# Math terms - please define/describe

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I have seen a lot of talk on here about math using these sorts of words: spiral, mastery, mental, conceptual....etc. Could someone please help explain what these are and why one would choose one over the other? Do you choose based on your philosophy of math or your child's learning style? Do the various math programs label themselves this way? (e.g., I use MM but don't really remember seeing one of these labels attached to it. Maybe I'm mistaken). I do know that I want to avoid the "Everyday Math" flavor of "new" math. I love MM but I may have to change math in near future.

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The discussion regarding "spiral" vs "mastery" is a common one at the moment. A math that is "spiral" teaches concepts and spirals around to those same concepts again, teaching them more in depth each time. A math that is "mastery" teaches a concept until it is mastered, then goes on to the next one. IMHO, spiral vs mastery is not the first decision that should be made; it's whether the math is process or traditional. A process math uses manipulatives to teach; examples would be Miquon, MUS, Making Math Meaningful. A traditional math...doesn't. :-) ABeka, BJUP, and R&S are traditional maths; they may use manipulatives or visuals, but they do not depend on them.

Mental math: you know what that is. :-) It's the math you do in your head. :-)

Some think that different math materials teach concepts better than others. Some think it's more important to understand the concept than to know/memorize math facts. I think both are important; furthermore, knowing only concepts without being able to actually use those math concepts in any real way is an epic fail.

It isn't always possible to figure out a child's learning style before choosing which math (or anything else) to use to teach him with. Not only that, but if you are teaching more than one child, sometimes you have to make compromises, because there's only one of you, and you have to pick something that you can actually teach. I never did figure out my dc's learning styles. They never matched any learning style tests. I know now that they are very visual, and that I am, but if I had waited to choose instructional materials until I knew their learning styles, they'd be illiterate to this day, lol.

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Spiral vs mastery refers to the organization of lessons. Every program is a little different and I see these terms on a continuum - there are sort of "degrees" of spiral or mastery. Spiral programs typically introduce a topic, do some problems with it, and then spend the rest of the lesson doing problems on other topics previously introduced; tomorrow's lesson might be on a different topic entirely, and the program will come back around to today's topic some time in the future (I'm sure others can quibble with this description - it all depends on the specific program in question). Generally, a mastery program focuses on one topic at a time, in depth, before moving on. Timing and amounts of review will vary among all programs.

Mental math in a math program would be strategies for, and practice doing, arithmetic in one's head (e.g., breaking large numbers into parts).

Conceptual just refers to concept instruction and understanding as opposed to instruction on procedures.

MM has a mastery organization. It has plenty of mental math and focus on concepts - MM always teaches the concept first and then gradually moves to the procedure (algorithm) over the course of a group of lessons on a topic.

If you feel you're missing some small aspect but are otherwise happy, sometimes it makes more sense to add in that missing thing from elsewhere, or tweak somehow, rather than switch programs entirely. It depends on the situation.

Fortunately, there aren't a lot of "new new math," also known as "fuzzy math," programs like Everyday Math that are often used by homeschoolers. (Everyday Math is not "new math" - that is something else.)

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Spiral - work on several different concepts at a time, introducing a new small piece of one each day, followed by small sets of the other concepts

Mastery - focus on one main topic at a time until it's been explored thoroughly, most will review old concepts at the same time

Most kids and teachers would be fine with either approach and it boils down to personal preference. There are some that have strong, adverse reactions to one or the other. (One of mine will struggle greatly with spiral, but succeeds with mastery.)

Mental - working problems in head without writing out the steps - Some curricula puts more emphasis on this than others.

Conceptual - teaches why the math works, not just how - Also known as the kewl kids club on this board.

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Conceptual and procedural can be a bit of a hot button here right now, but I still find the terms helpful in describing what works in our home.

Several months into homeschooling Grade 1 math for the first time, I realized that something was going very wrong in that subject. So I came to these boards, and folks recommended that I read Knowing and Teaching Elementary Mathematics by Liping Ma. I'm glad I did. The author goes in depth on the difference between what she calls a "profound understanding of fundamental mathematics" versus a procedural understanding. I recognized that I, despite all of my higher-level math, was teaching from my natural procedural stance, whereas what Ma called a PFUM was ultimately going to help me as teacher in the long run.

