Mathematical Reasoning

Reasoning is foundational for making sense of all mathematical skills. Reasoning can be thought of as the “glue” which helps math make sense. If this is true, then why do so many students lack mathematical reasoning? How do we develop mathematical reasoning when there are so many content standards to address at every grade level?
I’m incredibly grateful for gurus like Marilyn Burns who are smarter than myself, and who have developed tools for us to assess mathematical reasoning. Michael shared this tool with me last year, and it wasn’t until recently that I’ve been able to use it.
I was in a coaching partnership earlier this year with a 5th grade class where many, many of the students lack number sense and mathematical reasoning. I was able to identify these deficits without much thought, however the classroom teacher wanted to have something to be able analyze where her students are in their reasoning. Enter the Math Reasoning Inventory from Marilyn Burns

If you’ve read any of my prior posts, you know that looking at student work fascinates me!  The Math Reasoning Inventory is another tool that has continued to enlighten me about how a ten year old thinks about numbers.  Some of it is true mathematical reasoning, and other students really haven’t developed any reasoning, they are dependent on procedures without any thought.  So what do we do about this?

We start by having conversations to change the culture of our mathematics education. We need people, parents, educators, policy makers, to understand that procedures and memorizing rules are outdated and no longer serve our students!



When a Lesson Doesn’t Go “Right”

What do you do when a lesson doesn’t go “right”?  Recently I was asked to do a lesson in a 4th grade class and I just didn’t feel it went “right”.  It was a well written lesson plan though, so what happened?

Here’s how the lesson was supposed to go, (in my head):

Number Talk: Mrs. H is saving up to buy a present for her husband that costs $182. So far she’s saved $53. How much more does she need to save? 

It was evident by student responses that they the majority would do the standard algorithm and subtract to figure this out. A few attempted a different strategy of chunking, but incorrectly.  This was discouraging. It also was going on longer than it should have.

Engage students by asking them to turn and talk about where they like to go out to eat and what they need in order to go out to eat. Mini Lesson: Ask students, “How can multiplication help you count a collection of coins?” Read aloud Pigs Will be Pigs, Fun with Math and Money.  Look at a line plot with different coins and ask students how they could efficiently count this collection of coins. 

At this point, we’d been on the rug TOO LONG, even with a few stretch breaks!

Work Time: Students count a collection of coins in a bag provided to them. Record on a line plot and calculate the total.  (Hopefully using multiplication for multiple coins when too much to just count.) 

The work time wasn’t too bad as each student was engaged with counting their own bag of coins. The part that didn’t go as well as I envisioned was having them keep track of the collection on a line plot. (Thought this was a good way to practice line plots) and discovering that multiplication can be more useful when a collection of coins is large.

Discussion: How did you calculate your collection of coins? Did multiplication make it more efficient? 

When students were more focused on telling me the total, instead of how did you calculate your collection?  I knew I’d missed something. This is also when I realized discussion norms needed to be established. There was debate, however not in a productive manner. Maybe 2 students saw what I was trying to get them to think about  when multiplication might be helpful, while the rest blurted out, “Multiplication is too hard.” AHA, “they’re uncomfortable with multiplying the value of the coin by the number of coins”.

In a nutshell, there were 3 flaws I felt that made this lesson unsuccessful on the first take:

  1. Too much planned (read aloud, mini lesson, line plot, multiplication)
  2. Students were not seeing connections between counting a collection of coins, line plot (organizing the collection of coins), and use of multiplication to calculate a total.
  3. No class discussion norms established

In my debrief with the teacher, we decided a follow up lesson, but not a new lesson was necessary.

