**How to Analyze the Curricula**

If you have not read my last two articles in this series, I encourage you to take the time to do so before reading this article. This article only makes sense in light of those.

There are two main approaches to presenting mathematics concepts in textbooks. One type is a spiral curriculum in which the assumption is that children understand mathematics through repetition. Mastery of a concept is not expected until it has been reintroduced several times, sometimes over years. New concepts continue to be presented and then the previous concepts will be reviewed with each progressive lesson. A spiral curriculum touches on a wide number of concepts in a year in less depth with less time devoted to each new concept. Saxon Math is an example of this type of curriculum.

The other type of curriculum is content mastery. The assumption is that learning is sequential and concepts build upon each other. Foundational to this approach is the idea that children learn mathematics through understanding and mastering concepts. A concept or skill is presented in many different ways and is mastered before moving on to the next. Fewer concepts are covered than in a spiral approach and more time is given for each concept because it is taught in more depth. Mastery of a concept is required before moving on to a new one. Making Math Meaning by Cornerstone Curriculum is an example of this type of curriculum.

In light of the way in which we know how children learn math naturally as explained in part one of this series, one of these two methods seem counter to that. Can you identify which one? If you said the spiral approach you are correct. What this approach does is to take teaching the new mathematics concepts out of their proper real world context. If you take a concept out of its context you are taking away the one thing that gives the idea real meaning. Do you think your child would understand an abstract concept if you use another abstract concept to teach it? No, the natural learning occurs when you use something your child already understands to explain something abstract. Each lesson presents a new concept before your child has had the opportunity to grasp the one presented the day before. When your child is struggling to understand a concept, she experiences the physical effects of anxiety. If you ask any child how their tummy feels while they are in a state of confusion with a new concept or skill, they will tell you that they feel a knot or butterflies. When there is mastery of that difficult concept or skill, the anxiety is gone and there is a rush of relief and joy. Their perseverance is rewarded and their confidence is elevated. I have observed this over and over again in teaching children mathematics. With the spiral approach to mathematics, day after day the child is exposed to new concepts with no mastery of that concept. The amount of anxiety and frustration they are feeling is rarely relieved. After years of learning with this type of curriculum the student can lose heart and give up. They believe that they can’t learn math. It does not have to be this way.

Do you remember what Mason revealed as the practical value of a mathematics education? Those were the training in reasoning and the habits of understanding, a willingness to work, accuracy, and being intellectually truthful. Do you think this lofty goal can be achieved through the spiral approach?

So how do you know what a spiral curriculum looks like? You have to look at how the curriculum is set up. Look at the scope and sequence from kindergarten through 12th grade if it goes that high. Look for concepts that are taught, year after year. Look at a specific grade level text book. You can look at any grade level textbook table of contents at the Saxon Math website to see an example of how a spiral approach textbook is set up. Determine how often new concepts are presented and if mastery of that new concept is achieved after it is taught. Look at the assignments to see if they are loaded with computation problems with little or no word problems? A quick search on the internet for spiral math programs will also be a big help to you in identification of this type of curriculum.

Not all mastery programs are created the same. They all start with the correct foundation that children learn through mastering concepts sequentially taught, but the way the concepts are taught may be artificial and frustrate a student. When looking at mastery curricula, keep in mind the foundation of teaching mathematics that it starts with a problem that requires understanding before moving to the abstract symbolic representation of that problem. This is the way you help the child connect an abstract concept to something real that they already understand. Be aware, that there are many mastery programs that teach in the traditional method of starting with the symbolic representation of a problem then using objects to make the symbolic real. Remember that the objects themselves are a representation and can be abstract to the child. The proper context for mathematics is the real world math problem that needs to be solved first, with objects if needed, and then using the symbolic representation to show what they already know from solving the problem.

Before deciding on a curriculum, read the scope and sequence. Make sure that there is mastery of concepts before presenting a new concept. Fewer concepts will be taught with more time given to teach each one. Read sample lessons to see if you are comfortable with the teaching format. Look for a focus on word problems first, rather than the symbolic representation of the math concept. There may be some lessons with mostly computation problems, but this should be the case only after the concept has been presented in the real world context with word problems. At the website for Making Math Meaningful by Cornerstone Curriculum you can find a good example of what a mastery approach mathematics curriculum looks. It also adheres to the teaching of mathematics that corresponds with how children learn naturally. You can see the concepts and skills taught for each grade level as well as sample lessons to see how they are laid out. There are more curricula available like this one, so you have options and can chose one that best suits you.

