What is a whole orange ? A solution to teaching abstract mathematics.

We are teaching STEM subjects backwards. This inversion of teaching methods includes one of the most powerful tools for human thinking ever developed: Abstract Mathematics.

Let's consider abstract mathematics with the elementary math problem: "If Ray eats one orange today and 2 tomorrow, how many oranges has she eaten?"

Maybe you had a really easy time with this sort of math problem, however upon closer inspection, an insidious deception appears. Ray is not a real person in the physical world, there are no oranges, and there are no days except in the minds of those confused students sitting in class. Even if there really is a girl named Ray and she eats 3 oranges over two days the math question only refers to the mental representation of Ray and the oranges she ate. IF this were in the physical world it opens up reasonable questions like:

"Is Ray really Ray if her friends call her Rayray?"

"Did she really eat the oranges whole without peeling off the rinds?"

"Does this question only refer to those 2 days or her entire life?".

These types of questions could hypothetically go on indefinitely, while logically valid, these question have irritated school teachers for hundreds of years.

What is the issue? The issue starts with poor definitions and explanations of the simplification process and why this is essential to do calculations. We need to start by teaching students to convert objects or processes to mental representations that are absolutely certain. Ray is a discrete person, who ate exactly 2 oranges on one day and exactly 1 the second day. The question contains absolute things and absolute processes. We need to repeatedly clarify that these are simplified/absolute representations of people, things and processes in the real world.

Mathematics taught in schools across the world below the "high school level" deal with mental representations of idealized processes or objects (Abstract mathematics). They are not real and therefore, there is no uncertainty within the system. With these mental representations, it is possible to get a "perfect answer". This means that with the application of the right equation it is possible to get an answer with absolute certainty (correct) because the mental pattern itself has an idealized mental representation. This removes the need to rely on statistics and probability and is extremely powerful to use for simple processes.

For teaching math to students in kindergarten through elementary school.

We have it backwards. We teach 1+1=2 very early on, however, we often do this using pencil and paper and then use props if our students don't grasp the concept. We act like physical representations of real-life physical oranges, coins, or candies are the representations of the math problems on the paper. We often deplore the extra work of having to help students by using actual physical objects when if they were actually adept, pencil and paper would be good enough. BUT THIS IS BACKWARDS. The numbers written on paper are the simplified representations of the real objects. Nobody needs the number 5 to survive and be happy. The number 5 has absolutely no value by itself. However, to know that the number 5 is a simplified representation of 5 discreet units of something is indispensable in the modern world. To a child, they need to know what 5 oranges, 5 coins, or 5 candies are. Students need to be able to associate discreet physical units with representative numbers.

Practical Application: This means that whether physically or digitally (on a computer or tablet), numbers must be accompanied and associated with the things they represent. Rather than simple equations, problems should be presented as objects/picture-based problems accompanied by words and written equations.

Practical limitations: This makes everything more difficult, while listing a 30 equations is relatively easy, making questions that integrate objects, words and equations are much more complicated. However, with digital tools now readily available, there are tangible strategies to overcome these limitations.

Consider this picture and the question. "How many oranges are in this picture? Annotate the picture and show your calculations. Do your calculations assuming the parts of the oranges not in the picture are present."

There are 2 full oranges and 2 half oranges

Option 1: 2+1/2+1/2 = 3 oranges

Option 2: 2+ 2*(1/2) =3 oranges

Option 3: Further discussion

Making these questions digitally would take 3 minutes and could be copied and pasted for re-use in other contexts or other classes/years. Importantly this exercise opens up practical skills essential to analytical thinking that isn't present in conventional math classes. For example, going through each question in class is useful because they would include discussion rather than going over answers. It teaches segmentation skills (identifying an orange as a discreet unit) and it opens up questions about whether the orange juice could be included in the calculation. While much, much harder (computationally/logically) it exercises innate analytical skills that all humans (and many animals) have at some level. Importantly, it is practical. The practice of simplifying units and processes to be represented as numbers and equations are skills that can be used everyday, in any context or any profession.

In a comic strip of "Calvin and Hobbes" 6 year old Calvin complains about the difficulty of entry level abstract math problems. His solution is to become a hunter-gatherer that doesn't need society or a job. However, even a hunter-gatherer can benefit from knowing how many oranges they can carry home. This is the very first step to making mathematics accessible and practical for all of our students no matter their disposition or approach to math problems. Importantly, as educators, we can integrate mathematics into other classrooms. Once mental skills in abstract mathematics are learned in a tangible physical context, they are easily applicable to any other topic that contains physical context. This feeds directly into how we can apply thematic learning in our classrooms with learning revolving around our students learning to navigate situations and themes rather than trying to beat tough concepts into their minds. In the end, this is about removing unnecessary antagonistic forms of education to cultivate and create nurturing environments for learning.

Note: This post was written before reading the book "Limitless Mind" by Stanford researcher Dr. Jo Boaler. These concepts are included under the umbrella of "multidimensional" representation of mathematics questions. For further reading on these subjects we highly recommend viewing the work of Dr. Jo Boaler.

Boaler, J. (2019). Limitless mind: Learn, lead, and live without barriers.

https://www.youcubed.org/