Structured Maths
Low Floor High Ceiling or Differentiation through the use of Te Tau o te Rā/Number of the Day & See Think Wonder
The two approaches that are the basis for Tahi Maths which allow for differentiation are; See Think Wonder and Te Tau o te Rā/Number of the Day.
See Think Wonder comes from Harvard Graduate School of Education. When presenting a See Think Wonder, the teacher puts an image on the screen and allows time for learners to share their thinking about what they see, think or wonder about the image. The teacher can participate in the dialogue. All ideas are accepted and this sets the tone for maths time; freedom for creative thinking.
Te Tau o te Rā/Number of the Day is having a number as the focus for maths time. The Scope & Sequence used in Tahi Maths has numbers arranged in sequence from 1-10 for Year 0 then 1-20 for our Year 1, all the way through to 6 digit numbers in Year 6 and hundredths in Year 8. Having a Scope & Sequence reduces the cognitive load for both teachers and learners, knowing what will be coming in future lessons.
The first step of Te Tau o te Rā/Number of the Day is having concrete materials for learners to manipulate and find the different components of the focus number. Double-sided counters work well for single digit numbers then the use of place value blocks for the exploration of larger numbers is recommended. The teacher shows, or models, some of the ways the number can be decomposed, writing them on the board. Different concepts about number can be explored; fact families, repeated addition, adding on 10 to a number, subtracting 1 or 2.
At the beginning of a lesson the teacher writes equations to match the material objects and learners are given time to practise writing the number on a whiteboard and then recording something of their choice such as a picture to represent the number or equations. The following day the teacher repeats writing equations and learners record their thinking in an exercise book (simple notebook).
Differentiation occurs through the different levels in which learners enter into their participation with the See Think Wonder, and through their different recordings of their mathematical thinking within Te Tau o te Rā/Number of the Day.
The teacher is able to scaffold learning at an individual level when the learner shares their recording with the teacher. The teacher is able to bridge any gaps from any attempts that have been made that are not completely accurate.
For instance a learner may have attempted to write an equation that is not completely correct so the teacher is able to teach the conventional way of writing the equation. Or, if a child has not yet secured counting one to one, the teacher is able to work with that child and practise one to one correspondence.
Low Floor High Ceiling originated from the work of Seymour Papert and the central design of the Logo programming language in the 1970s which was referred to as “Low Threshold High Ceiling”. “Low threshold” means that anyone can get started and the “high ceiling” meaning that advanced users shouldn’t be limited by the programming language.
Low Floor is beneficial to all learners as they can see themselves as capable and competent. Te Tau o te Rā/Number of the Day has a low floor because everyone is capable of doing something with the focus number. It could simply be drawing the particular number of lines or circles corresponding to the number, or counting stickers (for those reluctant to draw). It could be drawing a favourite picture and representing the number within the picture and writing some equations. It could be listing equations based on a pattern or based on something that is of particular interest to the learner such as subtracting 2 or adding on 20 to the number.
Learners can challenge themselves. They can have a go at something that they are not completely mastered such as subtraction, writing 6-9=3, and the teacher can provide them with a quick lesson in what the numbers mean and how an equation is read from left of right.
Maths anxiety is reduced and learners have a sense of agency over the direction of their mathematical representations. They can explore many different components of the number and make attempts at doing something they are not completely competent with but knowing that the teacher is able to provide input if needed.
Using Te Tau o te Rā/Number of the Day & See Think Wonder means that all learners are doing the same thing, but in their own way, at their own level and achieving success. Teachers are not required to develop different tasks for the different abilities of the learners in the class, instead, the teacher provides individual support to meet the needs of each of the learners.
The use of Te Tau o te Rā/Number of the Day & See Think Wonder as a Low Floor High Ceiling task allows learners to show what they can do and are attempting to do. It allows for learning to happen through discussions with the teacher, or with other learners, and means that all learners can enter into the maths task at their own, individual level.
REFERENCES
Rotational Symmetry Year 1
Day 2 Number 7 Year 1 class folded coloured paper into quarters, cut an image then arranged them to show rotational symmetry.
This is an example of a Strand follow-up activity based on the See Think Wonder.
We’ll all have MOE funded workbooks for extra practise in the new year. Whoop!
Maths - No Problem!
On Friday I had some time to look at the maths resources on offer and attend a webinar on Maths - No Problem!
