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Thoughts on Teaching and Learning of Mathematics

Monitoring Numeracy #14This has become a focus along with Problem Based Learning for me. It is great importance to a teacher to know the student level and progression.

I will try and explain.

When we measure what Curriculum Level a student is operating in a subject or strand we are actually measuring their complexity of thinking. The thinking is wrapped in a developing cloud of knowledge, skills, attitudes, experiences, situation and upbringing. We all start out pretty much the same but pretty soon after we develop differently. All the time we are learning. Our thinking develops, is challenged, changes and adapts to our situation. This is the complexity we measure.

I measure Number. I could measure any of the strands and probably would prefer Geometry or Measurement but I measure Number. Students before getting to Year 9 get a lot of number so some are quite sophisticated in their thinking in Number. This measure also reflects their other strand thinking, and in different learning Areas with surprising agreement.

I use mainly the LOMAS test developed by Dr P Hughes and Dr G Lomas. The original way of determining a Stage in the Numeracy Project required what was called a NumPa Interview taking about 20 minutes. Secondary teachers preferred a faster way so Peter and Grigor generated and tested and selected some very clever questions. The test will quite rapidly, maybe 20 minutes for a student, or 60 minutes for a class. I developed a fast way to mark and also how to use the test in a formative manner. There are many other test options such as PAT and eAsTTLe or AWS.

Students are measured as being at Curriculum Level 1, 2, 3, 4, 5 or 6. I have noticed that a couple of tests will suffice to establish a broad indication and successive tests confirm that measure and help establish a trend.

Here is a cohort of about 200 Year 10 students. The HOD has tracked these students religiously from entry at Year 9. We are looking at 7 terms of data. There are no Level 1 Counting students at entry. The red bar is Level 2 and overtime has an annoying persistence as new students are enrolled and existing students continue to be reluctant. The grey bar is Level 3 and starts quite high at 30% but does reduce to only 5%. These bars are dynamic and there are students moving in and moving out so they tend to persist. The Yellow bar, my Multiplicative favorite is a good example of dynamic movement of data and remains around 30% over this time. The 20% in 10T3 are almost certainly none of the 35% in 9T1. The Blue and Green bars are teh students who have become proficient with fractions and decimals and are quite logical, strategic, critical and communicative students. From only 10% at entry they have grown to over 60% of the cohort. This means 60% of the following year NCEA candidates will easily gain Merit and Excellence. I asked if they should sit NCEA Level 1 at all and just study all year.

When a student starts learning anotherstrandlike geometry, or anothersubjectlike English, science or PE there head with their brain walks along with them. They do not miraculously become brighter, dimmer, more astute or less connected when they do this. They carry there hard earned complexity of thinking with them. They argue just as much in any subject. They deduce and connect in the same ways. They take their thinking to the sports field. The thinking is the measure. I am actually measuring the complexity of student thinking when I give them a test, any test. We, as teachers, just need to realise this and use the information. We do not need to test everything or every week or use many different tests. We need to know and use one test.

I often see a lot of data in a school in the form of tests, standardised tests, national tests, common tests, unit tests, exams, assessments and when I ask if the teacher has used the information I often get a "no" answer. The data from PAT and e-AsTTLe just get recorded and then ignored. No use is made of the information to improve thinking or inform learning. It should be and there is a lot of researched indicators within the tests that help to do this. But this is another problem.

FACT - You are actually measuring thinking ability, which is why SOLO is on the background on this page.

I will describe now at each level in terms of NUMBER. This we learned and described in the Numeracy Project 1999 to 2012. Overseas reseach projects confirmed these broad stages.

Level 1 -In my words, a very lonely place. This is the world of counting. I have 3 cats and 2 dogs, how many animals altogether? The fingers come out and we arrive at 5. What is I have 3 cats and 3 dogs? The fingers come out and we arrive at 6. 1, 2, 3, 4, 5, 6. There is no connection to any previous problem. The fact that a door opened and a dog entered are two separate events. An elephant might have entered.

Students at this level are plagued by no place value knowledge. How the number system works is a mystery. Every number has a name and none are connected to any other. I often hear parents boast claiming "My 3 year old knows all his numbers 20". The child can recite without error all names to these 100 numbers but they have no idea how the numbers connect and relate to one another, Just unconnected names. In reading we see the finger touching the words, one by one. We learn terms like counting on and skip counting.

L1 Key Features - No or not much connected thought. It is the ever regenerating, Generation Me. A counted world.

Level 2- Here there are some connections and the student will see 2+3 and 2+4 as "one more". They will start to understand placevalue or the valueof the number changes depending on where it is. They will know that 12 is a bundle of 10 and 2 more. The ten rule in place value is starting to develop. Why we only need 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to make numbers is starting to make sense. The image shows a rather sweet tool but the numbers shown are way to big for this level. In Level 2, 20 is a big number!

When presented with 136 + 45 the student will just fold and walk away. The numbers are too big. In reading we see a more smooth approach and more comprehension joining the words that are being read. Little words and little stories. Developing rapidly as language develops as well. These students are on the way.

Key Idea - Some connections in thought. Understanding of simple Place Value. Groups of 10 developing.

