Teacher’s guide
For the Multilingual mathematical thesaurus Tomorrow’s teachers should be prepared to integrate technology effectively into the curriculum and their teaching. Technology enables teachers to employ innovative teaching styles where appropriate and helps support the varied learning styles they are likely to encounter in their classrooms. This guide not only aims to give an insight into the teaching and learning approaches that underpin the production of the thesaurus but also to support improved preparation for teaching activities that integrate its use. The thesaurus project has involved the sharing ideas and has fostered increased collaboration across disciplines and among higher education, elementary and secondary schools. In many ways, what follows reflects this philosophy of a community of learning which engages collaboration and fosters inquisitiveness. As well as specific teaching and learning examples this guide outlines activities which can support: staff development, the inclusion of technology competence within teaching and learning, the use of technology-based teaching tools and resources, particular activities focused on teaching with technology. This guide offers a very few contexts in which the thesaurus can give opportunities for exploration of concepts and mathematical ideas both within the scope of the standard curriculum and beyond it. The aim is to encourage and support teachers and students to be independent thinkers who can apply mathematics in novel contexts and who wish to extend their own knowledge inspired by mathematical opportunities that ask many questions and offer many opportunities for exploration. The thesaurus is more than a dictionary The thesaurus not only gives definitions of mathematical terms but also connects mathematical concepts and ideas, encouraging the user to explore and discover new and related areas of mathematics. So, for example, the definition of a circle is linked to that of the concept of a locus and a locus to …. and so on. Each concept may have a number of descriptions or examples to meet the needs of different audiences. One of the most powerful aspects of the thesaurus is the ability to step into and move up and down definitions to find a level that is most accessible, and for any individual this can differ from concept to concept. After all, you may be excellent at Euclidean geometry but have very little knowledge of topology, in which case you would want to access the two areas of mathematics with more or less help. Although there is hold the answers mathematics, and this wider world. a wealth of information within the thesaurus it does not pretend to to everything, nor could it. There is the internet beyond, full of the thesaurus gives gateways into other mathematical resources in Once you have found the definition of a mathematical concept then
there is the opportunity to visit other websites that may contain further examples of the application of the mathematical idea or other definitions. The door is ajar and it is up to you whether you explore further. One of the features of the thesaurus is the concept tree. The concept tree is suitable for teachers and/or for individual learners, helping them to orient themselves. It can also be used in Mathematics clubs and master classes as a source of information, a starting point for an adventure, or a resource to support other mathematical activity. The thesaurus also contains a large number of images and interactivities that help explain mathematical terms. Managing access to the resource As well as the opportunity simply to discover and explore the thesaurus as an individual, students who are familiar with the thesaurus can support those who are less familiar or uncertain. The use of peer assisted learning can offer many advantages in terms of classroom management of the technology. When the task is delegated to students, they have better means to demonstrate and communicate their skills without being threatening or threatened. An approach that may also reduce teasing and bullying as pupils are encourage to show excellence or make use of peer support in a structured way. Assessment. Work delegated to students may require, or benefit from, the use of the thesaurus and other literature. The opportunity for students to look beyond standard algorithms and demonstrate their skills and knowledge, both taught and learned independently allows for the assessment of higher order thinking and problem solving skills. Using the thesaurus – some practical advice First of all the teacher has to decide what form of activity to choose. If the use of the thesaurus is to be part of a lesson, the main concern of a teacher is to develop a plan. The possible place and role of the thesaurus within a lesson is varied but what follows is just a few of the possibilities.
Example 1 - Exploring Percentage Increases A three part lesson might include the use of the thesaurus in the following way: A starter A starter tries to simulate conversation between students and/or students and teacher, or to invoke an intellectual conflict, to raise interest to the specific problem. For example, consider the following problem: A student wants to buy a coat in the sales. The price was 250 EUR before the sale and it is reduced by 60 % in the sale. Is it better for student to add VAT first, and then take reduction, or vice versa? The clash of opinions raises interest, and usually different opinions and views arise. A discussion on the best policy may take place followed by a time to explore and make calculations. The main teaching part Now it is time to consult the thesaurus, in order to agree a solution, and also to generalize the result. More examples might arise and more investigation and discovery takes place. For example: A student deposits 250 EUR in a bank with interest rate 5 %. How many years are needed to duplicate the amount of money? There are many ways to approach this problem including algebraic or iterative methods. The teacher can act as a coordinator and manager of ideas and suitable approaches to solving the problem. The conclusion The teacher may have many more related examples for lower grades. As for higher grades, the teacher may extend the middle part of a lesson to talk about multiplicative structures, and compare them with the use of additive structures.
Example 2 - Research Topic - Transformation and Scaling In order to assess how students understand enlargements and similarities, the teacher may ask students to engage in one or more of the following problem solving and investigative activities. Activity one Take a square, located on the line, and select one of its vertices, as shown in the picture:
Roll the square on the line and visualise the trajectory of the vertex marked in green.
Students might picture something like the image below:
Some students may need to manipulate a square or use an animation to come to this view. The teacher may ask students to calculate the area enclosed by the trajectory or its length. Students may respond differently. a) Some of them may produce calculations using the known formulae r 2 or 2 r . b) Other students may notice that the arcs on the left and right of the diagram are equal in length and enclose equal area. The central arc is related to the other two by a scaling factor 2 which results in the doubling of the areas. Some students might derive the expression: 1 2 1 2 1 r r ( 2 r)2 4 4 4
A student utilising the approach in case a) gives an indication that he or she has memorized and knows the formulae r 2 and 2 r and can apply them in appropriate situations. When the student takes approach b), he or she appears to be acting in a more conceptual and relational way. The given square may be rolled on the line, and enlargement factors applied to it involving ideas of similarity and other aspects of transformation. Students who have a sound understanding of the underpinning concepts can see the essentials of the problem immediately. Students who do not understand the problem but have memorised a process may proceed towards the solution in step by step manner but are unable to apply their knowledge in non-standard situations. Engaging with the thesaurus pupils may make their own connections and step into the problem without the teacher having to take the lead.
