# Big Ideas Concepts Procedures

## Current Big Ideas Material on the web

Here are some initial sources from a web search under 'Big Ideas, Geometry'. Most of them come from the NCTM standards and related documents.

http://www.a-plus.net/GSTA/grants/framework/cube/geometry-m.html gives a summary of the ideas/ geometry content etc. from the NCTM standards. It is not clear which are actually 'big ideas' - as many of them seem to be learning vocabulary, some basic representational skills, etc.

http://www.learner.org/resources/series98.html gives some videos on teaching k-8 mathematics, a couple of which are about geometry. I have not looked at them so it would be good to check them out and report.

http://www.wrightgroup.com/index.php/programsummary?isbn=007603500X&longCopy=Y describes a commercial product, but also lists what they think of as 'big ideas of 'Reasoning with Geometry':

```   * Attributes of Shapes
* Composing and Decomposing Shapes
* Spatial Reasoning
* Transformations
* Relationships Among Shapes
```

Some vagueness, as 'Attributes' could be lists of little properties, or it could include big ideas like symmetries which form the basis of reasoning.

A web site on 'Teaching to Big Ideas' http://www2.edc.org/CDT/dmi/dmicur.html identifies the following concepts/procedured for geometry:

1. Geometry: Examining Features of Shape • Participants examine aspects of 2D and 3D shapes, develop geometric vocabulary, and explore both definitions and properties of geometric objects. The seminar includes a study of angle, similarity, congruence, and the relationships between 3D objects and their 2D representations.
2. Geometry: Measuring Space in One, Two and Three Dimensions • Participants examine different attributes of size, develop facility in composing and decomposing shapes, and apply these skills to make sense of formulas for area and volume. They also explore conceptual issues of length, area, and volume, as well as their complex inter-relationships.

Overall, these do not seem to be the 'big ideas' as I know them in my work as a geometer, or in the history of geometry.

## Some Bigger Ideas!

What other resources / ideas do we have? Here is a preliminary attempt to list some procedures and some big ideas of geometry, as I (Walter) have learned them. It would be good to build these up / modify them / annotate and replace them:

Here is an initial list that jumbles these together. (BI) suggests there may be a big idea to extract:

### (1 BI) understanding isometries and congruence

This includes compositions of isometries (reflections, rotations, translations) - something we did work on in Introduction to Geometries: Math 3050

• This can become one of the introductions to 'invariance' - what does not changes under certain transformations.
• A related small idea would be connecting paper folding and mirror reflections in the plane.
• Also related it the connections to 'mental rotation' and 'movement' as well as how we actually detect symmetries in comparisons of objects, and within objects.
• this becomes one step in building Klein's Hierarchy, as we add additional transformations (e.g. similarities, affine transformations, ... )

### (2 BI) Symmetries of an object or a pattern

Includes composition of symmetries of an object (embodied group theory) and generation of objects by symmetries (Kaleidoscopes, wall paper patterns, crystallography, .... )

### (3 BI) working up / breaking down to compare 'sameness'

• criteria for testing when 'two objects are "the same" - congruent'.
• for example, for congruence of triangles:
```  SSS
ASA
SAS
```
• Note that we do not have such nice recipes for congruence of quadrilaterals, or for polyhedra in 3-space. We do have some recent results:
• the investigation of when, say, a set of lengths, is enough to guarantee congruence in a given dimension is a major are of research. The concept is called global rigidity.

In general, giving all lengths among all pairs certainly works. Sometimes, as in SSS, or SSSSSS for 4 points in the plane (and in 3-space), this is the best we can do. But often we can do better.

• when this is investigated within a class: e.g. within the class of convex polyhedram we encounter classical results like Cauchy's Theorem, and Alexandrov's Theorem.
• when we ask only that very small changes in position guarantee congruence, we have local uniqueness or rigidity. This is a really big area of research, with a lot of applications. See my home page http://www.math.yorku.ca/~whiteley/

With these connections, it does become at least one big idea. However, I fear that SSS, ASA, etc. (and their extensions to the sphere, such as AAA) are still a small idea.

### (4 BI) Decomposition of 3-D into 2-D parts / 2-D representations

This includes the 'faces vertices edges' Tayna mentions - but in a larger 'big picture' context.

• So some work with polydron and Platonic Solids helps this.

Also includes projections / isometric drawings, shadows, cross-sections.

