Number Bias - Yup, It’s a Thing!

Watching children learn fractions at our summer camp was fascinating, especially when they struggled and overcame number bias. Don’t worry, there is nothing nefarious going on - but it is a real thing that can hamper a student’s understanding of fractions. 

Number Bias2

What is it?

Number bias, or more precisely natural number bias, is when children hold onto assumptions about natural numbers that aren’t necessarily true for rational numbers. Researchers have identified three different types of bias: size, operational, and density.

 

During our summer camp, I was following four students very closely as part of a research study, and two of the boys exhibited a classic case of number bias as they tried to determine the size of some unit fractions.  In the fresh morning air, we drew a large chalk number line outside and created index cards with 0, 1, and a collection of fractions on them. When I worked with two rising 5th grade boys, they quickly placed 0 and 1 on either end of the number line and ½ in the middle.  But when it came to ¼ and ⅛, they weren’t sure.  They decided to order the  numbers as 0, ¼, ⅛ , and finally, ½. 

 

~ I asked, “So what is larger, ¼ or ⅛?” 

~ “One-eighth!” They declared confidently and in unison. 

~ “Why?”

~ “Because 8 is bigger than 4.” 

 

That is natural number bias.  They applied their understanding of natural numbers, eight is greater than four, and applied it to a new number system despite their conceptualization of fractions.  Only one day before, the boys exhibited a clear part-whole understanding of fractions. In other words, they understood that the denominator was the number of equal parts that you needed to divide the whole into and the numerator was the number of parts to count.  But those understandings didn’t compete with their number bias.  They certainly knew that 8 was larger than 4.  That meant the magnitudes of the fractions ¼ and ⅛ should behave the same way! 

 

Listening & Questioning

 

I needed to leverage their definitions of numerators and denominators to help them discover how to place the fractions.  I asked a series of questions:

Number bias
      • Why is ½ where it is?
      • What does that 2 mean again?
      • So why is it in the middle of the whole line?
      • Where are the two equal parts?

 

Boys + Jumping = Understanding

 

When I asked where the two equal parts were on the chalk number line, they quickly ran to the zero.  They made a huge, dramatic leap from zero to ½, and then jumped another half between ½ and 1.   I laughed and said, “Wow, it’s like you made two giant steps! So, are you telling me the denominator is the number of steps?” By their expressions, they thought I was very slow-witted.  I feigned any knowledge of fractions and then asked about the meaning of ¼.   Without answering, they made four large hops between zero and 1!  They bounced their way along to discover that 2/4 and 4/8 equaled ½, and then 2/8 could equal ¼. It was wonderful - they could bounce off energy and learn fractions!  Their jumps along the number line helped them to overcome their natural number bias and fall back on their part-whole understanding of fractions.

 

What does research say about Natural Number Bias?

 

Several studies (Siegler, et al., 2011; Van Hoof’s, et al. 2018) have determined that size (or magnitude) is the first type of natural number bias that children need to overcome.  Interestingly, children are typically able to overcome that bias and accurately determine the size of decimals before the size of fractions.  

 

The next number bias that children resolve is usually associated with operations.  For example, children assume that when multiplying two numbers, the product will always be larger than the factors (e.g. 3x4=12), but that isn’t the case when you multiply by a fraction less than one (e.g. 3 x ⅓ = 1).   That type of number bias is fairly persistent, but doesn’t last as long as concept of density.

 

Are Numbers Dense?

 

If you ask a child how many numbers are between 3 and 5, they will usually respond that 4 is the number, and they may think you are dense for asking!  When children begin to learn the rational number system, they don’t realize that there are an infinite number of numbers between 3 and 5.  In the world of mathematics, we describe rational numbers as dense.  In other words, between any two fractions you can always find another fraction.

 

When rational numbers are introduced in school mathematics, there is an increasing body of research that shows a natural number bias exists in young learners. In other words, children assume that rational numbers should behave like natural numbers. And, even though fractions and decimals are both rational numbers, children tend to develop more accurate understandings of decimals before fractions (Van Hoof et al., 2018).

