This Issue; WAR on ADDITION!, NCEA notes and Term Happenings.
Greetings All Teachers of Mathematics
I am always interested in good use of numbers. Here LINFOX (transport company) use zero as the target for fatalities, injuries, motor accidents, environmental exceedances, tolerance of
unsafe behaviour, practices and zero damage. This initiative starts at the top with a commitment from management to make “Vision Zero” work. Perhaps every school should have a similar public declaration agreed to by staff, students, parents and community. Thanks LINFOX.
HOGAN DECLARES WAR ON ADDITION
I have to say I am heartily sick of seeing 60 to 80% (or more) of Year 9 students arriving in secondary with an “addiction to addition”. How does this happen? What do we do to change this linear and lazy thinking behaviour? What is it instead that should be firmly established mathematically for students?
I can answer the last question. It is multiplicative thinking that should be the preferred method and first choice of every student in modeling situations and solving problems. Multiplication is first introduced as early as Year 4 and 5. This more efficient way to solve and explore mathematics is expected to become dominant by Year 8.
Multiplicative thinking is explained in my previous newsletters, but again very simply, if the strategy used by a student to solve a problem involves the distributive law of multiplication then the student is or is becoming a multiplicative thinker. This type of thinking is the target of adult numeracy and the new National Standards by the end of Year 8. Being a multiplicative thinker guarantees NCEA success.
The first question is difficult to answer because it involves investigating primary and intermediate mathematics programmes and practice. The second question about changing this unwelcome pattern is even more difficult to do and is related to the first question. Are teachers multiplicative thinkers? The numeracy project has focused on deepening number knowledge and the teaching and learning of number but must share some of the responsibility for this “addiction to addition” situation.
The “addition addiction” manifests itself as “many strategies used to do addition problems”. How many ways of adding does a student need? I say, not very many. Perhaps noticing “tidy numbers” and “doubles” then moving on to multiplication. If more strategies for adding are needed, in decimals for example, then return to addition and develop more. A good multiplier is also a good adder and a very good counter. Students can then become a proportional thinker.
So war is declared. In my numeracy project schools I openly ban “counted” solutions. “Thou shalt not count!” is clearly communicated to all students. This banishment of counting is effective and teachers see immediate higher level thinking by students. Lift the expectation!
But this war is on addition! The large group of lazy thinkers who choose additive methods get the blunt response of “OK, now do it another way (and use multiplication)”. By not accepting these simple approaches we value being a deeper and more complex thinker. Doing less problems in more ways is far better than doing more problems once each. Quoted from George Polya.
CASE STUDY OF CHANGE
A teacher in an SNP school had a very unruly group of about 15 Y9 students when I visited in the second week of T1 2009. The students did not know how to relate to each other and certainly had no intention of doing engaged class work. I returned in early June, 14 weeks later, and was astonished at the turnaround.
The teacher explained it took about two weeks to establish respect. Once trust and a relationship had been established, learning that slowly encouraged co-operation and sharing was put in place.
The students had Numpa tested as “counters” with very little number knowledge upon which to build. The Term 1 goal was to build number knowledge of place-value and addition basic facts and strategies.
As a result of the SNP workshops and the strong multiplicative message, the Term 2 focus was on developing multiplication tables and strategies. On my return visit the students welcomed me and I helped them with their daily problem solving start-up activity.
This wa a difficult multiplicative loopy that had several pathways and an unintentional error. Some students picked the error up and reported the mistake with great delight. They all enjoyed and engaged in this activity.
The lesson continued and I modeled for the teacher totaling the first 9 counting numbers using multi-link blocks. The students were able to develop the formula of n(n+1)/2 as being the sum but not as an algebraic formula.
We debriefed about the students and the lesson noting the vivid change in attitude and learning. These students had moved through to Curriculm Level 3 in Number in a very short time. We both agreed that a video of both before and after would have been very powerful record of learning.
I am looking forward to the next visit in two weeks time. How will all the strands of the mathematics curriculum be nurtured and developed for this group? Will they be ready to engage in the Math Week activities?
