Geometry
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The following is the next phase in the Maths in the Kimberley project and consists of some lesson experiences. It follows on from interviews with students at Kulkurriya and Yiyili schools.
The focus on geometry is based on an often stated assumption that the prevalence of direction words in some Indigenous languages implies that the learning of aspects of geometry may be closer to Indigenous students’ experience than the learning of number. The following is seeking to explore this.
If students do seem familiar with the concepts then it would make sense to use this aspect of the curriculum more actively to build confidence, success, and connection to mathematics and schooling. It may also be possible to incorporate visual approaches to other topics.
It also seems that geometrical questions are over represented in NAPLAN assessments so some attention to this topic is likely be useful for students and the schools.
An interesting aspect is that, in inspecting items in NAPLAN assessments, it seems that the majority of questions depend on knowledge of conventions as much as knowledge of space and location. One of the important conventions is the way that 3D shapes are represented on a 2D page or screen. Metropolitan based children are exposed to many such images and perhaps the remote students do not have such opportunities.
The approach proposed recognises that a significant challenge for teachers is the process of connecting English language, especially specific mathematical terms, with student experience. If concepts seem well developed in relevant language then the goal of instruction is to connect the English terms with local terms. If some concepts do not seem well developed then the goal would be to provide experiences in the use of the various concepts.
There are three aspects to the following. Note that these do not address all aspects of geometry required by the curriculum. Some key aspects not addressed are transformations (symmetry etc), and angles.
The three lesson sequence, each of which represents approximately one week of work, is not intended to be sequential and so can be taught in any order. They are written for upper primary classes but are probably suitable for secondary students as well.
The three aspects are:
· Perspective
· Nets and properties of 3D geometrical shapes
· Maps (this one is not included yet)
I might add another on isometric drawing, if the first ones go well.
For each aspect, the following presents:
· results from some data collection that has informed the emphasis as well as a sequence of the activities.
· a sequence of experiences, not written as scripted lessons, just as general ideas. It is assumed that teachers will adapt these to their own students.
The research questions at this stage are:
If lesson sequences are designed to address specific geometrical concepts as identified by student interviews:
· Which aspects of those sequences are engaging for students?
· Which aspects of those sequences improve responses of students to geometry items on NAPLAN style assessments?
Preliminary data collection
There are a number of items on NAPLAN that require students to interpret diagrams and drawings from different perspectives.
In an individual interview format, the students were presented with 2009 NAPLAN items on perspective and location that were relevant for their level, with the items read to the students if necessary.
In summary the results were:
· 13/15 of the middle primary students could choose the correct piece to go into a jigsaw
· Given a prism with an L shaped cross section that required students to count faces, 4/15 of the middle primary students could choose the correct response (5/15 counted only the faces that could be seen)
· Given a horizontal view drawing of an ice cream cone, 5/15 could choose a circle as the top view.
· For the upper primary, given the net of a cube, 5/13 could state which face is opposite a nominated face
· Shown a cube with corners cut off and asked to count the remaining faces, no upper primary student could write the answer correctly.
· Given a diagram of a rectangular prism, and told that the sum of the faces, edges and vertices is 26, 3/13 could choose the total of the faces, edges and vertices of a square pyramid.
There were also some comparative items that were presented using objects that could be touched or seen and not in the test item format. The results from this were:
· Given a small cube, 5/28 could name it, 4/28 could count edges, 18/28 could give the number of faces, and 17/28 could count corners.
· Shown a photo of a structure made from cubes, 4/28 could state how many cubes were needed to make it, 1/28 could say how many faces would be painted (one said all of them), 25/28 could make it (development was even across the grades), 23/28 could place a yellow cube to the left of the structure (7/11 grade 4s), 14/28 could put a red cube north of the structure (development was even)
· Asked to draw the bird’s eye view of the building they were in, 12/27 could do this.
· Asked to draw the water tank from above 17/27 could do this
· 6/17 Kulkarriya were a able to draw the shape of the homemaker building seen from above, all of whom were from the upper grades
In summary, nearly all of these items were answered correctly by some students, and most students were able to do some of the tasks, including replicating a shape in a photo, and some aspects of the properties of the cube. No student was able to answer all of the items. In general, the year 6/7 students did not seem better able to do the tasks than the 4/5 students.
Overall goals of this sequence of experiences on perspective and location
It is necessary to explain to the students why they are doing this. The fundamental goal, that you can say over and over, is “imagining”.
The capacity to interpret diagrams, drawing and photos is important for some topics in secondary level mathematics such as finding the volume of objects, and practical problems. It also helps in reading maps, interpreting drawing of buildings, communicating directions, describing objects and understanding instructions. These are important life skills.
Hopefully some experience at these tasks will also assist students in completing future NAPLAN items as well.
The following some activities intended to provide some relevant experiences.
GeometryPage 15
Explain the purpose to the students:
Explain that they are going to learn about interpreting and making drawings of objects, and how those objects look different from different perspectives, and that this will help them in their lives and also in their future school learning. Explain that one of the key issues for them is to learn the words that are used to describe the ways things look.
