Ma (2010) identified ‘Longitudinal Coherence’ as the final property of having a profound understanding of mathematics. If I am totally honest, this is the one that baffles me. I think this is because of my own fragmented experience of mathematics. When I was at primary school, I was never encouraged to link topics of learning, or reflect on more advanced learning, thinking about which concepts I had developed in order to get where I am now in my mathematics understanding.
I think I must have read the below definition of Longitudinal Coherence about 100 times:
‘Fundamental understanding of the whole mathematics curriculum and no limitation to the knowledge that should be taught in a certain grade. The ability to exploit an opportunity to review crucial concepts that students have studied previously and know what students are going to learn later and building the foundations for this future learning.’ (Ma, 2010, p.121)
After reading it 101 times and still feeling perplexed, I knew that I would have to do further reading to try and get different examples of what longitudinal coherence was in order to fully understand this property. Again, I found this difficult as every time I felt I was starting to get to grips with the concept, it began to feel like I was talking more about ‘connectedness’ than longitudinal coherence. I guess that it’s okay to have slightly different takes on the 4 crucial concepts of PUFM developed by Ma. I would say that connectedness and longitudinal coherence could have been combined as they do have very strong links with one another.
After a lot of research, I finally found some work which has helped me have a better understanding of what I believe to be longitudinal coherence from a teachers perspective:
“We produce many students who do not think globally – or to use a more common word these days, holistically- about mathematics. In the present context, teachers who come through such a training program may know the individual pieces of the school curriculum, but they are less adapt at seeing the interrelationships among topics of different grades. (Wu, 2002,p.19)
The above quote came with an example of helping students see the connections and coherent development of whole numbers all the way through to algebra:
Whole numbers ———> fractions —————> finite decimals, ratio, rates, percent, algebra (p.20)
Maybe Wu (2002) provided a simpler definition of longitudinal coherence than Ma (2010), or maybe because his description was accompanied by examples I was able to follow it better and have a clearer understanding. My role as a teacher is to continually encourage pupils to identify recurring themes and mathematical concepts when approaching new topics. Pupils should be able to see and draw on previous learning to help them develop new understanding. This should happen throughout the whole-school mathematics curriculum to enable students to see why previous learning was relevant and how it is supporting them in their current and future experiences.
Although this property initially baffled me, it is now the property which I connect with the most as I don’t feel I was given the opportunity to develop this at school. If anything, this places me in better stead for my future teaching pedagogies. I will always be able to look back on my own mathematics journey and ensure that I do the opposite to what I experienced at school.
Reflecting on my engagement with this module so far, I have found it extremely beneficial to breakdown the four properties of PUFM, (connectedness, multiple perspectives, basic ideas and longitudinal coherence), in order to develop my understanding of them. I feel that I can now engage with upcoming lectures with a different perspective and approach to mathematics. I want to be able to connect with the different topics we cover on a deeper level. I want to see how I can apply the 4 properties to help develop my own mathematics confidence and also my competence in developing positive teaching strategies.
Sources
Ma, L. (2010) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States. London: Routledge
Wu, H. (2002) “Longitudinal coherence of the curriculum” in What is so difficult about the preparation of mathematics teachers?. University of California: Berkley. Available at: https://math.berkeley.edu/~wu/pspd3d.pdf Accessed 31/10/15
