What is Actually Happening in this Problem?

I’ve kind of been on a number sense kick lately.  I know part of my focus is the time of the year (my honors class is working with rational numbers and my general classes are exploring ratios and percentages), but I think all of my experiences from last year (my first year teaching) are colliding with the experiences of this year in surprising ways.  I’m recognizing topics with a higher conceptual load and topics that require a solid understanding of number.

A prime example occurred in my honors class the other week.  Students were working with positive and negative decimals.  The basic skills of adding, subtracting, multiplying, and dividing were already in the minds of students.  The focus of the unit was applying integer rules to problems with negative decimals.

During a review day, students encountered problems with subtraction.  I kept hearing students use phrases like, “You make 9 into 8 and the 0 into 10,” or, “You make the 6 into 5 and the 8 becomes 18.”  Students were getting correct answers, but I was wondering if they knew why they needed to use a specific process when borrowing.

During the next problem (13.27 – 8.19), a student made a statement, “You make the 2 into 1 and the 7 becomes 17.”  I stopped him after this statement and told the class, “2 is not becoming 1 and 7 is not becoming 17. There’s no 2 or 7 in this problem at all, even though we do have those digits in the given number.  What is actually happening in this problem?”

Silence. Some students looked at me like I was crazy.  Other students looked like they were convinced I was pulling a prank of some sort.  After some wait time, I made the statement, “Think about place value.  Where is the 2 located in the number?  Where is the 7 located?”

Silence, but now I could see some wheels turning in the minds of students.  Finally, a student offered, “2 tenths is becoming 1 tenth.” YES!

The other half of the borrowing process required some explanation.  Students seemed unfamiliar with the idea of 1 tenth being 10 hundredths, so I needed to go down the path of equivalent fractions for a bit before we discussed adding 10 hundredths to 7 hundredths.

I was surprised at the length of this discussion (it took about 15 minutes).  In all of my classes, students are perfectly capable of reading decimals correctly; however, I wonder how many students were taught to explain algorithms with place value.

For example, how many students read a multiplication problem with decimals and work through it without any thought of place value?  Consider a problem like 3.7 times 9.8.  How many students begin the solving process thinking without place value (7 times 8) versus the number of students who use place value (7 tenths times 8 tenths)?  Do these differences in think explain many of the errors we see when students place a decimal point in their final product? The same idea applies to division.  How many students blindly divide 1 by 8 over viewing the problem as a series of place value statements?  I think many students read the problem as, “0 groups of 8 fit into 1, 1 group of 8 fits into 10, etc.”  instead of viewing the problem as, “0 whole groups of 8 fit into 1, but 1 tenth of a group of 8 does… so does 12 hundredths of a group… so does 125 thousandths of a group.”

Where and when should these understandings of algorithms using place value develop?  Is it a matter of consistently using mathematical vocabulary?  Is it a matter of comparing algorithms to manipulatives?

I don’t know if there’s any solid answers to those research grade questions, but I do know one better question I will keep using to further the algorithm understanding of my students.

WHAT IS ACTUALLY HAPPENING?

[cross posted to betterQs]

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