Mathematics Education with Digital Technology

Adrian Oldknow, Adrian Oldknow, Carol Knights, Sue Brindley

• Publisher: Continuum
• Year: 2011
• Print ISBN: 9780567250285
• eBook ISBN: 9781441129857
• Edition: 1
• Topic: Education
• Subtopic: Secondary Education

Mathematics Education with Digital Technology examines ways in which widely available digital technologies can be used to benefit the teaching and learning of mathematics. The contributors offer their insights to locate the value of digital technology for mathematics learning within the context of evidence from documented practice, prior research and of educational policy making. Key pedagogical uses of digital technologies are evaluated in relation to effective mathematics learning and practical ideas for teaching and learning mathematics with digital technology are critically analysed. The volume concludes by looking at future developments and by considering the ways in which ICT could be used as a catalyst for cross-curricular work to achieve greater curricular coherence.

Part One

Where Are We Now?

In this section we have contributions from France, the United Kingdom and the United States reviewing the extent to which digital technologies have become embedded in their respective educational systems, and what trends are now emerging. We have not attempted to get a systematic international survey as it is impossible to discount factors such as – the different attitudes shown by different cultures towards education and mathematics, how centralized the political system is, how schools are managed, what objectives the educational process is seen to serve, and so on. What does seem apparent is that the digital divide between teachers who have access to Information and Communications Technology (ICT) to support their role as a teacher and those who don’t has undergone a rapid change. This has occurred through access to devices such as broadband, the internet, portable computers, digital projectors, interactive whiteboards and virtual learning environments. We can see that teacher groups are developing curriculum resources which are being put into widespread use through digital distribution – even if the result is just a printed worksheet! What is emerging now, with the introduction of ever more portable, powerful and affordable student devices, is the challenge to integrate these into mathematical learning, and assessment. With regard to widespread students’ own use of technology as a research tool, an analysis tool and/or a presentation tool, it looks, in the words of Lewis Carroll, like a case of ‘jam to-morrow’. But we can see some very interesting approaches developing, and we can hope that, unlike the White Queen’s interpretation, ‘to-morrow’ may not be long in coming. Which is why it is also helpful to include a contribution reviewing what we know about how students learn.

Chapter 1

The Neuroscience of Connections, Generalizations, Visualizations and Meaning

Edward D. Laughbaum

Department of Mathematics, The Ohio State University, US

Introduction

Did you ever wonder whether teachers consider basic brain function of students when designing lessons or lectures? That is, do we ever think about teaching to be in concert with how the brain functions? Have we considered capitalizing on basic brain function that will improve understanding and long-term memory with recall? Are we aware that the brain requires neural connections to process understanding, long-term memory, and recall? Do we know whether the brain commonly generalizes through reasoning or through pattern recognition? Do we know why it is important for students to generalize a pattern on their own? Have we thought about whether using visualizations to confirm mathematical processes and concepts holds the same understanding/memory value as does using visualizations to teach processes and concepts? If we knew the answers to these questions, would we change the way we teach? Would textbooks change to facilitate such teaching? Would standards documents focus on teaching instead of providing topic lists?

There are a considerable number of basic brain operating functions that can be applied to the field of mathematics education, but in this chapter, the author will only reference research in brain function related to connections, pattern recognition, visualizations and meaning. Some may question the validity of a proposal to change education to be in concert with basic brain function since the brain is so complex. It is complex, but the proposal to change teaching is based on common neural function (no matter how complex on a cellular and molecular level), and the implementation is somewhat simple. Breakthrough ideas often emerge by applying ideas from one field to another.

Connections in Mathematics = Associations in the Brain

Neural associations are the connections among neural networks that the brain creates automatically and instantaneously when it learns something new. Donald Hebb discovered the creation of associations over 50 years ago. We commonly describe his discovery as ‘Neurons that fire together, wire together’. For example, suppose you want to create neural associations among the numeric, graphic and symbolic representations of a function. It is extremely simple to do. Graph a function on a graphing calculator (or computer) and use trace (with expression turned on) to trace on the graph, or use a graphic/numeric split screen. Both of these options present the brain with the simultaneous representations of a function causing the neural networks for the three representations to be associated (connected). But why is this important?

Current research shows that ‘ . . . the lower left part of the frontal lobe works especially hard when people elaborate on incoming information by associating it with what they already know’ (Schacter, 2001, p. 27). But there are issues when we facilitate associations in maths education. ‘This echo [neurons continuing to fire after the stimulus has stopped] of activity allows the brain to make creative associations as seemingly unrelated sensations and ideas overlap’ (Lehrer, 2009, p. 130). Do we really want students connecting addition of polynomials, for example, to concepts that are unique to each student? Doesn’t it make more sense for the teacher to facilitate the creation of appropriate associations that can be used later in the teaching/learning process? It is possible. ‘Being able to hold more information in the prefrontal cortex, and being able to hold on to the information longer, means that the brain cells are better able to form useful associations’ (Lehrer, 2009, p. 131). Outside of education, it is common to lead an audience to connections of choice. For example, ‘Advertisers don’t wait for you to develop your own associations. They go ahead and program you with theirs through television [like a cool-looking person smoking, or females showing interest in guys in cars]’ (Brodie, 1996, p. 25). Of course, what politician has not used the word ‘trust’ on the same TV screen with their name? Recall that it is rather simple to create associations. Simultaneously present the brain with the concepts/procedures you want connected. But again, why are connections important?

