Original post 25/12/2015

Part 1: The Challenge

This blog post explores how to plan an entire mathematics programme using PYP Planners. Part One describes the difficulties of using the IB PYP Mathematics Scope and Sequence document and the classic "one central idea" approach. Part Two describes how one school: Tokyo International School (TIS) uses the Common Core State Standards to systematically incorporate several conceptual understandings with their associated success criteria into each maths unit planner. This whole initiative was (and still is) led by Joey Kaufhold a Grade 1 teacher together with Greg Parker our PYP Coordinator. The opinions about the IB documents in this blog are my own and not necessarily representative of Tokyo International School, Greg or Joey.

## A Mathematical Myth?

Most IB Primary Years Programme (PYP) schools have a few good 'stand alone' mathematics planners to their name. Schools which have documented an entire maths programme onto PYP planners (in a which actually guides them to teach maths on a daily basis) are rare. Where these schools do exist, their trade secret is hidden from Google or at best a few examples are shown but never an explanation how all the pieces of the puzzle fit together. The comments section below would be a great place to expel my claim with some links!

## The Maths Problem

## What Goes Where?

When it comes to planning maths, one of the first questions PYP schools face is 'What goes where?' We all have scope and sequence documents be it the IB PYP Mathematics Scope & Sequence or one we have been handed from our school, district, state or government. These documents define what we want students to learn (the scope) and the order we have to teach it (the sequence). Logically then the contents should end up on PYP planners as this is where we plan our teaching and learning experiences.

## Our Wonderings?

But when we turn the cover and blink at the contents - it is not as straightforward as we may have been expecting. Questions spring to mind:

What is the difference between a conceptual understanding and a learning outcome?

What do we do with the conceptual understandings in the IB PYP Mathematics Scope and Sequence Document?

Where do we put the learning outcomes and how do we use them?

What happens if students are on more than one phase - how does that feature on our planners?

Our school has the MYP, what happens to the mathematics students learn after the students leave the PYP?

There are many great things about the IB's PYP Mathematics Scope & Sequence, but there are also some limitations, not least drawing a blank to the questions above.

One might assume that the conceptual understandings are to be used as central ideas for (stand alone) mathematics units of inquiry. This would be logical, after all they are written in the same format as central ideas.

Conceptual Understandings as Central Ideas?

But this begs the question 'How do we use them all?' Let's do some maths. Imagine you are a Grade 5 teacher and most of your students are working at Phase 4. That's four central ideas about Data Handling. Let's assume there are some students in your class at Phase 3; that's another four. You now have eight central ideas about data handling. There are 5 mathematics strands, so (using estimation) 8 conceptual understandings x 5 strands = 40 central ideas. If you were to use each conceptual understanding as a central idea on its own planner you would have approximately forty Grade 5 mathematics planners to write!

## Understanding an Outcome

Let's pause for a moment and think about what we mean by the terms conceptual understanding and learning outcome. A conceptual understanding is a thought process which exists inside our heads. We can only assess understanding if the student demonstrates this somehow. That's where good learning outcomes come in. Learning outcomes are most useful when they worded as actions which we can observe and measure. They tell the teacher what to look for. For this reason, well-written learning outcomes start with tangible verbs which we can see or hear such as: describe, explain, predict, illustrate etc. Learning outcomes are success criteria which specify how the understanding will be performed. The transferring and applying learning outcomes in the IB PYP Mathematics Scope and Sequence document are very useful in this regard.

If you look closely at the constructing meaning 'learning outcomes' in the PYP Mathematics Scope and Sequence document you will notice something - they too are phrased as conceptual understandings. If we try to use them as success criteria for assessment we run into a problem; they do not provide anything tangible to look for. For example what do you look for to gauge whether or not your students understand the difference between experimental and theoretical probability?

They are all important big ideas in maths, none of which I feel are disposable. If the alternative is to use them as (more) central ideas then that's a heck of a lot of maths planners!

Part 2: The Solution

Conceptual Understandings as Lines of Inquiry The simple solution of how to cope with so many understandings is to include more understandings per each unit planner. This is something which key educationalists have been advising the IB for some time and I have blogged about this in the past. Take this quote from Taking the PYP Forward (2010) or Lynn Erickson's keynote speech at the IB Conference (2011) in the Hague. I have faith that the PYP Review will react accordingly and modify the planner but in the meantime Lynn's advice is gold - use the understandings as your lines of inquiry. The Common Core The system I will describe could be adapted to any maths curriculum. At Tokyo International School we decided to adopt the Common Core State Standards (CCSS) rather than use the IB PYP Mathematics Scope and Sequence. Here are our reasons for using the Common Core:

We needed a curriculum which spanned elementary and middle school

The CCSS have observable, assessable outcomes (provided as examples)

There are lots of resources (such as MAP tests) which align to the CCSS

IB endorse and are working with the CCSS

We wanted grade level standards as a norm reference to evaluate how we and our students are performing.

The conceptual understandings and the learning outcomes in the IB PYP Scope and Sequence don't always correlate so we needed an alternative.

There are no related concepts listed in the IB PYP Scope and Sequence yet IB ask us to plan using these related concepts.

