“Knowledge” is a product of innate personal thought, social interaction — continues indefinitely — and knowers, ideally, employ a spectrum of approaches in gaining it. Our use of the ways of knowing is manipulated by a demand for accuracy in the knowledge we absorb, and though a knower may prefer reason and logic over intuition and emotion, the general formula appears to be that quantity equals quality.

How do we know that gaining information by using numerous ways of knowing guarantees enrichment of our understanding of an area of knowledge?

This "500 words on" refers to “networks” as the simultaneous employment of several ways of knowing at once, whereas the opposite means the use of them one at a time. By investigating mathematics through a critical lens it may be noted that relying on single ways of knowing still yields knowledge, but it is oftentimes flawed and hollow.

The discipline offers holistic knowledge as part of an exchange that occurs when a knower approaches them wielding a network of ways of knowing. Gaining well-rounded knowledge in mathematics is viable when language, memory, intuition and reason are involved.

Mathematics requires more than intuition of the knower to gain accurate knowledge. Intuitive judgements are often criticized for lack of explanation. During a mathematics tutoring session with an MYP4 -- the equivalent of 8th grade in a traditional American high school -- student, I asked her to solve an equation for me, which she instantly calculated correctly. The absence of a justification stood out to me.

My mathematics teacher reminds me to always show my workings, because even if my final answer is wrong, I prove that I have a method and a certain understanding of the subject. When I prompted the MYP4 student to justify her answer, she could not — it was “obvious” to her.

The intuitive method of learning sufficed in some chapters but malfunctioned frequently. Instead, I explained the reasoning behind her answer and the method deriving it, after which she could fluently apply it to other questions. The interaction of reason and logic are necessary supplements for intuition to be viable as a way of gaining knowledge.

On the other hand, knowledge in mathematics can be gained solely through intuition. College admission exams, such as the SAT Reasoning Test, assess a knower’s abilities based on 3 hours and 45 minutes. While gaining knowledge for the test I relied on reason. Although a “reasoning test," the limited 25-minute sections force a knower to intuitively apply the knowledge gained.

When preparing, I asked my mathematics teacher when struggling with conundrums, but his calculations were lengthy, and every time I had to ask whether there existed any quicker method. Studying for the test and sitting it proved to be two different worlds. “5 minutes remaining,” and I understood that gaining knowledge for the test required intuition. For my second sitting I had learned mathematics by rendering questions to an extent that allowed me to eliminate 4 out of the 5 options available quickly.

Using single ways of knowing at the time is occasion-specific, because the gap created by the sacrifice of accuracy is filled with the hints in multiple-choice options.