It reinforced to me that, if I was to choose a resource to support our maths programme, I would choose Maths - No Problem!
Maths - No Problem! have a NZ specific edition and Year 0-1 is play-based white Year 2-8 has textbooks/workbooks.
The workbooks are thin so that they last better and children are expected to take just 10-15 minutes to work in their books because it is consolidation of learning.
There is a money component and there are some games suggested.
It is Mastery-style, slowing down the content, teaching in depth and small steps.
There are Math Teasers available for $30 per class group for advanced learners.
Just thought I'd pass on this information because there is a lot to consider when making this decision for the school.
Provocation: I’m curious to read the research/evidence basis for your methodology.
Here’s some of the research/evidence I’ve used to develop Tahi Maths.
These links contain summaries I’ve made of the original document.
New Zealand Curriculum Refresh: Progressions Approach, Mary Chamberlain Charles Darr Rose Hipkins Sheridan McKinley Hineihaea Murphy Claire Sinnema, 20 May 2021
Report to the Ministry of Education “Mathematics and statistics skills and knowledge learners need to know by when, important cross-disciplinary links, and considerations in light of rapid changes and growth in computer science/ICT” Prepared by Robin Averill, Fiona Ell, and Jane McChesney, September 2021
Learning the work of ambitious mathematics teaching, Glenda Anthony, Roberta Hunter, Jodie Hunter, Peter Rawlins, Robin Averill, Michael Drake, and Dayle Anderson, with Tim Burgess and Roger Harvey, April 2015
Characteristics of Effective Teaching of Mathematics: A View from the West, Glenda Anthony and Margaret Walshaw, Massey University, New Zealand, Journal of Mathematics Education © Education for All, December 2009, Vol. 2, No. 2, pp.147-164
Addition, Subtraction, Multiplication & Division
Ākonga reveal their understands of number when engaged in Te Tau o te Rā Number of the day.
Someone asked a great question: What about Addition, Subtraction, Multiplication & Division?
When ākonga respond to the Number of the Day in their books they will engage in Addition, Subtraction, Multiplication & Division.
Teachers are able to model Addition, Subtraction, Multiplication & Division when they demonstrate how they would respond to the Number of the Day.
I’ll post photos of what I mean soon…
Meeting Diverse Learning Needs
Tahi Maths gives ākonga an opportunity to respond to Te Tau o te Rā/The Number of the Day at their own individual ability level.
Tahi Maths gives ākonga an opportunity to respond to Te Tau o te Rā/The Number of the Day at their own individual ability level.
Ākonga responses to Te Tau o te Rā/The Number of the Day provides the teacher with an insight into what ākonga know about numbers.
There is no right or wrong.
Teachers are able to make individual teaching points or take a concept to the whole group to teach.
Examples:
A child had written 4+2=5 so I used my fingers to show that when there are 4 and 1 more we get 5 and one more is 6 so that 4 and 2 more is 6.
A child had written 2-5=3 so I showed that we usually write 5-2=3 and if we did take 5 away from 2 we would get negative 3. A number line was drawn to show this.
A child had written 1,000+900=2,000 so I showed how to line up the numbers in columns to add together to get 1,900.
A child has written 05+2=07 so I showed how to line up the numbers in columns to add 50 and 20 to get 70. I also showed that 50+2=52.
A child had drawn a circle and drawn lines to cut the circle it into 8 pieces then coloured half of it. I took the opportunity to show that we could represent the drawing with the number equation 4+4=8 or the fraction ½ of 8=4.
See Think Wonder
See Think Wonder a thinking routine from Harvard Graduate School of Education.
See Think Wonder is a thinking routine from Harvard Graduate School of Education.
Thinking routines are tools specifically designed to help, support and guide mental processes or thinking.
Thinking routines originated from Project Zero’s Visible Thinking research initiative which included tools that teachers can use to find out more about what their children are thinking.
See Think Wonder is used daily as part of Te Tau o te Rā/Number of the day.
On the first day ākonga are presented with a picture of something to do with a strand, such as measurement. The first time the Year 1’s do this is with a picture of three objects. One shows a square with 1 Centimeter, a thin line with 1 Millimeter, and a baseball bat with a ruler with the word Meter. Children can talk to their Maths Buddy about what they see, think or wonder about.
