Kaprekar's Constant

 This number exploration, uses Largest, Smallest Four Digit numbers and practices subtraction. all an endeavour to find the Constant that many Four Digit Numbers end up at.

 

As the Card suggests all numbers will end up at the Constant within 1 to 7 steps.  I would challenge the students to see if there is one starting number that does not conform.

In the Classroom I would consider having 8 wall charts Labeled 1 through 7 the 8th Others!!  As Students check their four digit numbers and count the steps the record their discoveries on the appropriate chart.

Other students may wish to find out more about Kaprekar, who he was, what mathematical truths he found.

Prerequisites: Some understanding of four digit numbers and four digit substraction(or allow use of calculator) and possibly Flow Diagrams

 

Digital Roots, Patterns and Geometry

 I have long been an advocate of teaching mathematics from a known situation as well as in a context that helps students understand that mathematics is all around us.

We often do not connect Number with Geometry and Vice Versa but they are linked, intertwined, but we need to peel back a layer and see the connections.  I like the adage, "The Maths Behind The Door!" I encourage all teachers to help students find the connections.

Another area we do well, initially(Junior School) is with patterns.  We use shapes, counters and ask children to continue given patterns, create their own etc. This is a great foundation for exploring numbers later, but why do we forget patterns when exploring Number?

Digital Roots of numbers is a good way for students to practise their simple computation skills as well as look for patterns and then use them for Geometry explorations "Patterns in Circles" "Spirolaterals"

I hope these activities help you springboard the connections between Numbers, Patterns and Geometry.(There are more to come)

Indy Anna Jones and the Temple of Gloom

This activity seems logical to post as TV3 (NZ) is replaying the Indiana Jones Movies.

The idea was presented some years back at a conference I was attending, I have tweaked it for using with Teachers and Students I have worked with.

I hope you enjoy using this with your students.

Modulo Arithmetic (Clock Arithmetic) with an Art Component

 Modulo (clock Arithmetic) are Finite Number Systems, and for older students it is worth exploring a Finite System so that they may get a better understanding of an Infinite System like our Base 10(Decimal System)

Putting an Art Component with it, may encourage some reluctant mathematicians to become more interested and involved.

In the attached pics, I mention "Donald in Mathmagicland" (A Disney film 1959) I can remember showing often using 16mm film when I started teaching in '65. It gave me and my students a better understanding that maths is not just numbers but is intertwined in everything we do.

Donald in Mathmagicland

Other websites about Modulo Arithmetic for your interest and background information 

https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/what-is-modular-arithmetic

https://www.mathsisfun.com/numbers/modulo.html 

https://nrich.maths.org/4350   (this great Maths site, suggests students 14-18, I have worked with students 10+ on these activities. Often it is how the activity is approached)

Enjoy Exploring Modulo Arithmetic

An Interesting Question to follow on from a discussion about TIME

 This activity has been adapted from a Puzzle activity prepared by AIMS (Activities in Maths and Science) Fresno USA.

I can assure there are probably 3 solutions but students will come up with one or perhaps two.

Try it yourself first to see what you can come up with?

Counting Squares

Yesterday while entertaining my 6 year old grandchild we drew a picture with a number of squares on it. My question to him was 

• How many squares can you find? 
• His response was a cursory glance and counted the small squares but not the large one.
• With some help ad guidance he was able to start focusing on 1 x 1, 2 x 2, 3,x 3 squares.

I believe we need to do these types of visualisation Problems with all levels of students. 

Reaction Time

 A nice activity to help older Primary School students to check out how fast they can react, and whether through practise they can improve reaction times.

TIME: Loop Card Activity

 

Teaching the Analogue Clock!

 

I was somewhat dismayed when working with my grandchildren yesterday, that they (aged 6 and 9) could not tell the time from an analogue clock.

 

They are fairly fluent with telling the time with the digital clocks around the home, but not the Analogue. 

 

A short discussion with a quick drawn circle and placing the numbers then short and long toothpick, got them thinking about how to tell the time using this "round clock"

 

I don't believe we should discard the Round Clock at all, even though we are in a digital world, to me, it is like saying we can learn measurement by doing it all on a Tablet!

 

The Digital Clock is great for Telling The Time! but to be able to use this for measuring time needs an understanding of decimals to two places and the ability to add and subtract these decimals.

The Analogue Clock can be used to Tell The Time and it is easy to Measure Time by seeing how far the hands have moved, not much adding and subtracting required. 

