In the beginning of 2020, there were hundreds, perhaps thousands, of Orthodox home-schooling families. Many more have decided to home-school because of the COVID situation. Therefore, we thought it might be helpful if we presented some thoughts on choosing a math curriculum. We have taught and tutored students in math for more than thirty years and have used many different curricula. We have used Abeka, Saxon, Mortenson Math, Math-You-See, Singapore Math, Miquon Math, Math Mammoth, math from Key Curriculum, and probably others that have been forgotten.
Given so many curricula, it is understandable that we appreciated some curricula more than others. Thinking that this experience would be helpful to home-schooling parents, who should have the authority to make decisions about curriculum, we would like to share what we found to be the most important aspects to consider when choosing a math curriculum.
Our experience has revealed four criteria which we suggest home-schooling parents consider when choosing what curriculum to use. After simply listing these criteria, we will explain them in more detail.
When deciding which math curriculum to use, please consider,
1) Does the curriculum spend sufficient time on a particular concept before introducing other concepts?
2) Does the curriculum use pictures or manipulatives to demonstrate the concepts presented?
3) When teaching a concept, does the curriculum offer different approaches to the concept?
4) Is the curriculum “parent-friendly?”
Does the curriculum spend sufficient time
on a particular concept before introducing other concepts?
Some math curricula tend to spend a little time on one concept, then shift to another concept, and then to yet another. These curricula will eventually return to the initial concept, teach more about it, then shift again to another concept. This has been called the incremental approach. Although this approach sounds like it might be very effective, we have found that it is not the best way to teach math.
The idea seems to be that children can only handle a small amount of a single concept at a time. Spending more time and getting into more detail might be too taxing for them. Perhaps the authors of these curricula think that the students would be bored if too much time were spent on one concept.
While it is true that you can overwhelm a student with too much detail, it is also true that it takes enough detail and enough time to learn, to practice, and finally to absorb what is taught. Without this time, the teaching will not be effective. This is especially true in teaching mathematics in the early grades. What a child learns about math in the first six years of school is the foundation of everything else they will learn of the subject. If a child does not really understand the basics of mathematics, he will inevitably have trouble later in his studies.
In some ways, eating and learning have the same steps. We ingest the food, then digest the food over a period of several hours. Only when the food is thoroughly digested can we finally assimilate the food, making what we ate part of our own body. Similarly, when children are presented with a concept, (the ingestion stage), they need to be given the necessary time to practice it and experiment with it, (the digestion stage). Only when they have been able to thoroughly “digest” the concept, can they make that concept part of their understanding, (the assimilation stage).
Children may learn some math rules and skills quickly, but this type of learning is often superficial. Gaining a firm understanding of those rules takes more time. If they are not given enough time and experience with a concept, their understanding will not be solid, and will probably fail when they are expected to apply this understanding to more complicated concepts.
Children want to learn. They want mastery over the concepts they are taught. Knowing that they really know something is very important to children. This sense of mastery is an inspiration to learn more. The sense of barely knowing has the opposite effect. It is crucial for us to realize that children know the difference between these two experiences. They know when they do not really understand, although they will not easily admit it. If children do not understand something, they often claim that they do, thinking that not understanding is their fault. It is not their fault; the fault lies in the teaching method.
Some publishers of math curricula seem to be trying to sell their curricula as “academically advanced.” The incremental approach is a way of presenting many different mathematical concepts which, in the eyes of some, makes the curricula more complete, or academically superior. What is the benefit of using an “academically superior” curriculum if the students are not learning?
Does the curriculum use pictures or manipulatives
to demonstrate the concepts presented?
When reading a story, do you automatically form images in your mind? When trying to follow the instructions of how to assemble something, are pictures helpful? The answer to both questions is probably, “Yes.” This is because our thinking capacity depends heavily on our capacity to produce mental images. In fact, St. Theophan the Recluse teaches that, “the entire intellective aspect of the soul is imaginative.” (The Spiritual Life, pg 49). We are all familiar with the adage, “A picture is worth a thousand words.” Pictures help us understand the words. This is especially true with children, whose life experiences are far fewer than ours.
Although children often offer some amazing insights, seemingly beyond their experience, they generally think in very concrete terms. Their concrete experiences in life gradually allow them to think in more abstract principles, but first they must have these concrete, three-dimensional experiences. Without the concrete experience, the abstract understanding can easily be unstable, or untrue. This progression, from the concrete to the abstract, is very important to appreciate when teaching mathematics.
Some math curricula provide images in the instructions. This is helpful, but not enough. Since mathematics is actually a language which describes things like quantity, weight, force, velocity, etc., that is, events that happen in the three-dimensional, physical world, the best images are those which are three-dimensional. These three-dimensional images are called manipulatives, that is, objects that can be handled (the Latin word manus means “hand”) by the person learning.
The teaching of math and the teaching of science have much in common. It is probably known to all that science is based on experiments. How do you know if some scientific idea is true unless it has been tested, and affirmed, by a series of experiments? The entire scientific method is based on experiments. Yet, some poorly written science curricula present many concepts without any experiments. The students are expected to not only believe, but also to understand, what is being taught without any first hand experience. We have to tutored students who have suffered under these curricula. They understand very little.
