Swear words (in bold below) in any math class of mine include:
“I can’t do that.”
“That’s too hard.”
Yesterday as we worked on Puzzle #7 in Piles, I had a fourth grader swear. Very strongly. It was because she got the card (3×3)+(4×3). She was convinced that she would not be able to find a match for it because she didn’t understand the number sentence. So she guessed and just put the card “twenty-four” underneath it. I asked her to prove it, again at which point she swore that it was too hard. She then continued on saying that it made no sense to her and almost burst into tears.
This is the beauty of working with students who struggle. It’s all about asking the right questions to pull them out of what they feel is total despair and helplessness.
So all I did is put out a bag of square counters. I asked her and the other two with me to build it. I told them I won’t tell them the answers, but that there were things we could do to make sense of the problem. I asked them to talk to each other to see if they could figure out a place to start. Then, right before my eyes they began to build this:
We have to help our youngest mathematicians see their way out of the fog of not knowing where to start. I think this is true of every level of learner. At that moment, I was no longer needed. They began to do this one:
And this one:
I was beginning to feel a little useless, which I think means a job well done.
I have referred to the Concrete-Representational-Abstract instructional approach many times before, but I will say it again. This is research. This is how students brains WORK.
If you don’t already know about this approach, here it is in a nutshell:
- Concrete: When a student is introduced to a new concept or something unfamiliar, you allow the use of tools. Sometimes students become stuck here, and can be moved to the next stage by linking the two together.
- Representational: When the student can perform the task using tools, they move on to representing the concept with drawings or pictures of their tools. Again, when students become stuck here, we link the next step in with this one.
- Abstract: When the student can master the task with a drawing or a picture they move to using only numbers and symbols to represent their thinking. Many times they can visualize a model to represent the concept.
If we pull the tools before they are able to represent a math concept, how can we expect the abstract number sentence to make sense? I love that Piles is allowing these students to connect all three. Soon, I anticipate the tools staying in the bag as they begin to visualize what multiplication means, and hopefully their foul mouth swearing will stop.
Today with my 4th grade intervention group, I noticed a lot of giggling when we were working on Piles. This is a known fact about students who struggle. The more giggling and fooling around, the more lost they are.
I’ve been working on the concept of “rows” with these students because there is almost no understanding of what a multiplication sentence could stand for. There was a lot of quick matching going on, just putting things together that had similar numbers, hoping they were correct.
So I had to bust that right up.
I asked them to build the card with my square counters, and to make it equal to what they thought the array would look like if it actually had squares. So essentially I’m looking for them to build an array with 6 rows that has 5 squares in each row. We talk a lot about what the counters could represent, like seats in a movie theater.
Look what started to happen…an empty array!
After a brief discussion she and I came to the conclusion that she better fill that one in. Though I found out later that she didn’t understand why, and only filled it in because I told her to. She very proudly tells me this in the clip below. (Note: No matter how hard I try, I still occasionally slip and tell students what to do. STOP doing that Ms. Smith…)
Here’s another misconception 17 seconds into the video clip. “If you take out this, it’s still the same thing.” (Note: The pitch in my voice becomes noticeably higher, also need to work on my poker voice.):
This is very common when moving from concrete tools to representational models. They have been moving from tools to grids to the empty array models for several weeks now…all along I was assuming they knew what the empty squares and rectangles stood for. What an awesome thing to see the exact misconception right in front of my eyes because I chose in that moment to question the card.
Teaching point for tomorrow, check!
Sometimes ideas brew in my mind for a long while. They usually begin as this teeny tiny thought when I’m working with students of all ability levels, and I see misconceptions. Then, they grow and grow until I can’t hold the idea in any more. This one might be my favorite thing I’ve done in a while.
This is an equality sorting game in which students have to match cards that they cut out into piles of cards that belong together:
If you follow me, you know I don’t brag…so this isn’t bragging. I’m just going to break down why I want you to try this:
- Students do NOT understand the Equal Sign. They think it means “the answer is”. This activity will totally challenge that thinking.
- Students are way used to having a pair in a matching game/activity. This mentality won’t work for Piles. They won’t always make a pair, sometimes their pile may have 3 or even 4 cards.
- For years I’ve watched students thinking that the visual model, number sentence and words are separate things in mathematics. Piles will help connect this for them.
- This will get them talking! Even your struggling students will start talking about why things fit together, and why they absolutely do not.
- The activity can be done independently alongside your teaching. I’m coming out with some other concepts, multiplication is just the first of what I hope will be many. Fractions are next…
- You will uncover, and squash all KINDS of misconceptions that you didn’t even know existed.
I’m going to do my very best to get these ideas onto digital paper as fast as I can. I’ll always have a free one that is included in the preview of every set, which means as soon as I get enough free ones I’ll put out a whole free set. If you’re a follower you know that I do my very best to provide free as much as I can. Download the preview, try the free one and let me know what you think!