How do we take students to the top level of Bloom’s Taxonomy in math? Here’s how I wrote a lesson in which students look at their old friends: addition, subtraction, multiplication, and division, and then create a brand new mathematical operation.
All Operations Are Related
First, I’d help students recognize that (although they do very different things), all operations follow a basic rule: operations take input, do something to it, and then produce predictable output.
Most mathematical operations take in two inputs (this is true for adding/subtracting/multiplying/dividing as well as exponents/radicals and so on). We call them binary. Some operations only take one input (absolute value, factorial). We call these unary. And, yes, there are even ternary operations that take three inputs (but its unlikely your students will see those for a while). The inputs are called operands.
So we have our inputs or operands. Then we have to explain what the operation “does to” those operations. In addition, we combine them. In subtraction, we remove one set from another. Asking this question of even your most advanced mathematicians may lead to some interesting thinking: what does adding/subtracting/multiplying/dividing “do to” the two operands?
Then, out pops our “answer/sum/difference/product/etc” or (more generlaly) our output.
At this point, I want my students to understand that these very different tools are actually all related. And, since they’re just examples of an abstract idea, we could easily create a new example of an operation.
Create A New Operation
In order to make their own mathematical operation, students will need to define:
- The number of operands and their names (addends/subtrahend and minuend/ etc). Note that some operations require different names for their inputs (subtraction/division) and some don’t (addition/multiplication) – we’ll come back to this later.
- The symbol for their operation (+, –, ÷, and so on) and its name.
- An explanation of what the operation does to the inputs (my example would multiply by three and then add one).
You can scaffold this nice and slow and (of course) model each step by creating your own operation along the way.
What Was Boring Becomes Interesting!
Now I take my students back to some of their least favorite content: mathematical properties such as commutative and associative.
I’d point out that some of the operations have different names for their operands. Subtraction has “subtrahend” and “minuend” while addition just has “addends.” I’d note that the reason for this is that, for some operations, it matters what order the operands are in.
When we multiply, a × b is the same as b × a. So both inputs have the same name. With division, though, the operands have very different jobs, so they get different names: divided and divisor. This is all because some operations are commutative and some are not. a ÷ b does not always equal to b ÷ a, while a + b = b + a.
Now students determine if their operation is commutative.
We’d do the same for the associative property. Then, if it’s within the scope of your content (or your kids are just interested), you could continue with distributive and then have students find the inverse and identity elements for their operations.
Kids were suddenly way more attentive to these mathematical properties than when I was just lecturing about them!
From here, you can have students present and (kids might actually get here before you) they can try to combine their operation with a friend’s!” We can certainly have 3 + 6 × 7 right? So let’s combine Jessie and Juan’s operations and see what happens!
Interested in the full lesson? I created a series of videos over at Byrdseed.TV.