 

Magnitude is Big

 

Before advancing in mathematics, children must overcome all three number biases, but the first and most important is magnitude.  Siegler and his colleagues tested 6th and 8th graders and found that their understanding of the magnitude of fractions was correlated with their accuracy in fraction operations.  In other words, if children don’t understand the relative size of fractions, they are likely to be less successful in performing multiplication and division with fractions.

 

The Big Takeaway

After reviewing the research and watching our students learn last summer, the evidence is clear.  No matter what, bias is never a good thing, and we have to work to overcome it.  And finally, size matters, especially for fractions.


References:

McMullen, J., Laakkonen, E., Hannula-Sormunen, M., & Lehtinen, E. (2015). Modeling the developmental trajectories of rational number concept(s). Learning and Instruction, 37, 14–20. http://dx.doi.org/10.1016/j.learninstruc.2013.12.004.

 

Siegler, R. S., Thompson, C., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62, 273–296. http://dx.doi.org/10.1016/j.cogpsych.2011.03.001

 

Van Hoof, J., Verschaffel, L., & Van Dooren, W. (2015). Inappropriately applying natural number properties in rational number tasks: Characterizing the development of the natural number bias through primary and secondary education. Educational Studies in Mathematics, 90(1), 39-56.

Van Hoof, J., Degrande, T., Ceulemans, E., Verschaffel, L., & Van Dooren, W. (2018). Towards a mathematically more correct understanding of rational numbers: A longitudinal study with upper elementary school learners. Learning and Individual Differences, 61, 99–108.

I was preparing a workshop for the regional NCTM conference in Indianapolis on how to teach lessons with my math comic books when I couldn’t believe my eyes. I found an error in my graphic novel.

Certainly, we reassure our students, “Making a mistake means we are learning.” And yet, I always struggle with internalizing such a positive outlook when my stomach churns and my imposter syndrome flares.

Imagine my horror when I opened a digital page of my graphic novel and realized an error snuck its way into a drawing!  Certainly, I didn’t send this one to the printer. I crossed the room, picked up the hard copy of my novel, and flipped quickly to the page. My heart sunk. There it was next to the brilliant color of the art. The error. I had sent the wrong digital file to the printer.  The math was wrong!!

Who would make such a mistake?  I needed to be precise.  I was a fraction genius? Right?!?

My husband, an avid comic book collector, heard my exasperation and colorful language as I made the discovery. When I told him about the error, he was thrilled!  “Everyone loves to spot an error in comic books! Those are the best to collect!”  I felt little consolation. My masterpiece was ruined!

Should I get a sharpie out and correct each one of the graphic novels that were printed?  

Or is the error a teachable moment? 

Can you find the error on this page?  

An error in my graphic novel.

At the NCTM conference, surrounded by over 50 math teachers in my session, I decided to pose the question, “There’s an error on this page. Can you find it?”   Silence. A long silence. The session time was running out, so I told them where the error was and asked them what they thought I should do about it.

The consensus among the audience was to keep it as is. They suggested it would be fun for students not only try to find the error, but also to explain why it was mathematically incorrect. Could students discover what our heroes were thinking? What should they have written? This error could be a teachable moment.

Still. How could this happen!?!

In an early draft of the art for issue No. 3, I saw the error. The artist brilliantly drew our three heroes talking about the math problem under their covers with a flashlight. 

I was so excited to see it come to life.   I remember asking the artist to depict the children’s mathematical thinking on a piece of paper, as well.  On page 18, Ben was explaining that 2 was the same as 16-eighths, and the artist needed to create a drawing similar to the art on page 6.

Quint and Theo discuss the pattern on page 6 of issue No. 3.

We just needed to recolor the circles to represent three-eighths and replace all of the 2’s in the drawing with 16-eighths  He must have heard that he should replace the “2’s” with “16’s” instead. When I received the error on page 18, I chuckled and told the artist. He immediately sent me a new, corrected digital file, and told me to delete the old one. The art we printed for issue No. 3 was perfect. Why did I keep the old file? I should have listened to him.

