In the second issue, we join Theo’s nightmare as the evil witch weighs him. She checks her spell book and scowls as she complains, “30 kg! Too scrawny! My recipe calls for a 90 kg man,” setting the stage for taking a third of all the ingredients.

As the evil witch recounts her spell, she carefully takes a third of each item: 6 beets to root out his evil, 3/4 sage to steel his wisdom, and 1/3 yard of cotton because boys are rotten.

The first question for young readers is why does the witch need to take 1/3 of everything? Teachers and parents will quickly answer this question since 3 x 30 is 90, but children are often still developing proportional reasoning skills. In America, they are also just learning the metric system, so it’s a great time to explore how much they weigh in kilograms. It was pretty handy that a typical 10 year old weighs 32kg, which was perfect for my story!

For the recipe, I used specific problems to challenge children to think about what it means to take 1/3 of something:

- 1/3 of 6 – It’s dividing 6 by 3, but also multiplying 1/3 x 6. However, children might not recognize it as multiplication right away . Some might even think of taking away 1/3 from 6, which is subtraction!
- 1/3 of 3/4 – Theo solved this quickly. If children notice that 1/4 + 1/4 + 1/4 = 3/4, they will be just as speedy and be able to demonstrate great number sense.
- 1/3 of 1/3 – What a perfect time to explore models to allow children to conceptualize why that phrase is multiplication.

In the story, Theo goads the witch as she works over a boiling cauldron, bragging that a math genius never makes a mistake. We all know that’s not true, but many children feel that if they make mistakes they aren’t good at math. So of course, it was time for the old math-loving witch to make a big mistake!

*The old witch gasped as 1/3 of a yard of cotton fell into the boiling water. *

This scene is probably my favorite one in the series, which will be released in a new Kindle version this week on Amazon.

All this writing about recipes makes me want to bake something! Which reminds me, if you haven’t heard of the using brownies to teach fraction multiplication, here’s a nice research article by Debra Johanning at the University of Toledo that talks about brownies and “sticking points” that children have when learning to multiply fractions.

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“By allowing students to use and create comic books in math class, the students can bring their own language and cultural experiences into the mathematics lesson, thereby using their personal backgrounds and experiences to aid in the development of mathematics understanding.”

Reilly, 2014, p. 65

Inspired by the diversity of X-men, she tasked eighth grade students in an advanced math class to create comic books that explored a mathematical topic but also celebrated the differences in the characters. Her wonderful article that I encourage you to read features excerpts of a few of the collaborative teams’ creations including *The Subtractors*, *The Math Men*, and *The Adventures of Supercow and Frost*.

“While the students were brainstorming their characters and stories,

Reilly, 2014, p. 65, 73

the student groups had lively discussions about different abilities and

how those differences can be useful when working to solve a problem,

be it mathematical or societal.”

Comic books, which are successfully used in ESL classrooms (Gomez, 2014), are helping teachers reach diverse learners by using visually rich imagery that is accessible to all readers. As Mary Widdicks recently reported, psychological researchers are developing a body of evidence that demonstrate how comic books with their complex panels and visual language can be powerful learning aids in *all *classrooms.

The visual images that Reilly’s eighth graders created illustrates the true power of diversity.

Diversity brings together creative minds that think differently, and I’m thankful for Dr. King’s dreams that have inspired us all to celebrate diversity today and every day.

- Reilly, E. M. (2014). Superheroes in Math Class: Using Comics to Teach Diversity Awareness.
*International Journal Work and Days*,*32*(1&2), 61-74. - Gómez, M. V. G. (2014). Reading, speaking and writing through creative resources: Comics in second language teaching.
*Arab**World English Journal, 5*(4), 443-453. - Widdicks, M. (2020). The visual language of comic books can improve brain function. Quartz (January 2, 2020). Retrieved from: https://qz.com/1777533/reading-comic-books-can-improve-brain-health/

You might want to shop around, since I haven’t shopped for the best prices, but you can use some of the search terms or just use the links.

**Fraction Circles & Magnets. **Most teachers use fraction rods but don’t necessarily invest more money for fraction circles. One set is about $6, but I’d recommend at least 4-5 sets for a classroom teacher. A quick search on Amazon, and you can find these sets as Fraction Magnets, which are great for the home or for displaying problems on white boards.