• includes process of nets of polyhedra,
• When is Euler's formula v-e+f = 2 a big idea? When is it a small pattern?
• How about Descarte's Theorem: the sum of the angle deficits at the vertices (how much angle is missing from being flat, in the material at the vertex) is = 4π

This can be connected to the previous idea. So a 'net' can be a nice set of information for a single convex polyhedron in 3-space, up to congruence. This is actually related to Alexandrov's Theorem, and to Pogorlevo's Evolute Theorem.

• This extension actually says that you do not need to know the 'fold lines'. Just the outline, and clear labeling of which edges glue to which edges (ensuring the pairs are of the same length) along with checking that the total angle at each folded up vertex is less than 2π is enough to guarantee there exist fold lines and that they are unique!

### (5BI) Decomposition and Combining

Taking several shapes and combining them.

In general, decomposition and combining grouping, with processes like truncation and stellation, are key processes in the exploration of objects such as polyhedra.

Taking several shapes and combining them. See the related idea below on 'measuring'.

This can be used as a tool for exploring dimensions, and measures (e.g. perimeter with fixed area).

### (6 BI) Dimension

This groups a bunch of topics. Here are a few:

#### Stretching and its impact on measures

• Impact of 'stretching' of one, two, or ... dimensions on length, area, volume, .... More generally, scaling in geometry and its impact on measurement. This leads to one of the key properties of 'dimension': how does the measurement scale in various dimensions.
```   A nice exploration involves scaling animals and the impact on their functioning and abilities:
e.g.  Scaling a child to an adult:  why does the heart rate slow down?
e.g.  Scaling up a bird - why does it lose the ability to fly?
e.g.  Scaling up a whale: why can it dive longer (and deeper)?
```
• In a larger vision, one will look at fractional (fractal) dimensions, and see how measures change under scaling to assign a position in this listing.

#### Rate of Change of measures

• A number of examples suggest that the 'rate of change' of volume in 3-D is intimately related to surface area, and the rate of change of area in the plane is related to perimeter.
• there are obvious examples:
```  volume of sphere (with radius) to surface area
area of circle (with radius) to permiter
the fence on the river problem in calculus
the popcorn box problem in 3-D
```

Are there some others which are 'typical' of this connection?

### (7BI) Representing and Understanding one dimension in the context of the sequence of dimensions

• Taking a shape and imagining a series of cuts (truncations) to create a smaller shape;
• Taking a shape and imaging the various projections of the shape.
• It is a common problem solving method in geometry to shift down (or up) a dimension to get an insight into a given problem. Poly's video: Let us now teach guessing, is a wonderful example of this process.
• Liftings (e.g. finding 3D objects projecting to given plane images) is the converse of projection. Here is a thoughtful exploration of the themes (including lifting) by a master of the processes, including some nice exercises in spatial reasoning and proposals about representations:
• Janos Baracs (1992) Development of the spatial perception with the help of projections Structural Topology 19. PDF file from http://haydn.upc.es/people/ros/StructuralTopology/ST19
• Janos Baracs (1988) Twelve exercises of spatial perception, Structural Topology 14. PDF file from http://haydn.upc.es/people/ros/StructuralTopology/ST14
• This problem is addressed in areas such as scene analysis, as well related papers from the Maxwell-Cremona theory for rigidity theory. See for example:
• Henry Crapo and Walter Whiteley (1993) Plane self stresses and projected polyhedra I: the basic pattern (Structural Topology 20) http://haydn.upc.es/people/ros/StructuralTopology/ST20

Needs refining restructuring into a big idea of construction, representation, etc. . This can be grouped to a larger idea of dimension.

### (8 BI) Geometric Measures

#### Area, volume (and length) as additive for cuts in to several pieces

Use in deriving formulae for areas, volumes, etc. (E.g. The Greek derivation of the volume of a Hemisphere from the difference in volume of a cylinder surrounding the hemisphere and a cone with the same base. See http://scidiv.bcc.ctc.edu/Math/ArchimedesTombstone.html]

These are big ideas related to when integration measures something geometric, and connect on to situations in physics (e.g. density/mass) which connect to integration.