 

 

Apart from Time, what else could the Analogue clock help with?

• Fractions: the obvious with 1/4, 1/2, 3/4  but what about twelfths?
• Counting in Fives: 5, 10, 15, 20.. 
• Geometry: Rotation, parts of a circle(semi-circle), degrees in a circle, angles(between hands)
  
• Finite Number System: This is called a Finite System as it goes from 1-12 and then back to 1 not like our Decimal System which is Place Value and Infinite

Whether we teach some of these higher level skills or not, what the students are getting is an intrinsic introduction to some of these.

After posting an abridged version of the above on three FB Sites:  NZ Teachers (Primary), Mathematically Speaking, Developing Mathematical Inquiry Communities (DMIC)  I was pleased to see so many posts including a number which suggested than older kids did not know how to read an Analogue clock.  I have to admit all were in favour!!!

Please check the posts out for other teaching ideas, a few websites I have attached below.

 Please Teachers, if there is not an Analogue Clock in your classroom please acquire one, and don't forget the wooden teaching clocks as well, OR get the kids to create their own!!

An Interactive Analogue and Digital

https://www.visnos.com/demos/clock?fbclid=IwAR1FB5PJEQakuEAZbriF4zVkdLNSjbzfQbNDNVUdUcVXIwtXxMZZdHKQOAs 

 

 One approach to using the Analogue Clock

https://bookunitsteacher.com/math/measurement/telling_time.htm

 

 An Australian Educator, uses a hoop

https://www.teachermagazine.com/au_en/articles/teaching-students-to-read-an-analogue-clock 

 

A New Zealand site that has teaching ideas(and worksheets) but also a Time Line of telling time including the Sundial

https://www.twinkl.co.nz/teaching-wiki/analogue-clock 

 

 

Open and Closed Questioning

 Some thoughts and suggestions for changing CLOSED questions into OPEN ones!!

Questions to Promote Quality Thinking

 In mathematics we are too quick to use CLOSED Questions What is 23 + 47? These types of questions are just asking, in many cases to recall a fact or a skill, Little, if no thinking required!!  

So we need to "Open" up the questions(more about these in a future post)

We need to have a bank of good questions that will help us, as teachers,  so that we can encourage the Students, to reflect, Justify, and think about what they have been doing.

Here are a couple of lists that you may wish to use as you explore more Open Type Questions

Questions to promote Students Thinking

 Recently on a FaceBook Post a teacher was asking for activities to send home during New Zealand's Covid-Delta lockdown(August 2021)

I replied to the post suggesting that if we supplied parents with a list of questions to use when they were interacting with their kids, that may be better than a lot of busy work!!   This response got a number of likes and 💗's

This suggested to me that our institutions may not be communicating the best and easiest ways of helping their kids while in lockdown., so I resurrected a handout I created may moons ago!

I believe most of these questions can be used with all types of activities that kids get into, whether it be Lego Building, or playing on the swing, or looking for bugs in the garden

Hope they help.

We call it Streaming!! North Americans call it Tracking

 Jo Boaler and team "YouCubians" latest news letter has links to research about the benfits of removing Streaming in schools.

I encourage all teachers to take time to read the research.

I was encouraged to see the emphasis on Inservice support for teachers, something I believe here in NZ we have diminished over the years

Dear youcubians,
 
I hope you are having, or have had, a good summer break. With a new school year starting, I'd like to highlight an important question for education: how we group students so that they can have the opportunity to go forward in mathematics and take any classes they want. Traditional systems of tracking have not served our students well and have resulted in significant inequities in the school system. The new proposed California Framework, for which I am one of the authors, sets out different approaches that have been shown to be more successful.
 
To help educators with this complex issue we have –

1. Released an updated new paper that I co-authored with David Foster, which shares the evidence from multiple school districts engaged in de-tracking of middle school classrooms. Study 2, the new addition to the paper, shares significant student achievement increases, from across the achievement range, after a de-tracking initiative involving over 16,000 students. The paper also discusses the reasons for parent opposition and strategies to educate those who oppose such changes.
2. Designed new professional development to support teachers who are teaching mathematics to groups of students that are heterogenous, or “mixed ability,” a description that fits all classrooms!  We have planned an intensive one-day workshop to help teachers learn the different research-based options available to them, and the ways they work in classrooms.  We will also consider examples from teachers who work with heterogenous groups in schools without mathematics “tracks."  We are offering this workshop with two different formats and dates:

• On October 18 we will host the workshop in-person at Stanford
• On October 25 we will conduct a live, online version of the workshop

We hope these scheduling options, created at the request of many teachers who cannot travel to California, give you the flexibility to consider joining us this fall!  The cost for each session is the same— $595 per person— and you can register online by clicking on the dates above.  If you have questions, would like to enroll a group of more than 6, or cannot register right away and would like us to hold space for you, please email us at pdinfo@youcubed.org.