**Fraction Model Mulitpliers (Grades 5 +)**. I love this tool! Only $15, you can overlay these translucent graphs to visually represent multiplication with fractions. Look at the bottom middle image that takes 1/2 of 1/3.

**Books on Mathematical Mindset**. I love Jo Boaler’s work! My favorite is Mathematical Mindset, which does a great job teaching parents and teachers how to help children build confidence in solving math problems. She also has some great investigations in her series Mindset Mathematics, which has fantastic ideas for the classroom. Every teacher should have these.

**Clever Catch – Fraction, Decimal and Percent Ball. ** As a teacher, I was too cheap to buy one of these for myself ($12 for a beach ball!) so I’d buy a plain beach ball and put the questions on it myself. It’s a fun classroom activity that involves tossing the ball to students and having them answer questions that fall under their left thumb. Spoil a teacher and get the fancy ball! They’ll appreciate it!

**Fraction Comic Books**. A perfect winter gift for 5th grade teachers since fraction division is introduced in late December or early January, and 6th grade teachers love them, too. If you’re looking for a unique gift to help your children learn fraction multiplication and division, these are now available as a set for $8.00. It’s in a comic book format so stuff a stocking for any child from 4-8th grade.

Of course since I wrote the comic books, I think they’re the best gift idea! Just remember the comic books come with plenty of free resources, including a new lesson plan and activity set .

**Fraction Formula Game. **Honestly, I have no idea if this works, but it looks pretty cool and got good reviews on Amazon. I’d love to try it so I’m putting it on my wish list (to my family: hint, wink, hint!)

These are just a few things I found today, but I’ll add more when I’m done with my holiday shopping! Happy Holidays!

]]>Who would have guessed that creating this task would send me on a hunt through time and continents for the first definitions of *multiplication*, *multiplier*, and *multiplicand*?

If you think about what children need to know before they can connect the ideas that 12 divided by 3 is the same as 12 multiplied by ⅓, they need to have flexible models of multiplication in their mental tool box. So, before introducing why 12 ÷ 3 is the same as 12 x ⅓, I needed a task to check for their understanding of the meaning of a fraction and remind children of whole number multiplication models (or introduce them to children who didn’t have the opportunity to explore the ideas).

The underlying assumption is that one of the expressions ⅓ x 12 or 12 x ⅓ means 12 groups of ⅓ and the other means ⅓ groups of 12.

Of course, I’ve been known to roll my eyes and say, “Does it really matter? The commutative property of multiplication tells us A x B = B x A. Who cares!” But look closely at the models again. Which one would be the best model to use to explain why 12 ÷ 3 = 12 x ⅓?

**How do we define 12 x ⅓ ? **

If you look at the US Common Core Standards, they clearly expect us to define 5 x 7 as “5 groups of 7 objects” or 7+7+7+7+7. Therefore,

**12 x ⅓ is 12 groups of ⅓. **

But if you check the definition on Wolfram Math World it gives us this definition:

“multiplication is the process of calculating the result when A number B is taken B times…Multiplication is denoted A x B”

BUT, that means 5 x 7 would be 5 taken 7 times or 5 + 5 + 5 + 5 + 5 + 5 + 5. That’s 7 groups of 5. The mathematicians at Wolfram equivocate ⅓ x 12 as ⅓ taken 12 times, or 12 groups of ⅓. In other words:

**⅓ x 12 is 12 groups of ⅓**

Can it be true that the Common Core and mathematicians use different definitions!?! They both seem to agree that A and B are factors, but they disagree on which variable is the *multiplier *and which is the *multiplicand*. Steven Schwartzman (1994) wrote a book “The Words of Mathematics: An Etymological Dictionary of Mathematics Terms Used in English” published by the Mathematics Association of America, that examined Latin bases of terms. He defined:

Multiplicand– the number to be multiplied; In A x B, A is the multiplicand

Multiplier– the one that does; In A x B, B is the multiplier because B is doing the multiplying.

