Somewhat in support of old-fashioned math
While my pre-calc class has been digging hard into the world of technology and inquiry for their learning, I am feeling kind of guilty and kind of not-guilty for doing direct instruction and manual calculation in pre-algebra.
We’re working on a unit on decimal operations. Decimal operations are included in the 5th grade Common Core standards, but my assessments in our previous unit on positive/negative numbers told me most students aren’t 100% masterful with them yet.
I have only been a *tiny* bit constructivist with this unit. We did a brief activity using base-10 blocks to understand place value. I gave the students a pre-assessment on adding and subtracting with decimals, and based on the pre-assessment I could identify maybe 15 students over all three sections that needed some help with those operations. I gave most of the class some time on Khan Academy while I did small-group instruction with those kids. We just worked problems and focused on proper carrying/borrowing/place value. Every single kid was familiar with the basic idea, they just had some gaps when it came to the algorithm. The small-group work was just what they needed.
For decimal multiplication, I gave the students a worksheet I made up with some problems that had patterns in them.
I did not teach the students how to multiply numbers with decimals first… I just wanted to see if they could use the patterns in the worksheet to infer how big the numbers would be. We went over the answers together and then I asked the class if there’s a rule they started to pick up that told them where to put the decimal in the answer. Every class had multiple students that said something to the effect of: “however many decimals are in the problem, that’s how many are in the answer”. All we had to do was formalize that into a rule and then practice it.
As a unit project, I taught the students about how to write an invoice if they own a small business and need to charge their customers for items they buy. Without doing anything constructivist yet, I just taught the kids how to convert a percent into a decimal and then multiply to calculate discounts and tax. Creating an invoice involves decimal addition, subtraction, and multiplication, so it’s a great assessment to the unit.
For the students’ unit assessment, they will create their own virtual store that has multiple items I could buy, some made-up coupons involving percent discounts, and a made-up tax rate somewhere between 1% and 12%. I’ll shop at each student’s store and pick out some items and coupons, and they will have to make me an invoice by hand.
Now for the reflection on this unit so far: I have really mixed feelings about the unit’s topic, manual calculation of decimal operations. I am totally on board with the understanding that in a computer-driven world like ours, they can get by without knowing it. They can actually thrive without knowing how to do these calculations manually.
Why, then, does this feel so satisfying? Kids thanked me for teaching it to them. An intern volunteering in my classroom commented “this is really cool. I never actually learned how to do this. It’s great.” I’m reminded that I *can* plug a sudoku puzzle into a computer program and have it solved for me, but it’s satisfying to work through the sudoku puzzle myself and know I’m just as smart as a computer. Plugging numbers into a calculator is just as much of a rote task as large-number multiplication, but doing the math by hand, in the right mindset, gives you ownership of the puzzle.
Is that why manual math has value? Does it have value? I do not plan on emphasizing this kind of manual calculation once this unit ends – when we get into rates, ratios, equations, and expressions, we’ll use computerized tools as problem-solving aids most of the time to free up our working memory.
The other part of these lessons I’ve wrestled with is the direct instruction vs. constructivist. The lessons started out with a little pattern-finding and then I just taught the kids how to do the math. Over time, I have toned down the inquiry in some situations and I now believe it’s possible to overdo inquiry. Sometimes you are done trying to ferret out the method for yourself and you just want someone to show you how to do it. Some struggle is healthy but not unlimited struggle.
I would not have taught like this six years ago. If there’s a scale where 0 is pure direct instruction and 10 is pure constructivism, I would say this set of lessons is like a 2 or 3. What should be your criteria for determining when you do inquiry and when you do direct instruction? Lots of questions I don’t have easy answers to.
But this set of lessons, this unit, felt satisfying to teach and I expect good results from our little shopping assessments. I don’t have any regrets.