Framing #1: Fractions Are Hard.

Problem statement

In a world where mathematical division is easy... you can scale any recipe, divide any bill or calculate any tip, interpret statistics, you can even craft explosives from raw ingredients. Mathematical literacy is a pathway to greater understanding and agency in life. Ratios, percentages, rates, division, and fractions are all different names for describing the same relationship between numbers, that relationship is useful in every area of life, and is generally poorly understood.

A person who poorly understands division will struggle to use or interpret fractions, they will have trouble evaluating risks based on data, they will confuse the material ratios of their DIY TNT. Since division is one of the core mathematical operations it is an essential tool to working with numbers. When even simple division is an exotic concept then we get mistakes like this:

 

From Reddit r/theydidntdothemath

Some people have a fundamental misunderstanding of how the division operation works (for clarity: 44 billion dollars divided among 8 billion people can be found by $44,000,000,000/8,000,000,000 = $44/8 = $5.5. That's $5.50 per person). There was also a mildly well-known anecdote from the 1980s when A&W Restaurant started selling a 1/3 pound hamburger. "We were aggressively marketing a one-third-pound hamburger for the same price...but despite our best efforts... they just weren't selling" according to A. Alfred Taubman, who owned the restaurant at the time. After focus group research the restaurant discovered that most customers incorrectly believed 1/3 to be less meat than 1/4 pound, a worse value in their estimation. These customers didn't just embarrass themselves with a simple mistake- they also lost out on a real benefit, and would have no way of knowing how many other benefits they had rejected erroneously.

The problem is that mathematical division is poorly understood. Framing this problem involves more than just pointing at it, the problem statement is essentially infinite- so we must partition it within a frame and make the problem finite. Despite the importance of math in modern life the subject is a consistent groan-inducer among students and a topic of constant pedagogical meddling. One problem that is not included in this scope is the political or financial status of education in the United States. The problem of motivating a nation and government is well outside this authors interest. The problem here is to establish where a person begins to lose their grip on math, specifically where division leaves them behind. Why is division so hard? 

A depiction of the problem as a sequence

Sequentially this problem affects people when the foundational skills they are implicitly expected to have are not available. 

Root cause analysis

Is it... because this is an abstract concept, because students aren't motivated to learn it, because teachers have limited time to introduce concept and demonstrate the technique, because students don't know how to frame the problem?

Why don't students know how to frame the problem? 

Because they haven't seen enough examples, because they don't care, because they don't know the relationship between the parts of the problem?

Why don't they know the relationship between the parts of the problem? 

Because they don't care, because no one explained it, because they haven't related the parts to each other, because they don't know how to define the parts of the problem?

Why don't they know how to define the parts of the problem? 

Because no one showed them, because it's never obvious what parts a problem has, because previous examples didn't show where the parts came from, because they haven't been asked to define the parts of the problem before?

 Here the trail goes a bit cold. In the real-life examples cited above we saw two cases in which someone didn't know the meaning of the numbers they were using, in one case they reversed the numerator and denominator of the fraction, among other issues; in the other case they reversed the proportional relationship of two fractions. They didn't know how to frame the problems, so any further work was unfruitful. Rather than say that fractions are hard, we may find it more accurate to say that framing problems is hard...

This actually corresponds with data from the National Center for Education Statistics (a US Department of Education agency), in 2020 they released a report (PDF) on the varying levels of adult numeracy (ability to use, interpret and communicate mathematical information and ideas in a range of situations) in the US. According to that report only 69% of adults can use fractions or decimals in processes with more than one step (level 2 and above), and only 37% of adults could reason with unfamiliar or abstract information.

Why is it hard to identify the parts of a division problem?

As a math student and tutor I have the opportunity to pull the curtain back on math for some people, and the methods I use could someday help others beyond my reach. To tackle this problem will take more information and more insight, this may still not be the actual problem. Learning from others and gathering perspectives will be important to developing a holistic understanding of the problem. 

What did you learn or notice about the problem in this process?

Several times I found myself looking at causes outside the scope of the project, such as parental involvement with students or lesson plan pacing. I need to get information from more people since my speculation can only take me so far in seeking the root cause and potential solutions of this problem. I noticed that I tend to think of problems as statements, and that posing a problem as a question is already reframing it.