The Curse of Knowledge

Every once in a while there’s a lesson that makes me take a step back and think about how hard it is to truly learn and understand something. For example, fractions are not an easy concept to grasp conceptually. It’s one thing to learn that 1/3 is greater than 1/4 because the smaller denominator is the bigger fraction. It’s another thing entirely to understand conceptually why 1/3 is greater – each fraction has one part but the parts of 1/3 are larger than the parts of 1/4.

Fractions are even a difficult concept for adults to conceptually understand. In the 1980s, the A&W company tried to compete with McDonald’s quarter pounder. They created a third pounder which was cheaper and bigger than the quarter pounder. It beat McDonald’s burger in taste tests and was accompanied by a marketing campaign. So a company produces a bigger, less expensive burger, and the public likes the taste. Seems like a slam dunk, right? No. The public didn’t go for it. Why? During focus groups it was determined that customers thought they were getting less burger for their money because they mistakingly thought 1/3 is smaller than 1/4. A&W’s problem wasn’t a quality of product problem. A&W had a math problem – literally.

This is another reminder of how difficult it is to really understand some concepts. I can manipulate the numbers in my head and explain why 1/3 is greater than 1/4. It’s easy for me to forget that it takes time, modeling, and repetition to truly understand how to compare fractions with like numerators.

This struggle with fractions reminded me of a concept I first heard in the book Make It Stick. The idea is called the curse of knowledge. “The curse of knowledge is our tendency to underestimate how long it will take another person to learn something new or perform a task that we have already mastered.”

It is hard to remember how difficult it is to first learn something. Take driving for instance. An experienced driver doesn’t think about all the decisions he makes behind the wheel of the car. He has built a knowledge base over years that allows the process of driving to become almost second nature. Now put a 16-year-old behind the wheel and you’re reminded of how much a person needs to learn to drive a car. You’re also reminded that this knowledge is not obtained overnight.

I try to go into lessons with the curse of knowledge on my mind. It is important to remember that the 9- and 10-year-olds sitting in my classroom haven’t had the years of math exposure I have. It’s why thinking through the lens of our students is so important.

In an attempt to combat this curse of knowledge, I try to anticipate prior to a lesson which concepts will give students difficulty and why. My goal is to overcome the curse of knowledge as a teacher. In the process, hopefully, my students will conceptually understand difficult concepts like fractions and know the mathematical difference between a third pounder and a quarter pounder.

Numberless Word Problems

A few years ago I was doing a lot of one-on-one work with a student who struggled in math. We often met before class to do some mental math and try to improve his number sense. After school, we would revisit the skill we discussed in class that day. He made a lot of progress and was starting to feel much more confident about math in general and problem solving in particular. Then we introduced fractions.

Shortly after we started our unit on fractions, he put his head down in frustration. I asked him what was wrong, and he replied, “I was just starting to get math and then… then fractions happened.” It was as if he had contracted a deadly virus named fractions. We talked about how good problem solving doesn’t change just because we moved from whole numbers to fractions. Over the next month or so, we continued to work and his confidence with fractions improved as well.

My approach with that student would be slightly different today. Instead of moving him right into word problems with fractions, I would use numberless word problems to help ease some of the anxiety. The numberless word problems would be a sort of scaffold to the problems with fractions. This would help him see that we’re still using the same problem solving strategies we used with whole numbers.

Numberless Word Problem in 4th Grade

Numberless word problems are just that – word problems without numbers. Let’s look at the following problem.

Brian eats 1/8 of the pizza and Julie eats 4/8 of the same pizza. How much of the pizza did Brian and Julie both eat?

This is a pretty basic word problem involving fractions. Some students might struggle with this problem simply because they are intimidated by the fractions. So, why not take the fractions out all together?

Brian eats some pizza and Julie eats some pizza. How much of the pizza did Brian and Julie both eat?

Now we can have a conversation about what is happening in the problem. We can talk about how one person has some amount of pizza and another person has another amount. We want to know how much they have together. That hopefully leads students to see that this is an addition problem. Once they understand the context of the problem and have determined the operations they need to use (addition), then they can start looking at specific numbers.

Depending on the students and the problem, I might put some whole numbers in the problem before moving to fractions. This will give the student another opportunity to see the context of the problem before the fractions are introduced. I did this today with my class.

