One thing I’ve learned about teaching elementary mathematicians, is that you should assume NOTHING.  We think because our young learners know how to count, that they know what the quantity of those numbers are.  If you get a chance, watch Graham Fletcher’s progression video about Early Number and Counting. Actually, if you get a chance watch ALL of his progression videos. They are fascinating and full of information. I have watched each one at least half a dozen times. What hit home with me is that we must provide tools and visuals to all students.  I repeat, we MUST provide tools and visuals!

I was also inspired to write this post because of this amazing visual of a multiplication table that helps students understand quantity when multiplying numbers:

So here was the situation.  When working with some first and second graders, I noticed that they had just memorized the counting sequence. They were rattling off the numbers without thinking about their quantity. So when we asked how many tens or ones the number was made of, there was confusion about what that meant. Here’s an example of what happened last fall when I asked a student to get me 31 cubes. After she did that, I had to ask her what 13 cubes looked like.  She was focused on tens and ones, because that’s what the instruction was focused on that week, instead of thinking about how many cubes she actually had.

So imagine trying to ask her what ten more or ten less of 31 is…

After a recent assessment uncovered nearly 20 students who were struggling with the concept of 10 more/10 less, I knew we had to do something with a visual.  I introduced a visual 100 chart for the students that were struggling during our intervention block. (Except I only got to 60 before I ran out of paper which means technically it’s a 60 chart!)

I told the students that I wanted to put the tens and ones right on the chart (but all the pieces kept falling off!) so I drew them on instead.

After I explained how I made the chart, I asked them “What do you notice?”

Again this is why I say ASSUME nothing.  The things they said sort of blew my mind. One student excitedly said, “The numbers are growing by one dot!”

THEY DID NOT KNOW THIS ALREADY?

I was simply amazed at all of the things that I assumed they knew! They started to build on one another, saying things like:

• The ones stay the same when you go down on the chart!
• When you go across, the tens change in the last box.
• There are no ones in the last column!
• If you don’t remember the name of a number, you can use this to count and figure out it.

These were all things that we assume students would know, but because they’ve only memorized the counting sequence they weren’t visualizing what the quantity of these  numbers looked like. Instead of memorizing patterns with only the numbers showing, this gave them a visual model to anchor to.

The best part of it all was how easy it was to make that chart. It maybe took me 20 minutes, and can be used year after year. If you give it a try, I’d love to hear how it works for you.

I have a student that had many excuses for why math was not going to happen today. Here they were in this order:

“I’m tired.”

“I didn’t really sleep last night.” (Head down on the table, arms at the sides.)

“My hair hurts.”

“I ate lunch too fast. I can’t do math today.”

“I can’t function today. I just can’t function today. I am NOT doing math.”

Since I can relate to this Monday takeover of your rational brain, I decided to set aside my plans for the moment. I could have engaged in a power struggle, but instead I just started folding a paper fortune teller in front of her while the excuses continued.  I had no plan, and no idea if it would work…but I needed her to do some math today.

Pretty much immediately (but of course without showing it too overtly), she started to perk up and show interest. I am constantly reminding myself that our students are naturally curious and have everything in them to learn. It’s just a little harder to pull it out of some than others.

The next step was to write the questions on the inside. I tried to think of questions that might lead to a numerical answer.

After a while, she began coming up with the questions, and then she started to give all the answers.  When she told me 1,675 burgers, she asked, “How many burgers would that be in a year?!” So we had to divide it out and figure it out.

The height question led to us measuring her against the wall, so that she could get a sense of what a ridiculously short person would look like. The lottery question led to how to read a numbers in the millions, because of course she would win millions.

At the end of our 15 minutes together, she said, “We didn’t even do math!” What she ended up doing was much more difficult than anything that I had planned for her for the lesson!  I started to think about the open ended nature of this activity, and how our students bring it to a level that we don’t always expect.  This actually might be a great enrichment activity for your class to try out when you need to meet with a small group.

If you don’t want your students to do math next Monday, give it a try.

Sometimes I think we abandon the simple things that we know work because we are always on the hunt for new and better. Of course I’m not saying that we shouldn’t continually improve, I just don’t want to throw out things that still work well. I’ve seen a whole lot of “pretty” flashcards on Pinterest for students to practice math vocabulary. Learning words in isolation though, isn’t going to help make deep connections. Take a look at Robert Marzano’s steps for learning vocabulary.

We use mind maps in writing to help expand upon ideas.  They work amazingly well for brainstorming and thinking, students pages will fill right up.  Why not do the same in math with vocabulary that has multiple meanings and real world connections?

Take for instance the word “quarter”.  While working on a Fraction Equivalence Piles puzzle we came across that word, and it was apparent that for the whole group it was an unknown word.  Quarter is a word that goes deep and has many connections that students can relate to. Instead of putting it on a flashcard and practicing it in isolation, our small group took less than five minutes to think about all the different times we have heard the word quarter.  We laid it all out on the table (you know I love to write all over tables).  Check it out:

The beauty of taking those few minutes to explore the word means it brought it to the front of their minds. So the next day one of them came in and told me the night before they had heard his mom say while making dinner “I need a quarter cup of flour.”   He was elated to tell me that he knew that four quarter cups made one cup.

It is the simple things my math friends. It doesn’t always mean buying a pack, or a bundle or a worksheet.  It’s doesn’t always mean making it pretty from something we saw on Pinterest.  It doesn’t have to always be cut out or have a chevron background. Sometimes we simply have to trust that we can respond to what students need right there on the spot.  Trust in yourself to be the master teacher.

Swear words (in bold below) in any math class of mine include:

“That’s easy.”

“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:

1. 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.
2. 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.
3. 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:

1. Students do NOT understand the Equal Sign.  They think it means “the answer is”. This activity will totally challenge that thinking.
2. 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.
3. 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.
4. This will get them talking! Even your struggling students will start talking about why things fit together, and why they absolutely do not.
5. 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…
6. 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!