mathematics

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.
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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.

 Purchase  Free
 Hardware  PC
 Requirements  browser

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.
 Purchase  Free intro/preview, Tasks for  0,99 
 Hardware  iPhone, iPad, tablet
 Requirements  iOS, Android

Untangle

untangle-iconThe game we discuss here is (also) called Lazors, but a more suitable name could be Untangle or Network Points. This game is a typical example of a game that allows for ‘low floor – high ceiling’ activities. It is easy to start, yet very complex questions can be asked about the graphs.

Mathematical knowledge and skills that lie in the game are:

  1. spatial awareness
  2. geometry
  3. graph theory (topological characteristics of graphs).

The first task is to find out what the aim is. Don’t tell the students, but ask them to find it out and describe it. Perhaps write an instruction or guideline for a user.

After solving many levels students can think of new questions such as the ones below.

lazorsm5

 

How many different solutions are there?

 

 

 

lazorsm6

Is it possible to move all the triangles (and squares) to the outside so that no triangle lies within another triangle/square? When is this possible and when not?

Can you make a network that cannot be untangled in the way this game requires? If yes, how do you design such a network? What are it’s characteristics.

Can you predict whether a network can be untangled or not without trying it out. Evidence, proof!

The program GeoGebra can be used to draw the networks and discuss the reasoning and show the different options.

The game is suitable from primary school up to university level.

 Purchase  Free
 Hardware  PC, iPad, tablet
 Requirements  browser

The Moving Man

The applet The Moving Man enables students to experiment and learn about motion, position, velocity and acceleration. The movements of the man are plotted in charts.

  • Move the little man back and forth with the mouse and plot his motion.
  • Set the position, velocity, and/or acceleration and let the simulation move the man for you.

Moving man 1

Learning Goals

  • Interpret, predict charts/graphs on position, velocity and acceleration.
  • Describe, make sense of and reason about the charts.

If you register at the PHET website as a teacher, you have access to the information for teachers. The website offers examples of worksheets and questions for students at different levels.

Students can make a graph that fits a story, or make a story that fits a chart. At primary school level focus on one chart in the beginning. For example: What is the story behind this chart?

Do not underestimate the complexity of only the first chart. It shows a timeline, the position, negative numbers, and a man who covers a distance.

Moving man 2

 Purchase  Free
 Hardware  PC
 Requirements  browser, JAVA