CHAPTER VI CONCLUSION AND DISCUSSION
B. Local Instruction Theories
means to determine the unknown multiplication products so that the multiplication strategies cannot be elicited; they only saw the arrays as a source of the problem. Nevertheless, there will also a condition when there are students who get or find the introduced strategy from the designs or the previous activity but reluctant to use it since they are expecting more „formal‟
strategies that have been taught how to use it and how to write it. Therefore, the teacher needs to emphasize on using the strategies they found from the problems/the designs and also guide them to use a faster way to derive other unknown facts from a known fact using the introduced strategies.
Materials and Instruction (cont) Students‟ Possible Answers (cont) Material: a nearly covered array
(1) only objects in top and left sides are uncovered;
(2) use an array that represent big multiplication, like 9 × 10.
Instruction:
(1) conduct after the quick-image-activity above;
(2) ask students to work in-group to determine the total objects in the array.
There are students who will try to focus on fining the multiplication represented in the array by counting the total objects in the top and left sides representing the factors of the multiplication.
There are students who will find it difficult to see an object
simultaneously in a row and a column on their counting.
The second theory is an overview of the potential educational materials and the students‟ possible answers on introducing the commutative property as a multiplication strategy.
Table 6.2: A local instruction theory on introducing the commutative property.
Materials and Instruction Students‟ Possible Answers Material: quick images
Instructions:
(1) show an array as a quick image;
(2) ask students to determine the total objects in the array;
(3) rotate the array and ask students to determine the total objects in it.
There are students who can find the multiplication represented in each array, with no indication using the answer from the first array to determine the total objects in the second array
Material: two related arrays
(1) arrays contain the same number of objects;
(2) use rectangle-ish objects;
(3) all objects are uncovered;
(4) use an array that represent big multiplication, like 7 × 8 and 8 × 7.
(5) the total objects in the second array could be derived from the first array using the commutative property.
Instruction:
(1) conduct after the quick-image-activity above;
(2) ask students to work in-group to determine the total objects in the array.
There are students who still try to count one by one, but then failed to determine the correct answer since the design misleads it.
There are students who find the multiplication in the first array and then determine the product using finger-technique. After that, they will find the multiplication in the second array and then determine the product using finger-technique.
The third theory is an overview of the potential educational materials and
the students‟ possible answers on introducing the one-less/one-more strategy as a multiplication strategy.
Table 6.3: A local instruction theory on introducing arrays as multiplication models.
Materials and Instruction Students‟ Possible Answers Material: quick images
(1) use two related array;
(2) the multiplication fact in the first array serves as an anchor fact to determine the total object in the second array using one-less/one-more strategy.
Instructions:
(1) show the first array as a quick image;
(2) ask to determine the total objects in the array;
(3) tape the array in the whiteboard;
(4) show the second array as a quick image next to the first array.
(5) ask to determine the total objects in the array.
There are students who can find the multiplication represented in each array, with no indication using the answer from the first array to determine the total objects in the second array
Material: three uncovered arrays (1) the arrays are related to each other;
(2) the total objects in the second array could be derived using one-less strategy from the first array, and the third using one-more strategy.
(3) use an array that represent multi-digit multiplication, like 15 × 8, 14 × 8, and 16 × 8.
Instruction:
(1) conduct after the quick-image-activity above;
(2) ask students to work in-group to determine the total objects in the array.
There are students who will find the multiplication represented in the arrays and determine the products using short-multiplication.
There are students who will come up to the one-less/one-more strategies after they are asking to use using a faster way to determine the products, instead of using short-multiplication.
There are students who will find the multiplication represented in the first array and determine the product using repeated addition, and after that using the one-less/one-more strategies to determine the total objects in the second and third arrays.
The fourth theory is an overview of the potential educational materials and the students‟ possible answers on introducing the doubling strategy as a multiplication strategy.
Table 6.4: A local instruction theory for introducing the doubling strategy.
Materials and Instruction Students‟ Possible Answers Material: quick images
(1) use two related array;
(2) the multiplication fact in the first array serves as an anchor fact to determine the total object in the second array using one-less/one-more strategy.
Instructions:
(1) show the first array as a quick image;
(2) ask to determine the total objects in the array;
(3) tape the array in the whiteboard;
(4) show the second array as a quick image next to the first array.
(4) ask to determine the total objects in the array.
There are students who can find the multiplication represented in each array, with no indication using the answer from the first array to determine the total objects in the second array
Material: an partially covered array (1) some objects are covered by label;
(2) the label make the stickers are divided in two equally parts;
(3) all objects in the first part are uncovered;
(4) only objects in the left sides of the second part are uncovered.
(5) the total objects in the array could be derived from the total objects in the uncovered part using doubling strategy.
Instruction:
(1) conduct after the quick-image-activity above;
(2) ask students to work in-group to determine the total objects in the array.
There are students who will use the doubling strategy, but not as an indication to find the product of the multiplication.
There are students who will finding the multiplication represented in the array and then determine the product using repeated addition.