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.
