Before we get to the study, think quickly about these two problems. How would you solve them? The first problem:

*If 5 people equally share 15 cookies, how many cookies would each person get?*

That’s division: 15 divided by 5 is 3 or three cookies each. How about this second problem:

* If 5 people equally share 4 subs, how much does each person eat?*

If that made you pause, you’re not alone. Many might start by cutting up the sandwiches…

Okay, so three people get a half and hopefully two people will be happy with just a half. Maybe we should divide the rest into quarters…or wait. What if we cut them all into fifths?

Each person gets ⅕ +⅕ +⅕ +⅕ . Then each person can get four ⅕’s.

Peck & Matassa would call this *informal *modeling of the problem. That’s when you use images related to the context of the problem without using formal mathematical notation.

Notice I arrived at the correct answer, but I didn’t use the idea that the problem was equal sharing division right away. When it’s whole number, no problem! Equal sharing is division. Divide the two numbers you are given. But when the answer results in a fraction, hmmmm…that can be a bit of a struggle! I needed sandwiches to help answer the question.

That’s exactly what Peck and Matassa observed in their research study involving 12 high school students in what they described as taking a “support course” for Algebra I. Like many interesting studies, this one was inspired by something they noticed in their classrooms. Their algebra students reacted differently when trying to solve these two seemingly similar equations:

Their students could solve the first one, but not the second! Why? The first equation has an integer answer (2), but the second equation results in an “improper” fraction. The students wouldn’t see 9/7 as an answer.

I’ve seen this exact behavior in my college algebra classrooms. Students would look quizzically. Did someone make an error when writing the exam?! Clearly, 7 doesn’t divide 9! Determined students would resort to long division to find an decimal equivalent, quickly becoming frustrated with an unending number. Others would search quickly for a calculator, thinking, “What idiot would put a fraction as an answer?!?” Clearly, fractions don’t get any respect! They’re numbers, too!

Peck & Matassa surmised that it was their students’ lack of a depth of understanding of fractions, or more specifically, this idea:

A fraction can be used to represent, simultaneously, a division problem and the numerical result of the division problem.

Peck & Matassa, 2016, p. 246

Middle and high school teachers will recognize this problem when they introduce the concept of the slope of a line. Students learn “rise over run” or “unit rate of change” (in other words, for every one step to the right on the horizontal axis, how far up do you move?).

In the case of nine-sevenths or 9/7 , students should see the rise of 9 over the run of 7 or a 9/7 change for every 1 unit of change. Using real world scenarios, we want out our children to understand the steepness of a ramp that climbs to 9 feet after spanning 7 feet (rise over run), or the growth rate of a seedling that is nine-sevenths of a foot one week after sprouting (unit rates). After all, nature doesn’t happen in only in whole numbers!

Peck and Matassa (2016) observed their students struggling and wanted to learn how their students understood fractions and developed a concept of partitive (or equal sharing) division.

They used designed and adapted activities in a setting where students were used to sharing their strategies with a math congress. One of their activities was the sub-sandwich problem (abbreviated below; attributed to Fosnot 2007)

A class traveled on a field trip in four separate cars, which stopped for lunch to share sandwiches:

(Fosnot, 2007)

● The first group had 4 people and shared 3 subs equally.

● The second group had 5 people and shared 4 subs equally.

● The third group had 8 people and shared 7 subs equally.

● The last group had 5 people and shared 3 subs equally.

Was this fair? Or did everyone get the same amount?

Peck and Matassa found that students solved the problem with *informal *constructions (like my sandwiches), *preformal *models (using bars instead of sandwiches), or *formal *strategies (straight to the math – no context). Initially, students who could produce “preformal” constructions of the problem were the only ones who could solve it accurately and justify their reasoning.

Students who went straight to fractions (a formal representation divorced from the context) would sometimes guess ⅘ or 5/4 but wouldn’t be able to explain their answer. Those who used preformal not only arrived at the correct answer, but could connect their reasoning.

