ICT Integration: So how did it go?
I tried out this ICT task in class the today. The following are some notes on how it went and changes I would make.
- Engagement – Students were quite excited by how the task was introduced. I made a reference to schoolies week (and the connection that had to accommodation options on the Gold Coast) which gained their attention. A few students had heard of the Q1 building and at least a few students were familiar with its status as the worlds tallest residential building and Australia’s tallest building. They also were energised as I suggested that the textbook exercises they had been doing were simply the same question, with cosmetic changes, ad nauseum. When I explained that they could actually work out the heights of structures in Google Earth by measuring the length of their shadows, a number of students were audibly excited by the prospect. So far so good.
- On-task / Off-task – Roughly half the class was primarily engaged in on-task behaviour throughout the lesson. For the rest, any small hurdles to overcome quickly resulted in off-task behaviour. Some problems and attempted solutions are included;
- Inexplicably, some student’s accounts didn’t have Google Earth installed (although it is meant to be on every student account). This was relatively easy to deal with as they were working in pairs and having the other student log in usually sorted this out. If not, I was able to log the students in on my account.
- No internet credit. Yes, unfortunately the college I am placed at charges students for their internet access. I feel strongly about this inequity and had suggested to students before the task if they weren’t happy to use their own quota, I would log them in. This wasn’t an issue but it chewed through a little bit of time.
- Uncertainty about how to calculate heights. Some of the weaker students struggled to calculate scale factors, let alone apply them. I directed these students to a modeled example on the whiteboard and asked them to query me again if they were still unsure. Using an IWB in this situation would have been great.
All of these small issues rapidly resulted in off-task behaviour such as checking email. re-skinning their student portal page, or searching for the house they were partying at one the weekend.
For next time ….
If I were to run this task again I would make sure I could get access to a room with an IWB or data projector such that I could model the task. Presenting a series of escalatingly challenging problems as a guided lesson with modelling at each step may have helped to reduce off-task behaviour and student engagement. This would not necessarily detract from the exploratory nature of the task as some time could be left for more open ended task, after the guided tasks were completed. Students would also be better equiped to deal with an open exploration and more comfortable working in the Google Earth environment.
Despite the issues I experienced, I would definitely teach this lesson again. Those students who engaged with the task were enjoying themselves whilst doing maths problems (a major breakthrough for some students) and I feel they satisfied the learning objectives (though I did not assess this formally). Improvements in classroom management, and structuring the task to give students greater direction would allow a greater fraction of the class to enjoy and gain understanding from the lesson. As previously stated, an IWB or computer with a data projector would have been a useful tool such that I could model tasks and capture student attention when required.
ICT Integration: Potential problems
My mentor teacher suggested that he hated “computer classes” citing classroom management and on-task behaviour as the major reasons. In his experience, students are all too likely to spend their time surfing the internet, reading emails, or trying to bypass filters to access their Facebook accounts. This issue is real and if not managed, can lead to such a lesson becoming a waste of time, with minimal work completed by students, and no learning objectives met.
Fortunately, the physical layout of the computer lab I will be using will assist. The computers are on benches lining two adjacent walls of the room (in an L shape) such that the screens can all be seen at a glance, from a central position. I think that students will be willing to stay on task as it represents a break from the at-times monotonous task of mathematics book-work. As Yr12 students, I am able to speak frankly with the class on an adult level. They know what the learning objectives are, and know that in general, the tasks I set will help them accomplish them.
Another major concern is that students are simply not used to this style of learning and will have difficulty engaging with the task due to the medium. Survey data from the Australian Communications & Media Authority (ACMA) (2009) shows that 16-17 year old student (roughly the age of students in my class) use the internet primarily for socialising, recreation and entertainment. Will they be able to focus on the task when interacting with what they may consider an entertainment device?
Access to technology is an issue. Although the classroom I regularly teach in does contain almost roughly 20desktop computer, it unfortunately does not have an IWB. Gaining access to a room with both at an appropriate time may be an issue. Not having the IWB restricts my ability to model tasks and I must rely on a regular whiteboard and verbal instructions. Will it fly?
