This article provides a classroom resource for identifying, assessing and promoting one aspect of reasoning--analysing--through the Assessing Mathematical Reasoning Rubric. Practical examples of student work and rubric levels are offered.

Reasoning is the glue that holds mathematics together (Kilpatrick, Swafford, & Findell, 2001). Consequently, developing and promoting students' mathematical reasoning must be a goal in every mathematics lesson. Satisfyingly, reasoning has enjoyed a growing focus over the last decade (see studies by Bragg & Herbert, 2017; Clarke, Clarke & Sullivan, 2012; Herbert, Vale, Bragg, Loong, & Widjaja, 2015; Stacey & Vincent, 2009), possibly as a result of its place as one of the four key proficiencies introduced in the Australian Curriculum: Mathematics [AC:M] (Australian Curriculum, Assessment and Reporting Authority [ACARA], 2017). However, despite this focus and an assumed universal agreement on the meaning of reasoning (Yackel & Hanna, 2003), the definitions of reasoning in curriculum documents and research studies are numerous and often contradictory, thus leading to a lack of a shared definition of reasoning (Jeannotte & Keiran, 2017). Our research on reasoning has shown primary teachers hold diverse perceptions of mathematical reasoning (Herbert, et al., 2015). Hence, this lack of clarity can impact the successful enactment of reasoning in the classroom.

While we acknowledge the differing views of reasoning others hold, we hope to add clarity to the discussion by shining a spotlight on the reasoning action of analysing, and offer teachers opportunities to explore analysing with their students.

The importance of analysing

Analysing a mathematical problem systematically is key to forming conjectures, establishing generalisations, justifying a stance, and making a logical argument. As part of the reSolve: Mathematics by Inquiry collection of supportive resources for assessing reasoning (Australian Academy of Science and Australian Association of Mathematics Teachers, 2017) the authors and their colleagues developed the Assessing Mathematical Reasoning Rubric (see Herbert & Bragg, 2017 for Version 1). This is built in part from the following definition of analysing, and is based on the works of Lo (2012), and Lobato, Hohensee and Rhodehamel (2013), in exploring comparing and contrasting; and Bills and Rowland (1999) in generating examples to form conjectures. Our definition of analysing is as follows:

Analysing involves exploring the problem using examples provided or generating examples to form or test a conjecture. Analysing occurs by comparing and contrasting cases to notice:

* what is the same and what is different, and to sort and classify the cases.

* what stays the same and what changes and to recall, repeat or extend the pattern.

Analysing involves using numerical or spatial structure, known facts or properties when sorting cases or repeating and extending the pattern. Categories of cases and patterns are identified by labelling using terms, diagrams or symbols. (Vale, Herbert, Bragg, Loong, Widjaja, & Davidson, 2018, p. 4).

Understanding what constitutes analysing is the first step to raising one's awareness to notice students' enactment of analysing and supporting subsequent assessment of when and how analysing takes place. Figure 1 illustrates the five levels of the analysing action in the Assessing Mathematical Reasoning Rubric: Not evident, Beginning, Developing, Consolidating, and Extending.

To assist in identifying, assessing, and promoting analysing, we offer the Number Tower task (adapted from the Noyce Foundation, 2012), along with examples of students' responses against each level of analysing actions. Further examples of tasks employing a differing reasoning actions using the Assessing Mathematical Reasoning Rubric are located on the reSolve: Mathematics by Inquiry website: http://www. resolve.edu.au/

Number tower

A Number Tower is created by distributing the numbers 1, 2, 3, 4, and 5 along the bottom row of a blank tower (Figure 2). Each pair of adjoining numbers are added together, and the sum is written into the cell above, e.g. 1 + 2 = 3, 2 + 5 = 7, etc. Whilst on the surface this task appears to encourage simple addition, the goal of the Number Tower task is to create a tower which results in the highest possible number at the pinnacle of the tower. Hence, students are required to analyse the problem and trial various number combinations to form and test conjectures, ultimately establishing a rule (generalisation) for generating the highest number on top.

The Number Tower task builds on the following content areas from the Australian Curriculum: Mathematics (ACARA, 2017).

ACMNA054: Recognise and explain the connection between addition and subtraction. Demonstrating the connection between addition and subtraction using partitioning or by writing equivalent number sentences. ACMNA055: Recall addition facts for single-digit numbers and related subtraction facts to develop increasingly efficient mental strategies for computation. Combining knowledge of addition and subtraction facts and partitioning to aid computation.

Setting up the reasoning task

We introduce the Number Tower task using the Jane's conjecture scenario (Figure 3).

Figure 3. The Number Tower Task scenario.

