The-role-of-prediction-in-the-teaching-and-learning-of-mathematics

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/45164949 The role of prediction in the teaching and learning of mathematics Article  in  International Journal of Mathematical Education · July 2010 DOI: 10.1080/00207391003605239 · Source: OAI CITATIONS READS 29 8,774 authors, including: Kien Lim Gabriela Buendía University of Texas at El Paso Red de Cimates 31 PUBLICATIONS   197 CITATIONS    48 PUBLICATIONS   250 CITATIONS    SEE PROFILE SEE PROFILE Ok-Kyeong Kim Francisco Cordero Western Michigan University Center for Research and Advanced Studies of the National Polytechnic Institute 24 PUBLICATIONS   229 CITATIONS    164 PUBLICATIONS   1,439 CITATIONS    SEE PROFILE Some of the authors of this publication are also working on these related projects: Improving Curriculum Use for Better Teaching (ICUBiT) Project View project ¿Por qué se dice que enseñar y aprender matemáticas es difícil? View project All content following this page was uploaded by Francisco Cordero on 03 March 2015 The user has requested enhancement of the downloaded file SEE PROFILE This article was downloaded by: [Cinvestav del IPN] On: 07 March 2013, At: 11:09 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK International Journal of Mathematical Education in Science and Technology Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tmes20 The role of prediction in the teaching and learning of mathematics a b c Kien H Lim , Gabriela Buendía , Ok-Kyeong Kim , Francisco d Cordero & 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 c Department of Mathematics, Western Michigan University, 1903 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 To link to this article: http://dx.doi.org/10.1080/00207391003605239 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-andconditions This article may be used for research, teaching, and private study purposes Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date The accuracy of any instructions, formulae, and drug doses should be independently verified with primary Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 sources The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material 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 Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 Kien H Lima*, Gabriela Buendı´ ab, Ok-Kyeong Kimc, Francisco Corderod and Lisa Kasmere a Department of Mathematical Sciences, University of Texas at El Paso, 500 West University Avenue, El Paso, TX 79968-0514, USA; bCentro de Investigacio´n en Ciencia Aplicada y Tecnologı´a Avanzada, Programa de Matema´tica Educativa, Legaria 694 Col Irrigacio´n CP 29050, Me´xico, D.F.; cDepartment of Mathematics, Western Michigan University, 1903 West Michigan Avenue, Kalamazoo, MI 49008-5248, USA; dCentro de Investigacio´n y Estudios Avanzados del IPN, Departamento de Matema´tica Educativa, Av IPN 2508 Col San Pedro Zacatenco CP 07360, Me´xico, DF; eDepartment of Curriculum and Teaching, Auburn University, 5026 Haley Center, Auburn, AL 36849, USA (Received 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; mathematical activity; socio-epistemological practice; mathematical tasks 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 ß 2010 Taylor & Francis DOI: 10.1080/00207391003605239 http://www.informaworld.com 596 K.H Lim et al Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 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 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 visualizing 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, misconceptions and intuitions Studies on stochastic misconceptions 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 International Journal of Mathematical Education in Science and Technology 597 Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 Figure A curvilinear trajectory of a ball leaving a curved tube Lim [9] observed that prediction tasks, such as ‘Plugging x ¼ 127 into 4x À 20 3x À 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 x ỵ x ỵ x ỵ 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 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 ‘6/7 15/16 99/100 because in 6/7 you just have to divide the whole into 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 Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 598 K.H Lim et al 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: 10100 has 100 zeroes trailing the 1, whereas, 10010 has only 20 zeroes trailing the because 10010 ¼ (102 )10 ¼ 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 disequilibrium, 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 International Journal of Mathematical Education in Science and Technology 599 Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 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 certain applets (e.g Paper-Pool applet in NCTM Illuminations and 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 Prediction fosters learning in that it provides an opportunity for 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 either 4, 5, or ¼: 81405/67092 Ä 2884/3717 _ 81405/67092 (2) Is the following inequality always true, sometimes N is a natural number: 67/89 Â N N (3) Is the following inequality always true, sometimes N is a natural number: N Â 2/35 2/35 (4) Is the following inequality always true, sometimes N is a natural number: N Ä 11/25 N (5) Is the following inequality always true, sometimes N is a natural number: 32/23 Â N N true, or never true? true, or never true? true, or never true? true, or never true? 