Australian Communications & Media Authority (ACMA). (2009). Click & connect: Young Australian’s use of online social media. 02: Quantitative research report. Sydney: Commonwealth of Australia. Retrieved from ACMA website: http://www.acma.gov.au/webwr/aba/about/recruitment/click_and_connect-02_quantitative_report.pdf
ICT Integration: Why use Google Earth?
Why ICT?
It is increasingly important for students to gain exposure to computer use as part of their mathematics education. This statement is reflected in the Board of Senior Secondary Studies (2009) unit outline, which states that “Technology, its selection and appropriate use, is an integral part” of students mathematical achievement, and that “computational fluency” is an essential skill inherent in mathematics. Further, evidence suggests that students who regularly use computers in an educational content have better assessment outcomes (O’Dwyer et al. 2008)
In mathematics education, technology has long played a role (in terms of calculators and spread sheets) however, as the wealth offered by ICT swells, so must its incorporation into mathematics lessons increase. Goos et al. (2003) have proposed a model for ICT integration into mathematics education in terms of a series of metaphors of student and teacher relationships with technology. Namely,
Technology as;
- master
- occurs when users have only a limited limited abilities with the technology
- students may not lack mathematical understanding to make sense of computer generated content or output
- servant
- technology as an efficiency tool.
- replaces pen and paper calculations but doesn’t change the nature of the task
- partner
- creative uses of technology provide new kinds of tasks or new ways of approaching existing tasks
- allows exploration of different perspectives and allows deeper understanding
- an extension of self
- freedom afforded by an expert command of technologies
- permeates all facets of pedagogy and student learning approaches
I believe that this task represents a partnership with technology. The reformulation of the task to the Google Earth environment is not merely cosmetic but further, allows a new way of engaging with the problem and the course content. Although the task is structured, it allows some open exploration which will afford a deeper understanding of the mathematical concepts and their application.
Why Google Earth?
Google Earth is gaining recognition as an educational tool (see for example RealWorldMath.org, Google Earth Lessons, or Designing and Creating Earth Science Lessons with Google Earth^{TM}) and rightly so. The combination of an enormous dataset combined with an easily navigable visualisation environment, has made Google Earth a popular option for science and geography teachers wanting to integrate ICT into their lessons. Increasingly, mathematics teachers are using Google Earth’s built in ruler and protractor as tools to engage students in their learning of topics such as mensuration, coordinate geometry, plane geometry and trigonometry. I was however, surprised that I was unable to find any mention of lessons using Google Earth in a ‘shadow reckoning’ task such as the one described here.
The ruler tool allows students to quickly measure the length and true bearing of real world objects from their photos captured in Google Earth objects. Using the Google Earth ruler is valuable in itself as it provides experience using yet another tool for measurement, but further it models conventions of measurement such as units and significant figures. The opportunity for learning from the instant feedback given by computer-based tools such as this is noted by Goos, Stillman and Vale (2007)
The constant necessity for zooming in and out also gives students exposure to concrete examples of scaling and similar figures
The range of visualisation tools provides opportunities for students to view the problem from a number of perspectives and affords exploration and reformulation of the question into a form which is more intuitive for them.
An example is the inclusion of 3D buildings with Google Earth. This provides students the option of seeing the problem as it conventionally presented in terms of a horizontal view. Note, I have overlayed the ghosted-yellow right-triangle as a representation of the geometric visualisation that must occur in students’ minds such that they can formulate an approach to the problem. This direct visualisation aid is not available to students in contrast to the traditional formulation of this question as discussed in another post.
Board of Senior Secondary Studies (BSSS). (2009). Mathematical Applications unit outline. Retrieved from the BSSS website: http://www.bsss.act.edu.au/__data/assets/word_doc/0019/123319/Mathematical_Applications_T_0812_v2.doc
Goos, M., Galbraith, P., Renshaw, P. & Geiger, V. (2003). Perspectives on technology mediated learning in secondary school mathematics classrooms. Journal of Mathematical Behavior, 22, 73-89
Goos, M., Stillman, G. & Vale, C. (2007). Teaching secondary mathematics: Research and practice for the 21st century. Crows Nest, Australia: Allen & Unwin.
