mathematics

Turkey investigation problem

The Turkey Investigation project is part of a research program by Catherine Fosnot, dealing with inquiry-based learning in mathematics. Grade 3-5 students work on problems related to multiplication and division.

The problem is typically related to the American context. Here follows the short description.

Turkey Investigations, Grade 3–5: A Context for Multiplication invites you into Dana Ostrowsky’s third-grade classroom. Here children explore two problems that are posed separately by Dana. In Buying the Turkey, the first problem presented to the class, students grapple with the cost of a 24-pound turkey that is priced at $ 25 per pound. In the next problem, Cooking the Turkey, students think about how long to cook the 24-pound turkey if, as one recipe suggests, it needs to roast for fifteen minutes per pound. Because the numbers in each problem—the relationship between a quarter of a dollar and a quarter of an hour—have been carefully crafted to support the use of similar kinds of grouping strategies (e.g., grouping four quarters to make a dollar in Buying the Turkey and putting four fifteen-minute intervals together to make an hour in
Cooking the Turkey), there is the potential for students to model the problems in similar ways.
The challenges presented by these two problems to students who are making their first forays into multiplication push students to look for shortcut strategies and support the development and the discovery of specific mathematical big ideas (e.g., the distributive and associative properties of multiplication) and landmark strategies (e.g., repeated addition, skip counting, doubling and halving, etc.). As students struggle with these problems they also develop different ways of modelling them. This includes the ratio table, the open number line, and the double number line. (A. Cameron, S.B. Hersh, and C. T. Fosnot, 2005)

You can watch a part of the series of videos below. This may inspire you to look for problems that are interesting for your pupils and can be designed to challenge them.

Make paper boxes using Origami

Right now many people spend much more time at home. A good time to sit and make some nice boxes for gifts, small items such as jewelry, or for cookies you have made.

There are many examples on YouTube of slightly different techniques and results. Here just one example.

 

Link to a little more complicated closed box.

You do not have to think about school work when making such boxes, but for those who wish to see the mathematics there are many concepts that are being used and applied (division, fractions, angles, proportions, measurement, square etc.)

If you wish to make it a real challenge. Try to make a box out of one piece of paper where three ping-pong balls, or three golf balls, fit snugly. (It does not have to be a rectangular shape).

 

 

Critical on Maps

Rotating globe
Why all maps of the world are wrong. Or in other words: Why do all maps of the world present a wrong, distorted image.

Why do we use maps? How can we present the globe in two dimensions? What challenges do we face?

Start with one of the first two videos and then move on to the second one. The latter is spoken fast and uses a wide, scientific vocabulary.


The only correct representation of the world is a globe. Every projection serves a specific purpose. It’s interesting to explore the different projections and their use and purpose throughout history. It appears that projections and perpectives change over time and place and are culturally bound.

To get a good impression use the tool ‘The True Size. This tool makes it possible to drag a chosen country over the world and compare its (true) size with that of other countries. Visit the website  https://thetruesize.com

This topic can be addressed from many different angles: geography, politics. history, mathematics, ethics….

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.

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.

Additionally, 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.

 

 Purchase  Free
 Hardware  PC, iPad
 Requirements  Browser

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.