Here’s how Day 2 went:

I re-engaged students by passing back their papers they used to complete the line plot and calculate the coins in their bag. We began with a  brief discussion of students sharing their collection of coins and how they calculated the total, which I recorded on the board. One of the students offered up that he did repeated addition for 10 quarters. (That’s 25+25+25+25+25+25+25+25+25+25) I  followed up by asking the class, “Which is more efficient, to do repeated addition or multiply the value of the coin by the number of coins, 25×10?  It was at this point our discussion took on new meaning, a girl asked, “What does efficient mean?” Great question!!!! Once we moved past what efficient means, I re-read the story and while I was reading asked students to keep track of how  much money the pigs found around their house. We discussed how they might use  a line plot, an organized list, or just a try to count a running total.  At the end of the read aloud, we had about 4 differing answers for the total amount of money in the story, they weren’t too far off, and it brought to the surface that sometimes keeping a running total isn’t efficient or helping us be accurate.  I felt good about leaving them thinking about this!

I still left thinking some students weren’t making the connections we’d hoped for, however the discussion about counting a collection of coins efficiently was brought to the surface, and I left students continuing to think about when it’s more efficient to just count the coins and when multiplication might be more efficient. The word efficient was also brought to the surface. Do you teach students about efficiency? How do we continue to help students understand that math is about efficiency, not just calculations. (Ongoing conversations)

My final self reflection from these 2 lessons is this, sometimes I try to include TOO MANY ideas and students aren’t ready for all of them at once.  In this instance, the students weren’t solid enough in their understanding or use of multiplication, line plots (organization), and applying those to counting a collection of coins.

I left a game with the teacher to follow up, The Piggy Bank game from an issue of NCTM, where students have a collection of coins, one student is the banker and one student is the detective. The detective asks questions to the banker to figure out how much money the banker has. Here’s an example:

Detective: “How many dimes are there?”

Banker: “Two”

Detective: “How many nickels are there?”

Banker: “The number of nickels is three times as many as dimes.”

Detective: “How many quarters are there?”

Banker: “One”

Detective: “How many different types of coins are there?”

Banker: “Three”

Detective: “There are two dimes, six nickels, one quarter, and no pennies. The total is 2×10+6×5+25=75 cents.”

This game provides the opportunity for students to add a collection of coins, or use multiplication.  It also is a good way to have them discuss with a partner.

In the end, these 2 lessons left this group of students with some new thinking, and I’m proud of that.  We have to be okay with revising a lesson when it doesn’t go as planned.  This is also a good way to slow down and make sure we are focused on student understanding, rather than just covering material.  So, what do you do when a lesson doesn’t go “right”? I’d love to hear your ideas.

I Love Subtraction

I love subtraction. There, I said it. If I’m asked to come and model a number talk, I don’t care if it is a 2nd grade classroom or a 12th grade classroom, I will probably do some kind of subtraction number talk. Fractions, decimals, whole numbers…I don’t care. Subtraction allows for creativity, flexibility, and critical thinking if it’s presented in a way that releases students to follow their own intuition instead of their teacher’s strategies (or textbook’s).

In a 4th grade classroom, we were subtracting mixed numbers with like denominators (or like units). Here was the problem:

5 1/8 – 2 5/8

Problems like this are enough to bring 4th grade teachers to their knees. Not only do they have to subtract mixed numbers (a new concept), but they have to “regroup”. First, “regrouping” is only one way of solving this problem, which is the most traditional way. Here are some of the other ways we can solve the problem:

Adding Up: This is the one that makes the most sense to me if I couldn’t subtract by the given unit values (no need to regroup). To add up, I would start with the subtrahend and add up until I got to 5 1/8. One of the ways I model this strategy when students are sharing is to draw a number line.


The sum of the jumps will be the difference. The other benefit this strategy provides is the opportunity to look at subtraction in a different light. In algebra and many “subtraction” contexts, subtraction is used to find the distance between two numbers (or coordinates).

Instead of telling students, “Let me show you this strategy”, pose a problem to them that makes the strategy necessary. Here’s an example:

My aunt works at a bakery and she was making a large wedding cake. The wedding cake required exactly 5 1/8 cups of sugar. She has already separated 2 5/8 cups of sugar. How much more sugar does she need? 

Now, of course, students don’t have to solve it by adding up, especially if they’ve been taught that “how many more” means to subtract in previous grades.