On these boards, I now read "conceptual" as a short form for what Ma documented as "profound understanding of fundamental mathematics." It's significant that in her work, it's the teacher who leads with the material, and it's the teacher's understanding that is going to encourage a deep understanding of concepts, or a surface-level understanding of the procedures. It is not the program.

However, having used a variety of programs, I now believe that some make it easier for me as a teacher to draw out and emphasize the concepts. Their instructor's materials or the way they set up problems provide more support in developing a deep understanding of math. It's not that they don't include procedures -- it's just that the concepts are easier to reinforce if you follow the instructional materials. It's equally true that programs labeled as "procedure-heavy" DO contain concepts, but they don't take center stage unless the teacher is actively working to do that.

Programs that I've found that help me emphasize concepts, and stretch me as a teacher to push for a deeper level of understanding, are Miquon, Math Mammoth, Singapore, MEP, and Beast Academy.

I'm on the fence about Life of Fred. I think that some of the times algorithms take precedence over understanding what's going on.

I'm a bit nervous posting this because of the ire that this discussion can raise, but I know that I was really helped by these kinds of conversations when I was starting out. So I hope that this helps!

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I will have to check out that book. Selecting a math program is turning out to be more complicated than I thought. I'd always considered math to be straight forward and concrete so this is all very confusing! With MM there is not any instructor materials to speak of. If I'm not all that strong in math (granted, we are talking early elementary which I can handle for now) would a program that has an actual teacher's manual help? I never considered that I would need help explaining simple math problems and concepts but maybe I do :confused1:

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I will have to check out that book. Selecting a math program is turning out to be more complicated than I thought. I'd always considered math to be straight forward and concrete so this is all very confusing! With MM there is not any instructor materials to speak of. If I'm not all that strong in math (granted, we are talking early elementary which I can handle for now) would a program that has an actual teacher's manual help? I never considered that I would need help explaining simple math problems and concepts but maybe I do :confused1:

From what I've seen, Math Mammoth is written more to the student -- so in effect, Maria Miller is the instructor. Hence separate instructor guides aren't needed, apart from the notes she includes for each chapter. I've watched some of her videos, and they've been helpful for me as an instructor.

Beast Academy and Life of Fred take a similar approach -- the guide is written to the student. For programs like MM and BA written to the student, I often try to work along with (or ahead of, if I'm super-organized!) my student so that if they have questions, I'm at least up to speed on the topics they've covered and how they've been presented.

Singapore, Miquon, and MEP all have instructor's materials. I've used Singapore's and Miquon's the most, and both have changed how I approach math with my daughters.

The other aspect of math programs that was important for me was the spiral vs. mastery approach. I wanted something that I could easily compress or extend as we wanted, and I found it really hard to do that with some spiral programs. So I end up using programs that are labelled more "mastery" as a result, because I find it easy to switch gears with them.

ETA: If Math Mammoth is working for you, then you are in a great place. It's a strong, rigorous program that will serve your student well.

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I would love to stick with MM but my son is starting to dread it. He's good at math so I don't want him to develop a dislike for it. I suspect it's mostly because he hates writing so I'm going to stay the course for now and see if I can pinpoint if that is the problem before changing ... but in thinking ahead to other options I was not sure about where to even start looking. I never really considered other programs before now. This is all very helpful.

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I would love to stick with MM but my son is starting to dread it. He's good at math so I don't want him to develop a dislike for it. I suspect it's mostly because he hates writing so I'm going to stay the course for now and see if I can pinpoint if that is the problem before changing ... but in thinking ahead to other options I was not sure about where to even start looking. I never really considered other programs before now. This is all very helpful.

That is not an unusual problem to have - with any math program.

I would: (1) use the white board (more fun plus less friction while writing), (2) allow oral work when possible, and (3) try not to assign "too many" problems (MM has a lot on purpose - I'd assign some fraction of them, and then use the rest if there is difficulty or later for review).

Other frequent issues some have with MM are that the pages are crowded/busy (possible tweak: using the white board for a lesson or scratch paper for work) or that the lessons are broken down into too many steps (possible tweak: compress as you go, teaching the lessons in groups, or switch to Singapore).

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