Thursday, 27 August 2015

Four Sums in a Row

Why:                       Practice of addition Facts

Object:                   To cover four numbers in a row, horizontally, vertically, or diagonally

Students:                Pair players            N.Z. Numeracy Stage:            Counting on

Materials:              Game board, 2 sets of counters(two colours), 2 paper clips

How:                      1. The first player places two paper clips on numbers at the bottom of the board,
                                adds them together and then places one of their counters on the sum on the board

                               2. The second pair moves JUST ONE paper clip to make a new sum and covers
                               the appropriate numeral on the board.

                               3. Play continues with pairs alternating turns until one pair has four in a row.

Adaptations:            Change numbers on the board and use three  1-6 die or 0-9 die
                                  Adapted from Nimble With Numbers Grades 3/4

Note:                                    1.  I prepare these type of activities on one side of an A4 or Quarto Card.  This is so the rules and playing
                                             board are on the same side, often I have seen games disrupted because the rules are on the reverse side of 
                                             the page!
                                             2.  I encourage students to play as a pair against another pair. With two in a 'team' there is a need for
                                             maths discussion before a move is made. The more discussion the better the learning.

Ten Pin (Bowling) Maths

Materials:        Ten Pin Bowling format with Numbers 1 to 10, and  3 x 1-6 Dice (or 3 x 0 to 9 Dice)

Objectives:       To practice the four operations and order of operations to find solutions with 3                                    random numbers, as produced by the dice rolls

Goal:                 To ‘Strike Out’ as many of the numbers as possible

Organisation:    1. Can be played as a whole class warm up (but be aware some students may let                                 others do all the calculations)
                            2.  in small groups (of Four) each student has a Ten Pin Grid. They take turns to roll                             the three dice and then ‘strike out’ as many numbers as possible

Method:              1. One person rolls the three dice, records the numbers on paper, and then using                                  the four operations of (addition, Subtraction, Multiplication and Division), as well 
                             as brackets, and each number on the dice once they find a solution, which Strikes  
                             out a number. 
                             e.g. If 2, 2, and 3 are rolled  2+2+3  is equal to 7 so 7 is struck out. 
                                    then 2+2-3 is equal to 1 so 1 is struck out.
                                    2-2+3 gives 3 and 3 is struck out. 

                             Try to remove as many numbers as possible with each roll.
                             2.  After all solutions as possible are found the dice are rolled by the next player.
                             3. This can be played cooperatively, with each student in the group helping each                                  other, or competitively with each student only using their own rolls of the dice to                                  strike out numbers during their roll
Adjustments:    With younger/less able students focus on addition and subtraction.  More able all                                 operations

Note:                   When using dice I prefer to use ten sided dice (0-9) as it generates all the digits   
                             and has the spoiler of a zero as well.    1-6 Dice/Number Cubes are more suitable 
                             for younger students

Saturday, 22 August 2015

Explorations on a Hundreds Chart

Finding patterns on the Hundreds Board.

The Hundreds Chart is a wonderful piece of Maths Equipment that can be used for pattern discovery from the Early Years on. Ideally the use of a Flip Hundred’s Board is what I would encourage, but paper with coloured pencils will suffice.
With younger children the use of counters or Unifix type blocks can also help to model the patterns.

Please encourage the students to find the patterns, rather than tell them what the patterns are.  Mathematics should be student exploration, not practising what the teacher tells!

  • Our task is to be a guide on the side!
  • If we encourage lots of exploration students will not need too many directions,
  • they will find them for themselves, and be prepared to be blown away
  • by their discoveries, often they will find patterns that we least expect
  • or have never seen before!

  • The usual patterns of Odds and Evens, Adding 2, 3, 4 (multiples/Times Tables) are obvious starters
  •  Later, when Times Tables are known, an exploration of the Sieve of Eratosthenes to produce the Primes 
  •  Figurate Numbers (square, cubic.. numbers) can also be found on the board 
  • Niven Numbers; Any whole number that is divisible by the sum of its digits:  e.g. 4 + 5 = 9; 45 ÷ 0 = 5
  • Create Codes (Find the end point): ↑→↓←  (Each arrow means move one square in direction of arrow)                                                                                                              So 46 ↑→↓← means you move up one, right one, down one, left one so end at 46
  • Put a grid around any nine adjacent number. 
    • What Patterns can you find within the grid? e.g. what do the diagonals sum to?    
    • What do the mid number of each side sum to? What is the Mean of each row or column?….  
    • Do similar patterns occur in any group of 9? What patterns are there in a group of 4x4
              numbers? 5x 5? etc.