Our heroes discover the common denominator method on Page 18 of issue No. 3.

Although the perfectionist inside me was tempted to correct the error in the graphic novel, I decided to put away the sharpies and embrace growth mindset.  When we make mistakes, we can learn from them.

So, I decided to write a new lesson just about this error to make it a teachable moment.

What else did I learn? To stop hording digital copies of bad files!!  Now, I’m off to delete that old errant file and replace it with the correct one from issue No. 3!

Last week, I joined the team at the Columbus Regional Math Collaborative to host a summer math camp to help children study fractions after a year of virtual learning. It was filled with active learning, comic book surprises, and great problems to solve. I was able to conduct research on fraction learning while sharing the newly released issue The Mysterious I.D. Vide in Newton’s Nemesis No. 3 with the children. I’m excited to share what we learned as I dive into a summer of analyzing the data and blogging about our initial findings.

The first few impressions…

We planned a week of activities for our half-day camp that would allow children (ages 10-11) to think about fractions in different ways: as part of whole, as quantities, and as quotients. We had students work with Cuisenaire rods, physically jumping down number lines, and coloring with fraction tiles. The children were very sweet and rarely uttered anything negative as they struggled through problems. Of course, a few of the boys were VERY active and loved that our chairs had rollers.

During the last 30 minutes of the first day, I was worried. I watched one young man spin his chair around while making random engine-like noises and saw another talking speedily to a new found friend as he moved his chair excitedly back and forth from the desk. After two and a half hours of math, how was I going to get these children to read The Mysterious I.D. Vide in Newton’s Nemesis No. 1? I said, “Alright everyone, let’s read a story,” and we handed a comic book to each child. I was thinking through classroom management 101 strategies. I was certain that at least one of the children would ignore my request and begin to distract others.

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I was amazed.

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Only two children didn’t immediately pick up the comic book. Both sat quietly for a minute, looked around at everyone else reading, and then, began to peak inside. Both began reading through the pages. Not once did I redirect a child. They just read the comic book, silently, for fifteen minutes. The boisterous classroom fell silent. The younger boys curled up in their rolling chairs without movement. And they were still learning. It was teacher heaven!

By the end of the week, I invited the illustrator, Nathan Long, to come and talk with the children about the art in the math comic book series. Two of the young ladies were barely contained in their seats as Nathan explained how to draw movement and how colors are perceived by the human eye. Then he helped the students illustrate their math problems that we used for a parents’ gallery walk. So the big lessons I learned from our summer camp are:

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Lesson 1: Comic Books = Engagement

Student engagement is only half the battle. That’s why I conducted research with pre-camp interviews and collected data throughout the camp. There is a ton of data to sift through, and it will likely take me most of the summer. So, I’m inviting you on my journey as I learn more about how children learn fractions. There is one observation I can easily share now:

Lesson 2: COVID Math = Missing Manipulatives

The reason we hosted this camp was to help children who may have missed some math instruction due to the pandemic. We confirmed very quickly that our children didn’t have access to hands-on manipulatives that could help them master the basic concepts of fractions. Most could tell you that a fraction was a part of a whole and could locate one-half on a number line. But most struggled with other fractions, placing 1/8 as larger than 1/4. They exhibited natural number bias. More to come on that topic next time!

Summer Resources for Parents

If you know someone looking for resources, the Math Collaborative has this great page they call the Parent’s Corner.

I recently summarized research on how children think about fractions and described five different fraction concepts.  One of my followers asked me if I put any of them in my comic book, and the answer is, "Yes!"  The math comic books create opportunities for students to think about fractions - and teachers can help students explore different ways of thinking about fractions by using just few key questions.

Theo explains fractions to his little sister Leah
Theo explains fractions to his little sister Leah

Fractions as Part-Whole Relationship

When Theo explains what a fraction is to Leah in the first issue of The Mysterious I. D. Vide in Newton's Nemesis, he used the part-whole concept of fractions.  He was also careful enough to explain that the whole needed to be cut into equal parts, which is a critical part of the definition.  In his example, twelve is the whole.

