Algebra Before Arithmetic
So you watched some video at Crewton Ramone's House of Math, and you thought, "Oh my gosh, I want to do this with my kids!" After six months your Mortensen blocks finally arrived, and now you don't know what to do with them. You've tried to wade through Crewton's website, but it causes visual ADD. You wish he would just organize it better and give you a plan. Then you took my advice and tried to read Gattegno's ideas about algebra before arithmetic, but you got a headache just looking at the books. You want to do this, but it makes you crazy. Can't someone just give you a straightfoward plan to getting started? Sheesh!
I get this all the time. Up until now, I didn't understand the problem. I asked parents questions and assumed the problem rested in the parent's/teacher's inability to see the mathematical relationships in the blocks. Then I came across a great blog post about mathematical understanding by David Wees at the Reflective Educator Blog. This was a defining moment in math teaching for me. The clouds parted, and the angels started singing - I hear the Hallelujah Chorus as we speak.
Why do parents and teachers find Mortensen Math/Caleb Gattegno so hard to implement? Wees nails the problem. I've been dancing around the edges for months without actually "getting it". While it is true that parents can't see the relationships in the blocks, that is not the primary problem. The problem is that parents/teachers have an instrumental understanding of mathematics while Gattegno and Mortensen provide and expect a relational understanding of the subject. Mr. Wees' exploration of an analogy given by Richard Skemp gave me the insight I needed to grasp the fullness of the problem. I recommend you read Wees' blog post if you are having trouble learning or teaching math.
I lifted this directly from David Wees' blog:
“The kind of learning which leads to instrumental mathematics consists of the learning of an increasing number of fixed plans, by which pupils can find their way from particular starting points (the data) to required finishing points (the answers to the questions). The plan tells them what to do at each choice point, as in the concrete example. And as in the concrete example, what has to be done next is determined purely by the local situation. (When you see the post office, turn left. When you have cleared brackets, collect like terms.) There is no awareness of the overall relationship between successive stages, and the final goal. And in both cases, the learner is dependent on outside guidance for learning each new ‘way to get there’.
In contrast, learning relational mathematics consists of building up a conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point. (I say ‘in principle’ because of course some of these paths will be much harder to construct than others.) This kind of learning is different in several ways from instrumental learning.” ~ Richard Skemp, Mathematics Teaching, 77, 20–26, (1976)
Kalid Azad (Better Explained), in this fantastic interview about how he teaches, talks about two ways of understanding a difficult subject. 1. Moving from the big picture, which you can see but is slightly out of focus, to filling in all the details. Or 2. Starting in one corner, the observer examines and memorizes all the details before moving on the to the section immediately adjacent, without every backing up to see the whole image. Obviously, Kalid much prefers the big picture but slightly out of focus option. Both Kalid and Skemp are talking about the same thing.
Gattegno and Mortensen offer not only a relational understanding of mathematics, but they also require that parents teach math in the same way. This, as David Wees aptly points out, causes no small amount of both fear and consternation. Some parents are so paralyzed by fear they don't do anything at all, some ask for scripted lesson plans while most would just settle for at least a syllabus with the paths to completion highlighted in yellow. A clearly marked path with scripted lesson plans does not exist as that would defeat the whole purpose. What is a parent to do? Shall we pack up our blocks and go home?
Certainly not! The problem will remain whether you play with blocks or do not play with blocks. It's my firm conviction that base ten blocks are superior tools for developing a relational understanding of math. Not only that, Gattegno's position that algebra should be taught before arithmetic provides a student with a wide angle view of elementary mathematics. Details are easily filled in as the student grows and develops and more areas of mathematics can be added without the normal pain and fear that is part of nearly every child's mathematics experience.
Algebra Before Arithmetic - What We Need Is a Good Guide
I began this blog as a way to both chronicle our experience with base ten blocks, and to make what I've learned more accessible to others. As part of our work here (I don't do this by myself), we'll start webinar training shortly. In the meantime, we are not left in the dark. We actually do have some very good guides to lead us. But we must first get over the idea that there is a magic bullet for teaching/learning mathematics. There isn't one. But two hours with C.E. Chambers book The Cuisenaire Gattegno Method of Teaching Mathematics, particlarly chapters 2-4, and Gattegno Mathematics Textbook 1, chapter 2, along with a set of Cuiseniare Rods or Mortensen Blocks will do wonders for the struggling teacher/parent. After that, much of what needs to be learned can be aquired if the adult plays blocks along side the student.
What exactly is algebra before arithmetic when we are talking about a 5-year-old anyway? First, let's talk about what Algebra is. Most people think of algebra as that class they took in late middle school or high school called Algebra 1. They mean some kind of math problem with a letter in it instead of numbers. When we are talking about teaching algebra to small children, we mean teaching the big picture of arithmetic before we get to the details. "Mathematicians use algebra to represent and reason about relationships between mathematical objects and actions," Ian Benson, Primary Mathematics, Spring 2015. Gattegno's aims in teaching algebra before arithmetic is to provide the students with the tools and the time to explore those relationships physically by way of the blocks.
In elementary algebra, we are talking about learning the general rules and principles of manipulating the arithmetic symbols. The numbers can be anything under the sun, but the concepts and relationships between the symbols stay the same. When we are discussing 5- 7-year-old children, we are talking about providing them with a deep understanding of the 4 basic operations and fractions as operators. Starting with algebra gives the student a firm understanding of how math actually works. The student learns to read, speak and write the language of mathematics in its basic form before filling in the details with specific numbers.
What Outcomes Can We Expect After A Year of Teaching Algebra Before Arithmetic?
We are westerners in education. Even as a homeschool parent, I feel the need to quantify what my children learn and mark it off as if that constitutes knowledge. While there is no way to articulate what every child will "get" from a year of algebra before arithmetic, this short list that follows is a generalization for those who want or feel like they need a check list.
- Children can speak and write about mathematics as a language.
- Children fully explore relationships between quanitity and can articulate those relationships in mathematical language.
- Children can symbolically use the four basic operations and fractions as operators. Some students will have progressed to using integers.
- Children will be able to articulate the relationships between the four basic operations and fractions using spoken and written mathematical language.
- Children will understand their math activity as an exploration of concepts that can be approached from multiple directions.
If homeschooling parents aim to teach mathematics well, there is no shortcut whether or not you use the blocks. However, the blocks are a shortcut to a deep relational understanding of the subject. Gattegno and Chambers have provided anyone who wants to try, a path for getting there. It is not a quick and easy path. But it is a much quicker and way less painful than following the traditional route.