Dialogue and mathematical understanding what is it?
Amir Saei is a Maths Scholar. Why don’t you join him next year?
Understanding is an internal state of mind which has to be achieved individually by each student, and cannot be observed directly by the teacher. The fact that a student can solve a particular problem correctly does not necessarily indicate much about the level of understanding present (Cockcroft, 1982).
Suppose you have two students both of which answered correctly. The first student figures out by counting 38 further from 29 on a number line. The other uses column addition without any understanding of the mechanism behind it,. They are likely to have less of an understanding of the structure of addition and its various properties than a student who discovers (Webb, 2014). This is a sentiment that is also shared by Freudenthal, as summarized by Marja van den Heuvel-Panhuizen (1996) “…in Freudenthal’s eye, it is the way in which the student works on a problem that determines the level [of understanding].”
In the previous example both students achieved the correct answer. However, there are also cases where one student gets the answer correct but has less of an understanding. At least in my opinion, this understanding of the concept at hand can be less than a student who achieves an incorrect answer.
An example of this occurred in my year 9 class. I gave the class a starter of calculating the midpoint of 8 and 11. Then I asked students to provide some answers to the whole class. Student A answered 9.5. I asked him to explain his answer, he told me “you just add 8 to 11 and then divide by 2”. When I asked why that is the midpoint he could not explain. At no stage, did I inform him or the class if his answer was correct. Student B answered 9. Again, I asked him to explain his answer. He said: “the distance between 8 and 11 is 3. The midpoint is equal in distance from both 8 and 11. 3 divided by 2 is 1.5 so the midpoint is 9.5.” The student ended up providing the correct answer.
Hence, engaging in a conversation with the students and getting them to provide an explanation for their answer enabled me to get a truer reflection of their understanding as opposed to if I were to merely ask them to provide their answer. But the importance of dialogue and being able to explain answers goes way beyond mathematics. In engaging with students in discussions we help develop their ability to form a coherent argument, to reason, and to be precise and unambiguous when using language; the ability to say what you mean and mean what you say.
If you think that this kind of teaching approach inspires you to enter the classroom why not check out how the process begins.
Look at the Get Into Teaching website or check out the Maths Scholars website to see how you could apply for a £27 500 bursary to help you train.