But, if students want to act out exactly what is happening in the problem, they would start with a given amount (2 5/8) and add an unknown amount to get the total (5 1/8) (or 2 5/8 + ___ = 5 1/8.

Chunking: The greatest trick the devil ever pulled was convincing students that they have subtract everything all at once. By subtracting part of the subtrahend first, it allows us to subtract by unit value without even thinking about “regrouping”. For example:


In this example, I decomposed 2 5/8 into 2/8 + 2 3/8 so that I can use the associative property to go ahead and subtract 2/8 from 5 1/8 so that I can have 4 7/8. 4 7/8 allows me to subtract by unit value (4 – 2 and  7/8 – 3/8) without “regrouping”. This strategy requires students to plan ahead, craft a plan, and execute that plan.

It is important to use the names of the properties and identify them as they are being used. This is one way to record, but I would recommend recording (or acting out) in a way that makes the most sense for your kids. This isn’t the only way decomposition can work to our advantage. Here are some other ways:

Non-Traditional Regrouping of the Minuend: Sometimes, as teachers, we can’t see the forest for the trees. We are so consumed with how we were taught to regroup, we can only see regrouping as one way. This was made clear to me when I was helping students model this problem with fraction circles. We started with this:

How can we take 2 5/8 from this?

We started by taking away the 2 wholes (chunking, anyone?)


Here is where the new regrouping kicks in. A student suggested that we needed to take away the remaining 5/8. So I assumed that he wanted me to cross out the one eighth on the bottom row.

You know what they say when you assume…


“No, no, no! I cross them out on the third one. There are 5 already in that one, so take them from there!”


Students easily saw that we had 2 4/8 left. I was so consumed with trying to make his model reflect the traditional regrouping process, I almost missed out on a cool new way of thinking about subtraction. The discovery came when we began to model our picture with numbers:


If students have figured out how to subtract a fraction from a whole number, this strategy makes more sense than traditional regrouping (subjective, but I don’t care). This actually works even better when subtracting whole numbers. Take 67 – 39 for example. If we were to regroup the traditional way, we would end up subtracting 17 – 9 and 50 – 30 to get 8 and 20 to add to get our difference of 28.


One of the most difficult parts of this process is subtracting 17 – 9. Some students have invented their own strategies such as (17 – 10) + 1 = 8 or 17 – 7 – 2 = 10 – 2 = 8. However, many students still have to count back on their fingers or make tallies.

What if we regrouped a different way? What if, instead of decomposing 67 as 50 + 17, we decomposed it as 57 + 10? Would that allow us to “take away” by place value now?


Which would allow for fewer mistakes? 10 – 9 or 17 – 9? Is 57 – 30 prone to more errors that 50 – 30?

The point here is that traditional regrouping is a one way street. That is, there is one specific way to regroup the tens and ones in the minuend and any other way is wrong. If, instead, we explored different ways to rewrite both the minuend and the subtrahend, can we avoid regrouping? Look at some more examples:

Creativity exists in mathematics. We just have to give it an opportunity to flourish.

Fractions on Exhibit

It is no secret that fractions are a difficult concept for many of our young students to grasp, especially if they are still trying to figure out whole numbers. I’ve been coaching a 3rd grade teacher, and after she looked at her student’s formative assessment on 3.NSF.1 , we discussed the fact that her students could identify a fraction, but they couldn’t really explain a fraction. We decided to do a Fraction Museum with her class the very next day.  Here’s how it went.

First, I read a few pages of the book, Fraction Action by Loreen Leedy just to give students a few minutes to review what they’d already learned that week about 1/2, 1/3, and 1/4.  While reading I noticed our students were listening intently, and it occurred to me that we should read more math stories with our students, I don’t think we do it enough. (Or maybe in the classes I’m in we don’t, what’s that say?) I digress.

After the read aloud, I asked students, “Have you ever been to a museum? Which museums have you been to?  What did you see at the museum you visited?”