The curriculum should not require a lot of expensive manipulatives. Dried beans or centimeter cubes work well as counters, Unifix Cubes work well for making groups and fractions, and some type of place value blocks are sufficient for teaching addition, subtraction, multiplication and division. They can be found cheaply online, used, or in local stores so they don’t blow your budget.

With the growing popularity of homeschooling, there are new mathematics curricula entering the market every year. Your choices are vast, but don’t be intimidated. You do not have to be an expert mathematician in order to teach mathematics. All you need is determination and the knowledge of how children naturally learn mathematics. You have the knowledge you need to choose an effective math curriculum. I wish you well on your search and wisdom for your decision making.

I don't follow why a spiral program is inherently out of context? Thinking about the way my son has learnt maths so far (eg building a multiplication chart over a year and a half, in between playing with fractions, learning to read a clock and using more basic operations - all in the context of real-world problems- but in all cases building toward greater abstraction, for example he stopped needing any kind of manipulative after his multiplication grid was 2/3 complete) it seems to me that spiral-in-context is natural and therefore perfectly possible. Surely there are programs out there that do this?

ReplyDeleteThe problem to be solved in the real world is the actual context. For example, if I have to leave for an appointment at noon, and it is 10:20am now, how much time do I have before I leave? That is the real world problem that must be solved. One way to solve the problem is to look at a clock which would be the manipulative and count up from 10:20. You could choose to use the manipulative or not. The abstract representation of that problem would be a written form of how you solved the problem. The solving of the problem should come first. Eventually we use the abstract representation to help solve other real world math problems, but that is only after students have a full understanding of those abstract concepts. Often times we equate the manipulatives or charts we use as the real world problem but it is only the tool used to solve the problem. If there is a spiral program that presents the concepts in a way that children learn naturally, I have not found it. If you do, I would love to know!

ReplyDeleteHmmm. This is all very interesting! I just read your previous few posts on math, too. We're in our first year of AO (I found your blog through the forum) and are using MEP. (which is spiral as I'm sure you know)

ReplyDeleteI totally see what you're saying re: the spiral-type curriculum and I'm probably the kind of person who really needs to totally understand a concept before moving on to the next. That said, my oldest child (in Year 1) seems to be doing ok with MEP so far. We've only been doing it for a short time.

My question is, are you saying that with a spiral program, the concepts aren't being mastered because the teacher is moving quickly through the material? Or because not enough time is spent on one concept? So far, when I've taught anything new, like 'greater-than' and 'less than', I make sure she gets it before we move on. This means that sometimes one lesson takes two days. Which is the beauty of homeschooling. :)

Also, could you talk more about what you mean by children not being able to learn naturally with a spiral program? Since my kids were born, we've always added and subtracted things so when the day came to actually "do" math, my oldest already knew a few math concepts. So, if we're doing that sort of thing along with our curriculum, shouldn't that work? :)

Sorry if I asked the same thing the first commenter did! :) I'm just trying to figure this all out, I guess! :)

DeleteNo worries Catie (I love the spelling of your name). What really corrected my thinking about the teaching of mathematics was reading what Charlotte Mason taught about it. Have you read her teachings on the subject? It is so helpful. As to MEP, that is a little harder to pigeonhole. Yes it is a spiral curriculum, but is also has some elements of content mastery. Many children like the spiral curriculum, especially at the beginning. As the years move on though, in my experience they tend to burn out. Your child is young. It is really important to get this figured out now and pick a curriculum that you like and caters to how children learn naturally, in other words start with the real world problems and then move to the abstract representation of what they already know, and allows children to master concepts before moving on. Once you have mastered the concept and know how to do it, you don't need to continue to practice learning how to do it. Once you have learned to ride a bike, you don't need to leave the training wheels on it. Once you have mastered a math concept and really know it, you don't have to keep practicing it. Spiral programs often have workbook pages loaded with computation problems. The computation problem is a means for solving a real world math problem. When a curriculum continues to disconnect the computation from the word problem then it is backward in its approach.

ReplyDeleteI forgot to tell you this before, but I love the look of your blog. :) I love birds!

ReplyDeleteThanks so much for taking the time to respond! I'm going to do some more research. You've given me a lot to think about. Thank you!

Thank you for your kind words Catie. You are welcome. I am glad I could help.

ReplyDeletePerhaps this is partly why I have a bad taste in my mouth about Saxon. I taught at a private school that used it (I taught English not math), and I found that students who didn't start out with Saxon had a hard time entering successfully in later years.

ReplyDeleteCurious about the Life of Fred series - is that something you would recommend? We are just starting out on this journey - 6 1/2, 4 1/2 and 20 month old. I've only begun researching.

