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The role of prediction in the teaching and learning of mathematics

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International Journal of Mathematical Education in Science and Technology

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The role of prediction in the teaching and learning of mathematics

Kien H Lim a , Gabriela Buendía b , Ok-Kyeong Kim c , Francisco Cordero d & Lisa Kasmer e

a Department of Mathematical Sciences, University of Texas at El Paso, 500 West University Avenue, El Paso, TX 79968-0514, USA

b Centro de Investigación en Ciencia Aplicada y Tecnología Avanzada, Programa de Matemática Educativa, Legaria 694 Col Irrigación CP 29050, México, D.F

West Michigan Avenue, Kalamazoo, MI 49008-5248, USA

d Centro de Investigación y Estudios Avanzados del IPN, , Departamento de Matemática Educativa, Av IPN 2508 Col San Pedro Zacatenco CP 07360, México, DF

e Department of Curriculum and Teaching, Auburn University, 5026 Haley Center, Auburn, AL 36849, USA

Version of record first published: 24 Jun 2010

To cite this article: Kien H Lim , Gabriela Buendía , Ok-Kyeong Kim , Francisco Cordero & Lisa

Kasmer (2010): The role of prediction in the teaching and learning of mathematics, International Journal of Mathematical Education in Science and Technology, 41:5, 595-608

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International Journal of Mathematical Education in

Science and Technology, Vol 41, No 5, 15 July 2010, 595–608

The role of prediction in the teaching and learning

of mathematics Kien H Lima*, Gabriela Buendı´ab, Ok-Kyeong Kimc,

Francisco Corderodand Lisa Kasmere

Aplicada y Tecnologı´a Avanzada, Programa de Matema´tica Educativa, Legaria 694 Col

Investigacio´n y Estudios Avanzados del IPN, Departamento de Matema´tica Educativa,

of Curriculum and Teaching, Auburn University, 5026 Haley Center, Auburn,

AL 36849, USA (Received 6 September 2009) The prevalence of prediction in grade-level expectations in mathematics curriculum standards signifies the importance of the role prediction plays in the teaching and learning of mathematics In this article, we discuss benefits

of using prediction in mathematics classrooms: (1) students’ prediction can reveal their conceptions, (2) prediction plays an important role in reasoning and (3) prediction fosters mathematical learning To support research on prediction in the context of mathematics education, we present three perspectives on prediction: (1) prediction as a mental act highlights the cognitive aspect and the conceptual basis of one’s prediction, (2) prediction

as a mathematical activity highlights the spectrum of prediction tasks that are common in mathematics curricula and (3) prediction as a socio-epistemological practice highlights the construction of mathematical knowledge in classrooms Each perspective supports the claim that prediction when used effectively can foster mathematical learning

Considerations for supporting the use of prediction in mathematics classrooms are offered

Keywords: prediction; mental act; reasoning; learning; thinking; mathe-matical activity; socio-epistemological practice; mathemathe-matical tasks

1 Introduction

In the analysis of grade-level expectations (GLEs) in relation to reasoning in state mathematics curriculum standards in the United States, Kim and Kasmer [1] found that GLEs pertaining to prediction were the most prevalent across grade levels and across content areas Examples of such GLEs are predict what comes next in an established pattern and justify thinking and predict the effect on the graph of a linear equation when the slope changes Prediction, however, has received far less attention

*Corresponding author Email: kienlim@utep.edu

ISSN 0020–739X print/ISSN 1464–5211 online

! 2010 Taylor & Francis

DOI: 10.1080/00207391003605239

in mathematics teaching and learning, compared to other aspects of reasoning such

as justification and generalization

In this article, we aim to generate interest among practitioners and researchers in the topic of prediction This article is organized to answer the following questions: (1) Why is prediction significant in the teaching and learning of mathematics? (2) How is prediction being conceptualized in the field of mathematics education? and (3) How can prediction be used effectively in mathematics classrooms? We first present benefits of using predictions in classrooms We then present three ways of viewing predictions, based on existing research Our goal is to provide readers with an overview of the richness that prediction can offer both as a topic for research and as

an instructional strategy in the mathematics classroom We conclude with some remarks on the use of prediction in mathematics classrooms

2 Benefits of using prediction

Presented in this section are various benefits of using prediction in mathematics classrooms Prediction provides opportunities for students to be aware of and subsequently address their misconceptions Prediction complements other forms

of reasoning such as generalizing, conjecturing, abducting, imagining and visualiz-ing Prediction helps to draw students’ attention to structural and relational aspects

of mathematics and provides opportunities for students to experience cognitive conflict, to notice patterns, to generalize from specific cases and to expand the assimilatory range of a particular conception In addition, prediction can increase students’ level of engagement