The researchers observed that students didn’t seem to associate fair sharing with division when it involved a fraction. In other words, they didn’t immediately jump from, “If 5 people share 4 subs, then each must get ⅘ of a sub.” They might be able to understand equal sharing as partitive division with whole numbers, but not fractions.

Eventually with a sequence of activities and math congresses, students were able to understand fractions as fair sharing. They also developed activities that moved students from perceiving units of the numerator and denominator as totally separate ideas (people and sandwiches) to conceptualize the units of the answer (sandwiches per person). This shift is key, not just for mathematical slope, but most science applications.

These intricacies of fractions are beautiful and rich, and it’s not a topic that can simply be taught in elementary school classrooms. It needs to be layered like a sandwich as we weave in an out of algebraic concepts. It takes years of stretching our children’s minds as they play with models and build formal understandings before and during algebra. Yes, children need to understand fractions to learn algebra, but they can also use algebra as a tool to analyze fractions. That’s the beauty of generalized, formal mathematics! But which is the meat and which is the meat?

That’s something to ponder over lunch….

- Fosnot, C. T. (2007). Field trips and fundraising: Introducing fractions. Portsmouth, NH: Heinemenn.
- Peck, F., & Matassa, M. (2016). Reinventing fractions and division as they are used in algebra: the power of preformal productions.
*Educational Studies in Mathematics*,*92*(2), 245–278. https://doi.org/10.1007/s10649-016-9690-y

Every time I need to learn something (like my recent adventure changing a burnt out headlight in my car), I search for a video on YouTube. That gave me an idea! Why not create a video on that explains how use Newton’s Nemesis to teach fractions. But, after watching several long winded (but truly helpful) videos on changing a light bulb, I wanted to make sure my video covered things quickly. You can pause and rewind on things you find useful.

My headlight now works, and teachers have a new resource because of it!

The relationship between Theo and Leah echoes my relationship with my big sister: typically competitive but with moments of warmth. I remember sitting in the back seat of our car, listening to my mother quiz my sister on her times tables.

At one point I chimed in the answer faster than her, much to her annoyance. That just fueled my enthusiasm. I had no idea what multiplication was, but every time I could answer faster was a victory.

You might find Leah precocious and annoying, which I have to admit reflects my first grade personality, but she plays a vital role in the story. Since she’s learning how to divide whole numbers, it gives the reader the opportunity to recall different models of division using simple whole numbers. When children can make direct connections between whole number division and fraction division, they are much more likely to successfully divide fraction.

The interaction between the siblings is also an opportunity to see Theo explain concepts to his little sister. Children reading the story might reflect about how they would explain thing to Leah. That’s one of the reasons why I’ve included note cards in the e-Supplements that feature her. Teachers can ask their students to write a note to Leah giving her different examples to help her learn fractions.

When children create new examples for Leah, they are not only inventing story problem, they are generating a collection of concrete examples that they can use to make meaning out of fractions and division. Teachers should encourage their students to see connections between whole numbers division and fractions in their stories but to also reinforce their conceptual understanding of division. In Newton’s Nemesis No. 1, Leah is learning about partitive or equal sharing division. In Newton’s Nemesis No. 2, she learns about the measurement model of division.

Leah struggles with some of Theo’s explanations, which provides another opportunity for children and teachers to discuss mathematical terms that can sometimes be confusing. Think about the number 1/5. We sometimes refer to it as “one-fifth” but occasionally as “a fifth”. Conceptually, 1/5 can be thought of a quantity, a ratio, the fifth item counted or the fifth item in an ordered list. This is true for any unit fraction (i.e., a fraction with a 1 in the numerator).

“Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning.”

