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

]]>

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.

That’s why I was interested in the recent work of Pooja Sidney and Martha Alibali, which provided more evidence that teachers may want to activate their students’ knowledge of whole number division before introducing fraction division. Interestingly, our curriculum in the United States often moves directly from fraction multiplication to fraction division. But is that the best way to introduce fraction division?

Sidney and Alibali designed a simple but clever experiment to test this. They recruited 62 students who just completed their 5^{th} and 6^{th} grade year and asked them to use fraction towers to model and verbally explain how to add, subtract, multiply and divide.

*What are fraction towers? Think of colorful blocks that snap together. If four yellow blocks are snapped together to make a whole, then the yellow blocks are fourths.*

The scientists chose two types of problems: half were only whole numbers (e.g. 15 ÷ 5) and half used a whole number and a fraction (6 ÷ ½). Here’s the clever part. They randomly put the children in two groups and intentionally changed the order of the tasks.

Half of the children were given all the whole number problems first, then the problems with fractions. The other half received the problems that were ordered by operation. In other words, they first modeled how to add two whole numbers, then modeled problems with fractions. Second, they subtracted with whole numbers, then were given subtraction problems with fractions, and so on.

Their study revealed some interesting differences.

We found that children who demonstrated whole number division immediately before modeling fraction division were significantly more successful at modeling fraction division than those who had demonstrated fraction multiplication and other operations on fractions immediately before modeling fraction division. (Sidney & Alibali, 2017, p. 46)

They also noted that children who didn’t use the quotitive (measurement) model of division to explain whole number division were not as successful in explaining fraction division. Part of this may be because curriculum in the United States does not emphasize the partitive (equal sharing model) for teaching fraction division. I’ll leave the international curricular differences for another posting, but the bottom line is, if equal sharing is the only way a child conceptualizes division, then that may pose challenges when they learn fractions.

Of course, the study has many other interesting insights and I’d encourage you to read it in your spare time (what’s that?). Some may also critique the study since it only had 62 participants and is therefore not generalizable. But I’m giving in a “A” for effort just on tenacity. Each interview took 45 minutes! That’s 2,790 minutes (over a week!) of video tape to view and analyze. That’s a lot of work! Kudos to the researchers.

For teachers, these results suggest we should use warm up or launch activities that help students recall division with whole numbers before introducing fraction division. As for me, I’m hoping that Leah’s sassy explanations of whole number division will help Theo understand fraction division.

Reference

Sidney, P. & Alibali, M. (2017). Creating a context for fraction division. *Journal of Numerical Cognition* *3*(1), 31–57 doi:10.5964/jnc.v3i1.71

]]>

When I first envisioned using a comic book to teach fractions, I naturally went online to see if anyone had thought of the idea yet. There were plenty of resources on teaching the concept of a fraction, and even adding and subtracting fractions. But when I searched (and continue to search) for teaching ideas that truly get at why the algorithms for dividing fractions work, I was surprised. There are plenty of cute ideas of “Keep-Change-Flip” but not many ideas that tackled the “why”. That’s when I started to write and read more about the current research on fraction division.

There’s a lot of great research on division with fractions and how children learn math that has inspired how I approached the math in *Newton’s Nemesis*. But it takes a lot of time to digest. So, I thought I’d share the work that I’ve done by creating a blog about what researchers in the field are doing. I hope it not only can help others who are trying to find a better way to teach why we invert and multiply, but also give readers of *Newton’s Nemesis* a little background on why I put certain elements in the stories of Ms. I. D. Vide.