mathematics

Simulation: Diversity

DiScoro writes about inquiry-based learning, digital resources, and ways to encourage higher-order thinking. We focus on STEM education and the use of technology.


This blog is about two simulations based on the same concept namely Thomas Schelling’s Model of Segregation. The model tries to explain social phenomena and shows for example how difficult is it to build and sustain a diverse community. Schelling tries to explain when and why ghetto forming takes place and under which conditions this can be prevented or even reversed.
In other words, people with shared identities tend to cluster/group together. In most classes boys and girls form their own groups.

The first simulation by Frank McCown is named Schelling’s Model of Segregation. The second is by Vi Hart and Nicky Case and named Parable of the Polygons. The two simulations have different interfaces. Both simulations use two groups. The first simulation has four variables (and a interval timer) whereas the Parable of the Polygons offers different simulations for different variables.

The simulation by Frank McCown can be found if you scroll down on the page. The simulation generates a multitude of questions that can be explored.

    • When do communities remain diverse?
    • When and why does clustering take place even if people are relatively tolerant and open-minded?
    • Can segregated communities be tolerant?
    • Under what circumstances does segregation happen and why?
    • How can a segregated community become diverse?

The Parable of the Polygons contains a group of simulations and uses scaffolding to explore the concept. Contrary to Mc Cown’s simulation the Parable of the Polygons visualise if people are happy or not. Additionally the user can move ONE person and see what happens. The last simulation is a particular interesting one.

 

The Parable of the Polygons could be used as inspiration for the teacher. However, in our opinion the degree of scaffolding will limit the curiosity, thinking and reasoning by the students themselves.

The simulations can be used by policy makers, but also by students in relation with religion, geography/demography. It has been known in chemistry that seperate molecules and molecules in small quantities react differently than in mass. The same can be observed with people. Individual people can be tolerant and open-minded, but the large group will nevertheless become clustered under certain conditions.

As teachers we have to be careful how to introduce the simulation and how to discuss the issues. Minority groups in class can easily feel uncomfortable. It is up to the teacher to choose the context and vocabulary that suits the class. As you may have observed have we tried to use the word diversity instead of segregation.

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Technology and Design – Rocking horse

DiScoro writes about inquiry-based learning, digital resources, and ways to encourage higher-order thinking. We focus on STEM education and the use of technology.

We give an example of  a Technology & Design task. In several countries Technology & Design is a (project based) school subject for students (grade 6 to 10). This task comprises many subjects: arts, mathematics (measuring, scale and ratio, geometry) physics (friction, mass, centre of gravity, forces, movement), language (vocabulary, writing skills), technology. In addition, it is an example of co-creation and collaboration.

Task: design and make a rocking horse for young children
and test out it at a kindergarten.

The task is complex and challenging, yet at the same time has a very clear goal. We discern different steps.

(1) Research and design of rocking horses. Students may search the internet for ideas, examples, pictures. The students may check out different constructions, materials used. Students can search for technical drawings or draw the design from scratch. Others may want to build a miniature modell. At the end of this step the students have made their choice about what material they will use and about the type of construction.

Examples of rocking horses from the internet

(2) Planning. What are the steps to take in de construction phase? What do I need for the construction in terms of material, tools, and other resources. How much time do I need? What are the costs?

(3) Implementation. This will be carried out in the planned steps. A prototype made of cardboard or plywood (3mm or 5 mm) can be useful. If not a design on paper 2D/3D is required.

(4) Test at kindergarten. Students should decide in advance WHAT they will test out. (e.g. Is the construction strong enough? Is the the rocking horse attractive for the children? What age group does it suit best? Is it safe in use? Does it swing enough?) and HOW they will test this out.
The students will have to plan a visit to a kindergarten and explain what they wish to do. The results must be reported.

(5) Reporting and documentation. This should be done during the whole process and not just only at the end. Students may choose if they wish to use mainly written or oral, visual, or multi-media  documentation for reporting. The teacher could (or  should) specify what he/she expects and how it should be presented/delivered.

Of course the rocking horses could be examined, measured, compared in many ways and from different perspectives (physics, accounting/economics, maths).

DaVinci Kindergarten

DaVinci Kindergarten is a pilot project in which we design, develop and try-out inquiry-based activities for children in the age 4-8. We have worked with children age 4-5 at two kindergartens in Norway. The activities focus on concepts from science, and technology and foster mathematical thinking.

We present some of the activities that have been developped. Contact us if you wish a complete description of the activity.

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  1. Show-box– sight lines and mirroring.
  2. How big is the panther? – measuring, human-based measuring units e.g. foot, thumb(=duym/inch), span (=fathom), step.
  3. How do you get the light on? – electricity, battery, light, lightbulb, lamp, electrical wire, curcuit.
  4. What weighs most/least? – experimenting with balance scales and different materials with the same volume and different weight.
  5. Discover more about your toys. What kind of materials are they made of? – Categorise, recognise, examine the different materials and discover their characteristics.
  6. Bee-bot – programming a robot.

How big is the panther?

