Post-Compulsory PGCE Mathematics with Numeracy
Module 3: Mathematics and Numeracy Theoretical Frameworks and Curriculum Design
This module is a subject specialist module that combines key elements of mathematical knowledge with elements of teaching practice. The focus is on course and curriculum design, how the context of learning informs such design as well as how best to produce an inclusive learning environment. The module takes a particular look at statistics, algebra, geometry and proof and offers a range of approaches to these mathematical topics which will aid teaching and learning.
The main aims of the module are:
Module Learning Outcomes
By the end of the module trainees should be able to:
Module 3 Timetable
Academic writing centre sessions
|1st Thursday 11.10.18 10.00-13.00||537||Academic writing centre||AWC|
|2nd Thursday 18.10.18 10.00-13.00||537||Academic writing centre||AWC|
|3rd Thursday 22.11.18 14.00-15.30||537||Academic writing centre||AWC|
|4th Thursday 24.01.19 14.00-15.30||537||Academic writing centre||AWC|
|2.00 – 3.30||Tutor’s|
|1st: 01.11.18||537||Introduction to the module|
Course design – elements and
|AV||Discussions on teaching and learning|
– setting aims and objectives
|GG / AV|
|2nd 08.11.18||537||Situated and ethno- mathematics|
(numeracy as social practice)
|GG||Using ICT in teaching||AV|
|3rd 15.11.18||537||Numeracy in different environments, such as family learning,|
|JA||Discussions on teaching and learning|
- classroom management
|GG / AV|
|4th 22.11.18||537||Representations of data||AV||Academic writing centre||AWC|
|5th 29.11.18||537||Curriculum models: principles and practice||JA||Voice projection training session||Dr Sean Richards|
|6th 06.12.18||537||The process of solving|
|GG||Safeguarding / Prevent / GDPR||JA|
|7th 13.12.18||537||Working with geometry||AV||Discussions on teaching and learning|
– dealing with language
|GG / AV|
|8th 10.01.19||537||Nature of mathematical knowledge & implications for|
|GG||Table top discussions on course design for assignment.||GG|
|9th 17.01.19||537||Working with algebra||JA||Trainees choose their own topic Eg particular teaching approaches, core|
|GG / AV|
|10th 24.01.19||537||Mathematics, proof and the curriculum||GG||Academic writing centre||AWC|
This module will help you develop knowledge, understanding and skills related to the professional standards.
Professional values and attributes
Develop your own judgement of what works and does not work in your teaching and training
Professional knowledge and understanding
Develop deep and critically informed knowledge and understanding in theory and practice
Develop your expertise and skills to ensure the best outcomes for learners
In particular, the module will develop the following aspects of practice:
|Issues around curriculum development and inclusive practice for mathematics and numeracy teaching|
|1||Understand the range of contexts in which education and training are offered in the lifelong learning sector|
|2||Understand theories, principles and models of curriculum design and|
implementation and their impact on teaching and learning
|3||Understand the significance of equality and diversity for curriculum design, and take opportunities to promote equality within practice|
|4||Understand and demonstrate how to apply theories, principles and models to curriculum development and practice|
|Knowledge and understanding related to mathematics and numeracy|
|1||Demonstrate a knowledge and understanding of the origins of mathematics knowledge and the application of relevant learning theories on curriculum development, learning and teaching|
|2||Apply specialist pedagogical knowledge and understanding to promote and|
develop the quality of numeracy learning and teaching in a variety of contexts
|3||Demonstrate own mathematics skills, knowledge and understanding to an appropriate breadth and depth|
|4||Demonstrate knowledge and understanding of how to use specialist organisations and publications to develop own practice as a numeracy teacher.|
The following opportunities are available for trainees to feedback on the module:
All trainees are encouraged to raise issues with their tutors.
Summary of assessment for module 3
1. Written assignment
Analyse a mathematics or numeracy programme designed for an identified target group. Discuss the effectiveness of the course in providing inclusive learning opportunities for mathematics or numeracy learners.
This assignment has been devised to encourage you to conceptualise, amend / design, plan and analyse a short course of at least 20 hours.
The following are examples of possible themes for your course:
2. Professional Practice Portfolio
Develop your professional practice portfolio with three directed entries.
Continue to develop your teaching practice and plan for formal observations.
Assignment levels and word count
Full time trainees complete this module at level 6.
Part time, year 2 trainees may complete this module at Level 6 or Level 7 (this is to be discussed with your tutor).
The level of analysis required for each level is described in the outcomes and pass/fail criteria for each level. In essence, the discussions will involve more critical analysis and reflection using a wider range of sources at level 7 than at level 6.
Word count: Level 7 - 4000 words plus course description / scheme of work; Level 6
– 3000 words plus course description / scheme of work.