They also connect to 'decompose and recombine' proofs for Pythagoris, as well as the use of algetiles, and related 'proofs without words' in the plane. In the plane, any two polygons of the same area can be decomposed into the same finite set of smaller polygons. Equivalently, one of them can be cut up and pasted back together as the other. This was an important process / algorithm for 'measuring' area for the Greeks. They 'knew' the area of something if one could create a square of the same area. (In fact, from this point of view, Pythagoris' Theorem was primarily the technique for adding two squares to a single larger square. That may have been its core significance. )

It is significant to know that this process encounters major obstacles in 3-D. We can have to polyhedra of the same volume (even two tetrahedra) and not be able to to decompose one into a finite number of pieces and recombine to the other. This was the negative solution of Max Dehn to the third of Hilbert's famous problems: http://en.wikipedia.org/wiki/Hilbert's_third_problem

#### Measures as invariants

This definitely connects to the big ideas in Klein's Hierarchy of Geometries. Change the transformations, change the invariants, change what 'measures' are suitable.

• Lengths and angles, as well as area, volume etc. are invariant under isometries.
• Angles are invariant under similarity maps;
• ratios of areas are invariant under affine transformations (of the plane); as are ratios of lengths along parallel lines (or the same line);
• chirality (handedness) Orientation, and unorientable surfaces (e.g. the Mobius Band)

This is related to direct congruences (translations rotations) vs changes from reflection. This may belong as a bigger idea.

There are some studies, going back to Piaget, about the ages at which students understand conservation of key measures. Here is one:

• Thomas P. Carpenter; Ruth Lewis The Development of the Concept of a Standard Unit of Measure in Young Children, Journal for Research in Mathematics Education, Vol. 7, No. 1. (Jan., 1976), pp. 53-58.

The conclusion is that children do have difficulty with invariance or equivalence of measures. So when you change the 'unit of measure' they associate the larger number with the larger 'quantity', even for length.

### (9 BI) Klein's hierarchy of geometries

A geometry is a set of points, a group of transformations, and studies the properties unchanged (invariant) under the transformations. The larger the group of transformations, the fewer the properties (and the more 'objects' which are the same or equivalent under the group). See for example: http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/MOHR_TRIGGS/node40.html

• Note the plural: Geometries.

This is definitely a big idea, and historically has been one of the key shifts in mathematics. It is a reason why non-Euclidean Geometry is often proposed for high school.

Here are two articles which connect the use of Klein's Hierarchy and the analysis of Piaget:

• J. Larry Martin: An Analysis of Some of Piaget's Topological Tasks from a Mathematical Point of View, Journal for Research in Mathematics Education, Vol. 7, No. 1. (Jan., 1976), pp. 8-24. This observes that 'topology' as a full mathematical structure is not captured by children's activities, or Piaget's observations.
• F. Richard Kidder: Elementary and Middle School Children's Comprehension of Euclidean Transformations, Journal for Research in Mathematics Education, Vol. 7, No. 1. (Jan., 1976) pp. 40-52. This article has a nice chart of Klein's hierarchy, nested by inclusion of transformation groups, as well as by growing lists of invariants (as the transformation group shrinks).
• It also notes that, as mathematicians understand Euclidean Transformations, middle school students have serious difficulties. This suggests that, there are levels of precision and practice to be learned and taught here.
• Unfortunately, these seem to focus on 2D as the setting for the geometry, so it may not capture the capacities of children in 3D!
• Graph theory - the sense of what is connected to what, is an example of 'topology' in this general sense. I think there is a lot of evidence that this matches what children can, and do, do.

I understand this has even been used in a course for future elementary teachers. The point is that it is accessible to people, should we choose to access it.

• This Hierarchy is surprisingly invisible in the material on the web or in discussions of big ideas in geometry.

### (10) Spatial reasoning and connections to problem solving / cognition.

Probably a big idea but it needs to be more explicit to be learnable. See for example: Imagery, Spatial Ability and Problem Solving

### (11 P) Manipulatives / virtual manipulatives

• Process/Technology around hands on materials and the use of virtual representations in both 2-D and 3-D. This has the potential to be a big process / pedagogical idea, if we explore it.
• In a recent CMESG working group CMESG Working Group Page: Geometry, space and technology, a tentative conclusion was that experience with materials is an essential companion to the virtual representations. While we explored sequencing (physical first, then virtual), I anticipate that this is part of a cycle: the physical provides a basis for interpreting the virtual, and the virtual informs how we experience the physical when we come back. It is mistake to imagine a simple sequence, and to drop the sense-making, sense-embedding process of returning to our hands and eyes, from just our eyes alone.
• At a process level, there is evidence that activities like 'mental rotation' actually involve pre-motor planning in our minds - and we want to continue to activate / incorporate that portion of our cognition as we engage with virtual representations.
• There is evidence that the tools we use connect to the way we conceptualize the processes. Follow the link above.

```Giora Hon and  Bernard R. Goldstein: From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept