I had some great responses from the blog post – Dividing Large Numbers and teachers had some really good questions about dealing with obstacles. So here are some questions and suggestions:
What do I say to a student who knows how to do the long division algorithm already?
First, I would explain to all the students that the goal of the lesson is to deepen our conceptual understanding of what division means. The focus at this time isn’t on a procedure but rather we are developing our mathematical reasoning skills, problem solving, communication etc. Remember it is the important process of ‘trading in’ the blocks, or regrouping into smaller place values, that is often not understood by our students when dividing and the goal is to gain a solid understanding of why this happens and how it works.
Second, I would take a few minutes to sit with this student, or students and ask them to explain to me why they are doing each step. If they can’t explain the place value connection within the algorithm, then they likely don’t have conceptual understanding and will need this lesson as much as the students who cannot perform the traditional algorithm.
It seems like some students are playing too much with their blocks and are not on task. Do you think the blocks can be too distracting for some students?
I always give a minute or two (I would give a few minutes for younger students) for my students to play with the blocks before I begin. I tell them it is their opportunity to get it out of their system before we use the blocks as tools for math. I do find myself having to remind students throughout the lesson to get back on task as they are building something rather than dividing. However, I also know that when we teach traditionally, many students are totally off task, but you, as the teacher, have no idea until you mark homework or do a test. At least this way, you know immediately when they are off task and you can direct them back on task. Furthermore, it is worth it, pedagogically, to use this approach even if it makes my job a bit more challenging. Finally, I find when I use these blocks often, the students become used to my expectations and the off-task play reduces significantly.
As for some students, especially those who have special needs with respect to distractions, I have some base 10 blocks printouts (free on the internet) that I’ve laminated and put into envelopes. If a student can’t seem to handle using the blocks, I will give them the laminated version instead so that they can still benefit from seeing and doing the math but won’t have the blocks to distract them.
I didn’t have enough blocks for my students to do all the ‘trading in’ for some of the questions – what should I do next time?
You’ll notice in the Prescribed Learning Outcomes that many of the number strand outcomes ask students to demonstrate their understanding concretely, pictorially and symbolically. Because most brains generally learn in this order, the lesson sequence would also follow this and using the symbols of the blocks is a nice next step. Alternatively, you could create a set of laminated base 10 blocks as I mentioned in the above question and include more of each size block so that there are enough for the numbers you choose.
I did this lesson and it went well but then I tried to do division using just numbers and they couldn’t connect what we did with the blocks with the numbers, so what do I do now?
I’ve noticed the same thing for years. We need to explicitly teach students the connection between the conceptual and concrete (blocks) and the symbolic and procedural (numbers). There has been so much research conducted in the past few decades about whether we need to teach conceptually or procedurally but the most current and supported research I’ve seen indicates we need to do both, in a ‘hand over hand’ fashion. So, in this case, we are introducing the concept using the blocks, then we need to help our students connect the work they did with the blocks with an algorithm (either the traditional with meaning or the partial quotient method). I use the blocks and then record, using a document camera, what that looks like symbolically. I find I need to model this many times and then students develop a better understanding of the connection. Once they can use the blocks and then model it symbolically (with numbers), I ask them to do the symbolic first and then justify their solution using the blocks or pictures. After they have had time to be successful at this stage, the final stage is to be operating in the symbolic. This is not a quick or easy process. Many of us find that there is a big jump in getting the students from the concrete to the symbolic and that it takes explicit modeling to allow students to see this important connection.
Please send in other questions and suggestions regarding your adventures in teaching math with meaning We can learn so much from our collective experience and expertise. Kudos to all of you who put yourself ‘out there’ and are trying out using manipulatives in your classes, and remember that it takes time for you and your students to adjust to a new approach!
Educating Now was created due to teacher requests to have Nikki as their daily math coach. The site has lesson by lesson video tutorials for teachers to help them prep for their next math class and incorporate manipulatives, differentiated tasks, games and specific language into their class. Teachers who use the site can improve student engagement and understanding, in addition to saving prep time, by watching a 10 minute video tutorial and downloading a detailed lesson plan.