 

Key Questions: Will Theo's definition work for all fractions?  How would you define a fraction?  Did you see fractions explained anywhere else in the story?

 

If we limit our definition to Theo's, children may think fractions are always smaller than the whole, which leaves out all improper fractions (i.e. fractions greater than one).  But fractions can be difficult to define and the second question assesses what children already do and assumptions they may make.  The final question sends the reader on a treasure hunt through the pages of the comic book to discover another way we think about fractions.

Fractions as Quotient

But earlier in the same issue, I subtly snuck in a couple of concepts during the comical scene of Ms. I. D. Vide trying to rap about fractions.  When you think of a fraction as a little division problem (e.g. 3/4 is three divided by four) or the result of a division problem, you are thinking of the fraction as a quotient. Theo is thinking, "Two divided by one equals two," while Ms. Vide flippantly uses the word "over" in her rap.**

 

Key Questions: Is 2 divided by 1 really a fraction? If we think of parts and whole, what does the numerator of a fraction really mean? What does the denominator really mean?

Children may answer "no" to the first question, which ushers in a great conversation about the true meaning of numerators and denominators.  Since the denominator is how many equal parts a whole is divided into, we need to divide 1 into 1 part; the numerator tells us to count or enumerate two wholes. Another extension:  "Have you ever noticed that any number divided by one is it the number itself?" One is a special number in multiplication and division called an "identity".  Why does that name make sense?

 

Such a simple scene can generate very interesting math discussions! However, thinking of a fraction as quotient also has its limitations, as Ben finds out in the second issue.  He doesn't understand that a fraction is also a number that can be divided.

**A quick note ~ Math educators usually avoid using "over" when speaking about a fraction, but there were too many syllables in divide for the rap! So, I guess that is my "artistic license" ~ and don't you love how the word flippantly is so perfect for this situation!!!

Vide raps about fractions in Issue No. 1
Vide raps about fractions in Issue No. 1
Ben understands fractions as a quotient
Ben doesn't know a fraction is a quantity
Vide rapping about fractions in Issue No. 1
Vide rapping about fractions in Issue No. 1

Fraction as Operator

During the rapping scene, Ms. I. D. Vide also introduces fraction as an operator. Think about how common it is for us to say that we are taking a fraction of something.  When we do, we are using the fraction to transform another number, and it acts like a function or operator.  One math site explains, "A fraction can be seen as a division waiting to happen."

 

Key Question: When you take 1/4 of a number, what happens to that number?

 

We can use our part-whole conceptualization of fraction to rationalize that taking one-fourth of something results in a smaller number

Sometimes, researchers refer to the operation of a fractions as shrinking (when they are between zero and one) or stretching functions (when fractions are greater than one).  The shrinking caused by fraction multiplication is the first time children encounter multiplication resulting in something smaller! So it is important to explore what happens with concrete examples: What is 1/4 of 12? What about 11/12's of 12?  How about 13/12's of 12? Finally exploring 24/12's of 12 will reconnect with the previous panel in the comic book. Shrinking and stretching allows us to think of operators (i.e. addition, subtraction, multiplication, division) as transformers. For example 2 times 12 is taking 2 groups of 12, transforming 12 to 24. But this that leads to the next key question:

 

Key Question:  When you take 1/4 of a group of 12, what operation are you performing?

 

Some children will say the operation is division, which feels right. When you ask them to explain, they usually discover that they are thinking, "12 divided by 4."  Remind children that the key question asks about groups.  If 2 groups of 12 is multiplication, what do you suppose 1/4 group of 12 is?  Of course, this understanding brings us to a different conceptualization of 1/4.  It is also a number or quantity, which leads to our next concept...

