Uncovering Misconceptions One Array At A Time

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.”:

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!


Piles! My New Favorite Thing…

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!



The Biggest Mistake I Made in Math Class

This post is coming from a place of both passion and a little embarrassment. I was reminded of my mistake once again the last few weeks as I worked with a student in an intervention group.

If you’re a regular reader, then you know that I seek personal growth constantly. You would know that I readily (and regularly) admit when I’ve made mistakes in my practice. The reason for admitting my mistakes is not to shame myself, or shame others.  I think it’s important to recognize that we are constantly learning, and that we can often learn from our mistakes as much as our successes.

So here’s the biggest mistake that I’ve learned from in the last few years.

The challenging stuff in math is not just for the gifted kids.

The projects, the deep thinking tasks, the inquiry based all hands on deck tasks are for ALL kids, and especially for struggling kids. I have been blown away time after time with struggling students tackling difficult math tasks. They can do it, they may need extra supports, but they can do it. English language learners especially need to be participating in the rigor of these tasks. If you choose the right task, and implement it the right way, the task will be challenging for even your most gifted students.

Here’s a small case study of a student that I have been working with since first grade. She is now a fourth grader. The student came to us unable to identify the number of dots on a dice, unable to count or even identify numbers.  For the last few years, she has been receiving about an hour of intensive intervention daily in addition to her core instruction in the classroom. This wonderful girl is a fabulous reader, so we have been perplexed about why math is a struggle.  All through 3rd grade, when her classmates were learning multiplication, we were still trying to help her learn to add 5+2. Fast forward to this fall, I began speaking with a reading interventionist for more ideas. She suggested incorporating more reading into her intervention to see if that might motivate her.

So I took a big risk and decided to just try something different with her.  I wanted to see if she could learn to multiply through an experiential project.  Enter, The Cupcake Shop…which involves reading and physically making orders to learn about multiplication on a conceptual level.

So first we had to make the shop:


A little cardboard goes a long way…

Then, we started the work.  I was blown away. She knew how to do it. She made the order, she wrote the order slip, she was completely engaged…


Her cupcake were complete with blue “frosting”.


By the third day, this young woman who wasn’t making sense of numbers or symbols for the last several years, wrote the multiplication sentence for her cupcake order without ANY assistance.  To say this was a major breakthrough is an understatement.

I keep hearing these words, “Oh! That would be great for my gifted kids.”  If we want to really push every student, we need to set the bar high. We can’t just reserve these opportunities for gifted kids only, ALL kids can reach that bar. If the task is a good one, then all of your students will benefit from the thinking that is involved, even your gifted kids.

If you want to try out the type of task I’m talking about, give Doggy Dilemma a try for free. Over 55,000 teachers as of today have dowloaded Doggy Dilemma, which means it’s getting into the hands of a lot of kids.

Math Performance Task

If you’d like to try out The Cupcake Shop, you can head here and check out the preview:


The challenging stuff in math is not just for the gifted kids.




What’s the Point of Expanded Form?

OK, I’m going back in time to tell you something totally ridiculous that I actually said to another teacher. It was because of my inexperience and my inability to understand math at a deep level during my second year of teaching.  I was frustrated that my students weren’t understanding what expanded form (that 145 is 100+40+5) was, and had no idea how to teach it for them to understand it conceptually. To the other teacher I said, “What is the point of teaching expanded form anyway? There is no point in learning this!”

Insert forehead smack emoji.

Now, 11 years later I am seeing the error of my ways. Expanded form is a way to deeply understand place value, but because back then I was a young and desperate, I thought anything that was hard to teach was worthless. Expanded form isn’t hard to teach with the right tools, and it will greatly enhance students understanding of place value. Place value simply is a way of understanding quantity in parts or chunks.

So what I do now is use my favorite approach, the concrete-representational-abstract string of learning. The concrete part is the tools, the actual quantity of what we are using.  The representational part is a picture form of the tools/number, and the abstract part is the actual number itself.

I STILL learned my lesson the hard way with assuming things of children. I knew they could make these numbers with tools, show them with ten frames and write them with much practice…but get this. They STILL didn’t understand what the numbers stood for when the number was in standard form. In the photo below they made the number 158 with bundles of sticks (in 10s and singles), cubes (in stacks of 10 and singles), place value blocks, and ten frames.  I asked, “What do you guys think that the number 5 stands for in this number?”

“No! It’s for 50 tens!”
“What do you mean what does it stand for?”


So we did some practicing, I figured if they could make the number, they could identify parts of the number in what they made. So I asked them to put their hand on the hundred, what the 1 stands for:


Then we put our hands on the 50 (or what the 5 stands for):


We were getting somewhere! After practicing this a bit more, I was able to bring in the idea of expanded form with number cards, and then even write the equation:


Notice we kept as many tools in place as we possibly could. Making the links between the tools (place value blocks and bundles), the representations (ten frames), and the number cards finally seemed to solidify their understanding.  Eventually I began to remove the concrete tools, using only the ten frames and number cards…and then I could eventually remove the ten frames.

It was magical! Now I truly understand how wrong I was to teach it without the thinking behind it, just to get through it. I’m still learning, every day.


Subtracting Decimals Is Not Scary!