Then I told students we were going to turn our classroom into a “Fraction Museum” for the afternoon!  They got very excited and started blurting out all kinds of ways they thought we could do so!  This was great, because my partner teacher and I had discussed how she might need to give them more hands on experiences with fractions.

After modeling for the students how they would create a fraction using any materials from the Math Manipulatives shelf, write the fraction on exhibit, they got to work.  Here are some of their exhibits:


img_6049         img_6045

What was really interesting as we watched the students create their exhibits was how comfortable most of them were with using a numerator of 1, creating unit fractions. However, they needed to be asked, “Can you use another numerator, other than 1? What would that look like?” This was good formative assessment though.  After students created their exhibits they had to visit another student’s exhibit and talk about the fractions they saw.

The final part of the lesson was an exit slip. “Explain what a fraction is in your own words.”  Mrs. H and I met to look at those exit slips.  While only about 3 students clearly articulated a solid understanding of what a fraction meant, the majority of the class was beginning to make more sense of fractions than the previous day.  They have a better understanding of language used to describe the fraction because they had to explain their exhibits to others.

This lesson was fun, but more importantly it helped students put their hands on what fractions feel  and look like other than just this:



Make Me Better

Today I had the privilege of visiting four 4th grade classrooms at one of our elementary schools.  What I experienced in their classrooms today was different in several ways, while some core ideas remained constant.  There was evidence in every classroom that students are provided opportunities to use tools, drawings, and equations to represent their thinking. There was talking about math in every classroom. And each teacher held students accountable for, and expected them to use mathematical process standards.

This was the first day of their long division unit.  A unit that is often misinterpreted. All four of these classrooms began by formatively assessing  how students thought about division.  Two of the classrooms opened with the following task:

There are 24 students in Mrs. C’s class. When we go on the field trip to the museum, we will be allowed to sit 3 students to a seat. How many seats will Mrs. C’s class use on the bus?


What if we sat 2 students in one seat? How many seats will Mrs. C’s class use?




It was a good lesson, and the questioning of students is what drove the emphasis on modeling the mathematics.  It was evident that the goal of the lesson was for students to demonstrate their understanding of division from 3rd grade.

The other two classes opened with a different problem, it wasn’t based on basic facts, it was more open for students to really show how they approached the division situation. Even these two classes had a slightly different approach.

One class engaged in the problem in a 3 Act Task, Notice and Wonder approach, while the other was presented as just a word problem. The really cool thing was that students were provided the opportunity to not just show what they already knew about division, but it was evident that students were driving the strategies!

Notice & Wonder Approach:

Jaia ordered 84 red roses from the florist. She found this centerpiece arrangement online at Pinterest and is going to make hers just like it (fingers crossed). How many vases will she need?















Simple Word Problem:

Kristy bought 84 roses. If she puts 6 roses in each vase, how many vases will she use? Will there be any roses left over?

On the surface this is a really simple word problem, however the approach to just see how the students solve it without going over any vocabulary, strategies, or even talking about which operation to use made it powerful!

Then I got to see something really powerful and empowering for students! After having time to work it out, there was a gallery walk,  not just a gallery walk though. Music played that got the students moving, dancing, and looking at classmates’ work. When the music stopped students took a seat and started writing on a Post-It note. They were critiquing the strategy of whomever’s seat they took!






There were two rounds of this. I got chills watching it. Every student was getting feedback on their strategy from two classmates! I definitely will be trying this!  It was a new strategy to me on engaging every student, while at the same time integrating math and writing, specifically written critique. If you look closely at the notes, you will probably say, “This isn’t great feedback though.” But it sure is a start and a clear piece of evidence that students are able to critique one another! If we let them. It is only through sharing our strengths, knowledge, and creativity, while taking feedback from our colleagues that we become better learners.