Common Core Standards for Mathematical Practice

Mathematical language can be nuanced and precision can be developed by having children write about math to explain their reasoning. Sometimes it’s challenging to explain things to others, and in Theo’s case it pays off in the end. In Newton’s Nemesis No. 3, Leah’s uses what she learned from Theo to help our heroes solve the final puzzle and unlock the mystery that might save Newton!

]]>In mid February we started to raise funds to donate the first issue of our mystery math comic book to schools that need help raising math scores. Our Kickstarter Campaign will end on March 14th (Pi Day!), but we’ve already reached 100% of our goal!

The illustrator started the work to add color to the cover, and look at the results!

In addition to adding color to the first issue of Newton’s Nemesis, we’ll be donating 500 copies to low income schools in our area. I’ll be providing free professional development workshops to the schools, just like I did with the successful testing we did this summer.

But now, the comic books will be in full color!

So to all fraction comic book fans and our Kickstarter investors, “THANK YOU!”

]]>While there are many interpretations and connections, not all have to be made immediately since fraction concepts and proportional reasoning is developed over many grades (*not just 5*^{th}* grade)*. However, all are important concepts, not just to prepare children for learning algebra but to also build their foundation in proportional reasoning.

Flores wrote, “Some of the connections needed in division of fractions are fractions and quotients, fractions and ratios, division as multiplicative comparisons, reciprocals (inverse elements), and inverse operations.”

**Fractions and quotients.** That makes sense since all fractions are little division problems. We know that 2/8 is 2 divided by 8.

**Fractions and ratios**. What if 2 out of 8 bananas are brown and too ripe to eat? Ratios can easily be modeled by arrays.

**Division as multiplicative comparisons.** Now we get a bit more complicated. Recall the ‘as much as’ or ‘as many times as’ word problem. (I can’t think of any time in my life, outside a math classroom, that I’ve used those phrases). But, they are needed because they start our children down the path of thinking proportionally. I prefer to think of this in the context of rates. For example, a gold snail moved 2 inches, while the green snail moved 3 inches. The gold snail at moved 2/3^{rd} the speed of the green snail.

**Reciprocals.** I love anything related to identities!! Who doesn’t love the multiplicative identity, one! When you multiply by a number’s reciprocal, you get 1. That’s the best part of simplifying fractions, finding what numbers might “cancel out” because they are one. This is such a powerful tool! And fun!! My second favorite identity: the additive identity, 0. So cool! Then there’s the trig identities…wait I’m going off topic.

**Inverse Operations.** This harkens back to whole number connections where division and multiplication are inverse operations. Same is true with fractions.

Each concept can be found in interpretations of fraction division and connected example problems mentioned in a variety of research articles.

They all have different conceptual bases.

*THINK: If each problem was given to children, what algorithms would they build? What pictures would they draw? *

**Measurement:** This asks how many ½’s are in ¾? A concrete problem might be: “You have ¾ cup of flour and for each batch of brownies you need ½ a cup. How many batches can you make?”^{1,2}

**Partitive (equal sharing): **The easiest examples are when a fraction is divided by a whole number, “If you have ½ a candy bar to share among 3 people, how much of a candy bar does each person get?”^{1,2}

**Finding a Whole Number:** Conceptually similar to the partitive model, examples would look like, “If ¾ of a gallon of water fills 1/2 a bucket, how much fills the whole bucket?” ^{3} Or, as a unit rate: “If a printer can print 10 pages in 1/2 a minute, how many can it print in 1 minute?” ^{2}

**Missing Factor:** Going back to the relationship between multiplication and division, “If ½ times a number is equal to ¾, what is the number?”^{1}

**Inverse of an Area Model:** A fancier way to say it is “Inverse of a Cartesian Product,” and example would be, “If the area of a rectangle is 1 ¾
feet long, and the width is ½ foot, find the height of the rectangle.”^{2}

If I had the time, I’d draw a picture for each. Or perhaps you can add them in the comments? For now, I’ll close with a quote from Flores (2002), where he aptly explains the importance of the concept of fraction division and how in elementary grades we are forming the foundation for years of mathematical learning.

