As far as processes go, adding and subtracting mixed numbers is pretty tricky. Any given step is fairly doable, but knowing which steps to apply, and in what order is tough.
I find that my students understand each step independently, but need lots of experience discussing, practicing, and figuring out which step to do when and why. To promote these discussions, I had my students construct flowcharts.
Think, for a second, about how different the processes are for these computations:
In the first expression, you need to make common denominators, regroup the mixed numbers in some way, subtract the whole numbers, subtract the fractions, possibly regroup again, and simplify. In the second expression you only need to subtract the fractions.
I have found that my students struggle with adding and subtracting mixed numbers because they can't understand when certain steps are needed and when others are not. Trying to teach this through direct-instruction fails for most students. There are just to many if-this-then-this points. Students need to process those ideas on their own, you can't just tell them how and expect them to memorize the procedure.
So, I took this challenge as an opportunity to introduce my students to flowcharts. Flowcharts are a way of displaying algorithms and constructing them requires computational thinking. You must consider what possibilities exist and how your algorithm will process the inputs (problems) you give them. This was hard for kids, but they loved it.I find teachers write, "They loved it!" or similar a lot in their posts. But, really, there is no way to audit this. It's like a cooking show on TV. The hosts LOVE every dish they make, but they could really be terrible. #DeepThoughts
Each box in the flowchart is a command (ex: Make common denominators.) or a question (ex: Are the denominators the same?). Commands should only have one output while questions should split based on the answer. In the example below you can see that this group began by asking if the inputs were mixed numbers.
I had the students work in pairs on individual whiteboards. They used mini-post-it-notes for the boxes. This was CRUCIAL to their success. They are reworking, rethinking, and revising so often that they need flexible materials.
Groups also developed algorithms of varying degrees of complexity, detail, and sophistication. My goal in designing this lesson was to get the students to think through the different conditions encountered when adding and subtracting mixed numbers. Most groups were still revising and reworking their flowcharts when class ended and that was okay! They did not need a perfect flowchart to achieve the goal of the lesson.
The above group's flowchart lead to my favorite conversation. If you look at the chart, you'll see that they have MANY redundant boxes. Of course, that still works, but part of computation thinking is eliminating redundancy and promoting efficiency. After talking this over, this group scrapped their nearly finished flowchart with the goal of using as few post-it notes as possible!
Other groups, like the one below, had a harder time recognizing unneeded redundancy. They made separate charts for adding and subtracting fractions. Still, this allowed us to have conversations about whether one flowchart could have handled BOTH operations.
In class, we have been discussing different strategies for when the fraction you are subtracting by is larger than the fraction you are subtracting from (ex: \(\)). Some groups tried to build ALL of those strategies into their flowchart. But other groups realized that the always-convert-to-improper-fractions strategy is easiest to describe in a flowchart. That led to dead-simple flowcharts (see below) and also allowed us to discuss how optimal algorithms for humans and optimal algorithms for computers can vary.
I'd love to use flowcharts again in the future, but you need a certain type of algorithm for flowchart creation to be valuable to students. They are best for describing algorithms with many if-statements or conditional steps. I can imagine factoring quadratics or solving equations might be procedures that would benefit from computational thinking. Maybe you can think of other processes that would benefit from a flowchart. I know geometry courses use something like flowcharts for proofs.
Does anyone out there is flowcharts for other mathematical procedures?