I have spent the best part of today (the first day of much-needed Term 1 holidays, no less) at the MAV’s (Mathematical Association of Victoria) working party consultation meeting to draft a response to the draft curriculum for mathematics.
If you haven’t had a chance to view the draft curriculum you can go the ACARA website, click on the box with Australian Curriculum in blue, register and then you are able to view the curriculum. A word of warning: it is NOT an easy site to navigate. I would advise that you either download or print off the curriculum (with elaborations…these are only viewable when downloading or printing!!). You can also choose to view which areas of the curriculum are applicable to which proficiency strand (understanding, fluency, problem solving or reasoning) by selecting this strand then Apply Filters. The sections that are then coloured tan/orange are the ones associated with that strand.
Some common themes emerged at the working party session today:
- The initial framing paper’s rationale and aims and overview of the structure of the mathematics curriculum sounded promising and exciting:
- an emphasis on big ideas to allow for less content coverage and more depth
- a stated aim of ‘interaction of content and proficiency’ ie. content descriptors and how these relate to the proficiencies of understanding, fluency, problem solving and reasoning
- interconnectedness of mathematical concepts
- A feeling that the promises that were articulated in the initial framing paper were not evident in the draft curriculum statement ie. a general mismatch between stated aims and what was produced as the curriculum
- Big ideas were not articulated anywhere
- Throughlines in concepts weren’t evident from one year level to the next
- No (or very little) interconnectedness between the strands of content or, in many cases, within the same strands
- No allowance for differentiation of content to give meangingful access to classes who can have up to 5 to 7 years’ difference in ability levels within them
- A confusion as to the meaning of Level 10A
- ICT (or digital learning environments) not interweaved through the content in a coherent fashion or used to its best advantage for improving the quality of mathematical learning; usually referred to as only a calculating tool
- there was too much content
- good thinking displayed with the grouping of content areas to create the three strands
- that the emphasis on ‘financial arithmetic’ as, seemingly a separate topic almost, was misplaced and misguided
- an emphasis on processes at the expense of ideas
- a lack of students being required to create and develop rather than ‘just’ use a rule or formula
- that this was not a ‘futures-focused’ curriculum or a curriculum for the 21st century; that it, in fact, seemed to be taking mathematics education from a ‘teach for understanding’ focus to a ‘use of mathematics without understanding” focus
- no evidence of the ‘beauty’ of mathematics in the curriculum document
To me, it all just seems to have been pulled together too fast. Some of the English expression in the Mathematics – Rationale/Aims and Organisation etc overview at the start is appalling. The actual content of the curriculum also reads as if one group of writers did the Number and Algebra strand, one did the Stats and Probability strand and another did the Measurement & Geometry strand. And then each group didn’t communicate with each other.
Anyway, let me start at the beginning:
- A world class curriculum for the 21st century that is futures-focused.
Hmmm…someone from my school who has recently returned from an overseas conference has returned with the following information:
Tony Wagner, Co-director of the Change Leadership Group at the Harvard School of Education spoke of his ‘Seven Survival Skills for Preparing Students for the Future’. They were:
1. Critical thinking and problem solving; the ability to ask really good questions and not just get the right answers
2. Collaboration across networks and leading by example
3. Agility and adaptability to work across a number of disciplines
4. Initiative and entrepreneurialism; where you must be able to learn from failure
5. Effective oral and written communication
6. Accessing and analysing information
7. Curiosity and imagination
As Tony spoke about schools holding themselves ‘accountable for what matters the most’ he focused on the 3C’s: Critical Thinking, Communication and Collaboration and emphasised that to re-imagine our curriculum, we must make the teaching and learning more explicit to improve student learning.
I don’t think that the mathematics curriculum encourages critical thinking or how important it is to ‘not just get the right answer’. I don’t think it promotes agility in thinking, the taking of initiative or the development of curiosity and imagination. The emphasis on the use of mathematics’ formulas, or results, without students being required to develop these themselves, means students are denied the opportunities to cognitively engage with the ideas behind the formulas and thus have experience in designing their own creative solutions to new problems. The formulas can always be looked up – they are the ‘information’ of mathematics. What will be critical to future citizens will not be the way they can use formulas that might become outdated, but how to create new ones for new knowledge. It makes knowledge always secondhand, instead of firsthand…something that someone else created. Students will not develop their own neurological pathways by merely using someone else’s creation. To think critically, there needs to be some knowledge of where results come from. There needs to be some experience of justifying a position, providing proof. Someone at today’s session said that ‘surely this curriculum is a statement of content only, it is up to individual teachers to provide the appropriate slant such as developing proofs’. Maybe. But, in my experience, too many teachers will just teach the content and not worry about the rest if it isn’t articulated. This is our chance to ‘do it right’…or at least better than its current form.
2. Big Ideas
I would like to see big content-related ideas around John Mason (UK)’s mathematical themes of:
} Doing and Undoing
} Invariance Amidst Change
} Freedom and Constraint
For example, ‘doing and undoing’ could be the overarching theme for the Number and Algebra strand. ‘Invariance amidst change’ can be applied to the Geometry and Measurement strand eg. in transformations (what aspect of a graph/shape stay the same when a transformation or set of transformations are applied?). Freedom and Constraint could be the overarching theme for Probability and Statistics. These would focus teachers’, and students’, attention to ideas that can act as ‘throughlines’ for all the content and provide a lens through which we can determine essential learning that is at the heart of each content strand. Additionally, problem solving activities could be formulated that are based on these themes.