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] 600 K.H Lim et al Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 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 you know about this character that helps you predict what s/he will 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] 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 Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 International Journal of Mathematical Education in Science and Technology 601 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 arithmetic computation, symbol manipulation, graphical transformation, 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 602 K.H Lim et al Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 Viewing prediction as a mental act highlights the differences among individual learners in terms of their mathematical knowledge and understanding On the other hand, if a teacher wants to create learning opportunities for the students in her or his classroom, then it will be more meaningful to view prediction as a mathematical activity 3.2 Prediction as a mathematical activity When predictions are used in classrooms to foster learning, they are best conceived as mathematical activities or mathematical tasks Kim and Kasmer [1] present a list of prediction GLEs from state mathematics curriculum standards While those expectations can be categorized in terms of grade-levels or content strands, they also can be categorized in terms of the nature of prediction that is involved In order to make a prediction, one usually has to engage in various mathematical processes, such as visualization, estimation and generalization Table presents various types of prediction tasks with examples of GLEs and tasks These examples suggest that the kinds of mathematical processes involved in making a prediction depend to a large extent on the type of prediction tasks For example, in an extrapolation-prediction task, a person has to first establish a relationship or a model and then use it to make a prediction Certain types of prediction tasks are therefore appropriate for certain mathematical topics For example, visualization-prediction tasks, which tend to involve construction and transformation of mental images, are more common in geometry and in algebra than in arithmetic These categories of prediction tasks are not exhaustive Within each category, there may be sub-categories For example, prediction tasks in the generalization-prediction category may be aimed at (1) noticing a local commonality such as the relation between successive terms, (2) establishing a general rule or (3) reasoning with the established rule An example for each sub-category is as follows: (1) predict the next three terms in the sequence 2, 3, 5, 9, 17, 33, _, _, _; (2) predict the 10th term, the 100th term and the nth term and (3) predict whether 1001 is a term in the sequence Will students respond differently if the term predict in the examples in Table is replaced by the term find or determine? This is an open question The term find seems to suggest to students that there is a standard way for solving the problem The term predict, on the other hand, encourages students to use less rigorous methods, such as relying on intuition or making an educated guess, to foretell what the answer might be The lack of meticulous computations in predictions usually results in a certain degree of uncertainty The relief from having to be certain allows students to take risks by considering alternative ways Students are generally less afraid of making mistakes because predictions are not perceived as a direct evaluation of their knowledge In summary, asking students to predict seems to have the following benefits: (1) fostering intuitive reasoning and sense making, (2) encouraging non-routine approaches and (3) increasing student participation When prediction is used in a classroom as an integral part of the construction of a particular mathematical concept, its epistemological aspects are involved Especially with concepts that involve change, prediction-related activities can be designed to foster student development of certain mathematical knowledge, such as those in calculus Predict the relative size of solutions in addition, subtraction, multiplication and division of whole numbers Make reasonable