Fraction as Measurement or Magnitude

Like any number, fractions can be multiplied, divided, added or subtracted. In the second issue of my math comic book, Theo struggles to understand that a fraction is a number, just like any other number on a number line. This particular scene always make me chuckle, when he doesn't recognize that quarters are fractions!

 

In order for children to understand an expression such as 8 ÷ ¼, they need to know the ¼ represents a quantity, in other words a fraction is a number.  That number has a magnitude, represents a distance from zero, and/or is a measurable distance between two other numbers.

Theo in Issue No. 2 is about to remember that fractions are numbers.
Theo in Issue No. 2 is about to remember that fractions are numbers.
Theo discovers that fractions are numbers.
Theo discovers that fractions are numbers.

 

Key Questions: Theo is thinking about 8 ÷ ¼ as how many quarters are in $8.  Will that also work when 1/4 is a number on a number line?  What other ways could we represent 8 ÷ ¼?

 

By asking children to explore different models such as using number lines, fraction strips, and pattern blocks, we are giving them multiple tools in their mathematical toolbox to solve problems.

 

Fraction as Ratio or Rate

Since ratios and rates typically are not introduced until middle grades, I did not have the characters explore this fraction meaning.  However, there is an opportunity for teachers to use extension questions when two pizzas are delivered to our three heroes!

 

Extension Question: Theo's mother usually orders 3 pizzas for five people whenever his friends stay over. In the story, our three heroes ate 2 pizzas that were mysteriously delivered.  Did they each get more, less, or the same pizza compared to the amount they usually get when Theo's mother orders?

Final Thoughts

Throughout the series, teachers and parents can use key questions to help their students connect the story with the mathematics and assess how they are thinking about fractions.  As seasoned teachers know, you need to know what a child knows and understands before you can help them learn new ideas.  Our job is to help them build a strong foundation of fraction understandings so they can explore what it means to multiply and divide by this new and incredibly useful set of numbers.

 

I hope this helps you use The Mysterious I. D. Vide in Newton's Nemesis to motivate great discussions about fractions!

Over the last month, I have been working with the amazing master teachers at the Columbus Regional Math Collaborative, putting together a fun filled summer camp to help boost children’s knowledge of fractions before they enter middle school. Of course, if I want to boost their knowledge of the meaning of fractions, I knew I needed to reread some of my favorite research articles and books! What a great time to add to Half-a-Blog!

At the camp, we plan to start with the meaning of fractions before launching into fraction division – my favorite operation – and focusing on how these clever numbers are affected by operations. Hence, we named the camp….

Operation Super Solvers!

I’m so excited about the cool activities we are planning for the camp. We have to be pretty sneaky to make everything fun while also tackling the challenging topic of fractions. As usual, the background research is very interesting and a little daunting when it comes to the many ways children conceptualize fractions.

What are ways that children think about fractions?

Most of the time when we introduce fractions, we think of partitioning a whole.  Believe it or not, the part-whole relationship is just one of the five ways children need to think about fractions, according to most researchers (see Charalambous & Pitta-Pantazi, 2007 or Tien & Siegler, 2018). That’s just one 1 out of 5 concepts that children need to master.   And the part-whole relationship is more complicated than you would think!

Consider these two drawings:

In one, you have one whole area partitioned into equal parts.  In the other you have a collection of objects divided into equal groups. Think about what children need to master to fully understand what the numerator and denominator mean.

    • What is the whole we are talking about?
    • How are we sure that the whole is divided equally?
    • What are the total number of parts?
    • What are the number of parts represented by the fraction?
    • Will the part-whole relationship stay the same if I move the parts around?

Most of the time, the part-whole relationship is only discussed for fractions less than one.  Can you reimage the drawings above if the fraction was 15 twelfths?

The Part-Whole Concept is one the Beginning!

Some researchers have constructed a hierarchical chart to show the relationship between the different ways children think about fractions (Charalambous & Pitta-Pantazi, 2007; Behr et al.,1983; ). I’m actually designing a research project this summer loosely based upon this – especially since I’m not entirely sure I buy into this diagram.  But, it’s a good visual aid!

