What comes after nine?

What comes after nine

Arbitrary and necessary: part 1, a way of viewing the mathematics curriculum

Arbitrary and necessary: part 1, a way of viewing the mathematics curriculu

“Never carry out any arithmetic”: the importance of structure in developing algebraic thinking

The use of tasks which ask learners to find general rules for growing figural patterns is widely reported in the algebra research literature. Such tasks are sometimes seen as a way to develop early algebra thinking. This paper looks at examples of such activities from the literature and presents a theoretical argument against the common practice of learners creating a table of values and seeking patterns within the numbers. Instead an argument is made for learners to focus on more complex examples where learners are discouraged from counting and turning the original figures into numbers. Instead, it is suggested that learners seek structure within complex examples and express what they see without carrying out any arithmetic but instead just writing down what arithmetic they would do

Young students learning formal algebraic notation and solving linear equations: are commonly experienced difficulties avoidable?

This study looks at a mixed ability group of 21 Year 5 primary students (aged 9-10 years old) who had previously never had formal instruction using letters to stand for unknowns or variables in a mathematics context; nor had they been introduced to formal algebraic notation. Three lessons were taught using the computer software Grid Algebra where they began working with formal notation and were solving linear equations with some degree of success by the end of the lessons. The teaching was such that nothing was explained or justified by the teacher explicitly. The students appeared either not to meet, or to overcome quickly, some of the difficulties identified within previous research studies. They demonstrated remarkable confidence working with complicated linear algebraic expressions written in formal notation. A key feature of the software activities was that formal notation continually needed to be used and interpreted, and the software provided neutral feedback which enabled the students to educate their interpretation of the notation. © 2012 Springer Science+Business Media B.V

Arbitrary and necessary: part 2, assisting memory

Arbitrary and necessary: part 2, assisting memor

Designing educational software: The case of grid algebra

To be successful educational software needs to be carefully designed in order to provide constraints and freedoms (Greeno 1994) which relate to its educational aims, restrict or enable learners and teachers to carry out certain actions, and be based upon a clear set of beliefs related to the complex dynamic of teaching and learning. This paper seeks to show the contraints and freedoms built in to the design of Grid Algebra; software which offers an intuitive and novel approach to early algebraic learning, and how they have been informed by a clear set of beliefs and frameworks. The paper shows how the design has taken into consideration known misconceptions that many learners exhibit with early algebra work and how the three frameworks of arbitrary and necessary, subordination and embodied cognition have been fundamental to the design of both the software and some of the activities used with the software. The paper finishes off with considering limitations of the software as well as some strengths that its design offers, and poses questions to consider about the design of this and other resource

The role of subordination and fading in learning formal algebraic notation and solving equations: the case of Year 5 students

A group of 9-10 year olds, who previously had not met letters or formal algebraic notation, were taught over three lessons which led up to solving linear equations using software which produced formal algebraic notation as a consequence of making movements round a grid. Tasks using the software were understandable in terms of movements but the only information provided to carry out tasks was the formal notation. This feature of subordinating the notation to the required task was examined along with the way in which the strong visual support offered by the software was faded in activities. Students gained some success in solving linear equations but the greatest success came in the way students became confident with reading and writing formal notation

Forcing awareness

Forcing awarenes

Arbitrary and necessary: part 3, educating awareness

Arbitrary and necessary: part 3, educating awarenes

The economic use of time and effort in the teaching and learning of mathematics

I start with two statements: 1. The learning of very young children before they enter school is impressive. 2. The learning of those same children, later on when they are in secondary school, is less impressive. With respect to the first, newly born children cannot walk, speak in their first language, control their bowel movements, feed themselves, throw and catch things,…, etc. The list goes on. For the second, I look at the mathematics curriculum at the end of primary school and compare this with the end of high school and the difference does not seem so profound. Of course, there are many other subjects as well but overall I find myself far more impressed with the learning which takes place in a child’s first few years (see Hewitt, 2009, for how observation of this has helped me reflect upon my practice)