“Division of fractions also provides a setting to develop for proportional thinking, which is at the core of mathematics in middle school. “ p. 238

Sources:

1. Flores, Alfinio. (2002). Profound understanding of division of fractions. In *Making Sense of Fractions, Ratios & Proportions* (Eds. Litwiller, B. & Bright, G.) p. 237-246. NCTM, Reston, VA.

2. Sinicrope, R., Mick, H., & Kolb, J. (2002). Fraction division interpretations. In Making Sense of Fractions, Ratios & Proportions (Eds. Litwiller, B. & Bright, G.) p. 153-161. NCTM, Reston, VA.

3. Van de Walle, J. & Lovin, L. (2006). *Teaching Student-Centered Mathematics: Grade 5-8.* Pearson, Boston.

The beta test results for Newton's Nemesis No. 1 are in, and I am thrilled! The teachers and children loved the book.

I gave class sets of the comic books to teachers in four schools, three of which are low income and qualify for Title 1 funding. That means over 40% of the children receive free or reduced price lunches. Over 150 children read the book, and the teachers gave it 5 out of 5 on almost every metric.

These are direct quotes from the teachers (the all caps are theirs):

The students LOVED it.

My students loved the story and illustrations. They can't wait to see what happens next.

My students were HIGHLY engaged.

“When are we going to read this book again?" - I got this often, especially from students that are not usually engaged in math!

For the most part, they loved reading 'comics' during math and didn’t realize that they were actually learning math. SUPER COOL!!:O

Some of the children wrote charming notes to me:

I love your book. I can' wait to see part two of the book.

Can you come to our school to meet us and talk about the book?

I loved the book. I loved the part when you wrote the part where Leah didn't know how to say third. Instead she said turd.

I love this comment book but can you visit miss i divide's house

Are you the witch in real life?

Their comments are invaluable to me! Children, even those who don't like math, are getting engaged and enjoying the story.

Now for the next steps. It's time to get the comic book printed in color and distributed to more low income schools!

Dr. Boaler devoted an entire chapter on the power of mistakes and struggle, which presents compelling research that children are learning very little when they get every answer correct. However, when children aren’t hampered by the fear of making mistakes, they are more likely to develop persistence when working on more complex problems. One of my favorite quotes from that chapter provides a beautiful image, “The brain sparks and grows when we make a mistake.”

I needed a character that could explain this science and reinforce growth mindset as Theo struggled, so I created Theo’s scientific mother.

Whether children see her as a doctor, scientist, dentist or nurse, she’s a strong background character that introduces the idea that struggling makes your brain grow. If young readers pass over her comment, they’ll certain see it again in the next few pages. His mother’s statement, “Scientists have shown that your brain actually grows when you struggle to figure something out,” sets the stage for Theo’s nightmare. Fractions chase him down a dark hallway and cause his head to swell as they launch their attack.

In the teacher’s guide, I’ve added math journaling prompts to provide opportunities for children to engage in metacognition, or reflecting on how they learn. Below are two examples of math journal prompts that teachers may incorporate into their lessons.

Theo spent a lot of time feeling bad about making mistakes because thinks he’s not learning. Do you make mistakes when you’re learning? Can you describe a mistake you’ve made and what you learned from it? Write a letter to Theo that might encourage him.

Can you think of a time you struggled to learn something, but eventually did? How do you feel when you succeed at solving a hard problem?

But of course I couldn’t resist tying together the underlying narrative of productive struggle with an important fraction concept with this math journal prompt:

As Theo’s brain swells, the fractions ½ and ¼ seem to attack his head. However, if you take ½ of something or ¼ of something, that just a portion of something. His head would get smaller if you multiply it by ½ or ¼. What operations might cause his head to grow?