In our next newsletter we will be sharing the new Week of Inspirational Maths (WIM)! For those of you who are already back at school or planning lessons, the WIM lessons and mindset videos are always available for you to choose and use and can be accessed here.
 
As always, keep in touch, on social media, we always love to hear from you.
 
Viva La Maths Revolution!
Jo

Examples of Repeatable, Thinking Maths Activities

 These activities need to be considered in relation to my previous BLOG;  

"What To Do With The Others?

Note that each activity has the equipment needed as well as a playing board (if required) on the one side of the page. 

To make teacher easier for you, you need to collect, develop many of these types of Maintenance, Repeatable activities and have them organise in a way that suits you and the class.

What To Do With The Others?

Many of us have been encouraged to have groups within our Maths Classes, in fact it has been a basic organisation since the 60’s when I started teaching.  These groups were originally based on ability(much research says this is not the best way of grouping) now we can have all sorts of groups, 

•         Interest
•         Social
•         Groups of 2 or 4 etc
•         Small groups for interacting with the teacher, rather than a large class 
•         Groups of one
  

 

How ever we organise the issue is about the groups that are not under "Direct Supervision" so we here the catch cry

             “What Do I Do With The Others?”

For successful 2 or 3 group organisations to work then the most important thing is that the groups/children, not with the teacher, know what to do!!  

One organisation that has been used for a number of years is Groups of Four.  The class has a common open ended activity and groups of four work together to solve/attempt the activity.  After a period of time the groups report back their findings to the whole class. Sometimes roles are assigned to each child (with the roles changed for each new task)

• Reporter
• Recorder
• Gopher
  

For any groups/small group teaching we have to make sure that the students, not with the teacher, are generally on task and know what to do!!

This means:

• They can work Independently or in pairs, small groups, without supervision
• They know the organisation of the classroom:
• They know how to work quietly,
• They know what to do when they have completed the task,
• They knowwhere to get another activity,equipment and tidy up
  
• They know where to place completed work,
• They can Self Mark, as appropriate without cheating
  

They also need to Know the Mathematics, and I believe this is where most of us fall down: 

We teach something today/yesterday and then go and ask them to practise it.  

        In many cases students need teacher supervision to become masters of the skill/process. 

In the immediate aftermath of teaching a process/skill: 

        they may not know what to do or how to complete the worksheet etc.  

 

Please take time teaching repeatable Thinking type activities so that all students know what to do,

• Have repeatable activities printed on card and laminated, if possible, so students can write on them with Dry Erase and then wipe clean
• Have the activities organised by group, or content.  possible coded for topic/strand
• Some teachers have these in Plastic Bags-but if a die is missing someone has to check- so I advise that no equipment is with the activity but in a place where students can go and get the dice or pen or paper clips(organisation) 
• The equipment needed for an activity should be listed on the Card(see my examples)

I have advised all teachers to take time teaching and practising Independent activities and routines as a class, before trying to introduce groups.  The children’s knowledge of these routines and behaviours make your maths' teaching so much easier.

This time, organising students to work independently,  could be anything up to 3-4 weeks (a whole term) depending on the group of students.  We cant rush these things if we wish to enjoy the teaching as well as the kids learning the maths.  Remember KISS.

When ready have two groups and get those working before starting with 3 groups(and no more)

Without the appropriate activities, and behaviours by the children, we do make our work hard for ourselves.

Marking

Traditionally the teacher has used the Red and Green pens to say which one is correct or which is not.

• I prefer teachers and myself not to stand in judgement,( be the Policeman) but to ask the Groups/individuals to report back with reasons and explanations as to why they think they have ended up with the result. Other groups/children may wish to dispute the result so we should be asking the children and groups to JUSTIFY their answers.
• I can remember a situation in a classroom where a child gave an answer, say 4, and was told that it was wrong.  This upset the child so I went and said, “can you explain what you did?” During the explanation the answer was 4-for what the child did

Questions

In my time working with Teachers and Students, I often did not know the “answer/s” to particular questions, problems that I presented  This meant I could not “lead” the kids to what I thought was the answer, I used questions such as:

•     Did you try this?
•     Can you explain how you got that answer?
•     Can you show me what you did?
•     Would you like to teach /explain this to a group?
•      What does the problem tell you?
•      What do you have to find?
•      What Strategy approach will you/did you use?

