Using design to enrich mental constructs of a mathematical concept Dr Zingiswa MM Jojo Department of Mathematics Education University of South Africa

Introduction Instructional design is the process by which instruction is improved through the analysis of learning needs and systematic development of learning material (Carter, 2011) creation of effective meaningful lessons helping students to make sense of information cut through extraneous information The choice of instruction to be used in a lesson depends on:

The design of instruction and activities the teacher’s knowledge of the concept, preconceptions, misconceptions and the difficulties that learners could experience in learning the concept. To understand a particular mathematics concept or topic involves knowing the relationship between various topics and where a particular topic fits in the bigger picture

Curriculum Shifts Learner centred approaches and arguments have replaced examination focused, teacher- centred, and content driven approaches for deep understanding of mathematical concepts These bear evidence of teachers’ struggle in putting them to practice in their classrooms in South Africa due to both cultural and resource contexts.

Understanding a mathematical concept Wiggins: (1) explanation, (2) interpretation, (3) contextual applications, (4) perspective, (5) empathy and (6) self-knowledge.

In order to think about mathematical ideas there is a need to represent them internally in a way that allows the mind to operate on them As relationships are constructed between internal representations of ideas, they produce networks which could be structured like vertical hierarchies or webs.

With regard to learning mathematics with understanding: a mathematical idea or procedure or fact is understood if it is part of an internal network the mathematics is understood if its mental representation is part of a network of representations. The degree of understanding is determined by the number and strength of connections.

Initial genetic decomposition (IGD) IGD refers to the set of mental constructs which the learners should construct in order to understand a given mathematics concept. The genetic decomposition was composed in terms of mental constructions (actions, processes, objects, schemas) and mechanisms (contrast, separation, generalisation and fusion) learners might employ when learning inequalities.

APOS and Variation

Variation Interaction Variation interaction is a strategy to interact with mathematics learning environment in order to bring about discernment of mathematical structure. Variation is about what changes, what stays constant and the underlying rule that is discerned by learners in the process. Instructional design to enrich mental constructs from the lens of variation and how learners are brought to the schema level of understanding a concept.

Variation Based on Leung (2012) ‘the object of learning’ i.e. What is to be learnt? –discernment of what is to be learnt in the lesson Contrast presupposes that for one to know what a concept is, he/she has to discern and know what it is not. e.g. examples and non-examples Separation assumes that all concepts have a multitude of features, each of which give rise to different understandings of the concept. Generalisation refers to the verification and conjecture making activity that checks out the validity of a separation.

Variation continued Fusion is the simultaneous discernment of all the critical features of a concept and a relationship between them which allows a learner to make connections gained in past and present interactions

Research Question What is the nature of instructional support that can generate in students the kinds of mental representations that will enable them to think about these critical differences when engaging in symbol manipulation activity involving inequalities?

Questions to consider for instruction What are students’ conceptions of inequalities? What is typical correct and incorrect reasoning? What are common errors? What are possible sources of students’ incorrect solutions? What are promising ways to teach the topics of inequalities?

Challenges for Teaching the topic Inequalities are: taught in secondary school as a subordinate subject (in relationship with equations), dealt with in a purely algorithmic manner, taught in a manner to avoid the difficulties inherent in the concept of function. taught in a sequence of routine procedures, which are not easy for students to understand, interpret and control algebraic transformations are performed without taking care of the constraints deriving from the fact that the > sign does not behave like the = sign

Actions- Physical manipulations external to the mind Separation- discernment of the critical characteristics of a concept to differentiate it from others Objects-process encapsulated- Schema- reflection on process as a totality of knowledge Processes- Action interiorisation- transformations in the mind Fusion- Totality of actions Contrast- to know and discern what a concept is and what it is not Generalisation- verification and conjecture making activity on the separated pattern

An inequality is like an equation, but instead of an equal sign (=) it has one of these signs: < : less than ≤ : less than or equal to > : greater than ≥ : greater than or equal to

“x ≥ -2” means that whatever value x has, it must be greater than or equal to -2. Try to name ten numbers that are greater than or equal to -2!

Numbers greater than -2 are to the right of -5 on the number line If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right. There are also numbers in between the integers, like -1/2, 0.2, 3.1, 5.5, etc. The number -2 would also be a correct answer, because of the phrase, “or equal to”. -2

Solve an Inequality w + 5 < 8 We will use the same steps that we did with equations, if a number is added to the variable, we add the opposite sign to both sides: w (-5) < 8 + (-5) w + 0 < 3 w < 3 All numbers less than 3 are solutions to this problem!

More Examples 1 -2y ≤ y + (-1) ≤ 5 + (-1) -2y ≤ 4 All numbers from -2 up (including -2) make this problem true!

Example

Teacher 2 - Concept Map- IGD

Contexualising the problem The doctor instructed my grandfather to take no more than 3 pills per day My instructor advised me to run at least 10km per day to prepare for the marathon I ate at most two meals today

Example “x < 6” Teacher 2:What is meant by this? Are those the only numbers? To the answers given, the teacher probed by suggesting non-examples How can we represent this on the number line? Opened up the scope of knowledge- encouraging critical thinking The learner has to convince himself- Internalize the knowledge

Discussion-Teacher 1

Teacher 2 Contextualised the problem- using an authentic task as an instructional strategy make sense of some of the exceptional transformation rules used in solving inequalities properties underlying valid equation-solving transformations are not the same as those underlying valid inequality-solving transformations multiplying both sides by the same number, which produces equivalent equations, can lead to pitfalls for inequalities

Conclusions and recommendations Discernment of the concept - the learner to be aware of certain features which are critical to the intended way of seeing this concept Highlight the essential features of the concepts through varying the non- essential features Give the learners a chance to demonstrate argue and explain their solutions to others

See learners as constructors of meaning help learners to actively try things out, Experience the construction of multiple perspectives of mathematical concepts, Find components of the concept that are interconnected with each other

extend the original problem by varying the conditions, changing the results and generalize; (2) multiple methods of solving a problem by varying the different processes of solving a problem and associating different methods of solving a problem (3) multiple applications of a method by applying the same method to a group of similar problems.