Children might correctly answer that if you add fractions to any number, that would increase the numbers size, and some might make the more subtle connection that dividing by ½ or ¼ would increase a number. Teachers can leverage this scene to begin a deeper discussion of the effect of multiplying whole numbers by fraction less than or greater than 1, which are concepts often misunderstood by children. At some point children may also make the connection between the inverse operations by posing the following.

If taking ½ of a number is multiplying by ½, that makes the number smaller. Since dividing is the inverse operation of multiplying, will dividing by ½ make the number larger?

Of course, if your students complain that all this thinking is giving them headaches, that’s okay! It’s just their brains growing!

]]>In 2016, Katherine Lewis published some interesting findings from her dissertation on students with Math Learning Disabilities (MLD) and their knowledge of fractions. She completed the work at Berkeley under the direction of Alan Schoenfeld, a leader in mathematics education reform that I’ve admired since graduate school. Dr. Lewis conducted interviews of two young women with MLD as they compared fractions.

Children typically have fairly good number sense about 1/2, even in early grades, so they usually can determine if a fraction is more or less than ½. Children are also fairly successful in comparing fractions if the two fractions have the same denominator, such as 2/9 and 8/9. Since they know that 8 is more than 2, they can easily identify 8/9 as the larger number. But MLD students often find these comparisons difficult, and Dr. Lewis wanted to know why.

She interviewed two women with MLD, aged 18 and 19, who had persistent errors comparing the fractions such as 2/5 & 3/5 and 2/4 & 2/3.

When they compared fractions such as 2/5 and 3/5, they could create area or array models for their explanation. However, they would interpret the value of each fraction as its *fractional complement.* In other words, they would interpret a model of ¾ as the number ¼.

When comparing fractions such as 2/4 & 2/3, both women offered that 2/4 was equal to half, even though the interviewer didn’t ask them to make that equivalence. However, they both incorrectly thought 2/3 was equal to or less than ½. This error came about because the MLD students compared only the numerators (both are numerators are 2 so they must be the same) or only the denominators (there are more pieces if divided by 4, so that must be larger).

During the interviews, Dr. Lewis also noticed that the students perceived ½ as an action, such as halving something. They didn’t necessarily perceive ½ as 1 part out of 2 parts. She also noticed that one of the subjects sometimes wrote 5/10 as 5 1/10.

While these are only two students and we can’t generalize these findings, their perceptions of fractions are fascinating. How can we start where those students are and build a stronger foundation?

Interestingly in another study, Bugden & Anasari (2016) found perceptions of area can impact the intuitive recognition of numbers for students with dyscalculia. They tested 15 children with dyscalculia, aged 9-14, with a comparison group, and asked them to quickly compare two sets of dots that flashed on a computer screen. Which set was larger? The children with dyscalculia could recognize the larger set of dots with the same accuracy as children without dyscalculia, unless the area (size) of the dots was manipulated. If the larger number of dots had a smaller area, children with dyscalculia struggled to identify the larger number as larger!

*So what if area models or arrays are problematic if they aren’t drawn precisely?*

At the end of Dr. Lewis’ paper, she suggests that we need additional research to determine whether using length or weight models might be a better for MLD students than using area or array models. Her subjects struggled to compare and see the magnitude of the fractions with those models. When you combine this with the Bugden & Anasari 2016 study, she may have a really good suggestion for more research!

If we could leverage measurement models of division using length models that are halved or quartered then added together, perhaps we could make a better connection for our MLD students.

References

Lewis, K. E. (2016). Beyond Error Patterns. *Learning Disability Quarterly*, *39*(4), 199–212. DOI: 10.1177/0731948716658063

Bugden, S., & Ansari, D. (2016). Probing the nature of deficits in the ‘approximate number system’ in children with persistent developmental dyscalculia. *Developmental Science*, *19*(5), 817-833. DOI: 10.1111/desc.12324

When children connect the structural elements of whole number division and fraction division, they are reasoning by creating analogies. Analogical reasoning is the ability to perceive structural similarities among math problems that might not appear to be similar on the surface. Lindsey Richland and her colleagues have done interesting research on how teachers use analogies to teach math.

