2023 - Session 2 - Being Multiplicative and Proportional
The information on this page is my interpretation of what I notice and reason. It is designed to be of use to teachers to help them ponder, reflect and redesign learning programmes as required. Use accordingly.
For clarifications please ask. Email Jim Hogan ACC572 jimhogan2@icloud.com, Cellphone texts and calls 027 461 0702

Here is a quick SELF CHECK for your Math Department or the one you work within.
    1. Do you have a data measuring and monitoring system that allows you select a student and see what NZC Level in mathematics they are predominantly operating within? Y/N
        2. Do your current mathematics programmes for Y9/10 target NZC L4 and NZC L5 as the success criteria? Y/N
            3. Have you checked out the Ministry information for the new NUMERACY and LITERACY requirements for NCEA? Y/N
                4. Have you checked out the new NCEA Standards and Assessment Trial on the NZAMT website? Y/N

Session 2
A Year 9 class full of multiplicative students is a delight to teach. Most schools now have mixed ability however, thankfully, classes so there will be a range of NZC Level 2,3,4 students and possibly NZC 1 and NZC 5. There will always be three groups of students broadly described as lower, middle and upper referring to their general mathematics knowledge and thinking. It is a paradigm shift for many teachers to comprehend that being a mathematics teacher is actually to be a teacher of "increasing the complexity of thinking, using mathematics as a context." It is easy to get side tracked on "busy" and having students do endless repetitions, insisting on quiet and neat, and leaving them to do this all year. That will spell death to any mathematical curiosity a student may have. Get student voice and ask them! Teachers must have time in the classroom to walk around and engage in dialogue, ask questions, look, listen and in general notice. NOTICE is the driver of being a teacher. The art of creating a situation where students are engaged in CCCC is "to be a teacher." My favorite teaching mindset or space is to manage a lesson where students are engaged in teh problem and do not realise what it is they are actually learning. An example is the Exploration of Sum of Consecutive Numbers. Students leap into the investigation looking for patterns and finding some numbers can not be a consecutive sum. All the time they are using additive arithmetic. Changing the task to Exploring Factor of Numbers 1 to 100 causes a huge improvement in multiplication facts.

So this session leads to operating on fractions, decimals and percentages which is where ratio, proportion and scale factor reign. Be very aware that you do enter the Proportional world until you have mastered being Multiplicative.

Lets GO!
2. Being Multiplicative and Proportional.
These terms developed from the NZ Numeracy Project and helped to bring understanding of the NZC and improve communication between Primary and Secondary teachers in terms of mathematics. That was one of the major successes in my view of the NZ NP.

I have been pushing "being multiplicative" for a long time. Here is an extract from my 2008 Newsletter.


Critical to getting most, 80% or more, of your junior students to NZC Level 4.5 or better is a relentless focus on every student becoming multiplicative. This is across all strands as well. High expectations for everyone. These messages have been around for a long time. It is worth spending time on each strand unpacking progression and key factors which would indicate being multiplicative. A general indicator is "A student gives an answer AND the reason". An NZC L3 student will give an answer only. An NZC Level 6 student will probably ask "Why do you want to know?" This is the critical brain working and should not be mistaken for being a Smart-Alec.

Junior Programme Design
Now is a great time to reflect and redesign.

I have been trialing a very different approach to junior mathematics programmes in my project schools for some time now and I like the outcome. When I started teaching secondary in 1979 I was instructed to use a two or three week period to teach a topic like "Adding fractions" and assess what was taught with a test. This procedure continued all year unabated. Answers to questions had one answer and doing hundreds of problems in a graded way was thought to be the way. I very quickly become the AMC coordinator and made problem solving a weekly diversion. Along with projects and investigations I think my math students got a pretty good deal.

Teachers collected bucket loads of test data and never really used it. All that is needed to measure and monitor progression is a reliable measure once per term. You can do what you like with the other tests data. It is your time you are wasting, not mine. Hint - Monitor your Workload.

Here is a graphic showing how my current 2022 project schools are organising their maths programmes in y9 and 10. The theme for Term 1 is usually Geometry and all strands are accessed with knowledge, skills and problems in that context. Every lesson has a geometric flavour in Term 1. In Term 2 the theme becomes Measurement and in Term 3 it become Statistics. Term 4 is a strange term and has a habit of vanishing but projects, investigations, STEM and other hands on work are solid learning experiences.


An example of a context (using 45 mins or 100 minutes) would be around chords in a polygon or circle.  Below is a typical white board sketch showing a couple of examples (with a deliberate error to cause dialogue). 
A chord is any straight line joining two different points on the perimeter of a polygon or loop. We usually use convex shapes because the chords are seen to be inside but all the mathematics still applies to concave shapes.
At the start of a lesson the language of geometry could be a STARTER. Name these shapes, name the parts of a circle, convex, concave shapes and so on.   
Understanding the problem is important so making sure we are all dealing with a circle perhaps and points on a circle will help. A table should be an obvious way to see any pattern and need not be suggested. Generating the nth term a target. 

In this lesson is language around geometric shapes like chord and intersection (Geometry Strand). The table suggests a pattern and a general nth term (Algebra Strand). The pattern is the triangle numbers whhich of course links this lesson now to Number Strand. Measurement. Probability and Statistics are not strong in this example.

Another example is the tossing of three dice.