Teachers must create connections to improve the memory of the mathematics taught. ‘Memory recall almost always follows a pathway of associations. One [neural] pattern evokes the next pattern, which evokes the next pattern, and so on’ (Hawkins, 2004, p. 71). In teaching factoring of polynomials, one would connect the new maths being taught to the previously taught concept of zeros of a function. Using hand-held or computer technology, it is relatively simple to find zeros of polynomial functions expressed as rational numbers. By connecting the two processes, when students are asked to factor a quadratic polynomial at a later time, they are likely to think of zeros first (because of the visual methods used in teaching), followed by the factoring process.

The most important property [of auto-associative memory] is that you don’t have to have the entire pattern you want to retrieve in order to retrieve it. . . . The auto-associative memory can retrieve the correct pattern, . . . even though you start with a messy version of it. (Hawkins, 2004, p. 30)

So we have good odds that connected concepts will be recallable.

Teachers must also create connections to enhance the understanding of the concept or procedure being taught. That is, ‘We understand something new by relating it to something we’ve known or experienced in the past’ (Restak, 2006, p. 164). The word understanding seems to hold value in the minds of many educators. For example from Keith Devlin,

How many children leave school with good grades in mathematics but no understanding of what they are doing? If only they understood what was going on, they would never forget how to do it. Without such understanding, however, few can remember such a complicated procedure for long once the final exam has ended. . . . What sets them [those who ‘get it’] apart from the many people who never seem to ‘get it’ is not that they have memorized the rules better. Rather, they understand those rules (2000, pp. 67–68)

We will find that visualizations and pattern recognition also contribute to the understanding of mathematical procedures and concepts. They are discussed below.

Mathematical connections typically come in two forms. The first and most important connection is to previously taught mathematics, but also, ‘New information becomes more memorable if we “tag” it with an emotion [like a familiar real-world context]’ (Restak, 2006, p. 164). So we also need to connect new maths concepts to contexts that are familiar (evoke an emotional response) to students. For example, when teaching (not applying) the concept of the behavior of zero(s) of a function by modelling the amount of fluid remaining in an I.V. drip bag, we ‘tag’ it with the real-world meaning of the zero – the bag is empty. That is, the nurse must take action at the zero. If the nurse does not replace the bag, the patient may die. If the patient dies, the zero becomes important to the prescribing doctor and several lawyers, and so on. The result of tagging a mathematical concept or process with an emotional connection is improved memory. It turns out that the more connections to a mathematical concept/procedure, the more likely the correct recall. That is:

In general, how well new information is stored in long-term memory depends very much on depth of processing, . . . A semantic level of processing, which is directed at the meaning aspects of events, produces substantially better memory for events than a structural or surface level of processing. (Thompson and Madigan, 2005, p. 33)

Pattern Building to Pattern Generalizing

Using pattern building as a tool to help students generalize a pattern, like for example the first law of exponents, has a stained history. The pervasive view is that mathematics is understood through ‘reasoning’, and this is the standard to which mathematicians typically hold. This may be a noble thought, but it turns out that reasoning is NOT the brain’s dominate mode of operation. Gerald Edelman is a Nobel Laureate in medicine and makes an interesting point, ‘human brains operate fundamentally in terms of pattern recognition rather than logic [reasoning]. It [pattern recognition] is enormously powerful, but because of the need for range; it carries with it a loss of specificity’ (Edelman, 2006, pp. 83, 103). Of course, this loss of specificity is what concerns educators. In mathematics, we may want students to generalize the exact concept/procedure of our choice, and not other options that are open to the student’s brain. Yet given the evidence that the primary mode of operation of the brain is pattern generalizing; shouldn’t we capitalize on this? Might it improve understanding and memory? A good option for implementing pattern building is to use guided discovery activities because ‘ . . . the use of controlled scientific observation enormously enhances the specificity and generality of these interactions’ (Edelman, 2006, p. 104). Based on the author’s experience, successful guided discovery activities are short and lead directly to the desired mathematical generalization. The average brain will generalize on the third iteration. As you might expect, some students will generalize after the first or second – especially after using the process in class for a while, so one needs to think through the guided discovery activity questions that lead students to generalize. However, ‘After selection occurs . . . refinements can take place with increasing specificity. This is the case in those situations where logic or mathematics can be applied’ (Edelman, 2006, p. 83). Thus, it seems that guided discovery is a good choice.

There is more to the idea of tapping into common brain function through pattern building. We may not know at what point, if any, that the brain creates a long-term memory of a mathematical procedure through practice. But, in pattern building, ‘If the patterns are related in such a way that the [brain] region can learn to predict what pattern will occur next, the cortical region forms a persistent representation, or memory, for the sequence’ (Hawkins, 2004, 128, emphasis introduced). It is crucial that teachers not ‘tell’ students the desired generalization, but teachers must structure the pattern-building activity that gently leads students to generalize after a reasonable pattern-building activity has been completed. An excellent tool for knowing what each student has generalized is the TI Navigator. With the proper use of Navigator, every student must generalize, and...

Table of contents

• Cover
• Title
• Part 1 Where Are We Now?
• Part 2 What Does Research Tell Us?
• Part 3 Key Pedagogical Issues in Embedding ICT in Teaching and Learning Mathematics
• Part 4 Description of a Range of ICT Tools
• Part 5 Practical Ideas of ICT to Enhance Teaching and Learning
• Part 6 ICT Supporting Cross-curricular Work with Mathematics
• Part 7 Case Studies of Teachers Engaging with ICT
• Part 8 Implications for Professional Development
• Glossary
• Index