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Why is it harmful to teach the Computational Algorithms without understanding?
I coach basketball as well as teach. One thing I have seen is a lot of young kids that want to shoot the ball from the three-point line. The problem is they are not strong enough so they end up just throwing the ball at the basket instead of shooting with proper mechanics. These poor shooting habits have to be broken, as the kids get older so that they can learn to shoot properly. This is harder than just learning the correct way the first time.
With Math, a similar situation can occur. Students who learn algorithms without understanding the concepts behind the algorithm may develop some inaccurate ideas that will then need to be unlearned before they can learn the correct idea. In one Second Grade classroom, I wrote the equation 3 + 5 = 8 and asked the students if this was a true sentence. They all agreed it was. Then I wrote the equation 8 = 3 + 5 and repeated my question. This time the students all said it was backwards and wasn’t true. They had incorrectly learned to read the equal sign as meaning the answer is coming. This will need to be corrected so that they can work with more complicated equations like: 3 + 5 = __ + 4 in the future.
The biggest problem with teaching algorithms too quickly is that students learn the steps of the algorithm, but they don’t understand why they work. Math is logical and it should make sense. This is true when students are using paper and pencil to calculate answers as well. If they are just following a sequence, even if they get the correct answer we should not be satisfied. The biggest concept students need to understand before they begin to do multi-digit computation is the idea of tens and ones.
Thinking of Numbers as Tens and Ones: Once children begin to work with larger numbers, they need to recognize the underlying structure of numbers as tens and ones. To do this, children must be able to count groups of tens as though they were single entities. They must also recognize that when you know the numbers of tens and ones you automatically know the total. When they understand that numbers are composed of tens and ones, they will be able to add 10 and subtract 10 without counting. - Kathy Richardson (Bk5 p7)
This reminds me of what I see when I watch students working with coins. Students go through a stage where they can count coins, but not the value of coins. If you give a first-grader 3 dimes and 2 pennies, they are very likely to tell you they have five. When they have developed the ability to see the dime as one coin and ten cents at the same time, they can count that same set of coins as 10, 20, 30, 31, 32 = 32 cents. Using our number system is essentially the same thing. Students must be able to see tens like they see dimes. They are one object, but they do not have the same value as ones. That is why we see so many first and second graders making this mistake when subtracting.
13 - 7 14 Do you see the mistake the student made? The student calculated the difference of 7 and 3 without regard to the value of the whole numbers (The difference between 7 and 3 is 4).
Something very interesting happens when you present the same problem like this:
13 – 7 = ___.
Students are much less likely to make this mistake. They tend to see the 13 as a whole number rather than as two individual digits 1 and 3.
If you imbed the numbers in a story situation, even fewer students will make this mistake.
I bought 13 books at the book fair last week. I have read 7 of them already. How many books have I not yet read?
After all, in this context, it does not make sense that I have not yet read 14 books, when I only bought 13 to begin with. However, most students in Kindergarten and first grade will not choose to use a paper and pencil algorithm to figure this problem out either. They will choose a method to solve the problem that allows them to make sense of it. For some this will mean physically acting it out, for others it will mean drawing a diagram to represent the situation.
John Van de Walle and LouAnn Lovin suggest in their book Teaching Student Centered Mathematics Grades K-3 that for most students “the concept of a single ten is just too strange for a kindergarten or early first-grade child to grasp. Some would say it is not appropriate for grade 1 at all” (page 55).
When a student develops the ability to see tens and ones with a conceptual understanding, they are ready to begin to work on recording their thinking using algorithms. To ask them to do so before this understanding is achieved can actually hurt their development. The most important thing is that math makes sense. If it is not making sense to the student, we need to figure out why.
These ideas were taken from a talk by Mari Muri at Suffield Middle School on November 16, 2004. The danger lies in the fact that students that learn the algorithms by rote do not necessarily understand what they are doing.
Dangers of Teaching Computational Algorithms without meaning:
Benefits of allowing Students to use Alternate (Flexible) Algorithms:
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