math but well worth reading. I too teach 5th grade math. This might help in teaching multiplication & Division concepts. I just received this article today from our curriculum director and thought it was interesting.

Elementary Mathematics – Harder to Teach Than We Think

“All you have to do is add, subtract, multiply, and divide,” said one California legislator during a 1990s debate on mathematics education. “How hard is that?” In this thoughtful American Educator article, University of California/Berkeley professor emeritus and curriculum expert Hung-Hsi Wu totally disagrees, and proves his point by describing the pedagogical content knowledge needed to teach two topics: adding whole numbers and dividing fractions.

• Adding whole numbers – Wu asks us to consider the following second-grade math problem: 45

+ 31

76

Too often, students learn to solve problems like this by rote, without understanding what it means to add numbers, why it’s worth understanding, how it can be fun, and why it’s hard if it’s not done right. Wu suggests the following approach.

Ask students to imagine that a boy has saved 45 pennies and a girl has saved 31. Do they have enough money to buy some stickers that cost 75 cents? Give students two bags of pennies, one containing 45 and the other 31, and ask them to dump the coins out and count them to find the answer. Many students will mess up (it’s hard for second graders to count that many coins accurately), at which point you suggest a shortcut: put aside the 45 pennies (you don’t need to count them) and count on 31 more (46, 47, 48, etc.). More students will now get the right answer, but some will still mess up. “Then you get to play the magician,” says Wu, giving them a way to find the answer by doing two single-digit addition problems. How? Write the sum 45 + 31 and explain that this really means taking 45 and counting 31 more steps. Show that this can be solved much more easily than counting, by adding 4 + 3 and 5 + 1 to get 76.

But this needs more demonstration and the introduction of place value, one of the most important concepts in elementary mathematics. Collect the 45 pennies and put them in bags of ten – 4 bags with 5 stragglers. Do the same with the 31 pennies – 3 bags with 1 straggler. Then have students lay out and add up the bags of ten and the stragglers and they will figure out that there are 4 + 3 bags of ten totaling 7 bags and 5 + 1 stragglers totaling 6, which adds up to 76 pennies. To make the place values explicit, you can show that:

45 40 + 5 40 + 5 45

+ 31 + 30 + 1 + 30 + 1 + 31

76 ? ? 70 + 6 76

“Now, they will listen more carefully to your incantations of place value,” says Wu, “because you have given them more incentive to learn about this important topic… If we succeed in getting students to thoroughly understand addition without carrying, then they will be in an excellent position to handle carrying too.” What comes out of this procedure is students understanding that the most obvious way to solve the original problem – counting – is not the easiest. “Instead of tedious, error-prone counting,” says Wu, “you used the concept of place value to introduce the idea of breaking up a task digit by digit and adding only two single-digit numbers in succession… It teaches children an important skill in mathematics: if possible, always break up a complicated task into a sequence of simple ones.”

The beauty of arithmetic algorithms, says Wu, is that they reduce the most complex problems to computations with single-digit numbers artfully put together. “This is the kind of thinking students will need to succeed in algebra and advanced mathematics,” he says. They’re in a position to understand that even in a more complex problem, for example,

45723

+ 31251

76974

where the 4 stands for 40,000 and the 3 stands for 30,000, finding the answer is still a matter of adding 4 + 3 and the other single-digit sums. This principle carries over into upper-elementary math involving multiplication: problems can be solved by multiplying single-digit numbers (making it essential to master multiplication tables to automaticity). “One can flatly state,” says Wu, “that if students do not feel comfortable with the mathematical reasoning used to justify the standard algorithms for whole numbers, then their chances of success in algebra are exceedingly small.”

The beauty of place value and the decimal system, says Wu, is that it makes it possible to handle large numbers using only ten digits: 0, 1 2, 3, 4, 5, 6, 7, 8, and 9. The decimal system came from the Islamic Empire around the 12th century; the Arabs got it from the Hindus around the 8th century, and recent research suggests that the Hindus got it from the Chinese, who have had a decimal place-value system from way back.

• Dividing fractions – “The rhyme, ‘Ours is not to reason why, just invert and multiply,’ gets it all wrong,” says Wu. “With a precise, well-reasoned definition, there is no need to wonder why – the answer is clear.” He says he encounters many well-educated adults who have no clue why they invert and multiply when dividing fractions – and that’s the challenge that upper-elementary teachers face: getting their students to understand the logic. “Because fractions are students’ first serious excursion into abstraction,” says Wu, “understanding fractions is the most critical step in understanding rational numbers and in preparing for algebra.” Wu believes that starting in fifth grade, students need to be taught that each division statement (24/6, for example) implies a multiplication statement: 24 = 6 x 4. They also need to shift from the primary-grade methods for visualizing fractions – pizzas and pies – to seeing fractions as intervals on a number line. With whole numbers, students can count on their fingers. With fractions, the number line is the perfect way to making fractions concrete. Number lines also help put improper fractions in perspective: 1/3, 2/3, 3/3, 4/3, 5/3, 6/3, 7/3, etc., and visualize equalities, such as 2/3 being the same as 4/6. Once students know how to multiply fractions, they can make the leap (with explanation from an expert teacher) to understanding why they need to invert and multiply in order to divide fractions. A good example to drive the logic home is this problem: A 30-yard ribbon is cut into pieces that are each ¾ yard long to make bows. How many bows can be made?

Wu concludes by arguing that in the upper elementary grades, students need to be slowly acclimatized to three key characteristics that make higher-level mathematics learnable:

• Coherence – Math topics are not a collection of unrelated facts but a “whole tapestry where each item exists as part of a larger design,” says Wu. It’s essential, for example, that students see the close relationship between whole numbers, fractions, decimals, and percents. It’s also important that they understand the general principle of reducing complicated tasks to simple subtasks.

• Precision – “Children should learn about this mathematics tapestry in a language that does not leave room for misunderstanding or guesswork,” says Wu. “It should be a language sufficiently precise so that they can reconstruct the tapestry step by step.” For example, when fractions are taught without clear definitions, students never really understand them and have great difficulty when they begin algebra.

• Reasoning – “When reasoning is absent,” says Wu, “mathematics becomes a black box, and fear and loathing set in.” Students make mistakes like not shifting each row of answers over to the left in long multiplication, or adding numerators and denominators in fractions. “Reasoning is the power that enables us to move from one step to the next,” he continues. “When students are given this power, they gain confidence that mathematics is something they can do, because it is done according to some clearly stated, learnable, objective criteria. When students are emboldened to make moves on their own in mathematics, they become sequential thinkers, and sequential thinking drives problem solving.”

Because of the advanced content knowledge needed to teach this material, Wu advocates that schools should employ specialized math teachers starting in fourth grade.

“What’s Sophisticated About Elementary Mathematics?” by Hung-Hsi Wu in American Educator, Fall 2009 (Vol. 33, #3, p. 4-14)

http://www.aft.org/pubs-reports/amer...all2009/wu.pdf
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