Q: What's wrong with a spiral curriculum? Isn't distributed practice a good idea?A: Nearly all US mathematics textbooks are incoherent. That is, they don't teach that math builds upon in itself, block by block. For math to make sense, what you learn today should be completely derivable from what you did yesterday. You would never use a history book that taught history by teaching why the Puritans left Britain for two days, then spent two days talking about the Incas, then a unit on the geography of Antarctica followed by a bunch of other topics about the Ming dynasty, Reconstruction after the Civil War, and Medieval architecture, and the only several dozen or hundred pages later taught about what happened when the Puritans arrived at Plymouth. But that is what spiral curricula textbooks do in math. By jumbling up all of those concepts, kids can't keep track of why what they are doing is true, or how it's related to what they did before, anymore than they would keep track of why the society the Puritans built in America was about religious freedom when they hadn't seen nor heard about religious persecution for several weeks. Any well phrased problem that built on prior math would allow for distributed practice. But skills are built in order to build up other skills, and should do so in a sequential manner, not haphazardly. When we teach students that math is incoherent, they cannot learn how to reason mathematically. But algebra is all about reasoning. We must teach math coherently. What does an incoherent curriculum look like? Here's a list of lessons in Saxon Intermediate 5: (All lessons from Section 3): 21 Word Problems About Equal Groups 22 Division with and without remainder 23 Recognizing Halves 24 Parentheses and the Associative Property 25 Listing the Factors of Whole Numbers 26 Division Algorithm 27 Reading Scales 28 Measuring Time 29 Multiplying by Multiples of 10 and 100 30 Interpreting Pictures of Fractions, Decimals, Percents By comparison, here's a list of lessons from Singapore Math 5A, Standards Ed: Section 3: Fractions 1. Comparing Fractions 2. Fractions and Division 3. Addition and Subtraction of Unlike Fractions 4. Addition and Subtraction of Mixed Numbers 5. Multiplying a Fraction and a Whole Number 6. Fraction of a Set 7. Word Problems Section 4: Multiply and Divide Fractions 1. Product of Fractions 2. Word Problems 3. Dividing a Fraction by a Whole Number 4. Dividing by a Fraction 5. More Word Problems Until American students are taught in a sequential, coherent manner, they will continue to fare poorly at algebra and beyond. |