mathematics

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

Pattern problems

Pattern problems are a relatively new phenomenon in mathematics education. They can be used both for early algebra in primary school as well as in secondary school. At primary level students reason and come up with a description of how the figure or pattern grows using word formulas. At secondary school level, students can be encouraged to describe the formula for the nth pattern using symbols for the variables.

figure numbers

Two applets from the Freudenthal Institute make it easy to experiment with pattern problems:
Spotting number problems, if you wish to work with given patters
Spotting numbers, if you wish the students to design their own patterns

Introduce a pattern to the class and ask them to look at it first. Then ask them: How do you see the pattern grow?

For the example below we have used the applet Spotting Numbers and coloured in blue the different views students could have on how the pattern grows.

Thereafter students can explore:

  • With how many dots does the figure grow?
  • What about the 20th or 50th figure?
  • How many dots are required?
  • How many dots are on the base?
  • Can you describe a ‘rule’ for the growth?
  • Can you describe a formula for finding the nth figure?

Information on the applets: Spotting numbers and Spotting number problems.

 Purchase  Free
 Hardware  PC
 Requirements  browser, JAVA

Design math tasks

In the previous blogpost on how to Change math task so that they become more challenging, we wrote about guidelines on how to change existing tasks. In this blogpost we discuss how to design new math tasks. Challenging maths tasks give students the opportunity to learn, think, explore, discuss, be creative and learn (about) different strategies and respresentations or visualisations in the process.

The guidelines below are particularly useful when you design new, challenging tasks.

  1. Can you make it into an activity?

  2. Can you make it hands-on: use materials, equipment and tools?

  3. Can you make it into an experiment?

  4. Can you use problems from the real world?

  5. Can you integrate it with other subjects?

H) Fold an air-plane from and A4 piece of paper and measure whose air-plane gets farthest. Pupils work in groups. (They decide how many tries they are allowed, how they can measure in a fair way, how they can improve their paper plane etc.)

I) Use polydron squares. See the blogpost Cube 3D-2D and Make a cube and fold it out into a nett. How many different netts can you find?

J) Let the students measure the schools playground using ‘steps’ (or a rope). What is the shortest way to cross the playground? Ask them to draw the playground and their shortest route on cm2 grid paper.

K) Plan a trip from your home town to Oslo. Work out and compare different options.
When is the trip fastest? Which means of transport is cheapest? Depending of how much time you want to spend on the project, you can decide how much information you give: timetables, maps, pricelists.

L) The task in the blogpost on Balancing Act is both physics and maths. It offers students the opportunity to investigation, experiment, and to find the rule. The last step is generalisation and (early) algebra. Instead of using a computerprogram you can also use a real balance/scales or have the students construct their own.

weegschaal met schaal

Change maths tasks

A headmaster of a primary school asked us if we had some good maths tasks for the lower grades.

fishingrodReminding the expression:
Give someone a fish, and they’ll eat for a day. Give them a fishing rod, and they’ll eat for life.’
we decided to give some tools in order to be able to:

(1) change existing tasks into better tasks, and
(2) design challenging math tasks.

But what are criteria for a challenging math task? We refer to Prof. Jo Boaler and use her criteria and add some of our own ideas based on experience with realistic mathematics education (Freudenthal Institute) and other resources.

We strongly advocate Inquiry-based learning. Note that we do not write ‘teaching’, we write ‘learning‘, because the goal is not teaching, but learning. More about inquiry-based learning can be found in earlier blogposts: Research on inquiry-based learning and Inquiry-based learning in practice. Inquiry-based learning uses more open tasks. Closed tasks with more scaffolding are less exciting and challenging. If students get step-by-step instructions (cooking-book practices), they stop thinking.

Challenging maths tasks give students the opportunity to learn, think, explore, discuss, be creative and learn (about) different strategies and respresentations or visualisations in the process.

Jo Boaler (‘Mathematical Mindsets’, 2015) offers the following rules to open up an existing task, thus making it more challenging:

  1. Can you open up the task to encourage multiple methods, pathways and representations?

  2. Can up make it into and inquiry task?

  3. Can you ask the problem before teaching?

  4. Can you add a visual component?

  5. Can you make it low floor and high ceiling?

  6. Can you add the requirement to convince and reason?

A) If you take a simple task from a textbook such as  5 + 7   Ask the students how they came to their answer (before even asking what the answer is). This way you encourage pupils to think about their strategy, to express their strategy in words or visually, to use mathematical vocabulary, to see and learn different strategies from each other.

B) Take the simple task  24 x 3  Students have already learned different strategies, such as swop the numbers  3 x 12 which is probably easier for some. Or they have used the strategy to double one number and half the other number 12 x 6. Now ask the students why 24 x 3 is equal to 3 x 24. Can they prove that this is true? They could for example use rectangles on a grid.

C) Take a task like  28 :  4   Do not teach the way to check the answer, but first ask: How could you check your answer  yourself?

D) Take the following task  1/4  x  5 . First offer visualisation and ask: How much is 1/4 of the 5 circles? Or instead of asking for an answer, ask: How could you divide the 5 circles in four?

vijf cirkels

E) Take the topic average. Do not teach the different averages, but let the pupils work on average without specifying what it is. They will have a basic notion of what average is. Let them for example measure their own height and register this for all students in class. Let them come up with the class average. Discuss which ‘average’ they think is best/fairest.

F) Take a task about measuring. This task deals already with understanding, namely of units of measurement. We can change the task so that it becomes ‘low floor, high ceiling’ (accessible and challenging for all).

multi 5b measurement

  • Estimate height or length and write this down using a unit of measurement.
  • Measure the objects (except for e).
  • If the students in your group came up with different results, discuss how this may have occured.
  • Try to write the height/length in a different unit of measument.

Another task on measurement. Instead og asking What is the circumference of a given rectangle? ask: Draw different rectangles with a circumference of 12 cm.

Multi 5b circumference

G) See the task Beads on a string from FI-rekenweb. See also the blogpost Early Algebra on Beads on a string.

kralen applet

 

 

Early Algebra

Algebra is not about learning to write in symbols (although they become useful later), algebra is about generalisations and patterns.

Beads on a string is a game that one can solve with trial and error, but it is more efficient to find patterns and rules. The ten tasks offer children plenty of opportunities to think, explore, reason and discover the pattern or rule.

kralen applet

The applet is developed by the Freudenthal Insitute, University of Utrecht (NL). For more maths applets from the Freudenthal Institute see  Rekenweb (grade 1-7) or Wisweb (grade 7-10). The applet Beads on a string can be used with children in the age of 8-12.

Remarks:
You may want to add a few questions in between step 2 and 3.
If you have real beads in five colours and a string, the questions you and the students can ask are endless.

 Purchase  Free
 Hardware  PC
 Requirements  browser, JAVA