The date for uploading a formative draft of Module 3 assignment is 24th January 2019. Negotiate extensions of drafts with your tutor.
The date for posting the final submission of module 3 is: 28th February 2019. This submission date must be met. Extensions are only given through the university formal system with appropriate evidence presented in good time.
Semester 1 schedule of assessed work and deadlines
|w/b||Module 1 Assessment activity||Module 3 assessment activity||Portfolio||Observations|
|Module guide / assessment Information|
|ILP Stage 1|
|Module guide / assessment Information||FT Obs 1|
|FT Obs 2|
PT Obs 1 / 5
|Submission draft||DE observing another context|
|FT Obs 3|
|Submit final e-version (and email admin)|
|DE maths development||FT Obs 4|
PT Obs 2 / 6
|ILP stage 2 & DE progress|
|FT Obs 5|
Developing Problem-Solving in Mathematics
Mathematics, as a subject, is considered to be the “Father of all Sciences” and taught at every level of education. In spite of its wide popularity and reach, mathematics continues to be the most disliked subject for students worldwide, with an overwhelming majority of students finding the subject difficult and unpleasant.
Mathematics includes the study of number, quantity, structure, space, and change. Children are taught basics of mathematics like numbers at the playschool or nursery itself. Mathematics is a compulsory subject for all the students till class 10th. A strong mathematics foundation would help the student throughout his/ her life. It would help the students excel in the academic areas as many subjects (physics, chemistry, economics, finance and so on) have a mathematical base. If a student is good at mathematics, it would help him set his career (Bradshaw and Hazell, 2017). Be it a chef, designer, architect, doctor or engineer, everyone needs the support of mathematics in everyday life.
Apart from the academics, learning mathematics and being good at it helps the student in many ways. The problem- solving skills of the student is improved. He/ she would be able to analyze the problems in real life and critically think the solution outcomes to it. It would also inculcate the power of reasoning, creativity, spatial thinking and, abstract thinking. Therefore, mathematics must be taught and learned by all the students. Some students may develop a fear for the subject. It must be eradicated at the early stage (Kilpatrick, 1969). Through interesting and engaging training sessions and with the help of proper and dedicated teachers, mathematics can be made fun and easy to learn.
An alarming level of mathematical incompetency plagues our schools and it is imperative, that we take steps to change the way we teach mathematics in our schools in order to build a curriculum that encourages the development of mathematical skills by bringing in various, more child-friendly, learning strategies. In light of this need, more than a few groups of schools have adopted the technique of ‘Cooperative Learning' for middle school students. The technique aims to revolutionise the way schools teach mathematics and replace the current teaching methods with a more inclusive method of teaching.
Cooperative learning is a fairly generic term used to describe teaching techniques that are implemented with the aim of teaching academic and collaborative skills to students in small, heterogeneous groups. The technique has been shown to decrease peer pressure and isolation, promote academic development and promote the building of positive relationships (Kilpatrick, 1969).
Mathematical instruction through cooperative learning contrasts greatly from the traditional methods of teaching mathematics. In cooperative learning, the students work in groups of four or five to work out the problem and show how they connected various ideas and mathematical concepts in order to arrive at the solution. In this manner, students not only retain the information learned for a long time but also learn from their mistakes.
At the outset, it is important to understand that cooperative learning does not involve a strict set of methods to be followed rather, it is a lot more fluid style of teaching that can be modified and adapted depending upon the need of the students and the facilities available to the teacher (Leikin, 2003). As long as the teaching methods focus on learning mathematics while letting the students work in heterogeneous groups and utilising their knowledge to arrive at the solution through discourse and discussion, the methods fall under the bracket of cooperative learning.
The three-step method for cooperative learning consists of three major components, ‘lesson preparation’, ‘lesson instruction’ and ‘lesson evaluation’. Depending on the States’ standardized curriculum for mathematics students, the course’s learning outcomes and observations made about the class’s mathematical and social skills, a three-step plan is devised that works to maximise mathematical learning and teamwork among the students.
Lesson Preparation is the first step in the cooperative learning technique. It starts with the teacher defining the objective for the lesson which could be something as simple as “At the completion of this lesson, students will be able to solve quadratic equation problems with 90% accuracy.” Once the objective is set, activities can be designed around it to maximise learning efficiency.
The first step is to structure the activity. The students review the steps for solving quadratic equation problems that were learnt the previous week. After the review, groups of 4 to 5 students are made and roles are assigned to individual students, a student will monitor the time and keep the group on task, another will manage the material given to solve the problems (a single strategy cue-card should be given to each group to promote cooperation), a writer will make note of the group’s progress and a spokesperson will lead the group’s discussion regarding the problem. Individual tests can be given later, to test the progress of the students (Mulyono and Hadiyanti, 2018).