Activity two Similarity and scale are powerful ideas. Let’s investigate the following transformation. If large square is successively transformed into smaller squares by the scaling factor q what happens? Let’s start with the simple case of a single transformation:
Are there other ways of achieving the same transformation? What is the significance of the centre of enlargement? How could you construct the second square accurately, using straight edge, compasses and a set square? For many students this exploration may be enough, but many may then be ready to investigate what happens when the smaller square transforms into an even smaller square, which in turn transforms into a much smaller square, etc.
Eventually a student will have situation that can be represented as:
S 1 q q 2 q 3 q 4 ... or 2 Area 1 q q 4 q 6 q 8 ...
Extending this idea: The large dotted triangle ABC is similar to the small green one. corresponding sides are in the same ratio. Thus:
S 1 1 1 q
S 1 1 q
Therefore
This represents the sum of an infinite geometric progression (GP) with common ratio less than one. An interesting result and one that gives some concrete notion to the idea of summing an infinite series. Looking up sum or summation in the thesaurus will give you links to summation of a GP and hence to the summation of an infinite GP. The role of the thesaurus in this work is its ability to offer independent access to definitions and examples of animated transformations as well as summation of series can support the more able or engaged students in moving on. Links to other websites such as NRICH and KOMAL will give access to other examples where similar ideas are applied to different contexts. In this case links to the NRICH site can lead you to the Proof Sorter program which can be used to help generate the proof of the sum of a GP.
Example three - Probability Students are asked to throw two dice, calculate the sum and the difference of numbers, then take the last digit of the product of the sum and the difference. Visualize a table of results and describe what patterns appear.
The lesson is based on an idea of Tamás Varga. There are two dice of different colours. These dice are used as a random mechanism. All possible elementary events can be seen in the following table: 1 1 2 3 4 5 6 1,1 2,1 3,1 4,1 5,1 6,1 2 1,2 2,2 3,2 4,2 5,2 6,2 3 1,3 2,3 3,3 4,3 5,3 6,3 4 1,4 2,4 3,4 4,4 5,4 6,4 5 1,5 2,5 3,5 4,5 5,5 6,5 6 1,6 2,6 3,6 4,6 5,6 6,6
At the beginning students may focus on the sums of the outcomes on the two dice. How frequent are sums equal to 1, 2, 3, … ,12. The sum equal to 1 will never appear, it is an impossible event. The theory based on this table says that the expected frequencies (theoretical frequencies) can be represented by the frequency graph:
1
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5
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7
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12
This way may not only be used for sums but also for other operations as well. The resulting graphs, showing theoretical frequencies, can be obtained in each case from the table of outcomes of throwing two dice given above.
Some ideas for lessons based on this material, by age, follow.
Activity one - for ages 6 – 11. The teacher prepares in advance 10 parallel lines on the playground (or in the classroom), about 2 feet apart from each other. On the first line teacher marks the places by numbers 1 2 3 4 5 6 7 8 9 10 11 12:
On each numbered place there is one child, marked with the number, when dice are thrown by teacher. The sum is calculated and the child with corresponding number takes a step forward. The teacher might start by asking the pupils what they think will happen. After a few throws the teacher stops and asks the question. “Who is moving furthest?” There will be various guesses, but the child with number 1 will discover very soon that he or she has no chance of moving, and may even wish to stop playing! The question then might be “Who would you like to swap places with?” With the youngest children the first insights into discovering that not every outcome is equally likely and some outcomes are impossible may be enough. The extension of the activity to throwing a single dice or using different calculations may be useful in emphasising the notion of the probability of outcomes being unequal and some outcomes being impossible. The thesaurus can be used to find definitions of probability, equally likely, likelihood and many other terms related to probability. Using the links to the NRICH and MMP it is possible to find other related activities, probability games and problems. Activity two - for ages 11 – 13. In this scenario the pupils may not need to undertake the activity in the playground described above. The teacher may wish to refer to the frequency table and graph, to help pupils predict frequencies.
The activity can be undertaken in pairs. It starts on the imaginary river. Along the river there are places marked with the numbers 1 to 12. The two students are players and they are each a leader of a team of 24 students, symbolized by counters. All the students must go from one side of river to the other, but the team leader can only move one counter at a time. The die is cast and the sum indicates the place from which one counter will go to the other side. The teacher asks for a strategy from each team leader. How should the leader spread their team (including themselves) in order to cross the river first.
The student may choose to put all their team on the most probable place, that is 7. But how should they take account of the fact that the dice may result in a total of 3? Here is one possibility of spreading the team of 12 students along the river. Is it the best choice? The lesson might end with the pupils looking at the definitions of some of the terms they have been using in the thesaurus and commenting on how they could be improved to help understanding. The thesaurus has a facility that the teacher may wish to use to send in suggested amendments and improvements to definitions for pupils of different ages based on the feedback from their class. There are many similar activities and games that require a knowledge of basic probability and by using the thesaurus it is possible to locate other lesson ideas on other sites (such as NRICH) where problems that exploit and develop pupil’s knowledge of probability can be found.