Additionally, the listing of sets of content under each strand could be elaborated using the following verbs (also from John Mason) and an indication given as to which verbs relate to which proficiency strand.
} Specialising and Generalising
} Conjecturing and Convincing
} Imagining and Expressing
} Ordering and Classifying
} Distinguishing and Connecting
} Assenting and Asserting
3. The Use of Technology
Calculators and computers are reshaping the mathematical landscape, and school mathematics should reflect those changes. Students can learn more mathematics more deeply with the appropriate and responsible use of technology. They can make and test conjectures. They can work at higher levels of generalization or abstraction. We must make prudent decisions about when and how to use technology in both learning activities and in assessment activities and should ensure that the technology is enhancing students’ mathematical thinking. I do not see the explicit use of technology with regard to the learning and teaching of mathematics in a way that improves mathematical thinking in this curriculum document. In the “General Capabilities’ section at the start, the paragraph on ICT has to be one of the most appalling pieces of writing I have seen in this document. The general feeling of this paragraph seems to be that the main utilisation of the calculator is to do more complicated arithmetic. Not so. A statement related to improved pedagogy needs to be included here. For example: you can ask students to put in sq rt(8) and hit ENTER. The lesson then starts with the problem: why is 2 rt(2) the answer given? Technology opens up a whole new way of teaching mathematics, it is NOT just for more sophisticated calculations.
4. Connections of Concepts and Ideas between strands & Connections within strands between year levels
More connections need to be made within strands at each year level and across strands at each year level. For example, to link the Number & Algebra strand and the Measurement & Geometry strand, the investigation and use of Pythagoras’ theorem at Y9 level can be used to introduce surds and what an irrational number is. Discussion can then be had about exact versus approximate (another important ‘big idea’ in mathematics, especially when considering the use of CAS calculators). Decimals can be revised and developed through the Measurement strand in Y7 and earlier. This is important to set skills and ideas within authentic contextual content. The various ways of multiplying whole 2-digit numbers can be linked to the various representations of multiplying out two binomial brackets. And, incidentally, Chinese multiplication could be included in the elaborations for this to give an international perspective and show that mathematics is something that has evolved over time and, in many respects, is just as culturally dependent as other disciplines.
I cannot see links to historical perspectives of mathematics in the content strands. The development of the system of numbers (nowhere to be seen itself) is an important aspect of how ideas develop over time and how a closed minded approach (like the refusal to accept zero as a number or negative numbers) can halt the development of further mathematical thinking in certain cultures (and hence make a nice link to international mindedness education and the importance of remaining open-minded – something that seems to be also missing in our national curriculum).
6. General Comments
- Applications of essential mathematics ideas need to be just that – applications. Financial arithmetic is an application, not an idea or concept in itself. It needs to be removed as an area of content standing on its own.
- Matrices in Y10A? These are becoming increasingly important in various aspects of senior mathematics and should be included as requisite knowledge before students go into Y11 and 12.
- Number types and exact versus approximate are important ideas that need to be included.
- Strategy for implementation needs to be carefully thought through – how will the Y7 to 10 course adequately prepare students wanting to do alternatives to the Aust Certificate, such as the IB, for example? Also, what are schools meant to do with 10A? Will we be forced to offer a virtual streamed Y10 with a general course and one that leads into the higher conceptually challenging mathematics subjects in Y11? This means students will have to decide what mathematics they want to do in senior years at the end of Y9…and consequently perhaps lock themselves out of a future that needs higher order mathematics.
- How the various proficiency strands are related to the content needs to be articulated better. At the moment, the only way I can see of viewing which content aspects are related to te fluency proficiency, for example, is through the website when the areas are then highlighted in a different colour. This will be inappropriate for large-scale implementation. What about something like what Prof, Hojgaard from Denmark has devised below?
Prof. Hojgaard’s suggestion for the design of syllabus documents is as below:
||Number||Algebra||Probability & Statistics||Geometry||Measurement|
|Mathematical thinking competency||The body of the document would have sample questions, examples, suggested|
|Problem tackling competency||activities for learning etc.|
|Symbol and formulation competency|
|Aids and tools competency|
If you wish to send your feedback on the Australian Curriculum (and I highly advise that you do!) then you can either give feedback via the ACARA site (not what I would recommend as the portal for feedback seems to be highly orchestrated and doesn’t allow much opportunity for free-flowing comment) or through the MAV, who will be constructing a blog for the purpose, through AAMT or through the VCAA website. There may be other avenues for response I don’t know about. A word of advice for how to word your response: apparently the tool used to analyse the feedback will be a linguistic tool that picks up on certain words. So…use the words of the document and be very specific as to what you are commenting on.
Finally, the word is that the draft curriculum for the senior mathematics subjects will be ready just after Easter.
Hold onto your hats, everyone.