predictions using generalizations about patterns Make predictions and justify conclusions based on data Predict the position and orientation of simple geometric shapes under transformations such as reflections, rotations and translations Predict and evaluate how adding data to a set of data affects measures of centre Generalizationprediction task Extrapolation-prediction task Visualization-prediction task Concept-applicationprediction task Examples of grade-level expectation Estimation-prediction task Prediction task Table Different types of prediction tasks House-2 House-3 The average height of a group of 10 students is 150 cm Predict the new average height when the 11th student whose height is 170 cm joins the group Predict whether a given scalene triangle would be an identical mapping when it’s rotated by 120 around one of the vertices Predict the population of rabbits in 10 years using the data in the table that shows the rabbit population of the first years House-1 Predict the number of toothpicks required to make House-10 Predict which result is larger: 23 ỵ 18 ỵ 45 or 588/102 Examples of prediction tasks Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 International Journal of Mathematical Education in Science and Technology 603 604 K.H Lim et al Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 3.3 Prediction as a socio-epistemological practice Buendı´ a and Cordero [36] consider prediction as a practice that takes into account the epistemological and socio-cultural aspects of the mathematical knowledge This perspective is based on Cantoral and Farfa´n’s approach [37] that takes into account four dimensions – epistemological, socio-cultural, cognitive and pedagogical – to explain the construction of a piece of mathematical knowledge, say function in the context of variation In this approach, prediction is a considered ‘a fundamental tool for understanding variation’ (p 267) Cantoral and Farfa´n offer Newtons binomial expression, originally written as (P ỵ PQ)m/n, as an example to highlight the progression of knowledge, from ‘the system of socially shared practices linked to the solution of a class of situations that require prediction (italics added), where it is in transit until managing to take on the abstract form of the concept of analytical function’ (p 267) According to Cordero [38], the need of the scientific community to determine the change in the value of the dependent variable in relation to the change in the value of the independent variable motivates the development of predictive tools that eventually became transcendental mathematical concepts such as calculus Buendı´ a and Cordero [36] have used prediction tasks to help students construct the mathematical notion of periodicity In their historical review, they found that periodicity was initially used to bridge between empirical studies in celestial phenomena and predictive scientific theories in astronomy [39] That is, the periodic aspect was relevant as a certain repetitive form which favours the development of prediction techniques that eventually led to mathematization of certain physical phenomena In a socio-epistemological approach, the mathematical knowledge is reconstructed in a mathematics classroom through the intentional development of predictive tasks that allow students to acquire a significant meaning of periodic aspect, as a property that qualifies certain repetitive behaviour Many students today tend to rely on the presence of trigonometric functions to determine if a function is periodic; for example, they consider functions as such y ẳ x ỵ sin x as periodic functions and not attend to the way the function repeats Buendı´ a and Cordero [36] advocate a didactical approach in which prediction tasks are posed for students to grapple with the notion of periodicity In their study, students were given eight distance time graphs for the interval from t ¼ to t ¼ 12 and were asked to predict the position of the moving object at t ¼ 231 for each graph Student responses to two of these tasks are presented in Figure In both responses, the students could identify a ‘periodic’ unit that contains key information about the repetitive behaviour For example, the first student obtained 28 ‘repetitions’ by dividing 231 by and realized that the remainder corresponds to the output value of 1.3, using the first-period segment of his graph Through dealing with various kinds of repetitions in those graphs, students recognize the repetitive graph behaviour and progressed from noticing anything that repeats itself in any way to understanding how it is repeated Predicting allows students to study the repetitive patterns in those graphs in a meaningful manner Remarks for using prediction in mathematics classrooms The effectiveness of prediction use depends on the fit between the activity we design and the mathematics we want our students to learn In selecting or designing an Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 