Math rules. How often have you found that students are taught "tricks" to remember math rules? How often do they make procedural mistakes even though they've "learned" the rules?
I have taught decimal operations for more than 20 years, and I have seen, time and again, students who know how to add and multiply decimals but then follow the wrong "rule" for the operation they are completing. Line up decimal points when adding or when multiplying? "Jump" the decimal point over when adding and subtracting? Or is that multiplying? They don't remember when to use which method to place the decimal point.
So, this year, as we approach the decimal unit, I've been feeling like I don't want to talk about the rules for where/how to put the decimal point. I want to focus on logic. Today that feeling was reinforced when I asked my students to solve 35.2 + 7.489 and then explain why their answer made sense. Here are some of the answers and reasons (I didn't teach this yet, but they learned it last year):
"0.11009 makes sense because I tried my best and if I remember correctly, addition problems you don't need to line the decimals together"
"0.7838 makes sense because when I added I knew that it doesn't matter how it's lined up"
"78.42 - I added 9 and 2, then 1, 8 and 5. Next I added 1, 3, and 7. Finally I added 7 and 0 and I put the decimal in the middle."
"7.841 makes sense because with adding you only have to add the decimals on the top. Then you add and finally add the decimal back in."
"426.89 because I put the decimal point four spaces back because there are four numbers behind it"
"79.41 makes sense because you do it just like an addition problem (that's how I remember it anyway)"
"7.841 makes sense because you add like normal and take the decimal from the farthest out and put it with the answer"
A few correct answers, with reasons:
"42.689 - this makes sense to me because this is how I learned it. You do simple addition, but line up the decimal points"
"42.689 makes sense because I used what my fifth grade teacher taught me, line up decimals, add zeros so everything is lined up and then solve."
"42.689 - I don't know how it makes sense, but it's how I learned to do it."
Of the 120 students in my classes, only 8 said the answer made sense because "35 + 7 is 42" or because "I estimated" or "when we're doing addition, we know we end up with a bigger number."
Now, that doesn't mean that they didn't think about those things, but to them answers seemed to "make sense" when they followed the rules - even if the rules are remembered incorrectly; students got right and wrong answers and they all made sense because that's "how they learned it."
So, what is the point of teaching rules? Especially to those students who are a little weaker in math - if they can't remember the right rule, they can't tell if their answer is reasonable! They need to develop their number sense.
In the past, I have asked students to estimate the answer first, so they know if their answer is reasonable, and I have required them write these estimates on their tests. But we've also talked about the rules. I'm thinking that if I take the focus off the rules and put extra focus on the estimating/reasonable answer idea, students will be better able to identify reasonable answers and will feel less dependent on the rules.
I know that multiplication and division logic will be more difficult. Problems like
23.5 times 4.428, won't be as bad because there are whole numbers involved. This could be estimated as 25 times 4 = 100. So when placing the decimal point in 104058, it should be placed so that the answer is about 100 - not 10, or 1, or 1000.
Now multiplying 23.5 and 0.7 may be more confusing, but this will be the time to help them understand why the answer should be smaller than 23.5....but more than half of 23, since 0.7 is more than 0.5.
I think division will be the most challenging, as far as determining reasonable answers, and I need to think about this one a bit more. However, we have already done this activity I found on YouCubed -"Too Big or too Small Maze Board." In attempting to create the largest number possible (using a calculator to compute), many students have already made the discovery that dividing by a number less than one gave them a larger number, while multiplying by a number less than one gave them a smaller number. No rules were taught - they found this "secret" on their own. This will be great to reference and discuss when we begin working on the multiplication and division of decimals.
We'll see how it goes!