Sunday, 16 August 2015

Charlotte Wilkinson's personal view for the Direction of Maths Education

Where might we be heading with Maths Education

During June of this year I paid a visit to the UK and while there I visited a number of schools to find out what is actually happening in the teaching and learning of mathematics and how it has changed since I left 16 years ago.
New Zealand appears to be following a similar pathway in educational policies, albeit about 10 years behind.
What have we got to look forward to and are we ahead in some areas? We need to make sure we are not forced into making backward steps?
Firstly the UK have just been hit with a new curriculum which has no levels but in the maths there is a list of what should be taught at each year group level.
3 ©Copyright N C Wilkinsons Ltd 2015. All rights reserved.
A maths lesson consists of a whole class starter activity (about 10 minutes) followed by a whole class delivery of a concept then individual work with support from teacher or teaching assistant. All classes in this school and apparently many schools have a teaching assistant during a maths lesson. (A teaching assistant has received some training and is not a teacher aide nor a fully qualified teacher).
Lesson type is review -teach -practice which research data says while very common may not be the best method for teaching mathematics, particularly new concepts. (Teaching Maths in Primary Schools R Zevenbergen, S Dole & R Wright page 71)
The new curriculum focuses on mastery which was explained to me as meaning the bright students, instead of being introduced to new content knowledge are to explore the content assigned to their year group in greater depth through problem solving to develop higher order thinking skills.
So higher order thinking is only for the students who grasp new concepts quickly in the way they are taught.
Lesson type problem based learning is most beneficial when encountering new content - aligns with how people learn, use and apply knowledge in the real world. (Reference as above)
Another school told me that generally teachers teach skills and provide opportunities to apply them if there is time, but most often time runs out and the application of the skills doesn’t happen.
I suggested the NZ model of teaching through problem solving rather than only teaching for problem solving. This was a novel idea to the teacher concerned and she liked the idea.
There is no professional development to assist teacher to make this change and many teachers think problem solving is confined to solving arithmetic word problems. There is no longer an advisory service and no professional development courses.
I had the opportunity of having a long conversation with a teacher based at an SEN school (Special school) whose role involves going out to mainstream school to assist teachers with students who are struggling in mainstream. Firstly there is much more comprehensive support for students with special educational needs, many more being taught in dedicated special education units. In supporting the students who struggle in mainstream the new curriculum dictating what should be taught at each year group is causing problems as teachers are trying to teach, for example, fractions to students who are unable to work with numbers beyond 20. Many teachers are struggling with differentiation as concepts are taught whole class, only follow up work is differentiated.
On the way home we stopped in Singapore where they were celebrating 50 years of being an independent country. On visiting an exhibition I was very interested to read in a section on education a radical change they made in 2012.
They stopped all league tables comparing schools and even stopped naming their top students.
Reason: They found schools were becoming focused on rankings rather than real learning. They are trying to break down the “caste system” of education which ranks academic learning as superior to all other learning in recognition that they need people skilled in trades as well as university graduates to build their economy. Knowledge AND Skills is how they have achieved what they have in just 50 years.

Maybe a lesson could be learnt by looking closer to home rather than to the UK.

Doesn’t Singapore consistently rank higher than UK on international rankings?

Published with permission ©Copyright N C Wilkinsons Ltd 2015. All rights reserved.

I find Charlotte's Philosophy on Maths Education refreshing in an era where much is being driven by politicians.  
I would just love to see New Zealand, and the UK, as well as the USA and Australia, take a leaf out of Finland's Book.  A number of years back the Politicians unilaterally agreed that Education would be left to the specialists without political interference.  They decided that the teachers should be well trained educated and supported. In Mathematics they wanted a Problem Solving Approach to be the driving force.
It is my understanding that Finland is near the top of all International league tables, with overseas educators beating a path to their door to see what they are 'doing right'!     Len

Sunday, 2 August 2015

Some Number Magic?

This is a great way to practice addition and subtraction of two digit numbers.  I have found students will stick at a task like this for longer than doing a page of exercises from a Math(s) Book

•Choose any number between 50 and 99
•Add 62 to this number
                                                                                                         75 + 62 =137
•Cross off the left hand digit (hundreds digit) of the
result and add it to the units digit
                                                                                                         137; 37+1; 38
•Subtract the result from your original number
•Try it again with a different starting number

•What do you notice?

•Can you think why this always happen?

•Now work out how you can choose a number
(between 50 and 99) and add a special number, so
that when you cross off the left hand digit (hundreds
digit) of the result and add it to the units digit and then
subtract the result  from your original number you will arrive at 45.

Kaprekar's Constant

Dattaraya Ramchandra Kaprekar (1905–1986) was an Indian recreational mathematician who described several classes of natural numbers including the Kaprekar, Harshad and Self numbers and discovered the Kaprekar Constant, named after him. Despite having no formal postgraduate training and working as a schoolteacher, he published extensively and became well known in recreational mathematics circles Ex Wikipedia.

Image result for kaprekar 

• Ask your students to choose any four digit number

1. Now using the digits write the largest four digit number

2. Using the digits write the smallest four digit number

3. Subtract the smallest four digit number from the larger

4. Use the answer, just found, and repeat steps 2 and 3 until an interesting situation arrives
5265(one step)
3996(two steps)
6264(three steps)
• Investigate:
Which four digit numbers have the greatest number of steps to arrive at the situation found in 4 above?  (This is best as a whole class activity, with Butcher's Paper displaying 1 step, 2 step, 3 step etc students add their starting number on the appropriate sheet)

Which four digit numbers have the smallest number of steps to arrive at the situation found in 4 above?

Are there any four digit numbers which do not stop?

What happens if you start with a 3 digit number and follow the above steps?

Is there a similar situation with five digit numbers? 

Present a report to your class explaining what you have found justifying your results. A class report could be published in the School Newspaper