Pgde_statistics_v1

Front

Statistics Learning Part 1 ◆ students' concepts of average ◆ big ideas in statistics ◆ pedagogical considerations

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reflect

reflection on the studies and teaching concepts of average, is it simple? development of concepts takes time concept of mode, median; not just definition and use link to other concepts: variability, shape research and teaching students generated examples doing and undoing similarities to teaching of other topics: area, factorization, ... questioning, probing and task design ambiguity in the curriculum

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CDI

Key Stage 3 Data Handling Dimension

http://www.edb.gov.hk/index.aspx?nodeID=4905&langno=1 -4-

CDI (1999) p.26

Russell, Mokros

What Do Children Understand about Average? he statistical idea we come across most frequently is the idea of average. Children in fourth grade and beyond fairly easily learn to apply the algorithm for finding the mean, but what do they understand about the mean as a statistical idea? ' Many students do not have opportunities to learn about various kinds of averages as statistical concepts. They view an average as a number found by a particular procedure rather than as a number that represents and summarizes a set of data. Students may leam to find a mode, median, or mean - which technically are all averages even though average often refers to the mean - but they do not necessarily know how these statistics relate to the data being represented.

Avera,e as a Statistical Idea

sented by that number. Statistically literate readers can think about such a statement as "the median price of a house is $150 000" or "the average family size is 3.2" in terms of what it tells or does not tell them about the distribution of the data. Most students are unable to imagine what kind of data an average might represent. One student, who had had plenty of experience calculating the mean, said, "I know how to an average, but I don't know how to get the numbers to go into an average, from an average." Asking students to imagine what the data could be for a given average yields interesting insights into students' thinking about the relationship between data and the average of the data. They do not find it easy to use the memorized algorithm to construct data. They have to think about how the average represents the data. You might want to try the following problem, one we gave to students, before reading about how students solve it. We took a survey of the prices of nine different brands of potato chips. For the same-sized bag, the typical or usual or average price for all brands was $1.38. What could the prices of the nine different brands be?

Russell, Susan Jo and Mokros, Jan (February, 1996). What Do Children Understand About Average? Edited by Donald L. Chambers. Teaching Children Mathematics, 360-364. From: http://www.learner.org/courses/learningmath/data/overview/readinglist.html

We use the language "typical or usual or average" to keep the conversation open to any ways that students have to think about an average. When they show us one way, we ask them for other ways so that we get a view of the range of their thinking. Since these problems are administered in an interview, we are able to interact, ask questions, and probe students' thinking.

Average as mode

In interviews with fourth graders, many students consistently associated the "typical or usual or To investigate students' understanding of the inidea of average" Journalfor Research Mathematics Education value with the mode. In construction Vol. 26, No. 1, 20-39 average, we lise what we call 1995, "construction" prob- problems, they produced the data set by making all lems. Instead of asking students to find the average or most of the values the same as the average for a given set of numbers, we give students an value. They might have made a few adjustments to pushed, but despite probing for average and ask them what could be theCHILDREN'S data set it the data when OF AVERAGE CONCEPTS represents. This kind of problem is similar to situa- other approaches, these students stuck to a view of REPRESENTATIVENESS as the most frequent piece of data. As tions we often encounter in life; we read about AND a the average median or mean in the newspaper or come across it in one fourth grader explained, "Okay, first, not all our work and need to interpret what might be repre- chips are the same, as you told me, but the lowest JAN MOKROS, Massachusetts chips I everTERC, saw wasCambridge, $1.30 myself, so since the SUSAN JOtypical Massachusetts RUSSELL, price is TERC, $1.38, I Cambridge, just put most of them at $1.38, just to make it typical, and highered the SusanJo Russell directs a K-5 curriculum project called Investigations in Numher, Data, and Space. Her work focuses on how practicing teachers can learn more ahout mathematics and children's prices on a couple of them, just to make it Whenever to describea set of datain a succinctway, the issue of mathematical need mathematical thinking. Jan Mokros codirects TERC's Math Center and recently wrote athe book forarises realistic."

