As an education student I have become very familiar with the four categories of knowledge .  These include factual knowledge, conceptual knowledge, procedural knowledge and metacognitive knowledge (Drake, Reid & Kolohon, 2014). For those of you who are unfamiliar with the difference between the four categories of knowledge I will briefly break them down. Factual knowledge is having the knowledge of basic facts, such as all sides of a square are equal. Conceptual knowledge consists of understanding the underlying idea behind something. For example, if a student is given a mathematical word problem they use conceptual knowledge to recognize the type of problem it is (i.e. area or perimeter) so the student can then identify the relevant information in the word problem. Students then use their procedural knowledge to carry out the necessary steps to make the proper calculations. Lastly, a student uses metacognitive knowledge to reflect on their work and thinking process in order to help them learn form their strengths and weaknesses in order to become stronger learners.

When explaining the four categories of knowledge I used math examples because math is the subject where students are constantly asked to “show there work” and by doing so they display their factual, conceptual and procedural knowledge down on paper. Metacognitive knowledge is also very prominent in math because when students get stuck or make a mistake they must go back and review each step to find where they went wrong so they can fix their mistakes and properly carry out their work.  It is important that students “show” or “prove their work”.  so that if a student is experiencing difficulty in math, the teacher can follow their work to pinpoint what category of knowledge the student is experiencing trouble with. The teacher can then give the student effective instruction to help him/her overcome the obstacle.

"Fifty-Three times One Hundred Thirty-One" by Daniel Tammet
Retrieved from: http://www.danieltammet.net/artwork.php

However, teachers often expect their students to follow a very specific set of steps in order to complete a math problem. Sometimes students use a very different method to calculate an answer, and even if their final answer is correct, their work is often marked incorrectly if it is not what the teacher wanted to see. For example, look at the painting above. This is how someone answered the multiplication problem of “53 x 131”. I know if I was a teacher and a student handed this into me I would have thought that the student was doodling in class instead of completing their math work, when in fact, the student had a very unique way of thinking and showing his work that I just did not take the time to understand it. In the following video Daniel Tammet explains his different way of thinking and how he uses paintings to express numbers and different mathematical equations.

After watching this video it is very clear that Daniel Tammet has extremely strong factual, conceptual, procedural and metacognitive knowledge. However, the way he shows his work is so unfamiliar to us that in is not comprehendible without his explanation.

   As teachers, we should welcome different ways of thinking in our classrooms. I know if I ever have a student like Daniel Tammet, I want to know how to effectively assess my students thinking even though they show their work in a way I would never imagine. Stylianides (2007) explain a way that we as teachers can evaluate our students' work no matter what kind of method they use to demonstrate their different types of knowledge. If a student hands in something where he/she shows his/her work in a unique way there are four elements of proof they must be able to explain in order to help us as teachers to understand it. These four elements are foundation, formulation, representation social dimension (Stylianides, 2007).  Formulation is consists of the student explaining their factual knowledge and how it relates to their work. Formulation is how the student can explain how their formula or process was developed based on their factual knowledge. Representation is how the student expresses his/her work, whether it is in numbers, or pictures. Lastly, the social dimension, is how the student’s work fits in to the social context of the learning environment (Stylianides, 2007). When reading about the four elements of proof, it is somewhat difficult to understand. However, if you listen to Daniel Tammet explain how he sees each number in different colours and shapes, you can see how his mathematical artwork follows the four elements of proof.
As future teachers, we should always be open to the different strategies our students might use in their work. By teaching our students to explain their work by using the four elements of proof we will be able to understand their unique way of thinking and ensure that their four categories of knowledge continue to grow and develop. 

References

Anderson, Lorin & Krathwohl (2001). A taxonomy for learning; Teaching and assessing. New York: 

Longman.

Drake, S. M., Reid, J. L., & Kolohon, W. (2014). Interweaving curriculum and classroom assessment: Engaging the 21^st-ceturey learner. Tornonto, ON: Oxford University Press.

Stylianides, A. (2007). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65(1), 1-20.