Using manipulatives as a part of teaching mathematics is the equivalent of performing experiments when teaching science. Three-dimensional manipulatives allow the child to experience, not just hear, or read, the concept being taught. Experience is the foundation of learning both math and science.
As a child progresses in the study of mathematics, the subject matter becomes, in its nature, more abstract and therefore more difficult to represent with simple manipulatives. This is when those who do not have a firm understanding of the basics find that they have no idea what they are supposed to do. Their memory of rules is not enough on this higher level and their house of understanding, built on sand, is washed away. However, those students who were given enough concrete experiences with the basic principles of mathematics in elementary school, can apply the solid understanding they gained to what they are taught in high school and beyond. Their house of understanding was built on a rock.
When teaching a concept,
does the curriculum offer different approaches to the concept?
A beautiful diamond is fully appreciated only when it is observed from many different angles. Each angle reveals an aspect of the diamond that is invisible from other angles. Only after viewing the diamond from many angles does one “know” or “understand” the diamond.
The concepts of mathematics are like a necklace of diamonds, each concept being an individual diamond. Therefore, teaching a mathematical concept from different angles gives the student a fuller understanding of what is being taught. Simply teaching the rules, or mechanical skills, does not give this fullness.
Yet, mathematics is far more than a series of independent concepts, (or diamonds, in this analogy). In a diamond necklace, each diamond is independent of the others. One diamond can be removed and the necklace can still be beautiful, although a bit shorter. However, in mathematics, the concepts are, most often, not independent, since the understanding of one concept is often very dependent on the understanding of a previously taught concept. If a concept is taught from different angles, it will be understood more completely and, therefore, contribute to a firm foundation for future learning.
We have seen very few math curricula fulfill this criterion. Most seem to present the concepts without “playing” with them, that is, seeing the concept from different angles. In fact, when we saw a curriculum that did this, we were inspired, and started using it.
Based on this idea of playing with math concepts, we developed a few math games which explore the times tables from different angles. Knowing the times tables well is very important they are central to many concepts the student will need to know, not only in grammar school, but also in high school and beyond. A link to these games will be given at the end of this article.
Is the curriculum “parent-friendly?”
Many math curricula are written in a way which assumes that there is a math teacher, with some experience, conducting the class. Such a teacher has the experience to teach more than what is in the text of the book, therefore being able to make clear to the students anything that is not clear in the text. This assumption is very understandable since most curricula, math and otherwise, are designed for a classroom setting in an institutional school.
However, for the families who are home-schooling their children, this assumption will often be unfounded, making such a curriculum difficult to use. Home-schooling parents, (usually Mom), wear many, many “hats.” The math teacher “hat” is often the most uncomfortable. Many parents have had their own struggles with math - not their own fault - but are now expected to teach it! Therefore, this criterion of being “parent-friendly” is especially important for home schooling parents.
If a math curriculum fulfills the first three criteria, it is probably “parent-friendly.” It will be “parent-friendly” because such a curriculum will be “child-friendly.”
Many parents feel unqualified to teach their children, perhaps especially in math. Yet, with the right curriculum, the children can learn very well, and, glory to God, the parents can learn as well. A curriculum that is written in a way which a child can understand, especially with the addition of three-dimensional manipulatives, will also be understandable to a parent who may not have much confidence in his/her own math competence.
Before we conclude, let us make a general comment which applies to all areas of study, including math. Teaching your own children can be a wonderful experience, an enlivening educational and relational experience for both parents and children. Our experience of many years has taught us that teaching is a transmission in both directions - from the teacher to the student and from the student to the teacher. With the right educational approach, an atmosphere is created in which all involved are benefitted. Teaching is a wonderful way for the teacher to learn what perhaps they missed when they were students. This is nothing less than God’s mercy.
As noted in the beginning of this article, we have tried many different math curricula. The curriculum that we used at the end of our journey was Math Mammoth. We used it because we found it to be most consistent with our own approach to teaching mathematics. It fulfilled our criteria far more than other curricula. Although Math Mammoth does not include three-dimensional manipulatives, the images used in the textbooks were perfectly aligned with the three-dimensional manipulatives we were already using.
We were also impressed with the way that the curriculum approached story problems. We knew that the ability to solve story problems was a real test of understanding the math concepts presented. If a student had trouble with story problems, which are potential real life situations, we knew that he had not really understood what had been taught. Perhaps he knew some rules, but had little understanding to apply them to reality.
The author of Math Mammoth shows the student how to draw diagrams of the situation presented in the problem. We found this approach very helpful because it engages the imagination, a crucial part of our thinking abilities. Once the student can draw a diagram which illustrates the story problem, the solution is often very clear.
Another feature of Math Mammoth is that it is very flexible. The curriculum has two text-work books for each grade level, one for each semester. These books cover the material for the year. The books are written with the aim of being “self-teaching”, meaning the students can learn from the books without an expert math teacher. This means that they are “parent-friendly” as well.
In addition, the curriculum also has available separate books on individual concepts - fractions, multiplication, decimals, etc. These books provide extra practice in many areas. There are other supplemental parts of the curriculum as well.
The curriculum can be purchased as individual, printed books, or as a CD which contains all or many of the books. This feature adds to its flexibility. For more information, go to MathMammoth.com.
If you are interested in the manipulatives and games we added to Math Mammoth, you will find them in The School Store at stmichaelschool.us. These manipulatives and games would be a great addition to any curriculum.