Another activity for children age 4-8. This inquiry-based activity involves measuring up a big animal. The children will draw a big animal in its actual size, but the animal is in another room than where the animal must be drawn. Two children are sent to look at the animal and asked to come back and describe the animal. This process is repeated with the question to find out how big the animal is so that it can be drawn on the large sheet of paper.

Concepts
measuring, measuring units, human based measuring units, measuring tools, categorizing, ordering, serializing, relative size, proportionality, counting, member of the ‘cat’ family.

Vocabulary
size, height, width, big-bigger-biggest, large- larger-largest, small- smaller- smallest, thick, order, position, direction, shape, fur, skin, colour, tail, (girth).

Background
The world for young children is primarily three dimensional. Young kids play with three dimensinal toys. A drawing or a picture is a two dimensional representation of objects from the three dimensional world and therefor more difficult to grasp.

Measuring starts with the use of measuring units that are available. People have used measuring units related to their own body to measure length or height over many centuries e.g. foot, fathom/span, thumb/inch.

Show-box and Sight lines

This time we write about inquiry-based science and math activities we tried out in kindergarten, but this is definitely suitable for first and second grade as well. The first activity is about experimenting with sight lines using a show-box.

Concepts: sight lines, mirror, reflection.

Vocabulary: in sight, out of sight, hidden, position, sight line, eye, straight line, corner, behind, in front of, next to, around the bend …

The children worked in groups of three or four children (age 4 and 5) on one show-box. First, the children are presented with an empty show-box with four spy-holes. They are asked to furnish the room and place some dolls/animals using items they have in class. Thereafter we ask them explore what they see and what not and reason about it. We ask them to look through their spy-hole and tell each other what they see. We ask them why they do not see the same items.

There are many questions to ask that require experimenting, thinking and reasoning.
For example:

  • Can you place an item so that this can only be seen from one spy-hole?
  • Can you position an item that can be viewed from just two, three, or from all the four spy-holes?
  • Can you place an item in such a way that it cannot be viewed by anyone?
  • Build a half wall and place an item behind the wall. Choose a hole from which you cannot see the item. Now use the mirror so that you can see what is behind the wall.
  • One child take a picture though one of the holes while the other turn their back. Show the picture ans ask from which hole it was taken and why they thinks so.

FlaskFiller simulation

glazenFlaskFiller, or rather GlassFiller, is a simulation which enables teachers and students to experiment with and reason about the relationship between the shape of a glass, and the change in speed while filling it up (time vs height of the liquid in the glass).

 

The simulation program enables the user to

  • Select the horizontal axis’ quantity, which is one of height, time, volume, or rising speed.
  • Select the vertical axis’ quantity, which is one of height, time, volume, or rising speed.

Note that you can select the same quantity for both axes, which can make for an interesting topic of discussion. See some screenshots below.

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Read the information about the simulation and how to use it on FaskFiller Education.
Read the Research done on FlaskFiller software used with grade 5 pupils.

The simulation has been used with grade 5 pupils in a one-to-one setting. When using this simulation in class you might want to use a hands-on experiment first, whereby you use a measuring cup to fill up glasses with different shapes. Let the pupils measure, observe and reason about what is happening with the different glasses.

Afterwards let the pupils experiment with the simulation based on clear questions/tasks. Students should be encouraged to record their findings and discoveries on a worksheet.
Most grade 5 pupils understand the principle of instantaneous speed, but lack the vocabulary. After experimenting in small groups, you could start the simulation on your SmartBoard and discuss the findings. Here the pupils will learn to extend their vocabulary and express what they see and think. Vocabulary: (rising)speed, volume, height, shape, cola-flesjetime, timelap, decrease/increase in speed of height of liquid visible in the graph.
After experimenting and discussion, pupils should for example be able to match a glass (or bottle) with a graph and vice versa.

The simulation can be used from grade 5 up to grade 10 depending on the tasks given.

The simulation program is available online as an HTML file, but can also be downloaded for off-line use.

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Long division

long-division-touchEvery month DiScoro writes about resources that can be used in schools and about inspirational issues. See Services in the Menu for workshops, training etc.

Long Divison Touch is a well programmed App. This is really an App that makes use of touch technology. The instruction for different tasks is very clear, for example division with remainder, division with decimal, division with decimal in divisor etc.

A discussion that is carefully being started:
Should we spend time on teaching the long division in school?
Who does need this procedure in the 21st Century? Do you as an adult ever use the long division procedure anywhere else than in school? Are there any skills in learning the procedure that are important later?

Division as a numeracy skill remains important, and definitely the ability to estimate whether the outcome to for example 125 : 4,5 = … is a little less than 3, than 30, or  than 300.

A step forward could be to do the mathematical thinking, such as estimation, in class, and to use the App to challenge high-performers. It could also be a tool in Flipping-the-classroom.

Note: The format in which the long division is presented does match the way it is taught in schools in some countries, but in many countries the format used is different. It would be difficult to cater for all the different formats that are being used. The Germany, France, Norway all use a different formats. For students who already master the long dvision procedure it will be rather easy to figure out how this formats works.
Note: The App uses decimal comma, which may be an obstacle for countries where people work with decimal point.
Note: Students do not train multiplication and subtraction skills required in the ordinary long division.
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