As a group we talk together about the responses ākonga/learners have made. We talk about how small a millimeter is and show it on a ruler. We talk about the length of a centimeter and show it on the ruler and we bring out the meter ruler and talk about it being about the same length as a baseball bat.
On the second day ākonga/learners are presented with a picture of something to do with the number focus. The second day the Year 1’s do this it is with a picture of a grid of numbers which is an addition table where the number 1 is being added to numbers both across and down. We talk about the numbers and how the highlighted numbers on the diagonal are the same. We talk about the numbers that come next.
Research Articles
Some of the research articles we've collected along the way.
Five research-derived themes to consider when teaching maths - THE EDUCATION HUB
Is memorisation a good strategy for learning mathematics? OECD
How can schools improve students’ maths outcomes? | The Educator Australian K/12
The importance of pattern and structure Averill Lee
Day 1 for Year 1
Video - Count Dracula Number of the day - 1
Calendar - Look at the calendar and discuss the day
Today is Monday 1st of October
Clock - Look at the clock and discuss the time
The time is 1.40
It will be 2 o’clock in 20 minutes 5,10,15,20
Our number of the day, Te Tau o te Rā, is one Tahi
Here’s one circle on the tens frame
Here’s number one on the number line
I’ll put a one on the abacus
It’s in the one’s column
Children get whiteboards and Teacher Models
We’re going to write the number one
Let’s write it 10 times
What equations can we write for the number one?
We can write addition equations 1+1=2, 2+1=3, 3+1=4,...
We can write subtraction equations, let’s start with 10-1=9, 9-1=8, 8-1=7, etc.
Whiteboards Away
Slide - Measurement
What do you See, Think, Wonder?
I can see a baseball bat with a ruler that looks like a metre ruler
Discussion
We’re going to measure length
We can use centimetres to measure objects
Here is a ruler with 30cm
Here is a ruler with 100cm - this is called a metre
This rod is 1cm long, this rod is 2cm, 3cm etc - point to the rods on the wall as you do this
We’re going to find something that is shorter than 10cm and longer than 10cm and maybe even something that is the same length as this rod which is 10cm long
Look, my little finger is shorter than 10cm
What's longer?
The whiteboard is longer than 10cm, and what’s about the same?
My hand is about 10cm long
Children are invited to take a photo of their objects for Seesaw
Sharing time - show the Seesaw posts on the screen for discussion
Day 2 for Year 1
Video - Jack Hartmann
Calendar - Look at the calendar and discuss the day
Today is…
Clock - Look at the clock and discuss the time
The time is…
It will be…o’clock in 20 minutes 5,10,15,20
Our number of the day, Te Tau o te Rā, is one Tahi
Remember the tens frame looks like this with one dot
Slide - Number Pattern adding 1 grid
What do you See, Think, Wonder?
If we continued the pattern…
Discussion
Children go off to draw and write in their maths books
Sharing time - children share what they have done in their books
Daily Structure Explained
Day 1 Number & Strand
Number Video (Year 1 & 2)
There are many videos online to choose from. The Sesame Street Count Dracula one proved to be very popular with my Year 1 class. It’s a catchy tune, simple and repetitive. Numberblocks is always a great go-to. And Jack Hartmann is fantastic for children to get up and move to the beat before settling into their maths lesson.
Calendar and Clock
When the calendar has your birthday written on it, you are interested. By looking at the calendar, many concepts in our Maths Curriculum are addressed such as “identify how the passing of time is measured in years, months, weeks, days, hours, minutes, and seconds” as well as “name and order the days of the week, and sequence events in a day using everyday language of time”.
By looking at the clock we can identify where the hour hand is or how many more minutes until two o’clock and can practice counting by fives to work it out.
Number Visual - PV blocks, Abacus, Number Line
Starting with the tens board is a great way for ākonga/learners to consolidate their ability to subitize. The same if you look at a dice.
Cuisenaire rods are fantastic and children can start to see relationships between the rods such as two red ones being the same as four.
Whiteboards - write number & equations
The great thing about using whiteboards is that it doesn’t matter if you make a mistake, you can rub it out. It’s a great way for the teacher to see if a child needs help with number formation, if you want to go there, or with equations.
Strand - See Think Wonder
See Think Wonder from Harvard University is a powerful way for children to enter into maths where there is no right or wrong. The images used for this can be adapted to suit the ākonga/learners in your class.
