Throughout the past year, I learned time and time again the importance of having students climb the ladder of abstraction. When I taught high school students, I often started with numerical cases before moving to general cases for topics. Even in these instances, I usually found a couple students that this progression was still too abstract for them and I had to adjust my approach accordingly. This year, I walked into my classroom knowing that 11 and 12 year olds are just starting to be able to consider more abstract thoughts and ideas; however, I never fully realized the importance of moving from concrete representations to more abstract ideas. In many cases, this progression is critical to developing a solid foundation of a concept and avoiding the use of direct instruction.
I want to take a couple moments to note my favorite concrete to abstract progressions to help me remember for next year and propose some potential ideas for the future.
Fawn Nguyen presented a fabulous idea for visually representing fractions that I used as inspiration for teaching fraction operations to my students. To start the unit, I asked my students how we can compare fractions. A couple students recalled creating equivalent, albeit without technical language, but no one was able to tell me why we needed common denominators. I anticipated this difficulty, so we started comparing fractions using rectangles. I used a little guided instruction with a series of questions and answers, then set students loose on the rest of the problems on this document. Towards the end of class, I allowed students to use the numerical method of comparing fractions, but it was cool to see the idea finally click for some students. The following day, we reviewed adding and subtracting fractions. Some students recalled the process, but we began each of these operations by drawing rectangles again to see why we need common denominators. Multiplying fractions was a little difficult to model with the rectangles, but luckily students were fluent with this skill. Finally, I used Fawn’s approach as she presented it for division of fractions. This method took about 15 minutes of Q and A to introduce to students, followed by another 30 minutes of examples to really build the understanding of the concept. I even included mixed numbers. In both of my classes, students eventually said things like, “Hey, can I just flip the second fraction and multiply?”
These visual representations were a great way for students to finally develop a solid understanding of fractions, so the operations are not just a bunch of steps to memorize. The visual nature (and process of drawing) also makes fractions become quantities with meaning for students (how much pink is shaded out of the total number of squares in a rectangle) rather than more unrelateable ideas (Jimmy had 3/8 of a pizza. Who talks like that?).
When teaching integer operations, I use two color counters to introduce integer addition and subtraction. You can also use this representation for multiplication if you treat a negative sign as “the opposite of”. For instance, -4 x -5 can be stated as, “the opposite of 4 groups of negative 5.” Students would start with 4 groups of 5 red chips, then flip the counters to get the opposite. Along with counters, number lines are another way to visualize the process. I personally prefer this representation because it better illustrates the idea of additive inverses (zero pairs). I found this year that certain students preferred one representation over another. Once again, the goal of using these representations is to develop a concrete foundation, then moving to the move abstract process of computation without manipulatives. In the future, I want to move away from stating rules for subtracting integers. If students know addition and multiplication of integers, they can always manipulate a subtraction problem to become an addition problem.
The first time I heard of someone using Hands-On Equations, I listened politely and thought I would never use it. I never used this weird almost board game like tool, so I thought it was unnecessary. After spending two weeks on equations with my honors classes, I realized that few student were actually solving equations or using inverse operations. The vast majority of the class was just guessing and checking or using calculator kung-fu. I pulled out the Hands-On Equations kit reluctantly, hoping for a better week. I was surprised to find after we introduced the idea, most students were able to solve multistep equations with the process by correctly manipulating the pawns and cubes. Within a day, we transitioned back to the abstract process of solving equations using inverse operations and students finally realized the idea of balance in an equation.
When my general classes reached the equation unit, I began with Hands-On Equations for a day and a half. After the idea was developing, I moved to using pictorial representations of the process. Finally, the abstract idea of inverse operations was just the icing on the cake. Some students still resisted the idea of showing work on both sides, but a lot of students were able to tell me why we needed to add/subtract/multiply/divide on both sides.
I used nets to introduce surface area, then I moved to the point of view method I described in a previous post. I used nets without any annoying tabs so students would truly understand surface area as just the area of the surfaces (in mathematical reality, there are no tabs). Students glued a net into their spirals that they could fold, I explained the concept, then I told students to apply their knowledge of area to find the area of each face and add the areas. Super simple! I was surprised I was taught annoying formulas as a student when the idea is so much more intuitive.
On the whole, I used manipulatives to introduce concepts and eventually students worked with the more abstract processes. None of these approaches is overwhelmingly original, but the power of the methods was the understanding students gained from the progression. A lot of times even a simple reminder like, “If we were drawing rectangles, they would need to be the same size,” to a struggling student was enough to remind them why we needed equivalent fractions.
My plan for next year: Use snap blocks for volume and rectangles for percentages. For percentages in particular, I saw a lot of conceptual gaps this year related to the fact that many students never fully grasped the idea of breaking a whole into 100 parts. I think rectangles could remedy this gap. This method could also develop an understanding of benchmark percentages (10%, etc.).