Fraction Decomposition: More Than Meets the Eye

Since Kindergarten, students have had multiple experiences composing and decomposing (whole) numbers. They look at the quantity 8 as being 7 + 1, 6 + 2, 5 + 3, and 4 + 4 or compositions of multiple addends, including only ones. Any 1st or 2nd grade teacher can tell you that understanding and knowing all of the different combinations of 8 is more important that writing 8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1. The reason is that in order to add within 20 (and within 99 after that), they need to be able to flexibly think of addends and place value to add efficiently. For example, knowing all of the compositions and decompositions of 8 will help a student add 7 + 8 by thinking of it as 7 + (3 + 5) and using the associative property (7 + 3) + 5 to make a ten and some more to get 15. With weeks of practice and authentic opportunities, this decomposition becomes habit and students begin to use the decompositions intuitively.



In 4th grade, students continue their work with fractions by “composing and decomposing a fraction in more than one way, recording each composition and decomposition as an addition or subtraction equation” (4.NSF.3). This standard has always been interpreted in the same way the Kindergarten expectations have been interpreted:

3/4 = 1/4 + 1/4 + 1/4


3/4 = 2/4 + 1/4


3/4 = 4/4 – 1/4

There are benefits to this line of teaching. Students can learn about fractions as iterations of units and generalize about operations with whole numbers and operations with fractions. However, if we stop here, we are depriving students of opportunities to think more deeply and build a solid foundation for future fractional understanding.

For example, how else could we decompose 3/4 and other fractions?

3/4 = 1/2 + 1/4

3/4 = 1 – 1/4

5/6 = 1/2 + 1/3

5/6 = 2/3 + 1/6

5/8 = 1/2 + 1/8

5/8 = 1/4 + 3/8

1/2 = 1/8 + 1/8 + 1/8 + 1/8

1/2 = 1/3 + 1/6

Students work with equivalent fractions should allow them to recognize and investigate how these fractions can be made up of other units. In fact, it shouldn’t be foreign to them. In 2nd grade, (in South Carolina Standards), students are supposed to understand 345 as 3 hundreds, 4 tens, and 5 ones, OR 34 tens and 5 ones OR any of the other possible decompositions. This line of thinking is the same as the fractional line of thinking above. In both cases, we are looking at how numbers can be composed of other units.

The critical question here is why. Why do we need to investigate these non-unit aligned decompositions? For the same reason we do it in Kindergarten. Just like understanding decompositions of 8 can help you use an understanding of place value (namely, tens) to solve 8 + 7, you can use decompositions of fraction to help us solve 1/2 + 3/4. If we know that 3/4 = 1/2 + 1/4, then we can look at it as 1/2 + (1/2 + 1/4) = (1/2 + 1/2) + 1/4 = 1 + 1/4 = 1 1/4.

Think about all of the different ways this decomposition work can help us solve the following addition/subtraction problems:

5/6 + 1/3

1 1/4 + 7/8

3 5/8 -2 1/2

Now, I know that the addition and subtraction of unlike fractions are not expected until 5th grade. Therefore, this is not the baseline expectation for 4th graders. However, we are responsible for setting the foundation for successful fraction work in 5th grade and beyond.

Fraction Subitizing

The word “subitize” is derived from a latin word meaning “to appear suddenly”. In the lower grades, the ability to subitize (or instantly see quantity) is an important milestone in the development of number sense. In 5th grade, we can use this idea of subitizing to subconsciously (and sometimes consciously) teach the addition and subtraction of unlike fractions.

Let’s try one out


How many circles do you see? Technically, there are two parts of two circles, but if we were to combine them, how many circles would we have? Many students, even with no experience with combining unlike fractions, will be able to move parts around mentally to say “1 1/4”. Without the formal actions of making “common denominators”, students recognize that 1/2 is the same as 2/4 and can compensate for the missing quarter in the first circle.

Check out some others:






Try some of these in your classroom and let me know how they work.