Strand Activity (eg: measure using a ruler)
Every other day there is a strand activity that is a practical, hands on activity such as measuring, working with shapes or creating patterns.
End lesson with Sharing Time
It’s important to wrap the lesson up with some type of sharing, either sharing ākonga/learners Seesaw posts or something they have written in their maths books.
Day 2 Number Review & Book Work
Number Video (Year 1 & 2)
A different video or the same one as the previous day, depending on the children. Jack Hartmann’s videos are excellent as they have visuals showing the different ways numbers can be represented, such as on a number line or with tally marks.
Calendar and Clock
Having a daily routine lessens cognitive load and children love the predictable nature of knowing what they will be doing during Maths time. They know that there will be time to look at the calendar and maybe count how many more days until their birthday or another special event that is coming up.
Number Visual - PV blocks, Abacus, Number Line
Review numbers using the tens frame cards, abacus, number line, 100’s board. This is a time to subitize for the younger learners
Number Activity - draw and write equations in book (Teacher Model)
The teacher could model writing fact families or a pattern in their own maths book and ākonga/learners could do the same or make their own choice about what they write in their book.
Number Activity - draw and write equations in book
It’s empowering for ākonga/learners to choose what they will do in their books using the number of the day in their equations.
End lesson with Sharing Time
Share bookwork to show that it is valued and a way that we can all learn from each other.
REPEAT 2 DAY STRUCTURE WITH EVERY NUMBER
4 days a week for Number & Strand Focus
2 numbers each week
1 day for Maths Investigations such as Statistics or High Ceiling, Low Floor activities
Gradual Release of Responsibility
To With By is the same as I do We do You do.
I do We do You do was devised by Anita Archer.
In New Zealand the teaching of reading has always had the component To With By which is the gradual release of responsibility.
To is explicit teaching. The teacher might show how a number can be represented in different ways. Or the teacher might model how to measure an object, create symmetrical patterns or continue a number pattern.
With is what you do together with ākonga/learners. Teachers are listening to what ākonga/learners are saying or watching what they are doing and providing guidance or explicit instruction about concepts that are needed.
By is what the ākonga/learners do, either in a group, with a partner or alone. Ākonga/learners engage with the process of mathematics. They might be measuring or making reflection patterns with shapes. They might be creating their own number patterns. The teacher roves and provides guidance or explicit instruction as needed.
Scope & Sequence
The Scope and Sequence covers Year 1-6.
It is cyclic, repeating the maths strands and repeating the number focus throughout the year.
It comes in a powerpoint format.
Contact me if you would like a copy.
Every year group has its own Scope & Sequence:
Year 1 - numbers to 20
Year 2 - numbers to 100
Year 3 - numbers to 1,000
Year 4 - numbers to 10,000
Year 5 - numbers to 100,000
Year 6 - numbers to 1,000,000
Every week has a Strand focus:
Week 1 Measurement - Length
Week 2 Measurement - Width
Week 3 Geometry - Reflection
Week 4 Geometry - Turns
Week 5 Statistics
Week 6 Probability
Week 7 Algebra
Week 8 Measurement - Perimeter
Week 9 Measurement - Area
Week 10 Algebra
Te Tau o te Rā Number of the Day
It all begins with an idea.
Te Tau o te Rā/Number of the Day is a powerpoint presentation.
It contains a video (for the younger years) to get them moving so that they can then focus on their maths learning.
There are images for the thinking routine See Think Wonder.
Daily Structure
It starts with one.
Day 1 Introduce Number & Strand
Number Video
Calendar and Clock
Number Visual - PV blocks, Abacus, Number Line
Whiteboards - write number & equations
Strand - See Think Wonder
Strand Activity (eg: measure using a ruler)
End lesson with Sharing Time
Day 2 Review Number & Book Work
Number Video
Calendar and Clock
Number Visual - PV blocks, Abacus, Number Line
Number Activity - draw and write equations in book (Teacher Models first)
End lesson with Sharing Time
Repeat 2 day structure
2 numbers each week
First day introduce number and ākonga/learners engage in a strand activity (such as measurement)
Second day review the number and ākonga/learners draw a picture and write equations in their maths book
Third day introduce number and ākonga/learners engage in a strand activity (such as measurement)
Fourth day review the number and ākonga/learners draw a picture and write equations in their maths book
Fifth day is for Maths Investigations such as Statistics or a Low Floor High Ceiling activity