International Journal of Mathematical Education in Science and Technology 605 Figure A prediction task involving continuous and discontinuous graphs activity that involves prediction, a teacher should consider the following questions: (1) How does asking students to make a prediction contribute to their learning of a particular concept? (2) What conceptions and prior knowledge can this prediction activity elicit? (3) Are students supposed to experience cognitive conflicts, make connections, or abstract certain mathematical structures from their predictions? and (4) What kinds of student responses can we expect? Not all problems involving prediction are effective Teachers should use problems that are intrinsic to their students Students are more likely to engage in mathematical thinking if they understand the problem and are intrigued by it [40,41] For example, students may find the following problem interesting: ‘The boss says, ‘‘you remember when business was bad last year, I had to cut everyone’s pay by 10%? Well, business is better, so I can raise your pay by 10% now That will put you back to where you were before the cut.’’ Is the boss correct? ’ [42, p 193] This problem is appropriate for emphasizing the importance of attending to the referent base of a percent Ideally, we need to use problems that allow students to experience an intellectual need [40] for a particular concept Lim [41] discusses how tasks can be designed to provoke the intellectual need for mathematical concepts such as prime factorization and lowest common multiple In designing and planning a prediction activity, a teacher should consider many factors such as its purpose, students’ existing knowledge and beliefs, Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 606 K.H Lim et al epistemology of the concept to be learned, classroom norms and anticipated level of student engagement, availability of certain resources and the likelihood of students’ experiencing intellectual need Sufficient time should be allocated for students to predict, to discuss among themselves and to resolve cognitive conflicting ideas To implement a prediction activity effectively, a teacher must ‘predict’ students’ responses to the activity In implementing a prediction activity, a teacher should envision how the activity will unfold, anticipate students’ predictions and be prepared to respond appropriately In response to students’ predictions, a teacher may ask students to discuss and resolve the differences in their predictions, ask students to share their reasoning or their perplexity, pose follow-up questions to engender cognitive conflict among students, direct students’ attention towards a certain phenomenon, or proceed to next prediction task Through experience, a teacher will gradually be more effective in her use of prediction to optimize learning To recapitulate the main points in this article (1) prediction plays a significant role in the teaching and learning of mathematics; (2) prediction can be conceptualized as a mental act, as a mathematical activity and as a socio-epistemological practice and (3) effective use of prediction in classroom requires thoughtful planning and reflection In conclusion, when prediction is used effectively students are likely to progress from passive listeners to active thinkers and expand and deepen their mathematical knowledge References [1] O.K Kim and L Kasmer, Analysis of emphasis on reasoning in state mathematics curriculum standards, in The Intended Mathematics Curriculum as Represented in State-Level Curriculum Standards: Consensus or Confusion? B Reys, ed., Information Age Publishing, Greenwich, CT, 2006, pp 89–109 [2] J Garfield and D Ben-Zvi, How students learn statistics revisited: A current review of research on teaching and learning statistics, Int Stat Rev 75 (2007), pp 372–396 [3] L.S Hirsch and A.M O’Donnell, Representativeness in statistical reasoning: Identifying and assessing misconceptions, J Stat Educ (2001), Available at http://www.amstat.org/ publications/jse/v9n2/hirsch.html [4] D Kahneman and A Tversky, Subjective probability: A judgment of representativeness, Cogn Psychol (1972), pp 430–454 [5] A.B Champagne, R.F Gunstone, and L.E Klopfer, Effecting changes in cognitive structures among physics students, in Cognitive Structure and Conceptual Change, L.H West and A.L Pines, eds., Academic Press, Orlando, FL, 1985, pp 163–187 [6] R.F Gunstone and R.T White, Understanding gravity, Sci Educ 65 (1981), pp 291–299 [7] M McCloskey, A Caramazza, and B Green, Curvilinear motion in the absence of external forces: Naive beliefs about the motion of objects, Science 210 (1980), pp 1139–1141 [8] D Hestenes, M Wells, and G Swackhamer, Force concept inventory, Phys Teach 30 (1992), pp 141–158 [9] K.H Lim, Students’ Mental Acts of Anticipating: Foreseeing and Predicting while Solving Problems Involving Algebraic Inequalities and Equations, VDM