This week my second grade students have been exploring multiplication strategies. We started off early in the year looking at arrays and using doubling strategies. Then we moved to helper facts. These are still used to this day, but we introduced a new tool this week. Enter the area model. Hello!
Students transitioned from arrays to squares, but didn’t sit at that spot long. Through the area model, students take apart numbers and partition (yes, we say partition at second grade) the rectangle into parts. Each part is a partial product.
I’m fortunate in my position to see this strategy used at multiple grade levels. The rectangle evolves over time. As students progress, I find that place value and advanced decomposing strategies become more prevalent. You can learn quite a bit about a student’s understanding by checking out their math work with an area model. How they split up the numbers can also tell a story. Why did they split up the rectangle that way?
I find quite a bit of value in using this strategy. For one, it doesn’t immediately move students towards the standard algorithm and it helps build/show conceptual understandings. My 2-6th grade math students use it in a variety of capacities. My 5th grade crew has recently been using them to multiply fractions. Short story: It makes an appearance at every grade level. It’s also a a fairly smooth transition to using the partial-products strategy.
Even though it’s a useful resource, I find there are a a couple things that irk me about using this tool. Sometimes organization skills can hamper the effectiveness of drawing and organizing. I’ve had more than a handful of students draw boxes that overlap or numbers that might not be decomposed correctly. Also, it’s not to scale, but that’s not a game changer for me.
As students progress through elementary school they encounter a variety of math tools and strategies. Manipulatives are generally used to help students build a better understanding of math concepts. The CRA model is often emphasized at this level. Many tools are brought out to help fill gaps and others are continually used. At some point, I’m assuming the my students will rely on the standard algorithm to quickly multiply numbers (if they don’t have a calculator handy). They probably won’t understand why the algorithm works, but it just does. The area model shows multiplication in a concrete way. Don’t get me started on lattice.
My fourth grade group just finished up a unit on division. They spent a great deal of time exploring division and what that means in the context of a variety of situations. One of the more interesting parts of the unit delved into the use of the partial-quotients method.
You might be a fan of this method if you’ve ever wanted to know why the traditional division algorithm works. I was in that boat. During my K-12 math experience I never questioned what was introduced. I was always encouraged to use the traditional division algorithm. It didn’t make sense to me why I was dropping zeros or setting up the problem in a certain structure. I just played school and figured that it wasn’t worth trying to find meaning, but instead just pass the class and move on.
For the past ten years I have been introducing the partial-quotients method to students. This method brings more meaning to why the division process works. Students also seems to grasp a better understanding of how partial-quotients can add up to a quotient with a remainder.
This year I introduced something different to the students. Students were asked to use the partial-quotients method to divide numbers and then create an area model of the process. I had quite a few confused looks as most students think of multiplication when using area models. After modeling this a bit I noticed that students continued to have issues with appropriately spacing out their area models. This was a great opportunity for students to use trial-and-error.
I noticed that students had to first use the partial-quotients method to find the quotient and remainder. They then had to split up the area model into sections that matched the problem. This spatial awareness piece is so important, yet I find students struggled with it to a point that they were asking for help. Maybe it’s a lack of exposure, but estimating where to split up an area model to match the partial-quotients seemed to challenge students on another level.
This activity has me wondering how often students use spatial awareness strategies in the math classroom. How often are they given these opportunities? Reasoning and estimating strategies also play a role, but actually spacing out the partitions isn’t what students were expecting last week.
I’m looking forward to seeing how students progress in using area models moving forward.
Last week a few upper elementary classes started to explore different methods to divide multi-digit numbers. Many were familiar with repeated subtraction for numbers more than one and some even had experience with the long division algorithm. I asked students to explain their reasoning for the steps needed to divide numbers using repeated subtraction and long division. All of the students were able to explain the reasons for using repeated subtraction. Students gave quality answers and were able to communicate why each step was performed. Some students even related repeated subtraction to a number line. I then asked the students the reasoning for using the long division algorithm. I heard responses like “it’s quicker” or “that’s what I was told to use” or “you just do this and this.” I could tell that there was a disconnect between the shortcut and having a conceptual understanding. Students understood the steps but couldn’t provide solid reasoning to why you would bring down the next number.
The class then had a conversation about the importance of being able to clearly explain their mathematical thinking. The students that knew how to use the long division algorithm were getting correct answers, but couldn’t tell me why. Blindly following procedures can lead to holes in understanding. Explaining the reasoning behind completing a problem is important. Honestly, I don’t mind if the students use algorithms like the above picture if they already have a descent understanding. The problem I have is that sometimes this is the only way students are taught how to divide large numbers. The problem becomes steps –> answer without understanding.
After the class had a discussion about long division we explored the partial quotients algorithm. I explained to the students that this was another method to divide larger numbers. As we progressed through and explained the steps, students became more aware of how partial quotients seemed to make sense to them. For many, this was the first time exploring the partial quotients algorithm. The students were able to explain why each step was taken in the process.
I believe some of the students were also relieved that they didn’t have to get the partial quotient “correct” the first time. By breaking apart the problem students were starting to see the correlation between repeated subtraction and the partial quotients algorithm. More importantly, students were able to explain their reasoning for completing the problem with this method.
As we move forward, I feel as though students are becoming better at explaining their mathematical thinking. It doesn’t matter to me what method the student decides to use in the future as long as they are able to justify their reasoning. This thinking could also lend itself to just about any type of computation skill. Last week reminded me of the need to expose students to multiple strategies to complete problems. Providing these strategies can assist students in becoming better at explaining their mathematical thinking.
As my students progress through their fraction multiplication unit I came across another example of why using multiple strategies matters. In the past students learned how to multiply mixed numbers by 1) Convert the the mixed numbers into improper fractions 2) Multiply the numerators and denominators 3) Covert the improper fraction back into a mixed number. This is how I was taught to multiply mixed numbers. Although this method seems to work, students had trouble explaining why they completed each step. So early this week I decided to use a different strategy in class.
Students were able to visually represent this multiplication problem and the steps to solve seemed more logical.