2.1 Students’ predictions can reveal their conceptions

Prediction can be used to uncover students’ prior knowledge, schemes, misconcep-tions and intuimisconcep-tions Studies on stochastic misconcepmisconcep-tions typically require subjects

to predict or estimate the probability of an event in a given scenario [2–4] For example, most students predicted in a family of six children that the sequence BGBBBB (B stands for a boy and G stands for a girl) is less likely than the sequence GBGBBG although both sequences are likely equal This is because a three-boy and three-girl combination is more representative of the population than a five-boy and one-girl combination This example reveals that students make predictions based

on their judgment of representativeness – an event is more probable if it has some significant characteristics of its parent population [3,4]

The research in science education identifies students’ misconceptions by asking them to make predictions [5–7] For example, 51% of undergraduates in a study predicted that the path of a ball would be curvilinear (Figure 1) when the metal ball

is shot out of a curved tube at a high speed [7] The conceptual basis underlying their prediction is similar to the medieval theory of impetus, which claimed that ‘an object set in motion acquires an impetus that serves to maintain the motion’ (p 1140) Tasks that require students to imagine a scenario and then predict the motion are also found in the Force Concept Inventory – a multiple-choice test for assessing students’ understanding of basic concepts in Newtonian mechanics [8] Champagne, Gunstone and Klopfer [5] developed prediction-observation-explanation (POE) tasks to probe students’ conceptions in interview settings and subsequently designed

a POE model as an instructional sequence for teaching physics in schools

Lim [9] observed that prediction tasks, such as ‘Plugging x¼ 127 into 4x" 20 4 3x " 20, we get 361 for the right hand side What is the value on the left hand side? ’ (p 88), could not only help students attend to the structure of algebraic expressions, but may also reveal students’ inflexibility in interpreting expressions For example, an 11th grader, who was taking Calculus then (i.e., she was considered

an advanced student in mathematics), could only conceive 4x as four times x and not

xþ x þ x þ x

Prediction tasks can be designed to uncover students’ mathematical conceptions

in certain topics The first author found that asking students to predict the largest-fraction from a set of fractions may reveal certain misconceptions about fractions Predictions such as ‘all three fractions (99/100, 6/7 and 15/16) are the same because they are only one number away from being a whole’ suggest a conception that disregards the denominator Similarly, predictions such as ‘because 100s pieces is a smaller amount than 7s pieces you get a bigger chunk with 7 than with 100’ suggest a conception that disregards the numerator Prediction not only can uncover students’ mathematical conception, but also foster mathematical reasoning

2.2 Prediction plays an important role in reasoning

Asking students to predict has an advantage over asking students to find an answer Whereas finding an answer tends to reinforce instrumental understanding [10], predicting an answer can promote relational understanding When asked to find the largest fraction among 99/100, 6/7 and 15/16, students can instrumentally convert each fraction into a decimal or into its equivalent fraction with a common denominator, and then compare the adjusted numerators When asked to predict, students can think relationally For example, the first author observed that a pre-service middle-school teacher reasoned that ‘6/75 15/16 5 99/100 because in 6/7 you just have to divide the whole into 7 pieces while in 99/100 you have to divide your whole into 100 pieces and you get more of 99/100 than 6/7, [since] 1/100 is smaller area than1/7’

When students predict, as opposed to meticulously working through the steps, they are psychologically relieved from the need for precision and certitude

Figure 1 A curvilinear trajectory of a ball leaving a curved tube

International Journal of Mathematical Education in Science and Technology 597

By temporarily disregarding details, students can focus on essential features and structures For example, a person may predict that 1199 is greater than 9911 by relying on her familiarity with base-10 structure: 10100has 100 zeroes trailing the 1, whereas, 10010has only 20 zeroes trailing the 1 because 10010¼ (102)10¼ 1020 The role of prediction in fostering structure sense [11] is analogous to the role of estimation in fostering number sense [12]

Predicting complements other forms of reasoning such as generalizing, conjecturing and abducting The process of generalizing typically involves activities like identifying commonalities, finding a pattern, checking to see if the pattern holds true for ‘all’ cases, and formulating a general statement, and in some cases identifying the process underlying the pattern The act of testing whether a generalized pattern holds for other cases involves predicting, based on the conjectured pattern, the results for those cases The testing role of prediction is inevitable in the construction of new knowledge, as expressed in Peirce’s writings, where abduction is differentiated from induction and deduction [13]

Abduction is the process of forming an explanatory hypothesis It is the only logical operation which introduces any new idea; for induction does nothing but determine a value and deduction merely evolves the necessary consequences of a pure hypothesis Deduction proves that something must be; Induction shows that something actually

is operative; Abduction merely suggests that something may be Its only justification

is that from its suggestion deduction can draw a prediction (italics added) which can be tested by induction (p 216)

