Category Archives: 2.3 Pedagogical Theories & Practice

Profound Understanding of Fundamental Mathematics

If someone asked me two months ago what ‘profound understanding of fundamental mathematics’ was, all I could tell you it was something to do with maths. PUFM sounds very confusing and complicated, when in fact it is quite the opposite. Over the course of this module, my understanding of this statement has developed greatly, and I now feel much more confident in my abilities of teaching in a classroom setting.

Ma (2010) defines PUFM as “an understanding of the terrain of fundamental mathematics that Is deep, broad and thorough.” There are four principles in teaching and learning that represent a teachers understanding of fundamental maths in the classroom: interconnectedness, multiple perspectives, basic ideas and longitudinal coherence.

Interconnectedness – 

Ma (2010) defines interconnectedness as when “a teacher with PUFM has a general intention to make connections among mathematical concepts and procedures.” Interconnectedness occurs when the learner is trying to make a connection between the mathematical concepts and procedures. Here, it is the teachers job to stop the learning from being fragmented, and help the students to develop the ability to make links between underlying mathematical concepts, and showing how maths topics rely on each other.

Multiple Perspectives –

“Those who have achieved PUFM appreciate different facets of an idea and various approaches to a solution, as well as their advantages and disadvantages” (Ma, 2010). Multiple perspectives is when a learner can approach mathematical problems in many ways, as they understand the various perspectives when taking on a maths questions, and are able to look at the pros and cons of the different viewpoints. It is the teachers job to provide opportunities for the learners to be able to think flexibly about the thinking and understanding of different concepts in maths.

Basic Ideas –

MA (2010) says that “teachers with PUFM display mathematical attitudes and are particularly aware of the ‘Simple but powerful basic concepts of mathematics’ (e.g. the idea of an equation.”  Basic ideas is a way of thinking about maths in terms of equations. Ma (2010) has suggested that learners should be guided to conduct ‘real’ maths activities rather than just approaching the problem when practising the property of basic ideas. If a teacher is implementing this principle effectively, then they will not only be motivating the learner to approach the maths problems, but will also be providing a guide to for the learners to help them understand the maths themselves.

Longitudinal Coherence –

“Teachers with PUFM are not limited to the knowledge that should be taught in a certain grade; rather they have achieved a fundamental understanding of the whole elementary mathematics curriculum” (Ma, 2010). Longitudinal coherence is when a learner recognises that each basic idea builds on each other. When a learner does not have a limit of knowledge, it is impossible to identify the level or stage that a learner is working at in maths. the learner has instead achieved a holistic understanding of maths (fundamental). A teacher who has a profound understanding of maths is one who can identify the learning that has previously been obtained. The teacher will then lay the fundamental maths as a foundation for learning later on (Ma, 2010).

The four principles are vital to having a deep understanding of maths, especially when teaching the subject. I will make sure I have PUFM before I begin teaching mathematics, as without it, I cannot teach the subject as in depth and as thoroughly as possible.


Ma, L. (2010). Knowing and Teaching Elementary Mathematics. New York: Routledge.

Maths, Play and Stories

How could maths, play and stories intertwine? How could a child possibly be learning maths whilst playing with their friends, or getting read their bed time story? Maths is all around us, even where we least expect it.

Parents as Teachers

Parents may think the only way they can engage their children at home with maths is by helping them with their homework. Some parents may not even be able to help their children with their homework due to the maths anxiety that many adults suffer. As I have spoken about in a previous blog post on maths anxiety, the negative connotations that parents have about maths, can easily be passed down to the child, and create them to have their own worries about maths. It is vital that parents provide a rich learning environment for mathematics at home, so that children can be learning at all times and reaching their full potential. Studies looking at how often numeracy activities occurred at home found that they often happened less than once a day. They also found that the activities rated by experts as having specific numeracy focus occurred more infrequently than the activities with ambiguous numeracy content (Skwarchuk, 2009). Much of a child’s early mathematical development is enhanced through communication. This is why parent should ask open-ended questions to support and challenge their child’s thinking. For example, using varied mathematical language like bigger, smaller, fewer etc.


Importance of Playanalysis, blackboard, board

Lev Vygotsky says that the learning of a child takes place in the ‘zone of proximal development’ which represents the between what a child actually knows and what the child can learn with support from those who are more knowledgeable. He also believed that the teaching of maths can be influenced by relating the subject to the child’s own experiences. This helps us to understand why play is such an important aspect of learning maths, as it allows the child to personalise it and be able to relate it to themselves. Saracho (1986, cited in Saracho and Spodek, 2003, p.77) explains that when children play, they confront social circumstances and learn to collaborate,  help,  share,  and  resolve  social  difficulties. Play is extremely important to a child’s learning for some of the following reasons:

  • Helps children to make connections in their learning
  • Allows child to experiment
  • Provides a meaningful context
  • Promotes social learning
  • Encourages perseverance