• Jim Hogan, Adviser in Mathematics (Secondary)
School Support Services • 1358 Hinemoa Street • PO Box 935 • Rotorua
Phone 07 348 9079 • Fax 07 349 2214 • Mobile 027 278 5458 • Email
NCEA CONSULTATION
I do hope there was a huge amount of considered feedback from across regions. There was not a lot of time given for this but all involved were aware teachers are very busy people. The survey was not too daunting once begun and I added my peanuts worth despite being involved with some of the writing. The NZAMT selected committee that developed the standards, and the Ministry, published, in my view, a considered revision of the standards for this consultation. All feedback will now be interpreted and changes made accordingly. The Level 1 standards will be finalized this term and the Level 2 and 3 made ready for consultation in Term 4.
See NCEA latest report.
Assessments and exemplars. Keep a close eye on TKI and NCEA website for these. I know that resources are being written and schools are being asked if they would like to trial the assessments. Contact me if you wish to take part in this trialing. Likewise, if you have worthy assessments and exemplars then please send these for forwarding to NZAMT.
The ongoing issue of qualification for lower ability students will not go away so we must give some serious thought to constructing learning pathways that will give success in mathematics for these young people. Perhaps expecting them to progress from Curriculum Level 3 to 6 in three short years is excessive. They did take a long time to get to Curriculum Level 3 so is it fair to expect them to suddenly change? Why can we not put an extra year in the junior programme to help students become multiplicative with decimals and fractions. Are your contributing schools sharing data with you about the new student intake each year?
For the middle and lower ability studentearly identification of numeracy stage and appropriate learning is essential. All students can become multiplicative and so gain the necessary mathematical skills and knowledge for their chosen career paths. Look at your data, notice and identify the large “middle”. Monitor and move these students. Notice, notice, notice!
The media who publish the hugely fiddled “league tables” have a lot to justify. Just because it is written does not make it truth. Schools agonize over the exposure created by the league tables and make strange and unjustified expectations of both teachers and students. Are our schools designed for the large 60% group of “middle achievers”?
The concept of NCEA, to me, is acceptable and certainly more students gain credit for what they know. The interschool consistency is now more robust; calculators remain a concern; and there are more internals for better or for worse. I would like the standards to be selected from the subject description in the NZC and not the achievement objectives however.
INVITE SECONDARY NUMERACY 2010
I have a few places for schools wishing to join the numeracy project in 2010. Funding beyond that date has not been decided. If you are interested please tell me so that I can arrange a visit to explain all the details. Phone or text 027 278 5458.
The numeracy project is an effective professional development for mathematics teachers and does result in improved learning for all.
A Proof of Pythagorus’s Theorem
I often ask teachers for their favourite proof of this famous theorem. Usually the response is “I do not have one.” Here is a simple way and one of many available.
Coming up
Term 3
Maths Week Aug 9th to 16th
TAHI!
Register and be prepared! What activities do you have planned for your school to make everyone more aware of mathematics?
Census at Schools
You should have registered by now! Have your students take part in the survey and then use their data to learn statistical thinking. The site is “a must” to visit.
NZAMT Conference
“Pi in the Sky – Expanding Mathematical Horizons”
Palmerston North Boys’ High
28th Sept to 2nd Oct. The first week of the school holidays. See nzamt website for details and registration etc. The main conference is on the Tuesday to Thursday with NZAMT meetings on Monday.
Study It Website
Check out the mathematics queries and answers from students. This is quite a cool site to ask hard questions.
See the Auckland Maths Wiki at for a huge resource of tests and resources for junior mathematics.
Practice Exams.Four sets of practice external assessment tasks, Level One, Level Two,Calculus and Statistics and Modelling. $36.00 for each set. $144 for theset of four.
Orders are to be placed with Kohia Teachers Centre,
Lastly, a request. Do you have a special unit of work that really achieves something special and that you can share with others. One of mine is the 3D Geometry unit available from my website. I would very much like to collect more and make them available to other teachers.
Good luck and enjoy Term 3!
Jim Hogan
SEC MATHEMATICS ADVISOR