Liping Ma (2010) refers to ‘connectedness’ as being a key property of having a profound understanding of mathematics (p.122).

When I think back to my learning experiences at school, we were never made aware of, or allowed to explore, the interconnectedness of different mathematical concepts. Due to this, I now feel that I have a very fragmented view of the subject and I seem to approach different concepts as if they were isolated from all the others.

For my professional development, I think it is important for me to do some research around this area and to explore the importance of making connections; how does it help student’s learn? I hope this process will also allow me to have a better understanding of the underpinning network that runs through different concepts and begin to see mathematics as a coherent whole.

Education Scotland states “Within mathematics there are rich opportunities for links among different concepts.” (Mathematics Principles and Practice, p.4)

Before I did any research into the more complex connections between concepts, I asked myself, Where are the simple/basic connections?

I know that multiplication would be classed as a derived operation because it has links to the processes of addition. Then you have inverse operations; division is inverse (opposite) of multiplication, and addition is inverse of subtraction. These two processes, derived and inverse, are closely interwoven and tightly connect the four basic operations.

Internationally, the National Council of Teachers of Mathematics (2000) state that:

 “Mathematics is not a collection of separate strands or standards, even though it is often partitioned and presented in this manner. Rather, mathematics is an integrated field of study.” (p.4)

This notion of connectedness is promoted vastly throughout curriculums across the globe. Why? How do our students benefit from this approach?

If I had experienced this approach at school, I think I would have had a more coherent view of mathematics as a whole. I would have been able to connect my learning when dealing with different concepts and use these connections to investigate tasks and activities in other topics.

I believe that our students today are getting more than this. The are able to capitalise on the connections they build in mathematics. They use these connections to work through cross-curricular activities. For example, the use of data in tables to draw graphs and identify anomalies in science experiments. The opportunities to create mathematical connections in science is vast. These two subjects are tightly linked in school and further a field in employment and research-based jobs. There are links between shapes and symmetry in art activities, students may have to draw on previous mathematic concepts to make predictions or conjectures about the best way to draw a picture.

We recently had a lecture that focused on the process of tessellation. If I am totally honest, I had never heard that term used in my life and I considered myself a beginner learner in that class. (That being said, I am now learning to see this as advantage because, if I haven’t dealt with a topic before, I have no reason to feel anxious about it or let any previous experiences obstruct my confidence and motivation in the topic.)

Tessellation is the fitting together of shapes without having any overlaps or gaps between them.  We spent time discussing in groups and predicting which shapes we thought would tessellate (we had a selection of shaped on our table to use for our discussion). I never thought to think deeper into the activity and use my knowledge of regular and irregular shapes, polygons and congruent shapes to help me make an informed prediction about tessellation. I was pretty much taking a gamble for my predictions. As teachers we always encourage our students to use prior knowledge and experiences to help make predictions so that they have some meaning and thought behind them. So, in this situation I was contradicting myself and just jumping in without linking or drawing on prior learning.

Throughout the activity, it became clear that shapes would only tessellate if the angle they made when the vertices touched added up to 360 degrees. Therefore, I could have used this information to make a prediction about other regular shapes. (Regular shape has all its sides the same length and the internal angles are all the same size.) This approach would have linked to the process of connectedness and the underlying links between different mathematical concepts.

 

 

 

 

 

 

 

 

 

 

In conclusion, ‘connectedness’ is a crucial aspect of having a profound understanding of fundamental mathematics. It allows teachers, students and other professionals to see mathematics as a coherent whole and develop skills in connecting and linking different mathematical concepts together. These connections allow you to use prior learning and knowledge and apply it to new situations and contexts.

I end with the following clip which I was fascinated by. This teachers has a brilliant way of encouraging and supporting children to use and connect different concepts and apply them in new situations. There is a big emphasis on the vocabulary used, which is vital in making clear links in mathematics and the praise and feedback given by the teacher encourages motivation and problem solving within the activities.

Further reading and links:

Tessellation – Easy to understand!

Promoting connectedness in early mathematics education – This paper, written by Abigail Sawyer, describes the perspectives of two early years teachers involved in a new approach which encourages teachers to support students in making connections between mathematics and real-life experience. It is a very interesting read.

Sources:

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

Scottish Government (2009) Curriculum for excellence Mathematics principles and practice. Available at: https://www.educationscotland.gov.uk/Images/mathematics_principles_practice_tcm4-540176.pdf Accessed: 20/10/15

The National Council of Teachers of Mathematics (2000) Executive Summary: Principles and Standards for school mathematics. Available at: https://www.nctm.org/uploadedFiles/Standards_and_Positions/PSSM_ExecutiveSummary.pdf Accessed:20/10/15