 I like this approach that one school uses for getting students to explain how they tackled a problem.

 

Balloons Over The Wairarapa

 This is a great NIM type activity for all ages.

Can be played One to One or pairs (Two vs Two): I prefer the latter in a classroom and two places are more likely to discuss the moves before making a decision.

Dont emphasize the competition but after a few times get the students to think about patterns or special numbers of toothpicks etc.

A worthwhile quote for every classroom wall, OR Do, Say, Write

 Might need to put the "s" in.

I must admit I am a   Do, Say, Write Proponent

                        Do                explore learn through doing, using, drawing 

                                            when Proficient at doing

                        Say                explain in your own words what you have been doing

                                            when proficient and explaining orally

                        Write            first in long hand and then if appropriate in maths language

All for Maths and Maths for All

I have just received the Wilkie Way Newsletter and Charlotte's comments made me get on to my hobby horse:  I will copy her comments below)

Why is acceptable to say "I cant do maths, yet not socially acceptable to say I cant read!" 

Recently I wrote to "Seven Sharp" because both of their presenters said that couldn't do maths, which is perpetuating Society's attitude to maths. Needless to say I did not get a response and soon after stopped watching it.

Many years ago about 20 maths and science specialists were invited to a Learning Media Conference to discuss writing articles for the new Maths/Science Journals.  We were welcomed by the CEO who said to us all that, "he was amazed at us all with out interest in these topics as he could not do maths!"

At morning tea I was standing next to him and asked, (Paraphrased)

    "Why did you say you were no good at maths?"

        "Because I cant do it?" 

    " In your position to you set financial targets? Budgets? and try and meet or exceed these?"

        "Yes all of those"

    "Isn't this maths? What was your job before this one?"

        "I was an accountant!"

To me, this shows how many in society do not connect "Real World Maths" with "School Maths"

This is not a fault of the person/people it is the result of the education system that does not have a connect between the real world use of maths and what we are teaching in the classroom.  In the 80's an 90's there was a huge push for Problem Solving and Investigations to be part of the classroom curriculum, and I am pleased to reflect and say I was part of this push, after meeting and working with, the late, Murray Britt Auckland College of Education (writer of the NZ Maths Curriculum)

We need to make all our maths teaching and student learning Real and Connected.

All for Maths and Maths for All

Can you create an artistic masterpiece or are you happy to dabble, know about artwork, give it a go without a fear of failure? Can you appreciate art in the environment and the world around you without a negative attitude?

So why is it considered acceptable to have a negative attitude to mathematics?

Our New Zealand culture has allowed mathematics to become something completely separate from everyday life - something that is hard and fearful, that only a special few are allowed into the ‘Maths Club”It is a total myth that you are born either mathematically inclined or literacy inclined. This is actually a choice that you are making and perpetuating the myth.

Every human is born with a mathematical brain. Without it you would not survive. Within a few hours of birth, a baby can distinguish between same and different - the most fundamental mathematical concept. Most actions a child makes in their first year of life are grounded in mathematical concepts:

•   same/different - visual, auditory, tactile, pattern and rhythm
•   judging distances
•   exploring objects - for shape, size (length, area, volume), mass, temperature,

Through distinguishing between same and different, auditory, pattern and rhythm they learn the language of words to communicate by being spoken to. Their ability to distinguish between the patterns and rhythms of different languages means they are able to learn multiple languages at the same time. At no other time in a human’s life is this ability so strong.

The spoken word becomes a communication tool to enhance the ability for more conscious thought and more deliberate acts of discovery about the world around them. They learn to ask questions and seek information rather than having to find out everything for themselves. “Why?” is a word asked frequently by young children in their quest to learn.

The same/different concepts begin to be quantified as young children begin to use words of one and two to distinguish between these small quantities by subitising. Anything more is just more (one student in my class on entry to school recognised one, two, heaps.)