Teaching experiences:
It is assumed that you will use your usual strategies for revising and introducing terms. The strategies you use in AL may be useful. The students need to hear you modelling the use of the terms, and to say the words themselves many times.
Other strategies could include
- having labels on parts of the room to show north etc, left right, above, below etc.
- emphasising direction words when telling stories
- asking the AEWs or other locals what words would be used in Kriol or language and have them talk to the class about them. This could include asking specific questions such as “how do you show north? Left? near? above”
- any other strategies that you find work in literacy such as word searches
- connect the terms to what they know (“have you heard the word left before, where? How do you remember left”)
- asking what is similar and what is different about, for example, a square and a rectangle
Key terms for revision are left, right, above, below, under, on top, side, north, south, east, west …
Key new words for specific and formal emphasis are face, edge, corner, vertex (plural is vertices), 2 dimensional, 3 dimensional, 2D, 3D, perspective, bird’s eye view.
Note that it is suggested that you do some language activities everyday that you are doing this work.
As a prompt to this discussion, you could find some images that the students might recognise (perhaps like the ones here). Ask them what the logo or graphic might represent. How do they know? How would they describe the image?
Step 2: Drawing and making to instructions
Explain the purpose to the students:
Explain that it is often necessary to give instructions and descriptions, and it is also necessary to be able to interpret them. The following activities give experience at using and interpreting the necessary language.
Teaching experiences:
a. You will have to adapt this activity to the equipment you have. The ideal is the sets of Lego blocks with a base plate so that students can hold up the shapes once they have made them. However, you can do this with any materials.
Everyone has the same equipment (either individually or in pairs), including you. You make a shape but have it covered. Describe what you have made, with the students having to make what you say. For example, you might say, there is a blue block on top of the red block, there is a yellow triangle to the left of that, there is a another blue block to the north, and so on. Do this a number of times creating different instructions each time. If they are doing well, make the instructions more complex. The review of this is important. Have the students compare what they build with the one that you built. Are they the same?
You might ask one of the students to make a shape and give instructions. In fact they could give the instructions in Kriol if they prefer.
b. Give the students instructions, and ask them to draw them on a small white board. For example, “draw a square, draw a triangle on top of the square, draw a circle to the left, etc”. They can show you their whiteboards to see whether they match what you said. You can comment on their interpretation. Repeat, adapting to the level of success of the students.
As before you might ask one of the students to give instructions. In fact they could give the instructions in Kriol if they prefer.
c. Have pile of cubes. Ask the students to:
i. Build something using 15 cubes. Describe what they have built.
ii. Build something which is 3 cubes high and 3 cubes wide made with 15 cubes.
iii. Something like “make a tower of 3 cubes, put a yellow cube to the left, and blue cube to the north, …”
Repeat, adapting to the level of success of the students.
Explain the purpose to the students:
Explain that these activities are for students to experience the number of cubes used to construct a shape even though those cubes may not be immediately evident in the photo.
One of the tasks that the students found difficult in the NAPLAN test was imagining unseen but necessary aspects of objects.
Make a structure something like this photo using cubes, ideally they should be coloured. This will depend on the equipment you have. You may have linking cubes so you will have to adapt this.
Ask questions like:
“How many cubes did I use to make this?”
“If I painted the outside black including those on the bottom, how many faces would be painted black?”
Students can walk around your structure at first, but for later questions have them just look from one side.
Repeat this a number of times using different structures.
They could also do this in pairs, with one student making a shape and them asking the questions of the other.
Teaching experiences:
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Explain the purpose to the students:
Explain that sometimes drawings and photos show only one perspective of a shape, yet it is necessary to imagine what the rest of the shape might be. There is a particular view that is called “bird’s eye”.
Teaching experiences:
a. Ask them to draw what the school bus would look like to a bird flying directly above it. What would be school bus look like to a dog sitting on the road as the bus approached.
b. Show a picture from google earth of the community (on the next page – actually you might be able to get a better one). You might be able to put this on the smartboard. Ask them to name the various buildings or objects they can see. Ask questions like “how do you know that is the river?”
Explain that this image is old. What has been build since this photo was taken?
c. Draw a circle on the board. Sat “This is an object seen from above, What might be the object?“ (Note that there is a range of possible anwers to this question so consider all reasonable suggestions. Repeat this for different shapes such as rectangle, etc?
d. Have a photo of something around the school. Ask students to say where the photo was taken from. Ask them to draw what the building would look like if the photo was taken from the opposite side? … if the photo was take from above.
e. Have drawing of a building from above. Ask them to suggest what building might your drawing be?
GeometryPage 15
GeometryPage 15
GeometryPage 15
GeometryPage 15
Explain the purpose to the students:
Explain that this activity will give them the opportunity to revise the work from the previous lessons, and to match up some different views of the same object made from cubes. Emphasis that they will have to explain their reasoning.