I took a problem from our math book and covered up the fractions to create a numberless word problem. This is how I introduced the problem.
After we discussed the context of the problem, and determined the necessary operation, I put whole numbers into the problem.
Then, we looked at the original problem with fractions and talked about how the structure of the problem didn’t change because of the fractions.

This is a wonderful strategy because it helps students focus on the context of the problem. The focus on the narrative portion of the problem before worrying about the numbers. They can’t worry about the numbers, because they aren’t there yet.

Math Clothesline

We just wrapped up a unit on fractions which covered equivalent fractions, common denominators, simplest form, comparing fractions, and ordering fractions. Comparing and ordering fractions gave me an opportunity to use a strategy which I absolutely love: clothesline math.

As the name suggests, I put a clothesline up in my room (really just a piece of yarn) and students use it to create a number line. It’s a great strategy because it works with whole numbers, fractions, decimals, algebra, and on and on.

I fold a notecard in half and write the fraction on one half. This allows the fold of the card to hold the number on the clothesline. Since I’m writing the numbers myself, I can make the numbers fit whatever lesson or standard I need. For example, in Pennsylvania fourth graders are only comparing fractions with denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100. So, I create fractions cards which fit those parameters.

There are times when I will place cards on a table and let students choose their own number. Other times, I want to be a little more strategic with who gets each card, so I will distribute them to students. This allows me to ensure that students are getting fractions that match their math abilities. I give students who have a better grasp of the concepts fractions which are in the sixths, while struggling students might only get fourths. If a student has to draw a model of a fraction, it is easier to draw fourths as a visual representation than sixths.

Once all students have the cards, we discuss which numbers should be placed on the clothesline first. Yesterday, my students determined that 0 and 1 should be the first two numbers. Then they decided that 1/2 should go next because it is the benchmark we used in our lessons on comparing and ordering fractions.

Once we have 0, 1/2, and 1 placed, we start to have a discussion about the other fraction cards. I allow students to talk with a partner as we move through the numbers. When students go to place their fraction, they have to share three things:

  1. Say the fraction
  2. Share which two numbers their fraction goes between
  3. Explain, using math vocabulary why you placed it there (common denominators, common numerators, relative location to the benchmark, etc.)

Once we get started, my role is to simply facilitate the conversation. I don’t tell students whether they are right or wrong when they place their fraction. Yesterday, three of the fractions were placed in the wrong spot. Once every student placed their card, I asked if there were any changes we needed to make. Then the conversation moved into a bit of error analysis. After some discussion and debate, the cards were placed in the correct order. Once again, I let my students talk about what they saw and what needed to be changed.

Students were drawing models on whiteboards, comparing fractions using common denominators, and one student even asked to use the fraction tiles we have in our room to model his fraction.

One student models 5/6 and 8/10 with fraction tiles.

This is such a wonderful activity because it checks so many boxes. In addition to comparing and ordering, students are experiencing many of the Standards for Mathematical Practice. Students are constructing viable arguments and critiquing the reasoning of others, modeling with math, attending to precision, and looking for and making use of structure. I am a stickler about precision in this activity. One student said, “The number on the top in 4/8 is smaller than the number on the top in 5/8.” I will ask them what we call the number on the top and ask them to restate their reasoning. “The numerator in 4/8 is smaller than the numerator in 5/8.” If the students stops there, I will prompt them. “What does the numerator represent?” I will prompt the students until they explain that the numerator in 4/8 represents 4 parts, and the numerator in 5/8 represents 5 parts. The parts are the same size since the denominators are the same, so 5/8 is greater than 4/8.”

I also make sure there are some equivalent fractions included in the cards. When students have an equivalent fraction, they use a paperclip to stack the numbers to show they fall on the same point on the number line, or clothesline.

Each year I introduce this as a whole-group lesson. This allows students to experience those rich conversations with students of all math abilities. Then I use it as a small group activity. Students work in groups of 3-4 and deal cards to one another. They take turns placing their cards and have a similar discussion. As an exit slip, I’ll have students choose three fractions and write how they ordered them. This can also be done by taking a picture and using an app or other digital sharing tool.

This is a great activity that can be used in so many applications throughout the grade levels. The conversations are fantastic, and students are engaged in the activity. Hopefully you find it as valuable as I do.