Mokros, J. & Russell, S.J. (1995) Children's concepts of average and representativeness.

parents called Beyond Facts and Flashcards: Exploring Math with Your Kids (Portsmouth, N.H.:arises.The goal of this researchis to understandthe characteristicsof fourth representativeness Heinemann Publishers). numbersummarizinga data througheighthgraders'constructionsof "average"as a representative

Average as mediaR

set. Twenty-onestudentswere interviewed,using a seriesof open-endedproblemsthatcalledon Edited by Donald L. Chambers, Wisconsin Center for Education Research, University of childrento constructtheir own notion constructionsof repreFive on basic representativeness. Another groupofof students relied more reasonWisconsin-Madison, Madison, Wi 53706 sentativenessareidentifiedandanalyzed.These approachesillustratethe ways in which students ableness in constructing the data from the average. The action-research ideas in this article were prepared by Donald Chambers. are (or arenot) developinguseful, generaldefinitionsfor the statisticalconcept of average. They drew on what was realistic in their own lives TEACHING CHILDREN MATHEMATICS One objectiveof statisticsis to reduce large,unmanageable,anddisorderedcollections of informationto summaryrepresentations.The need to summarizedata is presenteven among young children.For example, in the surveys conductedby primary-gradestudents,we see movementfrom focusing on individualpieces of data("Ihave one brother")to highlightingand summarizingthe datain some manageableform("Mostof the class membershaveonly one brotheror sister").As soon as thereis the need to describea set of datain a more succinct way, the notion of representativenessarises:Whatis typical of these data?How can we capturetheir range and distribution? The word average often emerges during children's discussions about data. Youngerchildrenuse this wordin an informalway to referto typical,usual,or middle. Olderchildrenalso use the wordto indicatethe mean,median,or mode-terms they have learned in school. The connections that children make-or fail to make-between theirown ways of describingdataandthe "averages"thatthey are learningaboutin mathematicsclass is a focus of this article.

Journal for Research in Mathematics Education. Vol.26, No.1, pp.20-39.

BACKGROUND Althoughchildrenandadultsalike have underdevelopednotions of average,we know very littleaboutthe guidingconceptionsandmisconceptionsfromwhichchildrenbuild theirmodels of representativenessin a data set. It is often assumedby educatorsandtextbookpublishersthataverageis simplyanotherapplicationof division and thatchildrenwho understandthe notion of a fair or equal sharewill also understandthe notion of average.

The work reportedin this paperwas supportedby NationalScience FoundationGrant #MDR-8851114. All opinions, findings, conclusions, and recommendationsexpressed herein are those of the authorsand do not necessarilyreflect the views of the funder.

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potato chips

Consider how students might respond to this task ... Construction Task: potato chips We took a survey of the prices of nine different brands of potato chips. For the same-sized bag, the typical or usual or average price for all brands was $1.38. What could the prices of the nine different brands be? http://www.learner.org/courses/learningmath/data/session10/part_b/index68.html - 13 -

Untitled

The potato-chip task was presented to students in individual interviews to research students' understanding of average. Here are some of the students' responses: 1. Some students would put one price at $1.38, then one at $1.37 and one at $1.39, then one at $1.36 and one at $1.40, and so forth. 2. One student commented, “Okay, first, not all chips are the same, as you told me, but the lowest chips I ever saw was $1.30 myself, so, since the typical price is $1.38, I just put most of them at $1.38, just to make it typical, and highered the prices on a couple of them, just to make it realistic.” 3. One student divided $1.38 by nine, resulting in a price close to 15¢.When asked if pricing the bags at 15¢ would result in a typical price of $1.38, she responded, “Yeah, thatʼs close enough.” 4. When some students were asked to make prices for the potato-chip problem without using the value $1.38, most said that it could not be done. 5. One student chose prices by pairing numbers that totaled $2.38, such as $1.08 and $1.30. She thought that this method resulted in an average of $1.38.

