Over the years, math programs have changed.
No longer were we teaching kids how to do math, we were teaching numeracy skills. There used to be a simple set of review questions, present sample strategies for solving the simple questions, then an increasingly difficult set of questions. Finally, the new skill would be practiced with a number of problem-solving and more wordy questions that demand that the learner apply new skills in the right order, to one and two-step problems.
For example: subtraction
First the students learn the concept with manipulatives and counters on the carpet, on their desks. Once they have achieved abstract cognitive thinking, they are more able to manipulate the number on paper. They are taught the various columns, to give them a common understanding of the terminology. They understand the 10's, 100's and thousands columns. They are assigned a review, to ensure that they understand subtraction. They are taught a process and procedure for regrouping: borrowing from the 10's column, to regroups it into the 1's. The text book was divided up into two pages. The first page, called 'working together' presented sample answers to the target skill. On the 2nd page, they presented increasingly complex questions. I assigned kids homework from this page, did not require them to finish the first page.
Students then work alone, or in pairs. The teacher should take up a few questions to ensure that students are on track. Kids exchange their workbooks, kids are taught to correct their peer's work agreeably. They are invited to put in positive messages and give their peer an idea of where they went wrong, and provide extra support. At this point, some students might quietly approach me and suggest that their peer needs a bit more work on subtraction facts. I would assign the student questions on the 2nd page, with a 30-minute time limit at home to reinforce the skill. That way a student who was struggling could do 5 questions really well, and another student could move form the simple to the complex questions and word problems and each would work at their level of ability.
During the early years of the 2000's, retired teachers (numeracy experts) were hired by publications companies to produce new strategies for teaching math. At that time the 'leading the horse to water' movement seemed to flourish. A constructivist approach, best applied after basic skills have been mastered, did not work for some students. The text books no longer presented simple formats, they reduced the number of questions available for practice. They increased the images and graphics on the pages. They would present a problem, which might involved previously taught skills (or might not) and expect the student to come to an AHA moment and realize they needed to regroup to solve the problem. The texts were written well for deep-thinking, middle class students with a wide background knowledge and strong problem-solving skills. It did not work for ADHD, LD, ESL, or kids with limited knowledge or experiences with limited reading levels.
I found that once these texts were rewritten and flogged by those who were hired to write them, students began falling behind. Parents did not understand 'the new math', and everyone began to flounder. Kids no longer practised their skills. Parents gave up.
While I loathe the 'back to basics' movement, I think we need to redesign the texts to present simple operations and problems. We cannot assume, nor do we have the time to expect that students can reinvent mathematical notions, such as the operations (addition, subtraction, multiplication, division) or other more complex tasks (elementary algebra, plane and solid geometry, trigonometry) involved in working with numbers. It is much more effective to sit them down, demonstrate the skills, accept other means to solve the problem and work through the skill and then move on to apply it to real life.
I fear for those students, estimated 20 -30%* who cannot read well enough to interpret school textbooks, and fall behind and then fall out of the system. We need them to develop concrete skills and apply them to their lives. Each time a student successfully solves a problem s/he increases the dendrites (brain cell connections) and is able to use these connections to solve other problems.
Their model of text comprehension describes the complete reading process, from ...texts and the textual meso level of the math text.
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