Verlag, Saarbruăcken, Germany, 2008 [10] R.R Skemp, Relational understanding and instrumental understanding, Math Teach 77 (1976), pp 20–26 [11] M Hoch and T Dreyfus, Equations – Structure sense in high school algebra: The effect of brackets, Proceedings of the 28th Conference of the International Group for the International Journal of Mathematical Education in Science and Technology 607 [12] [13] [14] [15] [16] Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] Psychology of Mathematics Education, Vol 3, M.J Høines and A.B Fuglestad, eds., Bergen University College, Bergen, Norway, 2004, pp 49–56 J.T Sowder, Estimation and number sense, in Handbook of Research on Mathematics Teaching and Learning, D.A Grouws, ed., Macmillan, New York, 1992, pp 371–389 Peirce Edition Project (ed.), The Essential Peirce: Selected Philosophical Writings, 1893–1913, Vol 2, Indiana University Press, Bloomington, IN, 1998 J Piaget, The Equilibration of Cognitive Structures: The Central Problem of Intellectual Development, University of Chicago Press, Chicago, 1985 E vonGlasersfeld, Radical Constructivism: A Way of Knowing and Learning, The Falmer Press, Bristol, PA, 1995 M.T Battista, Fifth graders’ enumeration of cubes of 3D arrays: Conceptual progress in an inquiry-based classroom, J Res Math Educ 30 (1999), pp 417–448 J Bowers, S Nickerson, and G Kenehan, Using technology to teach concepts of speed, in Making Sense of Fractions, Ratios, and Proportions, B Litwiller and G Bright, eds., National Council of Teachers of Mathematics, Reston, VA, 2002, pp 176–187 B Tversky, J.B Morrison, and M Betrancourt, Animation: Can it facilitate? Hum Comput Stud 57 (2002), pp 247–262 M Hegarty, S Kriz, and C Cate, The roles of mental animations and external animations in understanding mechanical systems, Cogn Instr 21 (2003), pp 325–361 K.H Lim, Use of prediction items and classroom voting to address pre-service teachers’ misconceptions, in preparation L Kasmer, The role of prediction in the teaching and learning of algebra, PhD dissertation, Western Michigan University, 2008 O.K Kim and L Kasmer, Prediction as an instructional strategy, J Math Educ Leadersh 11 (2009), pp 33–38 D Ausubel, Educational Psychology: A Cognitive View, Holt, Rinehart and Winston, New York, 1968 O.K Kim and L Kasmer, Prediction and mathematical reasoning, Proceedings of the 5th Annual Hawaii International Conference on Education, Honolulu, 2007, pp 3016–3034 O.K Kim and L Kasmer, Using ‘prediction’ to promote mathematical reasoning, Math Teach Middle Sch 12 (2007), pp 294–299 R White and R Gunstone, Probing Understanding, The Falmer Press, London, 1992 C Block, L Rodgers, and R Johnson, Comprehension Process Instruction: Creating Reading Success in Grades K-3, The Guilford Press, New York, 2004 J.W Irwin and I Baker, Promoting Active Reading Comprehension Strategies, Prentice Hall, Englewood Cliffs, NJ, 1989 R.F Gunstone, C.M Gray, and P Searle, Some long-term effects of uninformed conceptual change, Sci Educ 76 (1992), pp 175–197 D.R Lavoie, Effects of emphasizing hypothetico-predictive reasoning within the science learning cycle on high school students’ process skills and conceptual understandings in biology, J Res Sci Teach 36 (1999), pp 1127–1147 D Palmer, The POE in the primary school: An evaluation, Res Sci Educ 25 (1995), pp 323–332 Merriam-Webster’s Collegiate Dictionary, 10th ed., Merriam-Webster, Springfield, MA, 1993 K.H Lim, Characterizing students’ thinking: Algebraic inequalities and equations, Proceedings of the 28th Annual Meeting of the North American Chapter of the Psychology of Mathematics Education, Vol 2, S Alatorre, J.L Cortina, M Sa´iz and A Me´ndez eds., Universidad Pedago´gica Nacional, Me´rida, Yucata´n, Me´xico, 2006, pp 102–109 J Piaget, Biology and Knowledge, University of Chicago Press, Chicago, 1971 A Bell, M Swan, and G Taylor, Choice of operations in verbal problems with decimal numbers, Educ Stud Math 12 (1981), pp 399–420 608 K.H Lim et al Downloaded by [Cinvestav del IPN] at 11:09 07 March 2013 [36] G Buendı´ a and F Cordero, Prediction and the periodical aspect as generators of knowledge in a social practice framework, Educ Stud Math 58 (2005), pp 299–333 [37] R Cantoral and R Farfa´n, Mathematics education: A vision of its evolution, Educ Stud Math 53 (2003), pp 255–270 [38] F Cordero, Taking a differential element: Its formation and meaning in the didactic discourse of calculus, Internat J Math Ed Sci Tech 22 (1991), pp 869–872 [39] A Pannekoek, A History of Astronomy, Dover, New York, 1961 [40] G Harel, Two dual assertions: The first on learning and the second on teaching (or vice versa), Amer Math Monthly 105 (1998), pp 497–507 [41] K.H Lim, Provoking intellectual need, Math Teach Middle Sch 15 (2009), pp 92–99 [42] J.T Sowder, L Sowder, and S Nickerson, Reconceptualizing Mathematics: Part I: Reasoning about Numbers and Quantities, preliminary ed., W.H Freeman, New York, 2007 View publication stats