The lesson instruction begins with a verbalisation of the lesson objective and the teacher's role in the teaching-learning process. The problem-solving strategy is discussed by having the students explain the strategy individually and in groups and solving a couple of problems with the students while explaining the strategy. It is important to explain the cooperative learning activity and focus on student roles and their individual contributions to their groups. Students must be advised to tackle problems with a step-by-step approach. Cooperative behaviour and teamwork must be reinforced and encouraged.
Lesson evaluation is the most fluid of the steps and a variety of approaches can be taken to evaluate student performance. The first type of evaluation is done during the activity, where evaluative comments about student performance are noted. The second kind involves the administering of an individual test to gauge student performance and the third kind involves letting the students evaluate themselves according to pre-specified standards.
Cooperative Learning also involves a series of activities aimed at creating an environment where students feel free to make mistakes. Some of those techniques include,
Rally coach is one such technique. Students work in pairs where one person talks and explains the problems while the other writes down the steps. If the writer disagrees on a step, they switch roles and continue. Once an answer is reached, it is compared with a neighbouring pair. This technique increases intelligent discourse and support among students (Neef et al., 2003).
Another technique is called ‘Hot Potato’. Each student in a group of four writes on a single paper in different colours. This way both the students and the teacher can keep track of each member’s contribution.
Proper training along with a dedicated teacher would help the students with the subject and also in developing the reasoning and critical thinking skills.
Let us take a numeracy program for the students of 13 years of age. The main topics taught to these students include mensuration, comparing quantities, introduction to graphs, linear equations, practical geometry, visualizing shapes and many more.
A numeracy program for the age group of 13- 14 years must include all these topics. An effective program must not only teach the syllabus and help the students in scoring good marks but should also help the students think beyond the books and help them understand the applications of mathematics in everyday life.
For example, students find the topic algebraic expressions and identities quite difficult. But it can be taught easily and effectively by using some techniques and methods. Engaging videos can be used to teach the students. This video would be an interesting way to learn and the students would not get bored and give their full attention to the video. Competitive games can be conducted which would help the students in understanding the concepts better and learning the formulas and math equations (Sahendra, Budiarto and Fuad, 2018). Word wall is another lucrative concept. Colorful papers are stuck on the walls of the classroom. On the papers, formulas and equations are written down. These visual reminders are an effective way to remember and help the students in memorizing numerous formulas. Real life examples can be given for word problems, which would develop an interest in the students. Another topic is practical geometry. In this, sub-topics like measuring angles, constructing quadrilaterals and so on are covered. Geometry can be made fun by combining it with the art and craft class. Origami is one of the best and easy ways to learn geometry.
Math yoga is another interesting activity. In this, the students are asked to form shapes or stand at particular angles. This would help them understand the geometrical concepts while playing and enjoying. For the topic, visualizing solids activities like the geometry scavenger hunt, quiz, state the examples from the classroom and so on can be carried out. Mensuration is one of the most interesting topics but many students find it difficult. This is probably because the topic deals with both 2D and 3D figures. This results in an increased number of formulas and concepts, to find out the area, volume of the shapes like sphere, circle, cylinder, cube, cuboid and so on (Sigurdson, Olson and Mason, 1994). The formulas can be easily memorized by the students through various activities. The formula chart could be posted on the walls. Competitive games and multiple- choice question sessions can be conducted. This would help create a fun and competitive environment for the children, helping them to learn quickly.
There are many such topics to be taught. All these topics find their applications in everyday life. These topics must be taught in such a manner that they are enjoyed and easily understood by the students. The three stages that must be followed by schools or tuitions for arranging numeracy program are design, methodology, and approach. The numeracy program must be planned according to the target group. The children, their physical, social and mental factors must be considered before planning the numeracy or mathematics session. Before starting the session, an assessment test must be conducted. This test would help in understanding the knowledge of each child and average knowledge of the whole class (Tambychik and Meerah, 2010). By knowing this, the tutors or the teachers would be able to understand each and every child and would be able to help them out. The numerical and problems of the topics must be solved quickly and efficiently by the students. This would not only help them in facing the exam confidently and gaining good grades but would also help them throughout their life.
Mathematics is not only an academic subject. It teaches many other important life lessons and skills. The main skill that is needed by every human being to survive in this world is the problem- solving skill and decision- making skill. All human beings, be it man or woman, rich or poor, or from any profession, face problems in life. These problems act as hurdles in the path of achieving the goal. The person must be able to handle the problem, find a solution and overcome it with confidence (Xin et al., 2011). A person can achieve success in life, only when he/ she learns to face the problem without fear and has the knowledge and will-power to overcome it.
Even nature follows the law of mathematics. The days, duration of each day, number of days in a year, everything is based on mathematics. Various topics of mathematics find various applications in everyday life. For example, statistics would help a person in sampling various options and gives a high certainty answer. Game theory would help in deciding the best strategic move. Logic would help the person in figuring out many things, even the complex problems for which, determining the solution may seem impossible. Mathematics helps in analyzing the problem in many ways. The solution can be obtained by not one or two, but infinite ways, depending on the person.
Professional Practice Portfolio
My own approach towards teaching mathematics has always been incredibly simple, and unfortunately not inclusive of students with different levels of learning ability. I would explain the theory, solve one problem with the class, have the students solve one problem themselves and move on to the next topic.
That was until I attended a chemistry lecture. The professor’s lecture was less about memorising concept and solving problems and more about linking the concepts to things we see everyday. His lecture did not focus on the properties of acetylene, it focussed on understanding why bananas when kept in a bunch in the open air, ripened faster than ones kept separately. Acetylene released from one ripening banana affects other bananas and causes them to ripen further.
That’s when it struck me, just like chemistry, mathematics is all around us and it only takes a bit of encouragement for students to be able to see and appreciate the beauty of mathematics. As is obvious, my teaching style has changed a lot since then.
A high school mathematics teacher I met at a conference, had an interesting teaching style. She taught mathematics in a troubled neighbourhood where most of the students came from broken homes; custody issues, abuse, homelessness, addiction and alcoholism were common among her students. The students were sensitive, and she needed to ensure that not only did the students learn the mathematical concepts but also did so without feeling stressed out enough that they felt the need to rely on their addictions. Her experiences taught me to look for tell-tale signs of stressed students in my own classroom, to slow my pace enough that even students who faced the most difficulty in the classroom could keep up and ensure that the students felt comfortable enough to raise their difficulties without feeling the pressure to excel.
Not all the teachers and tutors follow the same method nor technique to teach the students. I was initially just bound to the textbooks and taught the portion and gave the students sums to work on, from the textbook itself. But I realized my teaching method was not effective. Many students scored good grades and most of them passed the exam, but they were simply following the syllabus and regulation of the school. They were not able to apply mathematics to their real-life problems. I had to improve my practice and teaching methods. And one day, I saw my colleague, who is an English teacher, teaching a particular story. But instead of reading the story in a monotonous voice, she made the children enact the story. All the students were assigned the roles from the story, and a skit was carried out in the classroom. This helped the students interact with each other, enjoy the class and also understand the story better. It insulates teamwork spirit in the students. This inculcated many skills like communication, confidence and so on. My friend who is a science teacher explains the concept and later takes them to their school labs or planetariums to demonstrate the concept and its use in real life. This encouraged me to change my method of teaching. If other subjects can be taught in a fun way, then why must math be limited to solving problems with pen and paper? So, I included many fun and group activities for the students. This has proved to be beneficial for the students. They not only score good grades in the exam but also enjoy the learning process immensely. They are also able to relate the mathematical principles to real life. They are able to solve any problem, using the mathematical base. Hence, mathematics would help them in becoming better problem solvers.
Mathematics help a person with solving any issue, be it in a textbook or real life and improves the problem- solving skills. It is a mental activity which consists in carrying out, one after the other mental constructions. It allows a person to think in a more critical and reasonable way and increases the observation capabilities of an individual. People who are good at math are better problem-solvers as they can model the problem, analyze it and solve the problem using mathematical principles and equations (Xin et al., 2011). If a person solves many numerical problems, they begin to see a similar pattern between the problems of both textbooks and real life. It is like an arsenal of tools in your pocket. When a person faces any problem, he/ she can solve it using the mathematical tool or find a new approach altogether.
In all my years as a teacher, I have met a lot of brilliant teachers, but I have also met a number of terrible ones. My own mathematics teacher was, unfortunately, of the latter category. Once beginner’s theory was explained, he would hand us a bunch of questions and leave us to solve them. No explanation of the question was ever given, at best he would ask one of the students who had solved the question, to solve it on the blackboard and that was it. Fortunately for me, mathematics always came easily to me, and as result, I never struggled even with his teaching techniques or lack thereof. It wasn't until I started teaching high school mathematics that I realised the harm his teaching techniques did to students who weren't naturally good with the subject. The inclusion of students with various learning paces is essential, especially as a teacher of mathematics. If teachers do not practice inclusion techniques, they may end up doing more harm than good.
Mathematics Competency Levels today, are appalling, with one in four adults, in even the best-performing countries failing to do simple addition problems without a calculator. The reason behind this becomes pretty obvious once you dig a little deeper. Our teaching techniques are heavily lacking, especially when it comes to providing a learning environment where students of all levels of mathematical ability can learn comfortably. The number of students opting to pursue higher studies in this field is dropping and unless we change the way it is being taught in our schools, mathematics will soon lose its beauty.