I decided to explore the collection of books in my office, which is a mix of advanced mathematics, history of mathematics texts, and textbook designed to teach future teachers mathematics. The advanced mathematics didn’t provide any insights, but the future teachers texts consistently defined A X B as A groups of B. Then, I stumbled upon the text by Reconceptualizing Mathematics by Judith Sowder, et al. (an excellent book by the way – Mine is the 2010 edition), and she noted that US and British (and countries formerly colonized by the UK) define multiplication, multiplicand and multipliers differently:

A X B means A groups of B

multiplier X multiplicand

**United Kingdom **

A X B means B groups of A

multiplicand X multiplier

To me, the first way makes sense, since we want to define A X B as A groups of B as we try to translate the multiplication sign directly into words that empower our children to create models of multiplication. And certainly when we get to algebra, doesn’t it make sense that “3a” means a + a + a?

But why should be use the other definition ? Interestingly, I found some clues in my history of math texts. [A couple of fun facts: In 1631, an English algebraist, William Oughtred, was the first to use “X” as a multiplication sign, and one of the co-inventors of calculus, Gottfried Leibniz, was the first to use a dot or “a∙b” to represent multiplication.] It finally occurred to me that I should consult one of the first mathematicians, Euclid of Alexandria.

**Multiplication from Ancient Greece – Lost in Translations**

Just like in my comic book character Ms. I. D. Vide, I have a copy of Euclid’s Elements in my book collection. Math students usually associate Euclid only with geometry since planar geometry is also known as Euclidean Geometry. However, Euclid also demonstrated many results about number theory in his famous collection of books, *Euclid’s Elements*. His work on number theory begins in book number 7, which starts with a list of definitions. “Definition Number 15: Multiplication” was just what I was looking for! The perfect definition from the ancient Greeks!

Unfortunately, my English translation had the definition of “A added to itself B times.” That seemed odd. If you have 3 added to itself 1 time, wouldn’t 3 x 1 be 3 + 3 or six. Something was wrong with that translation.

Fortunately after a little more digging on the issue I found the work of Jonathan Crabtree a mathematician out of Melbourne Australia. He has analyzed many translations of Euclid’s work (e.g English, German, Italian, French), and noted that older English translations are incorrect. Dr. Crabtree argues that Euclid’s definition of multiplication is correctly translated:

**ab = a placed together b times**

So based on the ancient Greeks, ⅓ x 12 is ⅓ placed together 12 times or 12 groups of ⅓.

Bottom line is that both definitions are valid and thanks to the commutative property of multiplication, we can easily translate between the expressions ⅓ x 12 and 12 x ⅓. So what’s the correct answer to my task? It doesn’t matter! What is important is developing flexible understandings that allow us to move merrily and fluidly between expressions and models. Now let’s play with the task!

I see 12 groups with ⅓ shaded in each. 12 groups of ⅓.

I see ⅓ pictured 12 times, but each group three ⅓’s forms a whole. So, 12 groups of 1/3 form 4 wholes.

Now I see 12 dots divided into 3 groups, but I also see ⅓ of the 12 dots circled. Isn’t that ⅓ group of 12? Or ⅓ x 12? Or, depending on your translation 12 x ⅓ ?

This last model allows us to directly connect the concept of division and multiplication because of the great new way we can represent quantities: as fractions! It seems to me that unit fractions are a wonderful way to see the connections between multiplication and division. The models can be translated into English and mathematical expressions in a variety of ways.

Sharlene Kiuhara and her team wanted to know if the writing-to-learn hypothesis worked when teaching fractions to 4^{th}-6^{th} grade students who either had learning disabilities in mathematics or scored at or below the 35^{th} percentile on standardized tests for mathematics (Kiuhara, Rouse, Dai, Witzel, Morphy, & Unker, 2019). Using a randomized, pre- and post-test design, which is the strongest evidence I’ve seen to date, they found that students who learned fractions while learning how to write arguments out-performed their peers in the control group. They observed the largest improvement in the performance of students who were enrolled in special education.

The results show argumentative writing can be used as a learning activity for improving students’ fraction learning, as well as the quality of their mathematical reasoning.