Friedrich Froebel was another theorist that also emphasised the importance of play in children. He viewed play as the ‘work of children’ and believed that children’s best thinking took place during play. During quality play, children are:

  • making decisions
  • imagining
  • reasoning
  • predicting
  • planning
  • experimenting with strategies
  • recording

Susan Issacs had many of the same views as Froebel. She saw the value in play as a way to allow children to explore their ideas and feelings freely. Through play, children can move in and out of reality, and whilst doing this, she encouraged them to be curious and express their feelings. But how does this relate to maths? Nutbrown (1994) said that “mathematics is never far away from young children’s actions.”  All the different things that happen during quality play (as listed above) all link into mathematics. Predicting, experimenting, reasoning, making decisions are all needed in maths. Children using these strategies in play will help them to have a better understanding of the approaches they must use. It will also help them to understand why they are learning maths, as they can relate the problem solving and reasoning to their real-life experiences.


Maths in Stories

Another way that maths can be incorporated into a child’s everyday life is through White Teddy Bear With Opened Book Photostories.  Stories are something that are enjoyed by children, and can help eliminate that fear of maths that a lot of children have.  They can be used to introduce new mathematical concepts, or to build on ones that are already known. By showing children that maths can be fun and interactive, they will be much more willing to engage.

Here is an example of a math story book ‘A Place For Zero – A Math Adventure’. This story talks about the number zero and place value.

Stories like this one can help children to connect with, and understand the concept being portrayed in the book. Having a visual aid will allow the children engage in the book, and listen to the story line. At the time, they may not even realise that they are actually learning mathematical strategies.

Maths storybooks are not the only way to teach maths through stories. There are many popular children’s books that can easily be adapted to fit a mathematical story line. For example, ‘We’re Going On a Bear Hunt’ can easily be change to ‘We’re Going On a Square Hunt’. Using a familiar text will allow the child to acknowledge how maths can be related to them, and used in their day to day life. It may also make the experience more comforting, by having something the recognise, especially for those that suffer from maths anxiety. You should always remember to match the book and your discussions to the mathematical abilities and development of the children in your class.

From the workshop and my own research, I now understand the great importance of play in developing a child’s academic understanding, by allowing them to freely explore ideas and express their emotions. Not only in maths, but in every aspect of schooling, play can help children relate it to their own experiences. When working in classrooms in the future, I will make sure to incorporate all that I have learnt about play and stories to make the learning as fun and interesting as possible, as I now know the benefit of them.


Ehrhart, M. (2014) A Place For Zero A Math Adventure [Online].  YouTube. Available at:–wKA1yYQ&t=487s [Accessed 1st November 2017]

Nutbrown, C. (1994) Threads of Thinking. London: Paul Chapman Publishing Ltd.

Saracho, O.N. and Spodek, B. (2003). Contemporary Perspectives on Play in Early Childhood Education. Conneticut: Information Age Publishing.

Skwarchuk, S. (2009). How Do Parents Support Preschooler’s Numeracy Learning Experiences at Home?. Early Childhood Education Journal, [Online] 37. Available at: [Accessed 30th October 2017]

Valentine, E. (2017) “Maths, Play and Stories” [Powerpoint Presentation] ED21006: Discovering Mathematics [Accessed 31st October 2017]

Health and Wellbeing – Relationships

Yesterday, we had a health and wellbeing lecture about relationships. It was very interesting to learn about the kind of relationships children form at each age group and how they change as they grow older. To help further our understanding of the importance of relationships in the early years of a child’s life, we were asked to watch two videos,  one from Suzanne Zeedyk and  one from John Carnochan.


In the video, Zeedyk explains that human babies are born prematurely in comparison to other mammal species. This results in the human babies being born with an undeveloped brain. The brain is left to develop outside of the womb and the environment that the child is in can have a significant impact, positive or negative,on the development of the child’s brain.  The relationship’s that the baby forms in the first few years of its life are vital. The first four years of a child’s life are the most important years and can impact them for the rest of their life.

If a child is living in a household with domestic abuse, then their brain has to develop to cope with the threatening nature of this environment. As a result of this, they are using so much energy looking for their next threat that they can’t concentrate or learn. Carnochan goes onto mention that children need consistency in their lives. They may not be getting this at home in their threatening environment, so going to school can be their one happy place away from their troubles. As a teacher, it is valuable to recognise the importance of making your classroom a fun, safe and welcoming place for every child, especially if it is going to be an escape from their home life. Even children that are not facing difficulties at home need this environment at school.

After watching these two videos, it has made me more aware of the importance of relationships in a young child’s life and the valuable role of teachers if these relationships cannot be formed at home. Seeing how this affects a child’s learning and capabilities, it will allow me to have a wider understanding of every child and why they are acting the way they are. In turn I can accommodate my teaching methods to fit to every child and make them feel happy and safe inside my classroom.