We knew we wanted to include several conceptual understandings per a mathematics unit but we needed a system teachers could understand and follow. Such a system would maximize our chances of developing understanding and ensure understanding didn't just slip through the cracks. Here is how we went (and are still going) about it:

Step 1: Create An IB Friendly Scope & Sequence from the Common Core The Common Core is written as learning outcomes. The learning outcomes are expressed as assessable skills or processes the students must demonstrate. The Common Core doesn't include conceptual understandings so we have to tease them out.

(reference Understanding by Design Wiggins & McTighe)
Use the frame on the left to tease out the conceptual understandings. The conceptual understandings should be timeless, globally transferable, universal truths, central to mathematics. This takes a bit of thinking and the collaboration of your best mathematicians on staff.
Often the conceptual understandings from the IB Scope and Sequence are very appropriate, but be careful that they correlate exactly to each Common Core outcome. The readings from Charles Randal are amazing in this regard! I stumbled upon his work online.

Next find the mathematics concepts. The concepts are the mathematical, abstract nouns embedded in your conceptual understandings. These are the words for your maths word wall and the 'related concepts' you will later refer to on your planner. Tease Out The Related Concepts

Write User-Friendly Learning Outcomes Using the Common Core create easily assessable learning outcomes (success criteria). Success criteria should be written in simple language as "I can..." or "students can..." statements. These learning outcomes represent the normal depth of understanding for the grade. This is for benchmarking and reporting. Importantly (as we will see later) they do not represent the standard all students work towards. Each child should be challenged to their personal potential.

Put Them Together As A Scope And Sequence Lastly put together your concepts, conceptual understandings and learning outcomes to create your completed scope and sequence. Here is a section of the Tokyo International School Mathematics Scope and Sequence showing one section of the K2 programme.

Biggest Ideas: Central Ideas You will notice also there is a 'central idea' at the top. These are the biggest, most important ideas in each mathematical domain. These big ideas transcend grades and credit goes to the work of Charles Randal (2005). These big ideas become the central idea for each planner with each grade inquiring into the same central/big idea to greater depth year after year. In the Common Core 'Counting and Cardinality' is only featured Kindergarten so in this example above that particular central/big idea doesn't repeat after Kindergarten but this is an exception.

Step 2 Create a Mathematics Programme of Inquiry

Next we took all of the contents of our scope and sequence and transferred it onto our mathematics programme of inquiry. Our mathematics POI clumps together the content of our scope and sequence into units. The maths POI lets the teacher and the PYP Coordinator see an annual overview of the mathematics. The contents can be transferred onto maths unit planners. Each column is organized by the Common Core maths domain (e.g. measurement). These domains are also macro maths concepts so for those of you familiar with the work of Lynn Erickson the domain becomes the conceptual lens. Here is the K2 section. Looking at this section of the POI, you may wonder where the conceptual understandings are. As you run the cursor over the lines of inquiry the corresponding conceptual understandings appear. Watch this video which demonstrates this. The lines of inquiry correspond to and steer the inquiry so the students develop the conceptual understandings.

Some other user-friendly features of the TIS Mathematics Programme of Inquiry are the age appropriate vocabulary list and the integration hints which appear as notes as you move your cursor over the text. The maths concepts at each grade level are age appropriate but the words themselves are sometimes too highbrow for children. As a teacher moves her cursor over the concepts the POI displays the age appropriate equivalent for her maths word wall. We have also included some handy integration hints: Step 3 Transfer The Contents of Your Mathematics Programme of Inquiry and Scope and Sequence onto PYP planners One Size Doesn't Fit All

I began this blog with a claim to plan an entire maths programme on PYP planners. But what we found is that in the strands such as 'Operations & Algebraic Thinking' and 'Number Operations in Base Ten' that planning a whole unit of inquiry wasn't always appropriate or necessary. These domains last the whole academic year and involved numerous understandings. Instead teachers would use the contents of the maths scope and sequence and math programme of inquiry to teach groups or whole class lessons on mathematics (as opposed to sustained inquiries which necessitate one planner lots of understandings and one enormous summative assessment). In these cases one understanding is taken at a time lasting a few lessons.

26/05/16 We have just finished transferring our scope and sequences onto planners and are ready to go. Here is a sample. Notice each line of inquiry has its own section in Stage 4. Our next step will be to incorporate the summative assessment tasks from www.exemplars.com for both our pre-assessments and summative assessments. They dovetail with the Common Core. What About Integration?

At TIS we firmly believe that developing conceptual understanding about mathematics is just as important as developing conceptual understanding about other subjects including a school's transdisciplinary POI. As such maths planning needs to be arranged around it's own conceptual understandings.
Mathematics connections may be using and applying the skill of mathematics or showing students the conceptual connections maths shares with other subjects.Mathematics should be used to find out and sort out information about other subjects such as: using number to count events in social studies, using measurement to calculate weight in science or using data handling to sort out information about health. Similarly mathematics often shares the same concepts as other subjects such as pattern in mathematics, poetry and dance or shape in mathematics, design and art.
Where such connections occur a brief overview of the connection should be stated on the associated planner. But this doesn't replace the maths planner, it complements it. Also there is no need to be blinkered to the programme of inquiry as the sole place for integrating maths.
References:
Erickson L (2002) Concept-Based Curriculum and Instruction: Teaching Beyond the Facts, Corwin Press
IB (2009) PYP Mathematics Scope and Sequence
Randall I. Charles (2005) Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics in Journal of Mathematics Education Leadership, volume 7, number 3
Wiggins G and McTighe J (2005) Understanding by Design, ASCD