Blog 4 - Invented Strategies

Intro

     As was stated in the Van de Wall text, students do not spontaneously invent wonderful computational methods while the teacher sits back and watches.  The first step is to create an environment where students feel safe to explore numbers and where it’s okay if a strategy doesn’t work out because you can go back and rethink the steps and troubleshoot and you can confer with your classmates and share ideas on strategies that way.  The text also states that part of the role as a teacher is to help “move unsophisticated ideas to more sophisticated thinking through coaxing, coaching, and guided questioning.  This way you could help a student put their strategy into words and be able to apply the polished strategy to other problems.  Number sense, seeing patterns in number, feeling comfortable with manipulating numbers is a concept essential to allow students to develop strategies. 

     I’d like to focus on two problems from my math interview at Houston Elementary School; one problem without assistance and one with assistance.

Problem 1

The first problem asked the student to find the difference between two numbers:  “Jared has 13 pennies.  He loses 4 of them.  How many pennies does Jared have left? (separate result unknown)”

     At this point, the student had become comfortable in using the Unifix cubes and reached for them immediately; I think this is in part because when I would ask him to show me another way he could have solved the problem, I’d suggest he use the cubes.  The student counted out 13 pennies and connected them end-to-end.  He then removed 4 of them and stated the number 9.  I asked him if he could explain what he did.  He said that he counted 13 pennies and he took 4 away because the boy lost 4 of them.  I asked him if he knew another way to solve the problem.  He said that he could also subtract 4 from 13 in his head and also get 9.

From this problem, I gathered that the student was using the take-away subtraction strategy briefly described on page 221 in the text.  It states that the take-away strategy is actually more difficult than the add-on strategy, but that textbooks emphasize take-away as the meaning of subtraction.  Even though the student seemed to have used the cubes at this point to appease me, he clearly showed that he could count back in his head from 13 to 9.  I was not surprised that he was able to do this, but I found out later that the student had trouble dividing up numbers in anything that was not in solid tens.

     Another way I think the student could’ve arrived at an answer would be to use one ten rod and three single cubes to represent the 13 and 4 single cubes of another color to represent the four.  He could then cancel out the singles, followed by breaking down the ten rod to cancel out the final single cube.

     A second way the student could’ve solved the problem is to count up from 4 to 13 and keep track either in his head or on his fingers of the amount between those two numbers.

Problem 2

     The problem the student and I worked together on was a 2 digit by 1 digit multiplication problem.  I originally presented the student with a problem with slightly higher numbers.  I think he realized that he didn’t have enough fingers to count that high and he seemed unsure of where to start using the cubes.  I gave him a similar problem with slightly smaller numbers.  The student again seemed lost.  I offered to work with the student on solving the problem.  From my observations up to this point, I chose to use repeated addition to show the student a way to solve the problem.  The problem asked:

“There are 4 children going to the water park.  It costs 21 dollars per person.  How much money will it cost for all the children? (multiplication)

     I pointed out that if it costs 21 dollars for each student and there were 4 students going, we could add 21 four times.  I wrote out 21+21 in the standard algorithm and had the student add.  He added 21+21, told me the answer, and I wrote down the answer.  Then I asked him if there was another way of finding out 21 and 21.  He said we could go to the side and add 21 and 21 again.  I wrote out 21+21 again and asked him to tell me the answer.  Again he said 42.  I asked him what we could do next to find out our total.  He said we would add 42 and 42.  I wrote what he told me to write, added it and asked if 84 was correct.  He looked over my work and agreed.  I asked him then how much it would cost for 4 kids to go to the water park if it cost $21 for each student to get in.  He answered 42, then quickly corrected himself and said 84.  I asked him which one he was sure of.  He answered 84.

     Repeated addition is discussed in the text on page 227.  I have to agree that it seems like a very inefficient way of arriving at an answer.  And in this case, the student seemed unsure of what result to look to for a final answer.  I am not at all convinced that he knows why we added 21 four times to answer the question.

     To solve the problem, the student could have separated the tens and ones into 2 tens and 1 ones.  The student could then multiply the 2 tens by 4, resulting in 8 and multiplied the 1 by 4, resulting in 4. 

     The student could’ve also used two base ten rods and a single cube and decided that to find out how much it would cost for each student, he would have to have 4 of the same 2 base 10 rods and 4 of the single cubes to represent the cost for each student.