Her team did a brilliant job analyzing video-taped classrooms of 8^{th} grade teachers. The classrooms were in America, Hong Kong and Japan and were all part of the Third International Math & Science Study (TIMSS) in the year 1999. They chose to look at Hong Kong and Japan because their test scores are typically higher than ours and they wanted to know why. Interestingly, they found that all teachers used analogies at the same frequency in their instruction, * but *American teachers provided less support for analogical reasoning (Richland, et al., 2007).

**What was the di****fference?**

Here’s my imagined example from reading the paper. Suppose all the classrooms video-taped were teaching how to solve an equation such as “x + 3 = 5.” In algebra, you would subtract 3 from both sides to solve this, and a common analogy we might use is balancing a scale. Apparently both Japanese and Hong Kong teachers were more likely to use visual images or actual items (such as a picture of a scale or an actual scale), to ask students to mentally picture something (e.g. imagine a scale), or to gesture between the source and the target of the analogy (e.g. motion from the scale to the equation). The teachers tended to make the connections vivid, reducing the cognitive processing demands of the student, and to focus more on the comparison or relationships that existed.

In an earlier analysis of 25 U.S. classrooms, Richland, et al. (2004) observed that teachers were more likely to use analogies to teach procedures rather than concepts. Interestingly, if teachers saw that their students needed help, they tended to present very transparent analogies. Of course, that can have a positive short term effect. The students would be able to solve the problem quickly by repeating steps. But, they might be able to bypass understanding why a procedure worked.

This type of research inspired the idea of teaching through a comic book. I wanted great visual and mental images that helped students, but that students would look at over and over again. The more vivid the images, the better! Therefore, I’m currently working on more supplemental tools that ask students to come up with analogies. One of my tasks ask students to come up with more story problems for Leah to teach her why 1/3 is just like 1 divided by 3.

References

Richland, L., Holyoak, K., & Stigler, J. (2004). Analogy use in eighth-grade mathematics classrooms. *Cognition and Instruction,* *22*(1), 37-60. __doi.org/10.1207/s1532690Xci2201_2__

Richland, L., Zur, O. & Holyoak, K. (2007). Cognitive supports for analogies in the mathematics classroom. *Science*, (5828), 1128. https://doi.org/10.1126/science.1142103

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I was telling a math teacher friend about the comic book series I was developing about teaching fraction division, which really was based on how division is understood. She excitedly asked, "Which model of division are you using?" I said, "Both!" If you search for the two models of division on the Internet you'll quickly find the partitive model and the quotitive model. But what's this? A quotative model?

I'm sure you're wondering if I spelled quotitive incorrectly. Certainly, my spell checker is drawing a dramatic red line beneath the word, and many websites spell it as quotative. If you look up quotitive in most online dictionaries, it doesn't exist. According to Merriam-Websters, quotative means, "a function word used in informal contexts to introduce a quotation." The site also provides the example, ["like" is a quotative in "He was like, 'Oh, no! not again!'"]. So, like really, what does that have to do with math?

Why would quotitive be the correct spelling? As far as I have read, one of the first times the term quotitive was used to explain division was in a 1985 article by Fischbein and colleagues in the *Journal for Research in Mathematics Education.* The term is related to finding a quotient or the result of the division problem. Since the quotitive model (a.ka. measurement model or repeated subtraction) repeatedly takes an amount from the quotient. Therefore, I contend that quotitive is the correct spelling, and will add it to my personal spell checking dictionary. I'd like to know if anyone else has a good argument for quotative as the correct spelling? I could be wrong.

I provided a better illustrated explanation of both the equal sharing (or partitive) model and the measurement (or quotitive) model in my teacher's guide, and I've copied it below.