Kiuhara, et al. 2019, p. 1

Their paper, published in the Journal of Educational Psychology, was flush with details not only about how they measured the students’ improvements, but how the teachers changed the way they taught. But before going into the details, let’s take a step back to consider different ways that writing can be incorporated in the math classroom. Here are some I’ve used:

**Math journals and journal prompts**– A daily activity that allows students to write (and draw figures) about what they are learning, what questions they still have, and how they feel about the learning process. I treat these as a scientific journal as students make discoveries.**3-minute papers**– I’ve used these as an “exit ticket” for my classes to quickly assess if my lessons are effective or if there are questions or concerns that my quiet students may have.**Math reports**– When working with lengthy, real-world problems or problems that have many solution paths, I’ve asked students to prepare a full report explaining their solution and solutions they’ve seen other students or groups propose.

The purposes of writing in mathematics can vary from expository writing (explaining what is observed), to reflective writing (allowing for self-regulation or opportunities to think about the learning process), to argumentative writing (justifying the correctness of a solution and the solution process). Kosovo has published many papers on *Mathematical Argumentative Writing*, which he distinguishes from persuasive writing. In the latter, you’re trying to persuade someone to agree with your idea (Kosko, 2016; Kosko, 2014; Kosko & Zimmerman, 2019). However, argumentative writing is more akin to mathematical proof. You’re making the case that something is true (some might argue that “truth” is something that is socially accepted as true – but let’s not get too distracted with theoretical stuff here!). In any case, Kosko has completed some fascinating work in K-3 classrooms, demonstrating the strength of teaching writing with children.

Kiuhara and her team worked with 10 teachers, randomly assigning five of them to complete a 2-day professional development program on teaching argumentative writing. The other five reviewed the mathematics core content based on the district’s pacing guide. The teachers who learned about argumentative writing, taught their students to use R^{2}C^{2} model (I hope you just chuckled like I did – why couldn’t it be R2-D2?!?). Anyway, here’s how you use it for writing a mathematical argument:

**RESTATE:**What do I need to explain, describe, justify or compare? In my writing, did I use precise math words and transition words?**REASONS:**What is my reasoning or evidence for my answer? In my writing, did I use precise math words and transition words?**COUNTER CLAIM:**Was there another possible answer? Why was my reasoning better? In my writing, did I use precise math words and transition words?**CONCLUSION:**How can I wrap up my ideas? In my writing, did I use precise math words and transition words?

Before asking students to write argumentatively, teachers used a strategy filled with reflective questioning to help with metacognition and employed a graphic organizer called FACT to teach the children problem solving. Here’s how Kiuhara and her team explained FACT on page 9 of their paper:

**F – Figure out my plan:**What is my task? Do I understand the problem? What do I need to know? What tools do I need?-
**A – Act on it:**What are the math procedures? What reasons, evidence, and support will I use? What words will I choose? How will I interpret my results? **C – Compare my reasoning with a peer:**What is similar or different? What are my reasons? Does it make sense? Can I make improvements?**T- Tie it up in an argumentative paragraph:**Did I provide reasons, evidence, and support? Did I present the counterclaim? Did I choose good tools and words for solving the problem?

The teachers used these tools with their students to teach them to compare the magnitude of two fractions. When the researchers compared the pre-tests to the post-tests, they found that students learning from teachers who used argumentative writing in their lessons had greater gains on the fractions tests, the quality of mathematical reasoning, the number of rhetorical elements (e.g. statement and claim, counterclaim, reasons, etc.), and total word count. These were statistically significant differences, showing that argumentative writing is a tool that is more efficient than traditional lessons about fractions.

Argumentative writing improves performance on tests while improving mathematical reasoning, especially with struggling students.

So, I think I’ll spend tomorrow making sure the teacher’s guide to Newton’s Nemesis has plenty of writing prompts that not only reinforce mathematical mindset but can also be used to initiate argumentative writing. After all, shouldn’t all comic books have superpowers in them??