Jane's conjecture

Jane thinks to make the largest total at the top of
the tower, the largest number needs to be the first
number on the bottom row. Do you agree with
Jane? Why or why not?

Students' analysing actions and reasoning prompts

The Number Tower task offers many opportunities to explore reasoning through analysing, generalising, and justifying (see Herbert & Bragg, 2017). This section contains examples of students' responses that illustrate analysing at each level of the Assessing Mathematical Reasoning Rubric and might assist in assessing your students' work. The suggested reasoning prompts are offered to teachers for supporting or extending students' analysing based on their responses.

Level: Not evident

Student response: Creates a tower though does not attend to the rules for constructing each layer of addition. Does not add the adjoining numbers.

A student who does not notice the numerical structure of the tower would be classified at the Not Evident level. In this instance it is important to firstly ascertain if the child had simply misunderstood the nature of the task by asking them to explain their reason for constructing the tower in this way. This approach would assist to clarify any misunderstandings. Reasoning prompts to assist students might be: "What totals can you make using the numbers 1-5 on the bottom row?" or perhaps offer an enabling prompt such as a simplified tower with a reduced number of levels in the tower. As the focus is primarily on reasoning, maybe offer a calculator to students who are exhibiting difficulties with the addition aspect.

Level: Beginning

Student response: Attempts to sort the number towers based on a particular position of a number (Figure 4).

In this example, the student notices similarities across examples and attempts to sort cases based on a common property, i.e. the position of the 4 in the bottom row. Reasoning prompts to encourage an analysis of alternative properties might be: "Change something (e.g. the position of the number 4 in the bottom row) to see what happens to the Number Tower." "If we change this what will happen?" "Sort or organise the following according to ... (4 in different positions or a different number in the bottom left-hand corner)."

Level: Developing

Student response: The student numbers each tower and orders accordingly from lowest and highest totals to explore what stays the same and what is different (Figure 5).

This student commenced with trial and error via randomly placing numbers, then worked systematically to sort and compare the towers according to a common property (the low and high values of the top number in the tower). Reasoning prompts for this student's response might be: "What do you notice? Describe what stays the same and what changes."

Level: Consolidating

Student response: The student investigates how the position of the numbers impacts the distribution of subsequent totals and then the final total (Figure 6).

This student is demonstrating the Consolidating level of analysing, as he has noticed a common property (i.e. the importance of the central position of the highest number in the set) by systematically generating further cases and communicating a logical argument to complete the chain of reasoning. The student has reasoned, correctly, that the power of the highest number is reduced when placed in the outer blocks. To extend this student's thinking these reasoning prompts would be useful: "What happens in general?" "Are there other examples that fit the rule?"

Level: Extending

Student response: The student describes the impact the position of the number on the bottom row has on subsequent totals (Figure 7).

Through generating further examples using an alternative set of numbers (0.1 to 0.5) this student analysed their conjecture that the largest number should be in the middle of the bottom row and the lowest numbers to each end. Here the student noticed and explored relationships between the numerical structure of patterns. The student may be encouraged to clarify their thinking via the reasoning prompt: "How can you explain the rule to someone else?"

Further exploration of the task

As demonstrated in the work of the student in the Extending level, to further students' exploration of this task and promote analysis, the use of an alternative range of numbers might be employed.

* Repeat the task using any set of 5 numbers such as 4, 5, 6, 7, 8 or 2, 7, 10, 11, 24

* Use a set of numbers which include decimals, such as 0.1, 0.2, 0.3, 0.4, 0.5

* Increase the base of the tower.

The teacher might ask themselves: "Do the students apply similar strategies to those employed in the initial task to these follow-up tasks? Does the range of students' strategies in the initial task hold true for these sets of numbers?"

Concluding remarks

While mathematical reasoning is defined in various ways (Jeannotte & Keiran, 2017) and enacted from diversified approaches, this article has attempted to delve further into one aspect of reasoning, namely analysing, to provide primary teachers with a tangible resource for identifying, assessing, and promoting analysing in their classrooms. The purpose of utilising the Number Tower task was to encourage analysing via conducting trials using number facts and noticing additive properties of numbers. The analysing actions in the Assessing Mathematical Reasoning Rubric (Figure 1) is a useful guide to assessing current student actions and planning for future reasoning opportunities. The prompts offered for each of the levels of analysing are demonstrated in one task, however, we encourage teachers to adapt and modify these prompts for the multitude of activities undertaken each week in the mathematics classroom.

Acknowledgements

We would like to acknowledge the Australian Government Department of Education and Training funding of the reSolve: Mathematics by Inquiry project, and the project team from Deakin University, Colleen Vale/Sandra Herbert, Leicha Bragg, Esther Loong, Wanty Widjaja, and Aylie Davidson.