O’Dwyer,L,.M., Russell,M., Bebell,D. & Seeley,K. (2008). Examining the Relationship between Students’ MathematicsTest Scores and Computer Use at Home and at School. The Journal of Technology, Learning, and Assessment. 6, 5, pp. 1-45.
ICT Integration: Justifying the task
Mathematical justification
The use of ‘shadow reckoning’ to determine the height of an unknown object based on simultaneous measurements of the length of its shadow and that cast by an object of known height is a classic textbook question found in almost all textbooks on applied geometry. The technique apparently dates back to the 6th century BC when the Ancient Greek philosopher, Thales of Miletu, is said to have used this method to measure the height of the Great Pyramids (Swetz, 1994).
The selection of various incarnations of this problem depicted in the header graphic and the image below suggest not much has changed since then.
Problem Solving?
In all of these classic examples, the information is presented to the student in the same format; a side-on view clearly depicting a pair of similar triangles.
Whilst this may not immediately seem like an issue, this formulation of the question precludes students from engaging in any true problem solving. In mathematics, the crux of problem solving is identifying or creating techniques which can be used obtain the desired outcome. Once a method has been determined, everything else is merely mechanical. Yet, in the classic problems (see again, the header picture above) students are presented with a ‘problem’ scenario overlaid with two geometrical figures, a pair of similar triangles. Combine this with the fact that the question is likely found in the imaginatively titled “Problem Solving” section of the “Similar Triangles” chapter, it would take a remarkable student to ignore all the prompts for an appropriate method to solve the problem.
Using Google Earth as an alternative environment for this problem will not address the latter issue. That is, most students will realise that an exercise placed at the end of a week of lessons regarding similar triangles will likely be related to similar triangles. Yet the question of which triangles is left unanswered and students are required to formulate the real world problem into a mathematical problem for themselves, an importnat skill.
Further, this objection goes to the core of geometry as an abstract form of thinking and reasoning. The geometer deals with triangles, not made of steel, brick or wood, but of imagination. Lines have zero thickness, points are of zero size, and triangles emerge between any three points the geometer care to imagine. The visual representation of this problem given to students (a horizontal cross section, does not require imagination, or abstraction, the triangles are on the page and not in the mind. It is linking the abstractions of geometry to solid, real-world situations which we should celebrate as problem solving, not turning the crank on a pre-prepared exercise.
Learning in Geometry
Models of learning in geometry are largely based on the concept of van Hiele levels, loosely analogous to Piaget’s stages of cognitive developments. The van Hieles, were contemporaries and colleagues of Piaget and their work is similarly, a foundation for constructivist approaches to learning in the area of geometry. Pierre van Hiele (1999) has criticised teaching of geometry in schools as it is structured and presented in a way baed on axioms, theorems and other such generalisations. This approach necessarily assumes that students are operating at a formal deductive level (c.f. Piaget’s formal operations stage) yet many students lack foundational understandings about geometry that is achive through play, exploration and interaction with real-world geometric objects (Crowley, 1987). I would argue that although exploring geometrical objects using computer software lacks the tangibility of blocks or cardboard shapes, yet students’ use of computers, mobile phones and iPods has made the manipulation of objects through their depiction on a screen an everyday, ‘real-world’ experience. This task based in ‘Google Earth’ will give students an opportunity to explore real-world geometry.
Crowley, M. L. (1987). The van Hiele Model of the Development of Geometric Thought. In M.Montgomery Lindquist (Ed.) Learning and Teaching Gemretry, K12, 1987 Yearbook of the National Council of Teachers of Mathematics, (pp.116). Reston, Va.: National Council of Teachers of Mathematics.
Swetz, F. J. (1994). Learning activities from the history of mathematics. Portland, ME: J. Weston Walch
van Hiele, P.M. (1999). “Developing Geometric Thinking through Activities That Begin with Play.” Teaching Children Mathematics, 6, 310–316.