Prediction allows us to engage in thought experiments that have yet to be, or can never actually be, realized Consider the Zeno’s paradox, a person traversing half the remaining distance to a wall every minute Asking a student to predict whether the person will eventually reach the wall presents students with two conflicting answers, which essentially correspond to the two notions of infinity: potential infinity and actual infinity Activities that involve predictions can be designed for students to experience cognitive conflicts, resolving which can lead to learning of targeted concepts deeply

2.3 Prediction fosters learning

Prediction can be a useful pedagogical means to aid student learning in several ways

In terms of concept development, prediction allows students to activate and refine their existing knowledge In terms of affect, prediction can help to increase students’ level of engagement

2.3.1 Activating and refining prior knowledge

From a Piagetian’s perspective, learning involves cycles of experiencing disequili-brium, resolving cognitive conflicts and re-establishing a new equilibrium [14,15]

In the analysis of students’ construction of 3D arrays of cubes in an inquiry-based learning environment to determine the number of cubes in a rectangular prism, Battista [16] apprehended the power of predictions

Having students first predict then check their predictions with cubes was an essential component in their establishing the viability of their mental models and enumeration schemes and was thus crucial for the recursive development of these models Because

students’ predictions were based on their mental models, making predictions encouraged them to reflect on and refine those mental models Having students merely make boxes and determine how many cubes fill them would have been unlikely

to have promoted nearly as much student reflection as having students make and check predictions because (a) opportunities for perturbations arising from discrepancies between predicted and actual answers would have been greatly reduced and (b) students’ attention would have been focused on physical activity instead of on their own thinking (p 442)

Prediction also plays an important role in a computer-based learning environment, especially one that involves animation [17,18] According to Hegarty, Kriz and Cate [19], prediction induces people to activate their prior knowledge and to articulate their understanding of the phenomenon under investigation The effective use of

Frog-versus-Clown applet in SimCalc) would require students to make predictions before they observe the animation Bowers, Nickerson and Kenehan [17] propose an instructional sequence that requires students to play with dynamic graphs, predict and then test their predictions

students to account for the inconsistencies between what they predict and what they observe Lim [20] used five prediction tasks (listed below) and classroom voting, via a personal response system, to help pre-service middle-school teachers to overcome two misconceptions: multiplication makes bigger and division makes smaller

(1) Fill in the blank with either4, 5, or ¼:

81405/67092$ 2884/3717 _ 81405/67092 (2) Is the following inequality always true, sometimes true, or never true?

N is a natural number: 67/89% N 5 N (3) Is the following inequality always true, sometimes true, or never true?

N is a natural number: N% 2/35 4 2/35 (4) Is the following inequality always true, sometimes true, or never true?

N is a natural number: N$ 11/25 5 N (5) Is the following inequality always true, sometimes true, or never true?

N is a natural number: 32/23% N 5 N When students make a prediction prior to performing calculations, they are more likely to notice certain relationships, generalize from specific cases and expand the assimilatory range of a particular conception For example, having students predict prior to computing whether the result of multiplying 9.29 by 7/6 or by 0.64, is greater

or less than 9.29 can draw students’ attention to the effect of the multiplier Students may even advance their understanding of multiplication, from viewing multiplication

as an algorithm-to-follow and/or multiplication as repeated-addition to viewing multiplication as enlargement/reduction

Prediction can also function as an advance organizer [21,22] Ausubel [23] introduced advance organizers as introductory materials ‘to bridge the gap between what the learner already knows and what he needs to know before he can successfully learn the task at hand’ (p 148) Posing prediction questions prior to having students explore mathematical ideas associated with a problem helps students make sense of the problem context, provoke prior knowledge, identify related mathematical concepts and set the foundation for learning [21,24,25]

International Journal of Mathematical Education in Science and Technology 599

2.3.2 Increasing students’ level of engagement

Prediction can increase students’ level of engagement [2,21,25,26] This is because

‘the commitment involved in deciding on a prediction can have powerful motivation effects’ [26, p 63] Kasmer [21] found that when mathematics lessons were infused with prediction questions, students seemed more engaged in classroom discussions and problem solving She found that students in an algebra classroom where prediction questions were routinely posed prior to the exploration of a problem demonstrated a higher-level of engagement, compared to a similar class, where prediction questions were not used In the classroom, where prediction questions were posed, students were engaged in sustained conversations that were created by a culture precipitated by the inherent risk free virtue of prediction questions because of the absence of certitude in predicting In recommending some pedagogical principles for learning statistics, Garfield and Ben-Zvi [2] commented that ‘if students are first asked to make guesses or predictions about data and random events, they are more likely to care about and process the actual results’ (p 388)