Spatial Sense and Numerical Fluency in Early Elementary Classrooms

Numerical fluency is one of the most important developmental milestones in early elementary classrooms. Kindergarten, 1st, and 2nd grade students are expected to be fluent within 5, 10, and 20 respectively by the end of their grade levels. Many of those grade level teachers will quickly admit that getting them to those milestones is one of the more challenging parts of mathematics instruction.

What if there was a way to develop fluency within 10 without flash cards or mindless memorization? How can we get students to rapidly recall addition and subtraction facts without fingers (although the use of fingers shouldn’t be discouraged)?

Recently, I came across research by Robert Siegler and Geetha Ramani that demonstrated that linear board game (instead of circular ones) dramatically improved low-income pre-school students’ number sense. This has confirmed my hypothesis that spatial understanding and the ability to picture a number path spatially can dramatically improve student fluency.

I’m not aware of many linear board games, especially ones that have numerals on the boards, so I created some myself.

The Game Board

Using this board, I created two games: Target and Back and Forth. Both games require dice and Target requires playing cards or number cards.

Target: This game can be a little confusing at the beginning because there are so many steps to each turn. Each player (3 – 4 players) will put their marker (e.g. unifix cube) at 0. At the beginning of her turn, a student will draw a “target card” and place that card above that number on the game board. That number is now the “target”. She will then roll one die to determine how many “jumps” her marker will take. If the student lands on the target number she will get zero points. If she misses the target, she will get one point for every “jump” she is away from the target. For example, if the target was 6 and she landed on 4, she would get 2 points. The student with the fewest amount of points at the end of a pre-determined number of rounds wins. If a student rolls a number that would take her off the board in both directions (e.g. if she is on “5” and rolls a 6), she would just go to one of the ends that would be closest to the target.

Three 1st Grade students playing Target

Back and Forth: Back and forth is more simple than Target. The goal is to start at 0 and make it to 10 and back to 0 as quickly as possible. One catch is that you must land exactly on 10 and exactly on 0 to change directions. For example, if a student is on 8 and rolls a 5, she loses her turn until she rolls a 2 (or 1 to move to 9). A student gets a point every time he/she returns to exactly 0.

4 Students Playing Back & Forth

These games are incredibly simple, but provide students an opportunity to interact with numbers spatially. For example, in Target, students must think about how far they need to “jump” to get to their target and how far they end up being from their target to determine their points. This helps them determine sums and differences within 10. In back and forth, when moving up towards ten, they are practicing adding within 10 and when moving back, they are practicing subtracting.

Kids enjoyed the games and most importantly planted a seed that will hopefully blossom into fluency by the end of the year. If anyone is familiar with other linear games (or if I’ve accidentally stumbled onto someone else’s intellectual property), please let me know. Also, let me know your thoughts about the mathematical appropriateness of these games.

Conceptualizing Decimal Multiplication

In 5th grade, students are expected to multiply decimals to hundredths “using concrete area models and drawings” (5.NSBT.7 (SC) or 5.NBT.7 (CCSS)). Unfortunately, many classrooms begin by showing multiplication strategies such as the box method, area model, or even the algorithm. While these are all adequate strategies for getting the correct answer for a decimal multiplication problem, they do not allow students to grapple conceptually with decimal operations (e.g. why is 6 tenths x 4 tenths = 24 hundredths when 6 ones x 4 ones = 24 ones?).

Therefore, I created a context that allowed students to grapple with decimal multiplication:

“At my house this weekend, I dropped something on the ceramic tiles in my laundry room and they cracked. My wife, never missing an opportunity to redecorate, decided that it was the perfect time to retile the laundry room. So how many tiles do I need? I provided you with a picture of my rectangular laundry room here.”


“It is a scaled picture and our base ten flats are the exact same scale (funny how that works out!) as the tiles. So using the flats, tell me how many tiles I need to buy?”

Within 3 minutes of “go!”, hands shot up around the room. “Mr…Mr. what’s your name…we can’t fit these flats (tiles) in the rectangle!” Headache created.

“So, what does that mean? What do we need to do?”