Behr, Lesh, Post, & Silver (1983)

While some countries limit the concepts of fraction to partitioning and quantifying, several research studies that analyze how children discover fraction concepts rely on four constructs: measure, quotient, ratio/rate and operator. Teachers can use different tasks or questions to help children develop a deeper understanding of fractions.

Measure (or Magnitude)

Within this concept, we want children to see a fraction as a number.  Teachers can set up tasks that require children to find a fraction on a number line and to connect that a fraction can represent a distance from zero.    But, it’s also important that children understand that a distance between two numbers can also be a fraction. Placing fractions on clothes lines is a rich activity, and there are many strategies to explore when comparing fractions to benchmarks (e.g. ½, 1, etc.). For example, which fraction is larger, 7/8 or 10/11? A quick way to decide is to compare the distance each fraction is from 1. The first is 1/8 from one. The second is only 1/11.

Quotient

How many times have we heard, “A fraction is just a little division problem?”  We say three-fourths is three divided by four, but we don’t think it is a quotient (the result of division) as easily.  When I ask my pre-service teachers, “Show me how you could share 3 sub sandwiches with 4 people?” they typically begin dividing two subs in half and the last sub into fourths.  Understanding that ¾ is a more direct answer takes time.  “What if you divide 8 sandwiches among 7 people?”  That typically leads to a mixed number before an improper fraction becomes apparent.

Ratio/Rate

While we usually don’t dive into ratios and rates until middle school, both of these concepts are pervasive!  Four out of five rotten tomatoes? Costs per person? Miles per gallon? As one of the master teachers aptly observed – we use the word “per” without really introducing it as a rate. A famous task that brings out this concept is the “Orange Juice Task” posed by Gerald Noelting (1980) in which children were making orange juice for a party by combining glasses of concentrated orange juice and water. One child combined 3 glasses of orange juice with 5 glasses of water, but another combined 5 glasses of orange juice with 8 glasses of water. Which pitcher would taste more orange?

Operator

This concept is simple when you think of three-fourths as three quarters or three one-fourths.  However, the concept can also be more complex when you begin to think of a fraction as a composition of operations or dare I say a function that transforms one number into another.  When we think, “I need to take ¾ of the recipe” we reduce or shrink the amount of the recipe.

Fractions are Cool Numbers

Here I go again on the nerd train, but fractions are cool numbers! They open up the door for children to think about an entirely new number system! And I haven’t even started to talk about decimal fractions, rational numbers, and how there are infinitely many of them between each fraction!!

As I proofread this post, I found it fascinating to see the interplay of number concepts with operation concepts.  It made me wonder, when do we cross the line between thinking about a set of numbers, such as fractions, and thinking about operations, such as division?  I think there are some interesting research questions to study this summer, along with teaching all about fractions at the camp!

References & Research Articles

Behr, M. J., Lesh, R., Post, T., & Silver, E. A. (1983). Rational number concepts. Acquisition of Mathematics Concepts and Processes, 91, 126.

Charalambous, C., & Pitta-Pantazi, D. (2007). Drawing on a Theoretical Model to Study Students’ Understandings of Fractions. Educational Studies in Mathematics, 64(3), 293-316. https://doi.org/10.1007/s10649-006-9036-2

Noelting, G. (1980). The Development of Proportional Reasoning and the Ratio Concept Part I – Differentiation of Stages. Educational Studies in Mathematics, 11(2), 217-253.

Schumacher, R. F., Jayanthi, M., Gersten, R., Dimino, J., Spallone, S., & Haymond, K. S. (2018). Using the Number Line to Promote Understanding of Fractions for Struggling Fifth Graders: A Formative Pilot Study. Learning Disabilities Research & Practice (Wiley-Blackwell), 33(4), 192-206. https://doi.org/10.1111/ldrp.12169

Tian, J., & Siegler, R. S. (2018). Which Type of Rational Numbers Should Students Learn First? Educational Psychology Review, 30(2), 351-372. https://doi.org/10.1007/s10648-017-9417-3