Tag Archives: Liping Ma

Profound Understanding of Fundamental Mathematics

Four Properties – to ensure a profound understanding of fundamental mathematics

Liping Ma (2010) believes in four properties in teaching and learning which sum up the way in which a teacher’s profound understanding of fundamental mathematics can be represented in the classroom. By this, Ma means a teacher will demonstrate the four properties and if this is successful, he or she has a profound understanding of fundamental mathematics.
– If you are unsure about what this is – profound understanding of fundamental mathematics – scroll to the bottom to read quotes extracted from Liping Ma’s ‘Knowing and Teaching Elementary Mathematics’ (2010). This may aid your understanding of the Four Properties in this post. 
The first, connectedness, occurs when a learner has the intention of making connections between mathematical concepts and procedures. In pedagogical terms, the teacher will prevent the learning from being fragmented and instead, learners will develop the ability to make connections between underlying mathematical concepts that link.
The second property, multiple perspectives, is practised when a learner is able to take into account various perspectives when thinking in a mathematical way. This includes addressing pros and cons of all different viewpoints considered. In pedagogical terms, the teacher provides opportunity for their learners to have a flexible way of thinking and understanding concepts in maths.
Thirdly, basic ideas, is a way of thinking about maths in terms of equations. Ma refers to basic ideas as, “simple but powerful basic concepts and principles of mathematics” (2010, p. 122). Ma suggests, when practising the property ‘basic ideas’, learners are guided to conduct ‘real’ maths activity rather than just being encouraged to approach the problem. I suggest this means if a teacher is effectively implementing this property, he or she will be not just attempting to motivate the learners to approach the maths work, but instead, providing a solid and secure guide to the learners understanding the maths themselves.
The forth and final property, longitudinal coherence, is when a learner does not have a limit or boundary of knowledge. In other words, it is not possible to ‘categorise’ the learner or identify the learner as working at a specific level or stage in maths. Instead, he or she has achieved a holistic understanding of maths – a fundamental understanding. In pedagogical terms, Ma suggests a teacher who has achieved a profound understanding of mathematical understanding is one who is able to identify on demand the learning that has been previously obtained and will be learned later. Subsequently, the teacher will lay the fundamental maths as a foundation for later learning.

A few quotes extracted from Liping Ma’s Knowing and Teaching Elementary Mathematics (2010) that I feel define and summarise profound understanding of fundamental mathematics…

“The term ‘fundamental’ has three related meanings: foundational, primary, and elementary.”
– Ma (2010, page 120)

“By profound understanding I mean an understanding of the terrain of fundamental mathematics that is deep, broad and thorough. Although the term ‘profound’ is often considered to mean intellectual depth, it’s three connotations , deep, vast, and thorough, are interconnected.”
– Ma (2010, page 120)

“As a mathematics teacher one needs to know the location of each piece of knowledge in the whole mathematical system, its relation with previous knowledge.”
Tr. Mao (2010, page 115)

“I have to know what knowledge will be built on what I am teaching today.”
Tr. Mao (2010, page 115)




Ma, L. (2010) Knowing and Teaching Elementary Mathematics 2nd edn. New York: Routledge. Pages 115-122.

liping ma

The educator’s conceptual view – know what you are teaching!

Limited subject matter knowledge restricts a teacher’s capacity to promote conceptual learning among students. Even a strong belief of “teaching mathematics for understanding” cannot remedy or supplement a teacher’s disadvantage in subject matter knowledge. A few beginning teachers in the procedurally directed group wanted to “teach for understanding.” They intended to involve students in the learning process, and to promote conceptual learning that explained the rationale underlying the procedure. However, because of their own deficiency in subject matter knowledge, their conception of teaching could not be realized. Mr. Felix, Ms. Fiona, Ms Francine, and Ms. Felice intended to promote conceptual learning. Ironically, with a limited knowledge of the topic, their perspectives in defining the students’ mistake and their approach to dealing with the problem were both procedurally focused. In describing his ideas about teaching, Mr. Felix said: “I want them to really think about it and really use manipulatives and things where they can see what they are doing here, why it makes sense to move it over one column. Why do we do that? I think that kids are capable of understanding a lot more rationale for behavior and actions and so on than we really give them credit for a lot of times. I think it is easier for anybody to do something and remember it once they understand why they are doing it that way“.”
– Liping Ma, Knowing and Teaching Elementary Mathematics (2010, page 36)

The most important thing to remember when teaching maths – when teaching anything – as the teacher, the educator and the facilitator, is that you must understand what you are teaching. This is what Ma (2010, p. 36) is talking about here.

As a teacher and a professional educator, you are responsible for providing knowledge to your learners, not just passing it to them as information in a book or in the form of confusing statistics and facts, but as an understood conceptual view of the content. If you do not understand what you are teaching, this may invite opportunity for confidence to fall in your learners – you are the trusted educator in the classroom, on which your learners depend on to provide subject matter with an understanding you have thoroughly revised, in order to adapt the content to best explain it to them.

Outsmarted?… Imagine this. You are planning a lesson – a maths lesson. You have a vague and somewhat passive understanding of the content you intend to teach. And so you think your learners will trust that you understand what input they are going to receive, because, after all, you are the teacher. Right? That passive understanding you have, is only going to brush off onto your learners. Children are observant and will easily pick up on your mistakes, your struggles and perhaps your lack of confidence when you are teaching them. So, you plan your lesson, still intact with your passive understanding of the content you intend to teach. Then it comes to your lesson and your learner outsmarts you. Perhaps in the form of a question, that you cannot answer. Is this due to your negligence?

Your learners depend on you to know what you are talking about, and here, Ma, explains the profound importance to approach your intended learning content with a conceptual view – if you understand, you have more chance of your learners understanding!



Ma, L. (2010) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States. London: Routledge.