References

Australian Academy of Science and Australian Association of Mathematics Teachers. (2017). reSolve: Mathematics by Inquiry. Retrieved from http://www.resolve.edu.au/

Australian Curriculum Assessment and Reporting Authority. (2017). The Australian Curriculum: Mathematics. Retrieved from https:// www.australiancurriculum.edu.au/

Bills, L., & Rowland, T (1999). Examples, generalisation and proof. Research in Mathematics Education, 1(1), 103--116.

Bragg, L. A., & Herbert, S. (2017). A 'true' story about mathematical reasoning made easy. Australian Primary Mathematics Classroom, 22(4), 3-6.

Clarke, D. M., Clarke, D. J., & Sullivan, P. (2012). Reasoning in the Australian Curriculum: Understanding its meaning and using the relevant language. Australian Primary Mathematics Classroom, 17(3), 28-32.

Herbert, S., & Bragg, L. A. (2017). Planning and assessing mathematical reasoning in the primary classroom. In R. Seah, M. Horne, J. Ocean, & C. Orellana, (Eds.). Achieving excellence in M.A.T.H.S. Proceedings of the Mathematical Association of Victoria 54th Annual Conference, (pp. 48-54). Melbourne, Victoria: MAV

Herbert, S., Vale, C., Bragg, L. A., Loong, E. & Widjaja, W. (2015). A framework for primary teachers' perceptions of mathematical reasoning. International Journal of Educational Research, 74, 26-37.

Jeannotte, D. & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96(1) 1-16. DOI 10.1007/ s10649-017-9761-8

Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

Lo, M. P (2012). Variation theory and the improvement of teaching and learning. Goteborg, Sweden: Acta Universitatis Gothoburgensis.

Lobato, J., Hohensee, C., & Rhodehamel, B. (2013). Students' mathematical noticing. Journal for Research in Mathematics Education, 44(5), 809-850.

Noyce Foundation. (2012). Number towers. Retrieved from http:// www.insidemathematics.org/assets/common-coremath-tasks/ number%20towers.pdf

Stacey, K., & Vincent, J. (2009). Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks. Educational Studies in Mathematics, 72(3), 271-288.

Vale, C., Herbert, S., Bragg, L. A., Loong, E. Y-K., Widjaja, W. & Davidson, A. (2018). Assessing mathematical reasoning: Teacher's Guide. In reSolve: Mathematics by inquiry. Australian Academy of Science & Australian Association of Mathematics Teachers. Retrieved from http://www.resolve.edu.au/

Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics, (pp. 227-236). Reston: NCTM.

Leicha A. Bragg

Deakin University, Vic.

<leicha.bragg@deakin.edu.au>

Sandra Herbert

Deakin University, Vic.

<sandra.herbert@deakin.edu.au>

Aylie Davidson

Deakin University, Vic.

<aylie.davidson@deakin.edu.au>

Caption: Figure 2. Blank and completed Number Towers.

Caption: Figure 4. Student work sample of Beginning Level of Analysing.

Caption: Figure 5. Student work sample of Developing Level of Analysing.

Caption: Figure 6. Student work sample of Consolidating Level of Analysing.

Caption: Figure 7. Student work sample of Extending Level of Analysing.

Figure 1. Analysing actions in the Assessing Mathematical Reasoning
Rubric-Version 1 (Herbert & Bragg, 2017, p. 50).

Analysing

Not evident     * Does not notice numerical or spatial structure of
                  examples or cases.
                * Attends to non-mathematical aspects of the
                  examples or cases.
Beginning       * Notices similarities across examples.
                * Recalls random known facts related to the
                  examples.
                * Recalls and repeats patterns displayed visually
                  or through use of materials.
                * Attempts to sort cases based on a common
                  property.
Developing      * Notices a common numerical or spatial property.
                * Recalls, repeats and extends patterns using
                  numerical structure or spatial structure.
                * Sorts and classifies cases according to a common
                  property.
                * Orders cases to show what is the same or stays
                  the same and what is different or changes.
                * Describes the case or pattern by labelling the
                  category or sequence.
Consolidating   * Notices more than one common property by
                  systematically generating further cases and/or
                  listing and considering a range of known facts
                  or properties.
                * Repeats and extends patterns using both the
                  numerical and spatial structure.
                * Makes a prediction about other cases:
                * with the same property
                * included in the pattern.
Extending       * Notices and explores relationships between:
                * common properties
                * numerical structures of patterns.
                * Generates examples:
                * using tools, technology and modelling
                * to form a conjecture.