Students began to count the tenths that could fit and pretended to “cut” the flat to make a partial flat (consisting of tenths!!!). Some students began to think how they could split one of the wholes to fit to cover two of the necessary gaps to accumulate more tiles. Once it began to settle down, I gave them rods.

“But Mr. Geranamo (close enough…), now we need some small cubes! The tenths don’t fit in the corner!”

Me: “What do you mean? The width is 3 and 6 tenths and the length is 2 and 6 tenths…how does that create hundredths?” Headache #2

Me: “Why is it that we multiply 6 ones by 6 ones and get 36 ones, but if we multiply 6 tenths by 6 tenths, we get 36 hundredths? Shouldn’t it be 36 tenths, which is 3.6 wholes?Why is our unit changing when it didn’t change before?”

Silence. Confusion. Outrage. Struggle. A little bit of pre-lunch stomach growls.


After 5 minutes of group discussion, one girl said, “Well, isn’t 1 x 1 = 1? Also, isn’t 1 tenth of 1 tenth a tenth split into ten equal pieces, which is a hundredth?”

I wrote the following on the board (no classroom picture…too excited. Enjoy this picture instead)


“Would you agree with this?” A few more jumping on board…

I gave them some hundredth cubes and let them complete their rectangles

Complete! But where is their understanding? To figure that out, we must move to the most important part of the lesson, the debrief.

Me: (Writes 2.6 x 3.6 on the board), what did we do first?

S: “We put out 6 flats”

Me: “Where in this expression would we multiply and get six wholes? Do you see anywhere in here?”

S: “2 times 3! 2 times 3!”

Me: “So we made a smaller rectangle here with a width of 2 wholes and a length of 3 wholes right? Well, we have part of our laundry room covered (back to the context!!!), but we have these sections that couldn’t fit a tile. What did we do next?”

S: “We had to use the rods to fill in the gaps.”

Me: “Why did we have to use the rods? Why not the flats?”

S: “Because the flats wouldn’t fit, so we had to cut them up into tenths.”

Me: “So over here (the yellow in the pic below) we had 12 tenths. Where did that come from in the expression over here?”


S: “That’s two groups of 6 tenths, so 2 x 0.6”

Me: “What about these 18 tenths (green)?”

S: “Six tenths times three”

Me: “And these 36 small hundredths?”

S: “6 tenths times 6 tenths”

Me: “Hmmm…that sounds an awful lot like how we multiply whole numbers. Wouldn’t it be essentially the same thing if I used an area model for 26 x 36?”

(The rest is how I should have followed up)

Let’s revisit our context. How would you make the red part: the corner that couldn’t fit flats or rods? How many cuts would you need to make to a full tile?

A full tile cut into hundredth squares

I would need to cut 6 tenths vertically first to fit the width

1 whole cut into 6 tenths


And cut that result so that it will fit the length (6 tenths).

6 tenths of 6 tenths = 36 hundredths

So, contextually, the now have cut a part of a part of a whole. That is the most challenging, and most essential, part of decimal by decimal (or fraction by fraction) multiplication.

Highlights of the feedback:

Teacher: “That student over there is in the 98th percentile on MAP and he was getting so frustrated. He wanted the answer immediately.” In other words, he had to think deeply.

Student: “This makes the area model with whole numbers make more sense. I didn’t understand it before, but these have helped me get it.”

Teacher: “(Student) usually doesn’t get into any lessons, but he participated more than I’ve ever seen before.”

I would say it was a success. To close, I want to quote a blog post I read a few weeks ago:

“I tried hard to create powerful discovery experiences early in my career. An implicit belief embedded in that instruction was the idea that, if I could just find the perfect way to introduce students to an idea, they would remember it and be able to apply it in the future. At best I had mixed success. One activity, no matter how clever it is, never makes as much of a difference as I might think. The sum of student experiences — from the introduction of an idea, through practice activities, to opportunities to transfer that understanding to new contexts — are what make a difference for learning. I’m much more interested in focusing my energy on a range of activities that allow students to practice and extend the ideas we’re learning about than putting all my eggs in the “they discovered it so they’ll remember it” basket. Here’s my core value: it matters less how a student figures our something new than what they do with that knowledge in the future.” -Dylan Kane

Although I would fall a little bit more one the discovery spectrum he discusses in a follow up post, this is the part that is most important: One activity will not lead to proficiency. We must provide multiple authentic learning experiences where students have plenty of opportunities to discuss, apply, and construct their complete understanding.