Something about that decimal point really freaks kids out. So here we go back again to misconceptions and misunderstandings…

I’m working with a student right now who was taught procedurally how to subtract decimals, but gets all mixed up when regrouping.  The VERY BEST way to intervene with any student is to determine where the hangup is in the Concrete-Representational-Abstract approach. If a regrouping error is being made, you can guarantee that the conceptual understanding is missing, which comes in both the concrete and representational part of this approach.

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.

* If you have a little more time you can read about this sequence of teaching and it’s accompanying research here, and also here.

So with errors in regrouping, we backed up to the basics of the American currency. We looked at what each piece of currency represents and why:


So in America, we have many coins, but to keep it conceptual (whole, tenths and hundredths) I used dollars, dimes and pennies.  A dollar is the same as 10 dimes, and also the same as 100 pennies.  A dime is the same as 10 pennies.  To have a reference guide we made the key above to refer back to. Later this translated really well to using place value blocks when we needed to start using larger numbers, and when I was out of money. (Imagine that…a teacher without a lot of money! Ha!)

So for a while we added two amounts of money. Addition seems to come really easily to students. From K-4, I’ve noticed a pattern that adding two whole numbers seems to be a piece of cake, but subtraction is like HITTING A WALL.  So, I decided to start with subtraction slowly. This was the sequence:

  1. We subtracted over and over when we didn’t have to regroup, and we didn’t write anything down.
  2. Then we subtracted where there weren’t enough pennies, and he had to trade (regroup) a dime in for ten pennies.  We still didn’t write anything down.
  3. Next we subtracted where we had to trade in (regroup) a dollar for ten dimes. We STILL didn’t write anything down.

Why aren’t we writing anything down?

Well, that was the concrete stage. All we do in the concrete stage is play around with the manipulatives to be sure the student is understanding the concept behind the regrouping. It makes the next two steps SO much better.

Now that’s where we talk about noting what we did. We’ll use pictures along WITH the materials, which is how we link to the representational stage.

  1. We just kept subtracting when we didn’t have to regroup, and this time we drew what we did.  Since the student already knew the procedural shortcut to regrouping, we linked that in with the drawing.
  2. Then we subtracted where there weren’t enough pennies, and he had to trade (regroup) a dime in for ten pennies.  We still didn’t write anything down.
  3. Next we subtracted where we had to trade in (regroup) a dollar for ten dimes. We STILL didn’t write anything down.
  4. Once that felt good, we started writing, and then suddenly he wanted to try out the algorithm to see how it might match. Instant connections to the abstract stage…the procedure he had already “learned”. Now, it made sense to him, and he no longer had to memorize which number to cross off because he knew why.


Too often we are pulling manipulatives from our older students.  Whenever I’m working with a student who struggles, it seems to be the first place that I notice problems.  When we pull away those supports (or never start with them in the first place), math really is just a bunch of meaningless numbers and symbols without any connection to the real world.


Real World Examples For Teaching Fractions on a Number Line

Number lines go wrong in SO many ways. Let us count the ways:

  1. Students struggle to read number lines when labels are missing on the tick marks.
  2. When drawing number lines students do not evenly space the tick marks.
  3. Students count the tick marks instead of the spaces and are often off by one increment.
  4. When a number line is open (without labels or tick marks) it’s difficult for students to imagine a benchmark number.
  5. Not all number lines go up in increments of 1 like most students think.
  6. Number lines are so abstract that students are unsure of what they might represent.

So then for fun…let’s throw FRACTIONS on the number line, that won’t be difficult at all!

All kidding aside, number lines are incredibly important. We use number lines in our lives ALL the time. When we drive, a road is just a gigantic number line. We listen to our GPS and we know about how far 1/4 mile is to our next turn. We use number lines when we measure length, or even as we measure capacity in a liquid measuring cup. I use a mental number line when I’m trying to figure out how much sleep I’ve gotten the night before.  We use number lines in real life without thinking about their mathematical significance, and yet often in many books/math curriculum you will see a number line without any connection to real life with it. No connection whatsoever!

So maybe 4th graders aren’t old enough to drive, but they sure can pretend!  Since they could already identify benchmark fractions on a number line, could they use this idea to learn about equivalence? Could we find two points on the road that could be called two different things?


So we started our engines, the 4th graders quickly realized that our number lines represented a mile, and that a mile could be marked in different ways. They even talked about how they’ve seen signs for 1/2 mile or 1/4 mile on the highway.


After working for a bit, they began to see that this mirrored the previous day’s lesson of fraction equivalence with fraction strips.  The connections were forming, and even better they were splitting number lines in two ways without us having to present it that way.  Best of all, they made connections to why 2/8 was the same as 1/4, they were just cut into twice the number of pieces.  This was an awesome link to why equivalent fractions are related to multiplication and division.

Don’t have toy cars to use? Here are some other ways to make number lines real life for kids (all of which can be so precise that fractions are useful when reading the line):

  1. A tiny number line: You are watching an ant crawling on the sidewalk. How far has it gotten?
  2. An eating number line: A snickers bar is being eaten, how much has been eaten if you start at one end and start chomping?
  3. A measurement number line: How tall is the doorway to the classroom?
  4. A time number line: How long can you hang from the playground bars?
  5. A reading number line: How many pages have been read of your book if you stop in the middle of page 39?

When number lines are playful, and connected to real life they are not nearly as scary. Give them a try!

Start driving!