Algebra – Training A Mathematical Mind Series
This is the last post in this series on Training a Mathematical Mind. Today we are looking at algebra. First let’s define our terms. I went over to Wikipedia and stole this: “Algebra (from Arabic “al-jabr” meaning “reunion of broken parts”) is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. . . Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values.”
What I love about Cuisenaire Rods is that they are a semi-abstract model of math. The units are not delineated on the blocks which means they don’t become a distraction. Each block only has value in its relationship to other blocks. This makes “relationship seeing” super duper easy, even for small kids.
A. is 4 and loves to use a pencil, as opposed to P., age 5, who would rather die than write. She is not as adept at manipulating the symbols and expressing the relationships found in the blocks, but she is getting there. She can easily write those out on her own without my assistance. She is learning shorthand for rapid addition as she counts the number of white rods. Multiplying is a lot easier than adding a string of numbers one at a time. 3r and 6w make short work of r + r + r and w + w + w + w + w + w.
Algebra: The Starting Place – Not the Destination
It would seem like the easiest place to start children is the counting of objects. When children have trouble understanding arithmetic we say that abstractions are too difficult for them. But that isn’t true. Children deal in abstractions (and algebraic thinking for that matter) as soon as they learn to talk. The child learns that “I”, “me” and “mine” derive their meaning from the context of the person speaking. When the child speaks, “I” is a reference to self. If someone else is speaking, the word “I” refers to someone else entirely. A dog is a dog even though they vary in color, size, and shape. The dog may be a small, brown and white Shitzu or a very large, black NewFoundland; it is still a dog.
Why do we think children struggle with abstractions when it comes to math? I suspect that the problem lies in not having a good model. Dogs and people are good models. Numbers, themselves, are an abstraction. This is why Cuisenaire Rods make such a great tool for mathematical exploration and for examining the rules we use to express mathematical ideas. They make it possible to visualize ideas in a semi-abstract form. Algebra before arithmetic has proven to rapidly accelerate my children’s progress in math. Yet, we are not in a hurry. Some days we do math for 15 minutes and some days an hour. That depends on how my kids are feeling. Our days are relaxed and the forceful math conversations that once plagued home are missing.
In the above image, we have a dark green (d) and it’s equivalent: 2 light greens (g). If algebra is the study of math symbols and the rules for manipulating those symbols, let’s get to work on some of those symbols. When I gave this to my son this is what he discovered:
- d must be even as it can be divided into two equal parts.
- d = g + g
- d = 2g (Different symbol and much faster to both say and write.)
- 1/2 d = g (This is a fraction.)
- d ÷ 2 = g
- d/2 = g (This can be read as division or a fraction.)
- d ÷ g = 2
- d/g = 2
- d – g = g
In the above image, we have a purple (p) and it’s equivalent: 2 reds (r). Does the following look familiar?
- p must be even as it can be divided into two equal parts.
- p =r + r
- p = 2r (Different symbol and must faster to both say and write.)
- 1/2 p = r (This is a fraction.)
- p ÷ 2 = r
- p/2 = r (This can be read as division or a fraction.)
- p ÷ r = 2
- p/r = 2
- p – r = r
Now that we have exploited the relationships available to us, we can see that the same pattern exists in both images. Sure, we changed letters, but there is a pattern for even numbers. This becomes even more apparent if you repeat this exercise with all 5 even blocks. So can we make a master pattern for every even block based on what we know? Can this be done with children in the earliest grades? Of course. Most kids will figure this out on their own. But if not, we can simply ask questions and direct them where we want them to go.
How do we come up wth a master pattern for all even numbers? We will substitute the largest block with the letter “x” and the two smaller blocks with the letter “y”.
- x must be even as it can be divided into two equal parts.
- x = y + y
- x = 2y
- 1/2 x = y
- x ÷ 2 = y
- x/2 = y
- x ÷ y = 2
- x/y = 2
- x – y = y
What We Learn From This
Starting from algebra is just as simple, if not more so, than starting in arithmetic. Algebra is certainly not hard for children to understand. The Cuisenaire Rod makes the perfect rod for this type of discovery.