Level 3- Here there are some very good connections but it is all linear or straight line thinking. This , then this. It is the world of logic. A lot of our world works at this level. Xcel solutions are often additive. Accountants doing the books, engineers planning a project, programmers designing a program, buying an item at a shop, what we do when we get up, what the dog does every day, is all linear.

Students can add reasonable big numbers at this level and will master the efficient addition algorithm. They can double, triple, know a few basic multiplication facts and will derive the rest. "What is 4 x 6?" is answered with "2x6 is 12 so double that is 24." A student will choose addition to solve problems as first choice, every time. The methods are often inefficient and time consuming. "What is total of all the numbers from 0 to 100?" 1 + 2 + 3 + 4 + ...+ 98 +99 +100? The answer, if developed, will be by by adding.

Key Idea - Linear straight line thinking. The world is an added place. Addition rules. I have declared an War on Addition.

Level 4- This is my first target for Year 7,8,9 and 10 students. This is where a student develops the ability to think of two things at once. This is what multiplication is all about. What is 3 x4? 12. The student sees 3 groups of 4 at the same time. What is 4 x 6? 24. No hesitation. No issue the facts are known and used.

What shape is a square number? What is a factor, multiple or prime. What shape is multiplication? Many connections. The student at this level will also be able to give you a reason for their answer. This thinking is more advanced. More complex. More informed by better knowledge and more creative due to better problem solving experience and language.

Students here might not know 2/7 + 3/5 but are ready to find out. They may not be able to to figure if how to divide 288 in a ratio of 4:3. yet.

Key idea - Many connections, reasons are clear and conversation is structured. Reading is with meaning. Good comprehension. My first Target.

Level 5- The highly organised and connected world of proportional thinking is where students will not only answer but improve the question. These students debate and enjoy the mental challenge. They can operate with all operations on all fractions and decimals, percentages, rates and ratios.

This thinking is where we want people to operate most of the time in our technology driven, busy world. This is the plan. I saw a billboard on a trip to Auckland that said "Bepatientandwaitfora gapinthetraffic". I knew a proportional and creative student designed that idea.

How we figure out climate change, idiotic premiers of countries, electric cars and batteries, nitrates in our rivers, our own survival will be the realm of this thinker. We need as many as we can make. It is a learned event. Sue Lamon of the Rational Number Project once said "You do not become a proportional thinker by growing older!"

Key idea - Highly connected reasoned students. Very clear and powerful thinkers.

uipto here

Level 6- is where I depart from using Number as a way to measure complexity of thinking and instead use student argument and debate. These students can now specialise or further develop mathematics from a self motivated perspective. They are fun to teach. They learn quickly. These are the Merit and Excellence students.

It may be of no surprise to know that only about 15% of students entering Year 9 in NZ Secondary Schools are at Level 4 or higher. They rest are mainly at Level 3 and Level 2, in that order. This level has remained unchanged since it was first measured in 1999 despite the Numeracy Project, National Standards, ALIM and ideas including my own. I wonder why?

A Curiosity as Alice's Cat Suggested

It may be of curious interest to know that the current adult population is mainly at Level 3 with many professionals (engineers and lawyers) at Level 4 and a few at Level 5 or better. Society runs on addition. Look at your Tax Return and see if you can find anything harder than addition. Who amongst you can see that the sum of 0 to 100 is simply (101 x 100)/2 or 5050? If I travel to town at 30kn/hr and home at 20 km/hr, who can see my average speed is not 25km/hr? Who does cryptic crosswords?

Worse. The polls in an election are predicted by many different and usually correct methods. How linear and predicable is this? Propaganda and as Trump promotes "the false media" know that teh population is controllable or at least can be influenced. The news-media know this. They only promote what they want to promote and they know darn well what is going to happen. Oh well, perhaps this is the result of not improving thinking skills in our schools.

The Cohort view.

This shows all 230 students or so and the gains they have made over nearly two years at school, Year 9 and most of Year 10. The expected progress is 1 curriculum level in this time. Nearly 70 students have been accelerated. Why is this?

Being Multiplicative.

This cohort is going to be very successful in this school. Afgter entering Year 9 at an extraordinary 50%, nearly 3x normal, they have risen in a bumpy ride to nearly 85% and are certainly heading to exceed that target. Great to see. The dip at the beginning of Year 9 and Year 10 is the leaking of knowledge during our NZ Summer. The improvement over this time is only OK. The real data indicates no cohort acceleration but a definite improvement. They only gained .89 as a cohort when expectation was 1. I wonder why? Good data is not always the best.

The good news is that over 80% of this cohort will easily cope with our NCEA Year 11 courses. The issue for teachers and the school is how do we extend the better students so that we can maximise their efforts. No situation is ever easy or best. Improvement is always possible. Again, I wonder why many of these students will sit NCEAL1. Perhaps their time is better spent expanding and deepening their mathematical curiosity.

Hence Lesson 14Why is monitoring Number so important?

Can this be done in other ways?

How do you see thinking developing?

What messages does teh SOLO Taxonomy have for teachers?

Teacher TASK

Download the Lomas test and instructions and do the tests yourself.

Trial this test on some tame students.

Look at all the questions and ponder what they are seeking.

Circle three questions that are incorrect and use these to inform learning.