How do these students reason differently about average? http://www.learner.org/courses/learningmath/data/session10/part_b/index68.html - 14 -

framework

Conceptual Framework 1. investigate children's notions of representativeness 2. how new math knowledge grows from informal understanding 3. examine how children construct and describe data sets -> how they understand average 4. average is a tool for summarizing, describing a data set, comparing data sets Mokros & Russell, 1995 - 15 -

background

background ◆ 21 students (grade 4, 6, 8) were interviewed ◆ all taught how to compute an average as part of their regular math class ◆ most of the students' exposure to average consisted of practice with the algorithm, using examples such as finding the average score on a test ◆ the students had no special experience with data in their math classes

Mokros & Russell, 1995 - 16 -

approaches

students' approaches ◆ average as mode ◆ average as algorithm ◆ average as reasonable ◆ average as midpoint ◆ average as mathematical point of balance

Mokros & Russell, 1995 - 17 -

fig 1

With experience, students can begin envisioning the variety of data sets that might be represented by a median value of, for example, 48 inches.

Russell & Mokros, 1996, p.361 - 18 -

fig 2

... introduce the idea of asymmetrical balancing by asking what would happen if one family has 8 people. In this situation, the move from 4 to 8 cannot be balanced by a comparable move on the left-hand side of the mean ...

Russell & Mokros, 1996, p.363 - 19 -

compare data

http://education.uncc.edu/dkpugale/maed5040_spring2006/Proportional%20Reasoning%20Lesson%20from%20Research%20in%20Data%20and%20Chance.pdf

J A N E M. W A T S O N

AND

J. M I C H A E L S H A U G H N E S S Y

Proportional Reasoning: Lessons from Research in Data and Chance

P

RINCIPLES AND STANDARDS FOR SCHOOL

Mathematics (NCTM 2000) places proportionality among the major concepts connecting different topics in the mathematics curriculum at the middle school level (p. 217). What concerns us about many of the problems presented to students, however, is that they are often posed purely as a ratio or proportion from the start. Often the statement of a problem is a giveaway that a proportion is involved. For example, the question “If 15 students out of 20 get a problem correct, how many students in a class of 28 would we expect to get the problem correct?” does not tap the depth of proportional reasoning that is required for meaningful problem solving. JANE WATSON, [email protected], is a teacher educator and mathematics education researcher at the University of Tasmania, Australia. She is interested in all aspects of statistics education, particularly in students’ growth in statistical literacy and in students’ understandings of statistics in the media. MIKE SHAUGHNESSY, [email protected], is a teacher educator and mathematics education researcher at Portland State University in Oregon. He has a long-standing interest in students’ and teachers’ understandings of probability and statistics and in the development of conceptually engaging statistical tasks for both students and teachers.

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MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

In our view, the goal of the middle school curriculum is for students to recognize, without being told, that proportional reasoning is fundamental to problem solving across the curriculum, particularly in the areas of data and chance. We will discuss two contexts where proportional reasoning is needed for a thorough understanding of the problem. The first task presents two data sets of different sizes in graphical form along with a question about which a data set represents better performance by a class of students. In this setting, the focus is on the use of the arithmetic mean as a formal tool reflecting proportional reasoning, supported by visual intuitions of proportion. The second task involves proportional reasoning as students draw repeated samples from a mixture of colored objects and either attend to or ignore variability when predicting the results of the JANE M. WATSON and JONATHAN repeated samples. Proportional reasoning is funda-B. MORITZ mental to making connections between populations and samples drawn from those populations; in the THE BEGINNING OF STATISTICAL later grades, it provides a basis for statistical infer- INFERENCE: COMPARING TWO DATA ence. It is our belief that teachers and students maySETS overlook opportunities to make connections with applications of proportional reasoning in tasks such as comparisons of data sets or predictions of the comABSTRACT. development of school understanding of comparing two data position of samples.The Both of these typesstudents’ of tasks resets is exploredof through responses students in individual quire comparisons ratios, thus,ofproportional rea-interview settings. Eighty-eight students in grades 3 to 9 were presented with data sets in graphical form for comparison. soning. The word lessons in our title has a double

