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?
The Maths Problem
What Goes Where?
- 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?
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
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!
Conceptual Understandings as Lines of Inquiry
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
Tease Out Conceptual Understandings
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.
Tease Out The Related Concepts
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.
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
Here is a section of the Tokyo International School Mathematics Scope and Sequence showing one section of the K2 programme. The whole K2 section of the document is available beneath. Notice there is a correlation between each column. The concepts are found within the conceptual understandings and the outcomes are what we look for as evidence of conceptual understanding at a grade level expected depth of sophistication.
Biggest Ideas: Central Ideas
Step 2 Create a Mathematics Programme of Inquiry
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
How This All Looks in a Lesson
What About Integration?
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.
One Last Plea
Update - Finished Planner
Primary School Principal
Tokyo International School
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