- Kiuhara, S. A., Gillespie Rouse, A., Dai, T., Witzel, B. S., Morphy, P., & Unker, B. (2019, July 18). Constructing Written Arguments to Develop Fraction Knowledge. Journal of Educational Psychology. http://dx.doi.org/10.1037/edu0000391
- Kosko, K. W., & Zimmerman, B. S. (2019). Emergence of argument in children’s mathematical writing.
*Journal of Early Childhood Literacy*,*19*(1), 82-106. - Kosko, K. W. (2016). Making use of what’s given: Children’s detailing in mathematical argumentative writing.
*Journal of Mathematical Behavior*,*41*, 68–86. https://doi.org/10.1016/j.jmathb.2015.11.002 - Kosko, K. W. . (2014). What Students Say About Their Mathematical Thinking When They Listen.
*School Science & Mathematics*,*114*(5), 214–223. https://doi.org/10.1111/ssm.12070

An algorithm is a fixed set of step-by-step procedures for solving a (mathematics) problem

Fan & Bokhove, 2014, p. 483

I just finished reading an article by Lianghuo Fan and Christian Bokhove (2014) that provided an excellent literature review and argument that there is a role for algorithms in the mathematics classroom. But, as they noted, there is a negative perception of algorithms. Fan and Bokhove cited several studies that promoted this perception, including the TIMSS 1999 Video Study that described poor math lessons as “very algorithmic” and “rule-oriented”. Even Canada’s curriculum framework excludes the standard algorithms for the four basic arithmetic operations.

Personally, I think they’re fighting an uphill battle when it comes to fraction division. Any parent trying to help their children with how to divide by fractions will certainly go to the internet and find the following:

Even our public broadcasting system has jumped on board the Keep-Change-Flip algorithm . Of course the video reminds me of the old School House Rocks videos (that’s how I learned about conjunctions, memorized the preamble to the constitution, and more).

Algorithms are useful and efficient tools. They aren’t inherently bad. But if we’re not careful in how and when we introduce them in the classroom, we risk having our children miss out on developing rich understandings of mathematics. According to Fan and Bokhove, we need to consider the teaching of algorithms in the context of Bloom’s taxonomy (including updated versions of Bloom’s work).

If you were learning at the lowest cognitive level, you would just remember an algorithm because the teacher showed the steps or you read it in a book. At cognitive levels that are just a little higher, you could understand why an algorithm works and how it could be used. Teaching algorithms at such a level requires a teacher to explain why it works, provide a logical derivation or proof, and make connections with students’ prior knowledge. But at the highest levels, you can construct your own algorithm and evaluate alternative algorithms. Clearly, teaching at the highest cognitive level requires allowing students to explore and create their own algorithms.

Any topic can be taught at low levels (memorizing the constitutional amendments) or higher levels (applying the first amendment to case law). In the same way, mathematical algorithms can be taught at many different cognitive levels. We don’t have to begin at the basic level (e.g., drill and kill, imitate what I do, etc). We can let children explore and create their own algorithms, evaluate whether or not their strategy will work every time, and also, allow them to discover the beauty and efficiency of the standard algorithms (if they didn’t event those algorithms themselves)!

The key is to get children to understand why you “invert and multiply” and discover the reason “Keep, Change, Flip” works. ** If they can explain it, then they won’t have to memorize it! ** So maybe algorithms aren’t the arch enemy of mathematics.

Cindy Ticknor, 2019

Algorithms are Our Frenemies – While they might be our friends, they are often our rivals when we’re trying to teach mathematics conceptually.

Fan, L., & Bokhove, C. (2014). Rethinking the role of algorithms in school mathematics: a conceptual model with focus on cognitive development. *ZDM*, *46*(3), 481-492.

The first study I came across was Robert Siegler, et al. (2012), where the research team looked a data collected about large, but somewhat dated cohorts of children in the United Kingdom and the US. The same group of children completed math assessments in elementary and high school. In the UK the 3,677 children were 10-year-olds in 1980; in the US the 599 children were tested in 1997 when they were 10-12 years old. Their analysis revealed that the knowledge of fractions assessed during approximately the 5^{th} grade was the largest predictor of success in 10^{th} grade mathematics. Here’s what’s fascinating: The results held true even when the researchers statistically controlled for prior knowledge of whole number arithmetic, family income and education, race, ethnicity and verbal and non-verbal IQ. So, no matter what background the child was from or the predicted intelligence (if you believe in IQ scores), fraction knowledge had a significant impact on success in math.