If you want more information on multiplication of decimals (or decimal operations in general), click here.

The Power of Student Work Part 2

In my last post I shared some student work from a 3rd grade formative assessment. I mentioned how people tend to get stuck on the report card, rather than on what students can REALLY do.  The numbers and/or letters on a report card only report a general range of where students fall in a traditional labeling system. Test scores are another piece of data that tell us how a student measures up based on the norms of the test or the cut scores.  While these are factors that we, educators are evaluated on, they don’t have to define ALL of our work. The real power lies in student work.  By looking at student work we can dig into their thinking, without actually getting inside their brains.

When you first look at student work, ask these two questions:

What do you notice? What do you wonder? 

Let’s look at this student sample:

Screenshot 2016-08-21 14.49.32

“I notice this student drew tens and ones. I notice this student drew 7 tens, and 6 ones. I wonder why this student used tens and ones. I wonder if this student understood the problem. I wonder if this student is “stuck” in place value lessons, and that if that’s all he/she is thinking about? I wonder if this student could act out this problem?”

What instruction does this student need to move forward?

This student understands that objects can be grouped by tens and ones, but doesn’t understand this problem as 6 cupcakes in a pack, and 7 packs.  This student picked out the numbers, but I’m not sure why he/she chose 7 for tens and 6 for the ones.  This student needs some small group instruction with problems like this, first acting them out, then I would provide opportunities to represent the problem with manipulatives, and/or drawings. Finally I would use words and symbols to help this student get to the abstract level of understanding. Discussions and questioning about what the numbers he/she chose would help this student make sense of what the problem is asking as well as help this student reason about his/her answer. I would ask this student:

What does the 7 represent from the problem? What does the 6 represent from the problem? How can we act out this story? What tool could we use to help us represent the cupcakes and the packs? 

Let’s look at another student example:

Screenshot 2016-08-21 14.55.05

“I notice this student drew a picture to represent the cupcakes, but he/she put them in packs of 6, and then labeled some in groups of 10. I notice this student used repeated addition with 6. I notice this student wrote his/her answer at the top in a complete sentence. I wonder why the student drew the circles in groups of 6, but then labeled them with 10? “

This student has an understanding of multiplication as “groups of”. This student also appears to be in between the representational and abstract level of understanding. While this student didn’t write a multiplication sentence, he/she did write a repeated addition equation. This student also completely understood what the problem was asking, this is evidenced in his/her answer written at the top. I would ask this student:

Tell me about your drawing. What do the small circles represent? What do the 10 mean in your picture? Do you think you have to write out those sixes every time you add the same number over and over again, or is there an easier way to represent that thinking? This is how I would instructionally move this student to see that we can also represent it as 7 groups of 6, which can also be written as 7×6.

Now you try it with the next two student work samples: What do you notice? What do you wonder? What would be your next instructional move?

Screenshot 2016-08-21 14.53.27

Screenshot 2016-08-21 14.52.02

Next time you’re meeting in your PLC, bring a few student work samples. With your PLC members go through the exercise of noticing, wondering, and collaborating to make instructional decisions. I know we get anxious and feel like we just have to keep moving, but by sloooowing  down, looking at what students CAN do, you will gain MORE instructional time!  Remember to shift the focus on what students have learned, and what they still need to learn, NOT what you have taught and what you still need to teach. Spend your time doing powerful, intentional work and the report cards and test scores will affirm that work in the end.

The work of PLCs is hard, and it takes time, however it’s powerful and results in student success, which in turn makes teachers successful.