OPPORTUNITIES TO LEARN MEAN, MEDIAN, AND MODE AFFORDED BY TEXTBOOK TASKS

Karin Landtblom

Stockholm University

Sweden

Karin.Landtblom@su.se

Background

• Vertical approach of the textbook analysis: how mathematical concepts are treated through a deep analysis of the concepts.

Titel: OPPORTUNITIES TO LEARN MEAN, MEDIAN, AND MODE AFFORDED BY TEXTBOOK TASKS

• Opportunities to learn (OTL)–used to analyse the content promoted by the textbook tasks

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(Callingham & Watson, 2017; Charalambous et al., 2010; Floden, 2002; Husén, 1967; Leavy, 2010)

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Background

Non-contextual tasks:

• Do not treat, refer to, or contain any extra-mathematical elements
• Often without a context; so called bare tasks

Contextual tasks:

Many different definitions

• My choice out of the purpose how educational features are included in a task

(Charalambous et al., 2010; Greatorex, 2014; Kieran et al., 2015; Reinke, 2019; Watson & Mason, 2006; Verschaffel et al., 2000; Wijaya et al., 2015)

Background

• My purpose: How knowledge about mean, median, and mode is included in the tasks in terms of mathematical properties.

My definitions for non-contextual and contextual tasks:

• In the analysis of tasks for this study, tasks are categorised as contextual or non-contextual depending on whether they afford OTL about mathematical properties or not

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(Chick, 2007; Hong & Choi, 2018; Kieran et al., 2015; Leavy, 2010; Liljedahl et al., 2007; Reinke, 2019; Stein et al., 2007; Verschaffel et al., 2000; Watson & Mason, 2006; Watson & Thompson, 2015; Wijaya et al., 2015).

Background

• The used framework is build on the idea that a concept is based on central mathematical ideas that in turn are “built on a set of objects, transformations, and their properties” (Lithner, 2008, p. 261).
• Input object, transformation and output object
• The input object—the entity transformed—can be a number, a variable, a diagram, etc.
• Example: The input object of 2 (a number) is entered into a function f(x) = x³ (a transformation) to yield an output object of 8 (a number).

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(e.g., Burrill & Biehler, 2011; Byström & Byström, 2011; Lampen, 2015; Lithner, 2008; Strauss & Bichler, 1988)

Background

• Input object is the variable in the dataset.
• Categories: Nominal, Ordinal and Quantitative (including both interval and ratio level)

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(e.g., Byström & Byström, 2011; Leavy, 2010; Leavy et al., 2009)

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Background

Categories for transformations:

• For mean: Calculation
• For median: Rank ordering
• For mode: Grouping.

Other approaches:

• Reverse calculation: To find a missing value in a dataset
• Debugging: To interpret a mathematical activity or formula such as identifying errors in a calculation.

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(Glasnovic Gracin, 2018; Groth, 2007; Groth & Bergner, 2006; Konold & Pollastek 2004; Watson, 2006)

Background

• Equal distribution: Compare to fair share. Builds on the mathematical property that the sum of deviations from the mean equals zero.
• Validate values: Focus on an answer, the output object, and affords the opportunity to explore the relation between the calculated measure and the context of the given data set.
• Example: A task with an output object that equals a mean of 2.3 siblings per family give us a possibility to explore how a discontinuous input object changes because of the transformation.

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(Konold & Pollastek, 2004; Strauss & Bichler, 1988; Watson & Moritz, 2000; Watson & Thompson, 2015)

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Background

Output object: the result of transforming the input object–here a measure of central tendency with corresponding mathematical properties = categories of the output objects.

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Mathematical properties are described in the article

I will come back to mathematical properties in the result

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Aim and reserach questions

The focus of this study is on analysing the mathematical properties inherent in specific tasks. By examining what opportunities to learn (OTL) measures of central tendency are presented in Swedish textbook tasks for students aged 10–13, we can gain insight into their potential for learning. Focusing on representative textbooks from a country can be considered a unique signature of the textbooks for this particular country (Charalambous et al., 2010).

RQ:

1. What is the distribution among non-contextual and contextual tasks?
2. What opportunities to learn (OTL) about a) input objects, b) transformations, and c) output objects do textbook tasks afford, and what does the distribution among input objects, transformations, and output objects look like?

