Math Education Rant - Part 2

What Can be Done 

The previous post Math Education Rant - Part 1 divided young math students into three groups: the top students for whom math makes sense and is generally pretty easy, the middle students, usually the majority of a class, and the poor students who find math extremely difficult. My opinions about math education in elementary school take into account all three groups.

1. Maria Montessori based her educational methods on child development. She understood the prerequisites for a child to be ready for math (and other subjects) in a classroom. Her prerequisites were "First a child has established internal order, Second the child has developed precise movement. Third, the child has established the work habit. Fourth, the child is able to follow and complete a work cycle. Fifth, the child has the ability to concentrate. Sixth, the child has learned to follow a process. Seventh, the child has used symbols." Once a child reached this maturity of mind and a readiness to work, it was time for them to work with concrete math materials. There are far too many students who start kindergarten without these prerequisites. These children need to be given time and a curriculum to make sure they have "a readiness to work." 
2. Speaking of time, there is a continuous pushing down of the curriculum. When five-year-olds are expected to do first grade work we are pushing them to learn concepts they may not be developmentally ready to learn. There are complaints about the number of students who require remedial math classes when they start college. Perhaps if children were given time to learn at the beginning of their academic careers, they wouldn't need as much at the end. I have seen first graders who were having difficulty with double-digit addition. When tutoring these children it was apparent that they were still having trouble identifying written numbers such as 11 or 12!
3. Most children in elementary grades need concrete materials to help them understand abstract ideas. Using Unifix blocks, Cuisenaire rods or other math materials needs to be a continuous activity. One day of demonstrating a concept doesn't give children enough time to work through the process on their own. Many teachers stop the use of concrete items in the upper elementary grades, but most children still need concrete items to understand fractions and decimals.
4. Teachers need to understand that manipulating concrete items like beads, while necessary for young children, doesn't mean the children transfer that knowledge to the abstract symbols of numbers. The symbols need to be taught with the items, but that still does not mean a child will totally understand an abstract concept. 
5. In a effort to insure that children "understand" the underpinnings of math we overwhelm them with explanation. Some children who might be good at math are hindered by this overuse of words. An explanation of how to do a complicated process, even with a demonstration, is very difficult for some young children to understand. Requiring students, themselves, to explain the process is even more difficult if they have poor language and/or writing skills.
6. While in theory teaching a child more than one method to do basic arithmetic sounds good, that approach adds a layer of confusion to some children. They may decide to combine the methods, and do so incorrectly, or never thoroughly learn any one method. Adults may readily see how the methods are related, but an eight-year-old may not have the language, or cognitive ability to see this.
7. The visually busier the page (lines, boxes, circles) the more difficult it is for some children to comprehend what they are suppose to do. Lattice multiplication is an example of this. I have tutored children frustrated by required lattice multiplication because the spaces were too small for their large handwriting. A child with visual perception problems can even be confused by graph paper!
8. Some children need more time to learn than others. We have no problem understanding a young child wanting to hear the same book read again and again. We need to give the children who need it time for repetition, or a break to do something else, and then come back to the concept. I worked with classes of children who had difficulty with math (as you can see in the example I gave in Part 1) Parents raved at the progress their children made in my class. I often felt it was just lucky that I taught a child the year he or she was finally able to understand the lessons. When I tutor I meet young children who obviously can't understand something (glazed eyes, a look of panic and complete confusion, etc.) who three months or even three weeks later find it suddenly makes sense. Educators have to be aware if something is beyond a student's ability at a given time. If we can't give an elementary school age child the necessary time, there is something extremely wrong with our education system.

The above eight points concern the need to look at children's ability to learn rather than what is being taught. Too often curriculum ignores the learning differences in children, the various developmental stages of the young student, and the variation of abilities in the new student. The next paragraphs concerns how to implement teaching a curriculum to groups of young children. 

I had the privilege to do student teaching in a school with an excellent math program. The entire school (k-5) had math the first period. The math curriculum had been divided into weekly modules. Each module had a pre- and post test. This allowed a child to test out of a module. The school had specific teachers who worked with children who had to repeat a module. Repeat modules tended to include more hands-on materials and more individual attention. Because everyone had math at the same time, the student who tested out of a module could move up right away. Math classes might consist of children in 3rd, 4th and 5th grade. A child who found math easy was not held back, but still could repeat a module when she came to material that gave her difficulty. A student who might have difficulty in subtraction could still move up if he found it easy to measure perimeter. Some modules were reviews of pre-requisite skills for new concepts. 

The children seemed to love math. The math-adept second graders got to go to class with fourth graders. All children knew that if they didn't "get something" they would have a chance to try it again. No one seemed to feel bad about having to repeat a skill because everyone had to repeat now and then, and you got a different teacher when you repeated a module. Worksheets tended to have large print, simple instructions, and plenty of white space since there was a limited amount of material and it was understood that there might be very young children in a class with subject matter often taught only to older children.

The U.S. tends to have a math curriculum that is very broad (many topics) and fairly shallow (not much depth to each topic.) Modules allowed students to get the depth they needed.  I am not suggesting that all schools use this method but I believe elementary schools should have math programs based on the idea that children need to be able to work at different speeds and move up or down at their own pace. The emphasis should be on children and how they learn, rather than on speeding through a curriculum, holding children back who could readily move ahead under the guise of saying, "Why those children learn while they are helping others," and refusing to recognize that individuals might find some areas of math easier than others.  Such a program would give all children a change to get a firm foundation to build on, rather than a shaky platform to start middle school.