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Defining Math Programs...


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I'm sure this has been done, but I am looking for it all in one place. So can y'all either refer me to a thread or help me out here.


Can we please define the different math approaches - spiral, mastery, mental...what else???? and then give curricula examples of each?

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I'm going to tackle this, because I have never seen so much confusion over these terms as I have lately on these boards. :) Everybody is trying to redefine terms, including many elsewhere on the internet. A math program is defined by the approach to mathematics education that the author/publisher takes. Do they believe that a concept needs to be mastered immediately, do they believe that a concept should be taught in pieces and looked at repeatedly and mastery will eventually occur, do they have another idea altogether?


Here are the traditional meanings:


Mastery - one chapter on one topic, then another chapter on another topic, may be a small review section at the end of the chapter with previous concepts, but there is not daily review of concepts taught in other chapters


examples: most traditional textboks we grew up with, BJU, MCP, SF Exploring Mathematics, Singapore


Spiral - includeds a well-thought-out daily implementation of review of previous concepts taught, new concepts may be taught a chapter at a time or they may come willy-nilly, usually a topic is dropped for a short time, then picked back up, then dropped, and so forth, but this may be more or less the case depending on curriculum, when defined in theory, spiral math would actually be Saxon, which is instead called incremental (confusing, huh?)


examples: Horizons, R&S, A Beka, Everyday Mathematics (Chicago Math) (though Everyday Mathematics could be called incremental)


Incremental - Saxon is the only program that calls itself incremental, I believe, this is a form of spiral with another twist: the concepts are not taught completely in one lesson, they are broken down into pieces and taught over a period of lessons


Other - new (or not so new) programs that take a completely different approach


examples: MUS, Miquon, RightStart, TouchMath, Developmental Mathematics (Dev Math and MUS are related to mastery programs in ways)


Mental math is a concept, not really a type of program. All programs teach mental math to a varying degree.

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This is my best understanding of mastery vs. spiral.


Picture a pie, down each cut is a different math concept- adding, fractions, subtraction, decimals, ect.


Mastery starts down one of those "cuts" and stays on it. Very obvious in a program like Right Start, but also something like Miquon. When you are studying fractions, all of the problems on the page are fractions. These programs jump to another cut, only fractions, then jump to only subtraction, ect.


Spiral programs are like you took your knife and spiraled around the pie. So you just touch addition, then drag your knife to the subtraction line, dragging along some addition, and on to the place value line dragging along the addion and subtraction parts, ect.


Of course this is all confusing because math isn't really that clearly cut, I mean you can't study percents with out using some of those earlier learned ideas. And some programs are a mix, maybe teaching mastery, but still doing lots of review every chapter, or something else.


Good luck I don't think this is ever going to be clearly defined and split up!

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Base 10...Right Start, MUS

Am I right?


Base 10 is not a pedagogical method, it's how our number system works.


All math we normally use is base ten. So...


1,2,3,4,5,6,7,8,9 are the individual digits, then we bundle them up (figuratively) and add a digit on the left that we define as one "ten", but we use the same 9 single digits to represent this "ten" concept till we get to "ten tens", when we bundle up those ten tens and call it "hundred".


So in Base 10, the "ones" place represents 10^0, the "tens" place 10^1, and the "hundreds" 10^2, and so on.


Number systems can be based on any number of digits. Binary, upon which all computers are based, is a base two number system.


so, 0= zero, 1=one, 10=two, 11=three, 100=four, 101=five, etc.


Each place still represents an exponential value of the base.

The "ones" place represents 2^0, what we think of as the "tens" place here is "twos", or 2^1, and the "hundreds" place is really "fours", or 2^2. The "thousands" place would represent 2^3, or "eights".


You could also have a number system based on a number larger than ten, but we'd have to invent more single digits to represent the numbers we're adding before we bundle. What we think of as "ten" would be represented by a single digit. A fun explanation of a base-12 system is found on Multiplication Rock's 12-times song, about how if humans had 12 fingers and toes intstead of 10, we'd most likely have a base-12 system.


So, 10 = twelve, or 12^1, and 100 = one-hundred forty-four, or 12^2. In base twelve, 14 = sixteen, or one twelve plus four ones.

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These are my definitions. I know others disagree with me.


Mastery: Teach one concept thoroughly then move on.


Spiral: Teach one concept to a point, then move to another concept. At some point, either during the same school year or in a subsequent year, the concept is briefly reviewed and then expanded upon. I think of spiral more as a slinky, with the same topic being revisited at a higher level each time it comes around.


Incremental with continual review: This is a type of spiral, sort of a supercoil. Each concept is broken into smaller chunks and the chunks are distributed at intervals throughout the year. The concepts are reviewed and to some extent, built upon between the actual lessons on the material. There is also review of material from year to year.




Mastery: MUS


Spiral: Singapore. The spiral is from year to year, but not within the same year.


Incremental: Saxon, of course!

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