Thanks for any input and glad I stumbled on your blog. :)

Hi Christie,

ReplyDeleteI am so glad you are trying to figure this out now. This is the time to do this rather than years down the line of failure and frustration.

As to Life of Fred, I have seen it but I have not analyzed it on a deep level. Honestly, I was so bothered by the story I rejected quickly. The thought of the orphan boy not adopted into a loving family was a story line I did not want to live with year after year. My son is adopted. Most importantly, I was not impressed by the way in which the concepts were taught either.

The only curriculum I would recommend is Making Math Meaningful. I have been using it from the beginning, we are now in level 5, and the longer we use it the more impressed I am by it. The way every concept is presented is frankly the most brilliant way to present the concepts that I have ever seen. It all makes so much sense. I keep saying to myself, "Now that makes sense, why have I never seen such and such concept presented this way?" This curriculum teaches children to think mathematically in the way that children learn math naturally. I don't own stock in the company, nor get any money from them for advertising but I do highly encourage others to use this curriculum. I have yet to find anything nearly as well written as this one in all of my years as a math teacher.

Wow, I had no idea that was the premise of LOF. I will definitely check out your rec, and thanks for the reply!

DeleteYou're welcome Christie. Any time. Many blessings to you on this wonderful journey with your children.

DeleteSo here we are in Beta level of Math U See and I'm wondering if we should transition to MMM. I know you talked about not switching but there are not nearly as many word problems as computational problems in MUS. Would just doing the word problems first solve my problem?

ReplyDeleteThe reason I am thinking of switching is that my son seems to have a problem with the computational problems. For example 4, 953-385. He will subtract in the ones, tens and sometime hundreds, then he routinely switches to adding in the house farthest to the left. When he goes to check problems, he sees where the problem is, but often fails to see that he has performed the wrong operation. Sometimes there are even tears. I don't want math to be sadness to him. He does tend to be anxious and a bit of a complainer so I don't want to switch on a whim either but he does seem to grasp word problems better.

Any advice is appreciated. I did read the articles. I still don't know what to do.

Thanks, Jennifer

Hi Jennifer,

ReplyDeleteThe reason that he is confusing the operations is two-fold. First he has no context for the operation. They are just numbers that he is acting on by rote memorization. He is trying to follow steps and therefore can get confused when the steps are not memorized. He hasn't mastered the steps and now with the larger numbers it is impossible for him. Second, he is not visualizing the magnitude. A magnitude is an amount or quantity. 3 can be visualized in different ways depending on the context. It can be a linear distance of miles, the number of gallons of water in a vessel or the number of birds on a bird feeder. He is not visualizing what is happening and why the numbers are being subtracted. What does the 953 and 385 represent? Why is the 385 being subtracted? There is nothing for his mind to make these numbers anything but abstract symbols with nothing to ground them in. The actual story problem can provide that context and help the child visualize what needs to be done. He is frustrated as a result and hence the tears. I don't think it would hurt to switch him to MMM. If you do, I would recommend you place him at least one level lower than where he is now. This way he can pick up any concepts he missed and also, so that he can experience success and build up his confidence. If he has truly mastered a concept then you can move him on to the next. We want him to have success with math.

Blessings,

Rebecca

Thank you Rebecca, I ordered level 1 even though he is 9. I figure we can move as slowly or quickly as we need. What is your take on the longer lessons on fewer days a week?? Jennifer

DeleteYou're welcome, Jennifer. I am so glad I could help. The length of the lessons is completely dependent on the attention of your ds. Once his attention wanders, the lesson must come to an end. For my special needs 13 yo child, lessons are only around 10-15 minutes. I don't worry that he goes more slowly. He is doing what he needs to be doing.

ReplyDeleteHi Rebecca,

ReplyDeleteCan you please explain the equalization concept? What I mean is in the problem would say in 5 plus 8 equals box that we are equalizing 5 and the box. But in the problem Eight minus box = 4 are we equalizing 8 and 4 or the box and 4?

Thanks, Jennifer

Imagine that your equation is a balance scale. The two sides of the equal sign need to balance or equalize. The 5+8 on the left side of the equal sign is equal to the unknown amount (box) on the right side of the equal sign. What would need need to put in the box to make the balance scale equal out? If you put 13 in the box, both sides would equalize. With the problem 8-box=4, what would you need to but into the box to make your balance scale equalize? 8-4 is the same as or equal to 4. I hope that helps!

DeleteThank you, it does!

ReplyDeleteWonderful. You're welcome.

DeleteWonderful. You're welcome.

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