Student responses were analysed according to a developmental cycle which was repeated in two contexts: one where the numbers of values in the data sets were the same and the other where they were different. Strategies observed within the developmental cycles visual, numerical, or a combination of the two. The correctness of outcomes associated with using and combining these strategies varied depending upon the task and the developmental level of the response. Implications for teachers, educational planners and researchers are discussed in relation to the beginning of statistical inference during the school years.

Copyright © 2004 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. were This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

The need to develop an understanding of statistical inference is acknowledged in recent national curriculum documents which include chance and/ or data handling as part of the mathematics curriculum. In Australia for example, A National Statement on Mathematics for Australian Schools (Australian Education Council [AEC], 1991) has a separate heading under Chance and Data for the middle school years onward titled ‘Statistical Inference.’ While leading to the ‘use of estimates of population parameters and confidence intervals’ in the senior years (p. 185), in earlier years it is suggested that students ‘draw inferences and construct and evaluate arguments based on sample data’ (p. 179). Other countries make similar suggestions (e.g., Department for Education and the Welsh Office [DFE], 1995; Ministry of Education, 1992). There are many points in the statistics curriculum where the idea of inference can be introduced, for example using summary statistics to compare a data set with an hypothesised model, or looking at the relationship of two variables with correlation. Usually in formal statistics courses, basic hypothesis testing begins with a single data set and an hypothesised population model rather than with a comparison of two data sets to look for differences in populations. This is likely to be due to the relative complexity of the formulae involved in testing hypotheses in the two situations. For young students, however, it may be more sensible and relevant to compare two groups with each other than to compare one against an hypothetical model. Being close to many out-of-school applications, comparisons of

Watson, J.M. & Shaughnessy, J.M. (2004). Proportional reasoning: lessons from research in data and chance. Mathematics Teaching in the Middle School. 10(2) pp.104-109.

Watson, J.M. & Moritz, J.B. (1999) The Beginning of Statistical Inference: Comparing Two Data Sets. Educational Studies in Mathematics. vol. 37, pp.145-168.

Educational Studies in Mathematics 37: 145–168, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.

Article: EDUC 779/DISK Pips nr. 202353 JM.WEB2C (educkap:humnfam) v.1.15 educ779.tex; 1/06/1999; 19:34; p.1

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part a

Task: Comparing Two Data Sets Two schools are comparing some classes to see which class is better at quick recall of 9 math facts. Consider the blue and red classes. ... Did the two classes score equally well, or did one of the classes score better? Explain how you decided.

Watson & Shaughnessy (2004) - 21 -

part b

Task: Comparing Two Data Sets Two schools are comparing some classes to see which class is better at quick recall of 9 math facts. Consider the green and purple classes. ... Did the two classes score equally well, or did one of the classes score better? Explain how you decided.

Watson & Shaughnessy (2004) - 22 -

part c

Task: Comparing Two Data Sets Two schools are comparing some classes to see which class is better at quick recall of 9 math facts. Consider the yellow and brown classes. ... Did the two classes score equally well, or did one of the classes score better? Explain how you decided.

Watson & Shaughnessy (2004) - 23 -

part d

Task: Comparing Two Data Sets Two schools are comparing some classes to see which class is better at quick recall of 9 math facts. Consider the pink and black classes. ... Did the two classes score equally well, or did one of the classes score better? Explain how you decided.

Watson & Shaughnessy (2004) - 24 -

4 cases

Task: Comparing Two Data Sets In each case, was there a class that you think did better or did the classes do equally well—and why? What do you think your students would say for these comparisons? How do you think they would decide which class did better?

Watson & Shaughnessy (2004) - 25 -