Visiting the grade 3 math class again, I observed students continuing their work on equivalent fractions.  They were doing some written desk-work to answer problems from the Smartboard e.g. ¾ x 12 and 2/6 x 18.  They were to build the needed  patterns and write the answers in their books.  At the end of 10 of these questions there were three “Math Challenge” questions based on some of the patterns they had used to answer the above questions.  E.g. (1/4 x 16) +  (1/3 x 9) + (3/5 x 20)

It should be noted that each child had their own large box of rods, in sufficient quantity to create several patterns to keep on their desk while they worked out the answers.

Most of the kids were getting correct answers, some very quickly.  Others were working slowly and needed some guidance.  I worked with one girl, new to the school, by asking some more basic questions such as which pattern do we need to look at to find the answer, what does the number on the bottom of ¼ tell us, etc.  She was then able to get an answer.  There were two other boys that were having trouble who asked for the teachers help.  I did notice that the teacher did not ask students who were finished, to help those that were having difficulty.  I think this would be part of Gattegno approach, but there may be some practical reasons why this was not used here. (I did not see students helping other students often in other classes either!) It would be interesting to hear from other teachers about how useful this is or problems with it (i.e. problems with students helping others who are having difficulty).  I have certainly seen that students helping students can be very effective in my small tutoring classes.

It should be noted again that the teacher had most of the students fully engaged in their work.  They all seemed very enthusiastic about the challenge!

After most students had finished the questions, the teacher asked them to cover up their patterns.  They were then asked to imagine the patterns and answer a few questions without looking at the rods.  If they can do these fractions without the rods, they have made an important step in moving towards a mental perception of the math and away from dependence on the rods.  I overheard some of them say “This is easy!”, which is I think, the mark of a successful lesson.

On to another grade one class….

They continued to work on multiples, but this time also with subtraction, first with the teacher using the Smartboard and magnetic rods (I must get some of these), with one student coming to the board to show how to solve a problem. (eg. 2P – 3r = ___ )  Again I watched the class get increasing restless – without anything in front of them to keep their hands busy – until they were asked to write down a list of questions using multiples.  Once they had written them down, they were allowed to get their individual boxes of rods and start working on the problems.  They were much more involved in their work once they had the rods, and were busy building the trains and patterns.  Some needed help, but most worked busily away and constructed  correct answers.  The students whom I helped would call me back for more help often; I tried to give them some direction, but then the next question they would have the same problem.  They had not “learned” how to solve the challenge.  I wondered if they were newer to the rods than other students, or if they just needed more time.  Maybe we needed to go back to the previous lesson with these kids ( on subtraction or difference) before we tried these problems again.  Without students helping other students, it certainly puts more work on to the teacher.

What was obvious in both these classes (besides the “subordination of teaching” approach) was the incredible usefulness of manipulative’s in keeping the students focused and giving them a way to understand and solve the problem!