The next year, Siegler & Pyke (2013) published the result of tests given to 120 US children in 6^{th} and 8^{th} grade on fraction arithmetic. While 8^{th} graders performed better than the younger participants, they answered only 57% of the questions correctly overall, with the greatest success in adding and subtracting fractions. But, they only correctly answered fraction multiplication problems 48% of the time, and fraction division problems only 20% of the time. The researchers found that 55% of the errors on fraction division problems stemmed from co-mingling procedures between the different arithmetic operations. For example, when trying to solve a problem, if a child came upon a step with multiplication that had a common denominator, such as 3/5 x 4/5, he or she would errantly produce 12/5 as the solution. Braithwaite & Siegler (2018) later termed this as one of those “spurious associations” that our smart little students make. If they saw common denominators, they predicted the problem must be addition or subtraction. The smart little buggers were trying to predict the operator because, get this, that’s the pattern you see in their text book!

Siegler and Lortie-Forgues (2017) reported on several suspected causes of these difficulties which included, among other issues, the lack of attention that fraction division receives in text books. They used a compilation of studies to craft this argument by comparing some of our US textbooks, in particular *Everyday Mathematics *and *Saxon Math*, to a Korean textbook (Son & Senk, 2010). Overall, they found that US texts have much fewer fraction division problems when compared to Korean textbooks, which provide many more opportunities for students to work on division than multiplication. Take a look:

Siegler and Lortie-Forgues (2017) also noted several other possible issues including teacher preparation and reliance on memorization of procedures rather than developing conceptual understanding, which we’ve been working to improve for years! They also made a strong argument that understanding fraction magnitude (e.g. knowing where they fall on a number line) is strongly associated with success in fraction arithmetic.

To test the theory that understanding fraction magnitude is key to understanding concepts in fraction arithmetic, Siegler and Lortie-Forgues (2015) gave uniquely designed multiplication and division problems to preservice teachers, middle school students, and math and science majors at a university to learn more about how understanding fraction magnitude effected conceptual understandings. Their findings had implications for teachers:

Students could be encouraged to first judge whether the problem involves dividing a larger by a smaller number or dividing a smaller by a larger number. If the problem involves division of a larger by a smaller number, for example, 5/8 divided by 1/8, students could be encouraged to think of the problem as “How many times can N go into M” (e.g., “How many times can 1/8 go into 5/8”). If the problem involves division of a smaller by a larger number, such as 1/8 divided by 5/8, students could be encouraged to think of the problem as “How much of N can go into M” (e.g., “How much of 5/8 can go into 1/8”).

(Siegler & Lortie-Forgues, 2015, p. 916)

I can’t say it much better than the Washington Post, so here’s a link to his article critique Monopoly’s decision to move from counting money to using bank cards. Seriously?!?!

**Professor Robert Siegler weighs in on Monopoly bank cards**

- Braithwaite, D. W., & Siegler, R. S. (2018). Children learn spurious associations in their math textbooks: Examples from fraction arithmetic. Journal of Experimental Psychology: Learning, Memory, and Cognition, 44(11), 1765–1777. http://dx.doi.org/10.1037/xlm0000546
- Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., . . . Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23, 691-697. doi: 10.1177/0956797612440101
- Siegler, R. S., & Lortie-Forgues, H. (2015). Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology, 107(3), 909–918. https://doi.org/10.1037/edu0000025
- Siegler, R. S., & Lortie-Forgues, H. (2017). Hard Lessons: Why Rational Number Arithmetic Is so Difficult for so Many People. Grantee Submission.
- Siegler, R. S., & Pyke, A. A. (2013). Developmental and Individual Differences in Understanding of Fractions. Developmental Psychology, 49, 1994-2004. doi: 10.1037/a0031200
- Son, J.-W., & Senk, S. (2010). How reform curricula in the USA and Korea present multiplication and division of fractions. Educational Studies in Mathematics, 74, 117-142. doi: 10.1007/s10649-010-9229-6

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 of decimals places. 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!

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