In a purely playbased environment students are likely to develop sound foundations in the fundamental mathematical concepts through opportunities to explore and experiment. However if these experiences are not verbalised and mathematical language acquired by the students to communicate, describe, explain and reason with these concepts, they will remain unusable foundations for building further knowledge and therefore developing the concepts to deeper understanding.
These foundations are very important and while schools are trying to compensating for development opportunities often missed in our rushed modern world, schools cannot ignore their essential role in teaching the knowledge required for developing the man made created mathematics. The mathematics we use in our daily lives did not just happen - it was invented to be used, it shapes the world in which we live

Counting is the first concept that requires direct teaching, it involves

•the learning of the words,
•the assigning of a word name to a collection of objects and understanding the last word said tells you how many. (Cardinal aspect of number)
•the understanding that the next counting number is the result of adding one more
•the numbers form a sequence where each number has a set position (ordinal aspect of number)
•recognising the number symbols (0 - 9)
•recognising the symbols are repeated in particular positions to represent other numbers

Do not underestimate learning to understand counting:

Achieved Level 1 Patterns and Relationships is when a student has generalised the concept of counting

•Generalised that the next counting number gives the result of adding one object to a set and that counting the number of objects in a set tells how many

Most adults see mathematics as about numbers - while it could be argued to be true, I would define mathematics as about mankind’s ability to use numbers to describe and model situations that already exist in the real word (using and applying) or to use the patterns and relations in the known mathematics to predict and create what could exist.
Much mathematics is created and never used, like an art canvas that has been abandoned. Much mathematics is created and then put to one side and picked up by someone else and reworked - like an art canvas that has been painted over.
Whatever mathematics is created, it is mostly built onto an existing body of knowledge - explored and experimented with just as humans do with those fundamental concepts. A new idea can be created from something already known that can revolutionise mathematics. An historical example is the idea that nothing is not nothing. If numbers represent something then what if you had nothing of something? Zero can now be classified as a number - from that idea a complete new number system was created.

I am not creater of mathematics in the same way I am not a creater of artistic masterpieces. However I like designing very nice quilts and I use and apply mathematics to do it.

Most people use and apply mathematics in their everyday lives so why do people perpetuate the myth that mathematics is something separate, that is hard, is not accessible to all, is to be feared and is allowed to generate so many negative emotions.

Listening to a speaker at a conference last week, she probably didn’t even realise she was separating mathematics from real life when she could not see how mathematics could be included in a Christian view of education when collaborating in PE was given as an example.   I hear public figures say they were no good at maths as if it is something to celebrate.

What do people mean when they say I am no good at maths?
If being good at art means being able to create artistic pieces, then being good at mathematics must mean being able to create new mathematics.

Most people do not create fantastic artistic pieces likewise most people do not create new mathematics.

We need to change the culture about mathematics - change your words

I am good at using and applying mathematics. I have the ability and inclination to use mathematics, at home, at work and in the community. I am numerate.

 
If you put your mathematics lenses on you will see how much you use and apply mathematics and how much mathematics surrounds you in your world.
Enjoy and appreciate it as if it were a masterpiece.

Charlotte Wilkinson  www.wilkieway.co.nz

Please check out Charlotte's website.

    

DIFFY - A subtraction Investigation

 Instead of regular worksheets, it is often more exciting for students if the activity has an investigative or a surprise result.

Diffy, from Nimble With Numbers is one of these activities.

Four In A Row Multiplication

 Recent FB Posts have been asking for ideas on how to help students memorise (and use) multiplication facts.

There are many game type activities that help with this and one of my favourites is the one that follows.

The beauty of this one is it requires thinking and strategy development to win, which in my view is better than more worksheets that do not require this sort of thinking, problem solving.

This game could be played with two teams(pairs). Encouraging teams to play means students will need to discuss the possibilities for placing the paper clips, perhaps blocking opponents etc.  Communication in maths is very important.

The Temple of Gloom

 I have just been asked for this activity.  It was one I used extensively while working with teachers in an endeavour to use more investigations and problem solving in the maths classroom.

With the recent pronouncements by the Govt about our declining Math's abilities at year 8-10 perhaps it is time for more of these to be included in our programs.

The beauty of this activity is that:

1. it can be used with individuals, pairs, small groups.
2. it can be acted out with children, and hoops, in the playgound
3. it can be modelled using stones and circles.
4. it can be done with pencil and paper (and lots of crossings out!!!
   
5. it may not have one unique answer
6. it lends itself to students to create similar problems(work backwards etc

 

Please post some of the ways your students solve the problem.  Teachers-not too much direction please.