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! *

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. 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