Understanding and Teaching Spatial Reasoning through the Australian Curriculum Proficiency Strands

Spatial thinking is defined as the mental processes of representing, analysing, and drawing inferences from spatial relations (Uttal, Miller & Newcombe, $2013$ , p. $267$ ).
It is essential for comprehending the world, beginning with first interactions with the environment and increasing in complexity throughout life.
Foundational examples of spatial understanding include:
- Constructing $3\text{-dimensional}$ shapes from nets.
- Interacting with smart devices.
- Driving vehicles, which requires multi-faceted spatial skills.
In the school context, spatial thinking involves several sophisticated processes according to Hegarty ( $2010$ ):
- Constructing visual representations.
- Choosing when and how to use mental imagery.
- Mental manipulation of both seen and unseen objects.
- Using mental representations to solve problems.
- Positioning objects in the environment.
- Navigation.
- The communication of visual stimuli.

Spatial reasoning is a foundational predictor of success in mathematics and other STEM (Science, Technology, Engineering, and Mathematics) subjects.
Higher student achievement in mathematics is linked specifically to spatial skills (Mix & Cheng, $2012$ ).
Despite being essential for mathematical concepts such as number lines, mapping, patterning, measurement, graphs, fractions, and arrays, spatial thinking is only explicitly mentioned in the Australian Curriculum: Mathematics in relation to the numeracy capability.
The Australian Curriculum: Mathematics can be understood through a grammar analogy provided by Sullivan ( $2013$ ):
- Content Descriptions (Nouns): These provide the fundamental information and skills of the mathematical canon.
- Proficiency Strands (Verbs): These describe the ways to develop effective mathematicians and act as the "verbs" that make the curriculum meaningful.
The four proficiency strands—Problem Solving, Understanding, Reasoning, and Fluency—provide the necessary tools to develop spatial reasoning.

Problem solving requires students to make decisions about mathematical methods, deconstruct problems, find solutions, and assess the adequacy of answers.
Effective spatial problems should be complex, multifaceted, and related to student worlds (Kilpatrick, Swafford, & Findell, $2001$ ).
The Equidistant Activity:
- Students are challenged to place themselves equidistant from a given object (Penner & Lehrer, $2000$ ).
- Through measurement and discussion, students discover they can only achieve this by forming a regular shape.
- This task allows abstract concepts of shape regularity to be represented concretely.
- Extension of this task involves creating paper or geoboard representations of shapes with differing numbers of vertices while maintaining equidistant restrictions.
- Adding a second or third object further challenges upper primary students as configurations change.
Digital Integration (Minecraft):
- Digital tools like Minecraft allow exploration in virtual worlds free from the limits of physics.
- Students solve spatial problems involving navigation, positioning, and the construction/deconstruction of complex $3\text{-dimensional}$ shapes.
- Teachers can promote mental manipulation by asking students to replicate objects in different orientations.
- Rich questioning can extend thinking (e.g., "Why did you build it that way?", "What benefits does your choice of shape have?", "How could you represent these buildings outside of the game?").

The Understanding proficiency involves recognizing natural relationships between mathematical concepts to avoid superficiality (Kilpatrick et al., $2001$ ).
Conceptual Expansion:
- In the equidistant activity, using string to join participants or chalk drawings on asphalt exposes students to interrelated concepts.
- Adding or removing students/vertices changes the polygon's name (e.g., changing from a triangle to a square or pentagon).
- Older students can move vertices to investigate the effect on internal angles and shape classification.
- Applying a grid to a surface allows for the exploration of enlargement, reduction of area, and transformations (translations, reflections, and rotations) on a cartesian plane.
Dynamic Geometric Environments (DGEs):
- Tools such as Geogebra or Cabri allow for digital representations of shapes that can be manipulated and viewed from multiple viewpoints.
- DGEs represent a $3\text{rd}$ dimension not easily accessed by physical movements.
- They allow for independent exploratory tasks where students can measure area accurately in a virtual space.
- DGEs can be programmed with mathematical constraints (consistent length or angle) to avoid logistical distortions often present in physical activities.

Reasoning allows mathematicians to develop complex concepts from basic axioms and involves evaluation and generalization (Clarke, Clarke, & Sullivan, $2012$ ).
Spatial reasoning can be elucidated through open questions such as "What happens if…?" or "Do you think this will always be true?" (Sullivan, $2011$ ).
Rotational Symmetry Task:
- Students can conjecture that only regular shapes have rotational symmetry.
- Testing the statement involves using cardboard shapes or moving students to recognize factors affecting rotation, such as the placement of the rotation point and reflectional lines of symmetry.
- Deeper reasoning is indicated by the use of causal conjunctions such as "because," "as," and "in order to" (Bragg & Herbert, $2017$ ).
Reasoning with DGEs:
- DGEs allow students to record and playback actions to explain their reasoning and develop visualization skills.
- Students can rotate shapes easily on-screen and move the rotational point to determine different effects.
- "Dragging" vertices in DGEs helps students identify shape invariances (e.g., discovering the four right-angles defining squares).
- Conjectures are tested via a cyclical process of noticing invariance and evaluating its effect (Sinclair & Robutti, $2013$ ).

Fluency involves procedural efficiency, fact recall, and the constant assessment of method efficiency (Sullivan, $2011$ ).
Practice and class discussions regarding method efficiency help students determine the correct processes for specific problems.
Refining mental prototypes of shape is aided by sharing alternative perspectives and solution paths (Presmeg, $2013$ ).
Integrating Digital Tools for Fluency:
- Digital tools reduce the need for complex calculations, allowing students to focus on solving problems fluently.
- True fluency is achieved when technology is integrated seamlessly into the learning environment (Goos, Galbraith, Renshaw, & Geiger, $2000$ ).
- Students should be taught to choose the most efficient tool (e.g., Microsoft Excel for statistics versus Computer Aided Design for $3\text{-dimensional}$ shapes).
- In DGEs, students might find that positioning shapes next to each other or overlaying them is more efficient for determining congruence than using digital measurement tools.
- Decreasing the effort needed to understand a user interface enables greater mathematical fluency.