real textbooks. The test administrator lists the covered topics so you can cross reference the material in textbooks. Unless you already have a firm background, the test prep books alone will not cut it. In my opinion, it is better to prepare properly the first time around and get it done in one confident shot with recognition than to take shortcuts, fail multiple times and lose your drive. Here are books that I've used for test prep. You can probably get used older editions for a bargain - you don't need the newest shiny editions. http://www.amazon.com/Algebra-Colle...3543/ref=sr_1_1?ie=UTF8&qid=1290561161&sr=8-1 http://www.amazon.com/Understanding...=sr_1_1?s=books&ie=UTF8&qid=1290561318&sr=1-1 http://www.amazon.com/Precalculus-U...=sr_1_1?ie=UTF8&s=books&qid=1290561374&sr=1-1 Study smart and good luck!