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Method (Very consice)

• Sample: 17 printed textbooks from seven different Swedish textbook series
• School years 4–6 (ages 10–13)
• The unit of analysis: a task (every written marked division of a proposed student activity)
• 1 392  tasks
• Deductive content analysis

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Results RQ1

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(1) What is the distribution among non-contextual and contextual tasks?

Table 2. Frequency of observed (and expected) non-contextual and contextual tasks, chi-square statistic for each cell, percentage of observed task type by measure, and chi-square test results

* p < .001

Nearly two-thirds of the tasks

The statistical tests provides evidence of differences in the distributions of the two categories of contexts among the three measures with the mode components contributing disproportionately more to the chi-square statistic value

Definition:

A contextual task afford OTL about mathematical properties (appr. 61%)

Less OTL about mathematical properties related to the mode

Align with previous research that mode have a less prominent role in teaching/learning

Here–many contextual tasks

However – all properties are not explicit.

In the article I discuss them as explicit contextual or implicit contextual.

Results RQ2a

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(2a) What opportunities to learn (OTL) about input objects do textbook tasks afford, and what does the distribution among input objects look like?

Table 3. Frequency and percentage of total of input objects

Focus on whether the measure is appropriate to the given data—no calculation was asked for (validate values)

Input objects—variables predominantly data on a quantitaive level also for the mode

Ordinal values treated as quantiatavie

Possibility to practice mode on a nominal level appears totally 57 times (appr. 4% of the total) [unique property]

Remember:

7 textbook series and 17 textbooks

Prevalence of quantitative variables for all measures–Supports the idea that mode applies only to numerical data.

Results aligns with previous research

Results RQ2b

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(2b) What opportunities to learn (OTL) about transformations do textbook tasks afford, and what does the distribution among transformations look like?

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Table 4. Frequency and percentage of total of transformations

1007 (appr. 71%) of all transformations are linked to the definition of the measure

Mode and median: higher percentage of transformations linked to the definition

More other approaches of transformations for the mean

Equal distribution – uniqe to the mean

Reverse calculation–most frequent–especially for the mean

Debug–least frequent

Mostly procedurers–aligns with previous research indicating that statistics instruction primarily focus on calculation for measures of central tendency

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Few alternative transformations–few opportunities to develop a deeper understanding by exploring connections between the transformation and input or output object.

Limit students' OTL key concepts.

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Previous research state an overweight on procedures

79% of all tasks apply to a given procedure (Jäder et al., 2015)

Results RQ2c

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(2c) What opportunities to learn (OTL) about output objects do textbook tasks afford, and what does the distribution among output objects look like?

Table 5. Frequency and percentage of total of output

n/a = not applicable

Most properties related to the mean (appr. 73%)

Some properties only applicable to the mean

Many properties not applicable to the mode

Most frequent property

Second most frequent property

Unique property to the mode – appears in 16 tasks in totally 17 textbooks (in average once per textbook)

Mathematical properties are identified both in the context of a task but also through solving the task. The implicit properties appeared when calculating the measures.

The theoretical framework: to consider mathematical properties in the output object, allows us to identify implicit properties.

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Task design:

Tasks need to explicitly focus on the mathematical properties or the transformation.

Awareness to “make” implicit properties explicit.

Example: The mean of flower petals is 8.16. What does that value mean?

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Conclusions

• A lot of implicit properties: Tasks serve solely to develop procedural knowledge without proper identification of the properties.
• Mode: The analysed tasks do not give students the opportunity to perceive deep knowledge about the mode.
• Overall, more attention needed to mathematical properties in task design to make the properties explicit.
• Ordinal level values in tasks: Textbooks do not provide sufficient guidance on specific properties of ordinal data.

(Brousseau, 1997; Groth & Bergner, 2013; Kitto et al., 2019; Leavy et al., 2009; Mayén & Diaz, 2010; Star, 2005)

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Future studies

• It is important to consider the impact of textbook tasks on students' learning, and future studies could explore different types of tasks and analyse students' mathematical reasoning to determine how they use different mathematical properties in their solutions.

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• It is worth noting that although the data analysed in this study came from Swedish textbooks, the methodology and analytical approach used can be applied to textbooks or other tasks from other countries. For instance, further studies could analyse how TIMSS and PISA treat measures of central tendency.

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Limitations

Overarching goal: Examine the OTL about mean, median, and mode afforded by Swedish textbooks and to determine how this information was presented in a country-specific way

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• Altering the sample could lead to slightly different results.
• Despite this limitation, interesting patterns can still be observed.