Ditches are long narrow passages made in the ground by digging. People dig ditches for various reasons, at varying depths and varying purposes. Someone once compared the educational system to men digging a ditch. How deep is your ditch?
If you think about it, ditches have different lengths and depths, depending on their proposed purpose. Some ditches are for laying pipeline, necessary for water or sewage. These have to be dug at a certain depth so that they avoid problems with weather, but not so deep that the diggers have to dig through rock and other obstacles that are unnecessary for the purpose. Some ditches are short but deep, like a trench used by soldiers on a battlefield or a trap used to catch animals. Other ditches are designed for directing run-off water, and the depth of those depends on the amount of water expected at any one time. The purpose of the ditch decides the depth and length which it will be dug.
So how does this compare with education? The depth of a person’s education shows how much that person understands a particular skill. The length of that education shows how many individual skills in one particular subject that person learns. If someone spends a great deal of time and energy understanding a particular topic, then we can say that person has a deep understanding. If that same person knows many things about a special subject, then we can say that person has a long “ditch.” People only have a limited amount of time in school, whether homeschool, public school, or private school, so how “deep” and how “long” should a person’s educational “ditch” be?
This person who compared today’s schools to a ditch (sorry, I cannot remember who it was) said that if America’s Educational System were a ditch, it would be 2 inches deep and 2 miles long. Wow! That is a really long ditch! There are many purposes which could be satisfied by a ditch that long! But, only 2 inches deep? Really? What use is that to anyone? Good question. And that’s where you should be examining your own educational program.
How deep are you teaching the students in your life? Whether you teach one child or thirty, teaching them a lot of things on a checklist with no depth to each item is not really valuable, is it? Just like a 2-inch deep ditch, there is no purpose for it. Instead of teaching a little of a lot of things, teach a few things with a lot of understanding. Like a person digging a deep ditch, you may spend a lot of time in the same place, doing the same thing, but you will end up with a product that can be of value later, once this “ditch” is finished.
What does a “deep understanding” look like? Let’s take something many teachers teach a lot of but frequently don’t teach with depth – addition. Typically, children will learn to count their numbers, then they will learn how to use either a number line or their fingers to add single digits. From there, they go on to double digit addition, and then harder problems. (Yes, I am skipping subtraction skills, which are usually taught with addition, but subtraction is not my focus right now). Many children fully understand that when you have five apples and your friend brings you 2 apples, then you will end up with 7 apples. However, most math programs may only have one or two questions that are written that way. Usually, the child sees 5+2=___. What does that mean? Some children see it as the same type of problem, but other children do not. Are these children behind? NO! These children are exactly where they should be developmentally – they understand things concretely but not abstractly (see my other article “Not Just a Little Adult” to understand more about concrete and abstract).
If a child is given a problem of 5+2=____, the teacher should use the concrete, pictorial, abstract method for this problem. First, for the concrete part, the teacher should get 5 items and 2 related items (these could be 5 red grapes and 2 green grapes, 5 stuffed animal dogs and 2 stuffed animal cats, 5 red cars and 2 brown cars, or whatever you have available). Have the child take them out and put them in 2 separate piles, then push them together and count the entire group. Write the answer. Then get another set of 5 items & 2 items, so they can see that you can apply this type of problem to many things, not just what you took out. Do this 2 – 3 times. Then, go to the pictorial method. Have your child draw 5 circles in one box, then 2 circles in another box. Have them put the plus sign between the boxes and put an equals symbol after them. In a box after the equal symbol, have your child draw 7 circles, crossing off one circle from each of the other boxes as they put them in the equal box. Then count what is in the box and write the numeral 7. Do this again, only this time, cross off the squares (or whatever you have them draw) and count aloud. This time they do not need to draw them in a box on the other side of the equal sign – only write the answer in numeral form. Finally, after a couple of different problems, have the child write “5+2=____.” Then they can answer this problem.
That seems like an awful lot of work to solve one problem! It would take FOREVER to have the child do an entire worksheet full of those addition problems, especially when it is so much easier to teach them to count on a number line or their fingers. If this was your thought, then you are in the habit of digging 2 inch, 2 mile ditches. Yes, it would take too much time to do an entire worksheet full of these problems (doing 3 concrete examples, 2 – 3 pictorial examples, and 1 abstract notation for each problem), so don’t! Only do 3 – 5 problems like this or possibly only one problem if the child becomes very frustrated. However, where only a few students understand what they are doing when they count by number lines and fingers (an abstract concept), almost EVERY student would understand what they are doing with the depth method (which begins concretely). THAT is your goal, isn’t it? To make sure they understand what you are doing. As a 4th & 5th grade teacher, it was very frustrating to find students (very intelligent students, by the way) who did not really understand what they were doing when they were adding. Since multiplication is simply repeated addition, it meant even less understanding of the multiplication tables. These students were on a downhill slide of education which is harder to stop the longer they are allowed to continue. If they don’t know what they are doing when they are multiplying (just writing down something they memorized), then how can they understand Algebra? Geometry? Trig? If they can understand the basics, then the “harder” work will be much more readily understood.
Math is not the only subject this is true for. If children have a chance to play with water, then they can be ready to understand the three forms of matter (solid, liquid, gas). If they take field trips to a nearby lake or cave or mountain, they are more able to understand landforms and varying bodies of water. Education should be useful, not just philosophical.
How deep is your student’s ditch? Spend time in the same place, doing the same thing over and over. When you feel they understand one part very well, go to the next step (instead of 5+2, learn 7+6). Just like when you dig deeply in one place it makes it easier to dig deeper in the next step, when your student deeply understands one thing, the next step is much more easily understood, and probably won’t take so much time. Dig ditches that will be the foundation for a valuable education – your student deserves it!