Currently at my school we are in the process of revising our maths curriculum. Schools are typically in a cycle of curriculum revision and ours is no different. It's been an interesting process so far and we still have a way to go. To put things in context, last year we created a K-12 continuum of learning outcomes for Maths, Language, Science, Humanities, PSHE, German, Music and PE - I think that's all (phew!). This was obviously a lot to achieve within the timeline of one year and our Curriculum Director has indicated that these will need tweaking as we work through them.
Our Maths Coach (a new appointment this year) is settling into his role and has brought with him a new approach to the design of our maths units. In previous years we have focused on a core resource for informing our mathematics practice and supplemented this with other resources where required. Along with our newly created outcomes we revised this resource and decided that it wasn't the best fit for our program. We recently received our new core resource and are now familiarising ourselves with this.
Through the self-analysis of our program in the Elementary School, one of our main criticisms of our work was that we don't revisit mathematical concepts throughout the year. This is referred to as 'spiraling', as concepts are visited and re-visited throughout the course of the year. The belief behind this is that students are able to build on their knowledge throughout the year and develop deeper understandings. This also aligns with the constructivist approach to learning that the PYP values. Conversely, a curriculum based on 'mastery' places emphasis on mastering specific concepts of mathematics before moving onto the next one. Mastery is the end goal for all mathematical concepts, spiraling is one way of achieving that. Some educators achieve a spiraling curriculum by addressing concepts during each year of a students' school life. In this example the students learn certain parts of concepts (eg. fractions) in grades 1, 2, 3 etc, with each part becoming developmentally more challenging. Spiraling can also occur within a grade level and this would involve students re-visiting the concept at different times of the year. This is what we are looking towards in our school.
To encourage this approach our maths program has been re-designed around five units: Counting, Partitioning and Calculating; Securing Number Facts and Understanding Shapes; Handling Data and Measures; Calculating, Measuring and Understanding Shapes; and Securing Facts, Calculating and Identifying. I believe that these are similar to (or the same as) those used in the National Curriculum for the UK. Each unit has multiple mathematical concepts within it, for example, Algebra, Measurement and Number, and our outcomes have been placed accordingly. In order to spiral these we visit each unit twice - once in the first semester and again in the second semester. In this structure we spend around 3-4 weeks on each unit per visit and in the second visit are able to build upon the outcomes that we initially addressed in the first half of the year. Because the units are structured like this, we can choose to place them at any time of the year that we wish. This lets us link our maths units more authentically with our Units of Inquiry. For example, in our recent unit on Forces and Motion for How the World Works, we decided to run the Data Handling and Measures unit alongside it as it provided a natural trans-disciplinary link.
The difficulties we are facing at the moment is that it is nearly impossible to create a central idea for a maths unit if it contains so many different mathematical concepts. The Data Handling unit is unique in this setup as it contains concepts that are only linked to data and probability, instead of many other concepts, but the others have many differing ideas all covered under the same umbrella.
This leads onto the next obstacle. With the many different concepts being grouped in each unit there is a lot to cover. It has been suggested to us as teachers that we should aim to cover some element of every single aspect of the unit during the first visit. The result is that teachers are rushing through the work in order to complete it within the given time frame. There is little time for genuine inquiry as the students are being pushed through the topics at a fast pace. One solution to this that is being trialled is the development of rich maths learning engagements that ask the students to connect their skills, knowledge and understanding of several areas in order to complete the task. The one we trialled in Grade Four (centered around 'volume') worked well and was more geared towards inquiry-based learning. It gave the students the opportunity to experiment with different arrangements (measurement outcomes) test out their own theories and construct a formula (algebra outcomes) that could be used to determine the volume of any cuboid. They also had to calculate their answers along the way (number outcomes). It is envisaged that these tasks will then lead into inquiries in some of the areas of maths that were included in the task. Time will play an important role in this strategy as these tasks need to be developed for five units across six grade levels - and then again for the second visit for that year!
The irony of all this is that by developing a spiralled curriculum within each grade we run the risk of making the maths less meaningful for our students. I'm curious to know how other schools handle this paradox. If you do create units based on certain concepts, such as fractions, do you ensure that they are being re-visited again later in the year? If you don't, do the students have a thorough enough mastery of the concept to hit the ground running the next year or does time need to be spent on refreshing them at the beginning of the topic? If you do re-visit them, how do you make sure you've had enough time to give them their due course initially and are they addressed through the principles of inquiry?