ICT Integration: Learning outcomes
This teaching episode falls within a unit of teaching for a Year 12 Mathematical Applications class. The unit “MA Financial Modelling and Trigonometry” includes a four-five week segment focusing on the content areas of ratio & proportion, and applications of geometry and trigonometry.
The ‘specific unit goals’ related to this content area are reproduced below from the Board of Senior Secondary Studies unit outline (BSSS, 2009).
This unit should enable students to:
- apply an understanding of ratio and proportion to practical situations
- apply geometric and trigonometric procedures in real-life contexts
Further, the teaching guidelines for this content area explicitly calls for a Focus on applications of the techniques in contexts where direct measurement is not feasible e.g. shadow reasoning” (BSSS,2009)
This task and supporting teaching episode clearly falls within the scope of this scope and appropriately support students achievement of the unit goals.
Specifically, this task is designed such that by the end of the lesson, students should be able to;
- Make measurements using software-based tools when direct measurement is not possible
- Calculate the height of unknown objects using a ‘shadow reckoning’ method.
- Identify opportunities to apply knowledge of similar triangles to solve problems.
- Interpret information from a ‘top-down’ graphical representation and represent it in a more conventional geometrical setting.
This lesson will go towards building the skills described in the BSSS(2009) unit outline (relevant skills are reproduced below)
- Mensuration
- employ appropriate techniques and a variety of technologies, tools and formulae to determine measurements in various contexts to suitable degrees of accuracy
- Computational fluency
- confidently use computational technology
- employ efficient and accurate methods of calculation
- Problem solving
- formulate different kinds of mathematical problems by various means – including extensions of existing problems
- apply and adapt a variety of strategies to solve problems
- Communication
- communicate their mathematical thinking coherently and clearly to peers, teachers and others
- use appropriate representations to express their mathematical ideas precisely.
Board of Senior Secondary Studies (BSSS). (2009). Mathematical Applications unit outline. Retrieved from the BSSS website: http://www.bsss.act.edu.au/__data/assets/word_doc/0019/123319/Mathematical_Applications_T_0812_v2.doc
ICT Integration: Description of teaching episode
This entry will provide a description of a teaching episode designed for a Year 12 Mathematical Applications class. The teaching is focused on similar figures and scale factors, and their applications, mainly in mensuration. The task I will describe provides an opportunity for students to engage in a task designed to allow a series of guided investigations using computer software.
Specifically, students will be using measurement tools in the Google Earth software package to determine heights of buildings based on the lengths of the shadows that they cast. In this kind of mathematical problem, an object of known height is necessary (typically a meter stick or a persons height is used).
In this task, the known height is that of the Q1 buliding, Australia’s tallest building, and the tallest residential building in the world (according to wikipedia) however any known building or structure height (possibly near by the school) could be used. I chose this building for a number of reasons.
- It has that ‘Guinness Book ‘ feel to it and is relatively well known.
- The link between the Gold Coast, apartment accommodation, and ‘Schoolies Week’ (which is fast approaching) makes this task thematically interesting for students.
- Information on the height of the Q1 building is readily available on the internet
- The Q1 building is available to view as a ‘3D Building’ in Google Earth
- This geographical area features a high density of high-rise structures with clear, obscured shadows cast onto relatively flat ground.
As seen in the screen capture above, the necessary features (shadow lengths) are clearly visible and relatively well defined.
The teaching episode will take place in a one hour lesson at the end of a unit on similar triangles. I will revise the theory and model a problem on the board (unfortunately I don’t have access to an IWB for this lesson, which would be ideal), after which students will be working in small groups on computers to solve a series of problems which have been set, and to create and solve some problems of their own. Students will be encouraged to extend this method to measure objects of interest to them. Suggestions of buildings around the college, monuments in Canberra, and a tower in a local shopping centre will be made but are suggestions for the unimaginative and not strict guidelines. Students will be encouraged to communicate their ideas to me and their peers via a table on the board, where the height of various objects are tabulated.