Prediction can also increase student engagement in other subjects such as reading [27,28] and science [29–31] For example, predictions questions such as ‘What do you know about this character that helps you predict what s/he will do next?’ and ‘Given the situation in the story, what will possibly happen next?’ were found to improve student comprehension in reading [28]

Prediction plays a bridging role in helping students make connections between a physical phenomenon and associated scientific concepts In biology education, Lavoie [30] found that the addition of prediction-discussion phase to a three-phase learning cycle (exploration, term introduction and concept application) could improve students’ process skills, logical-thinking skills, science concepts and scientific attitudes In physics education, the POE instructional approach, which requires students to predict prior to observing a demonstration or performing an experiment and account for the discrepancy between their prediction and their observation, was reported as effective [29,31]

3 Perspectives of prediction

The mathematics education research on prediction conducted by different researchers has different emphases In our attempt to consolidate various theoretical frameworks, we offer three complementary ways of viewing prediction: as a mental act, as a mathematical activity and as a socio-epistemological practice A cognitive perspective of prediction, as a mental act, emphasizes the conceptual basis of one’s prediction, namely one’s schemes A curricular perspective of prediction, as a mathematical activity, highlights the spectrum of prediction tasks that are common

in US mathematics curriculum A socio-epistemological perspective of prediction underscores the construction of mathematical knowledge in classrooms Each perspective supports the claim that prediction when used effectively can foster mathematical learning

3.1 Prediction as a mental act

To predict is to declare in advance In the Merriam-Webster’s Collegiate Dictionary [32], four synonyms are differentiated: (1) ‘predict commonly implies inference from

facts or accepted laws of nature’, (2) ‘foretell applies to the telling of the coming of a future event by any procedure or any source of information’, (3) ‘forecast is usually concerned with probabilities rather than certainties’ and (4) ‘prophesy connotes inspired or mystic knowledge of the future’ (p 456) Predicting, foretelling, forecasting and prophesying are similar in that they tend to be ‘verbal’ acts of declaring something before it happens or before it is known for sure They differ, however, in terms of the cognition that leads to an expectation prior to the declaration of the expectation For example, the act of forecasting seems to involve more computational effort, whereas the act of predicting seems to involve more inferential reasoning The cognitive aspect of arriving at a prediction is more important in mathematical reasoning than the verbal aspect of declaring one’s prediction Hence, we consider predicting a mental act

In the context of solving a mathematics problem, predicting means having an expectation of something prior to working out the details Lim [33] defines predicting

as ‘the act of conceiving an expectation for the result of an event without actually performing the operations associated with the event’ (p 103) An event could be an

equation-graph translation and so forth Lim’s [9,33] notion of prediction highlights the difference between ‘to predict’ and ‘to perform’ When one predicts, as opposed

to perform, one is relieved from the need for certitude and precision

An act of predicting involves a certain amount of cognitive effort on a continuum from a mere guess to an elaborate prediction The amount of cognitive effort depends on many factors such as the object of prediction, the person making the prediction and the basis underlying the prediction For example, predicting the larger fraction between 4/9 and 8/25 is cognitively less demanding than predicting the difference between them Someone with good fraction sense, such as recognizing that 4/9 is larger than 1/3 and 8/25 is smaller than 1/3, will find it much easier to predict efficiently than someone without Predicting by capitalizing on certain mathematical understanding such as converting the fractions into equivalent fractions with a common numerator (e.g 8/18 is larger than 8/25 because 1/18 is larger than 1/25) requires less cognitive effort than converting them into decimal equivalents or into equivalent fractions with a common denominator

What and how a person predicts will depend on the knowledge the person has From a Piagetian perspective, a person’s prediction depends on the scheme(s) that is/are enacted Prediction is possible because of our ability to assimilate situations into our existing scheme(s): ‘anticipation is nothing other than a transfer or application of the scheme to a new situation before it actually happens’ [34, p 195] A scheme,

as outlined by von Glasersfeld [15], involves three components: the perceived situation, the activity and the expected result The expected result component provides the anticipatory feature of a scheme For example, a student with a multiplication makes bigger scheme is likely to predict that the product of multiplying two multi-digit numbers is going to be larger than each factor This scheme may interfere with a student’s choice of operation for problems involving decimals, such as finding the cost of 0.22 gallons of gas at a rate of £1.20 per gallon [35] For example,

a particular student predicted that the cost would ‘be under the £1.20, so obviously it’s 1.20/0.22 or something like that’ (p 405) This student’s choice of division to obtain a smaller value is considered to be influenced by her or his multiplication makes bigger, division makes smaller scheme

International Journal of Mathematical Education in Science and Technology 601