Cling to the main vine, not the loose one.

Kei hopu tōu ringa ki te aka tāepa, engari kia mau ki te aka matua

Thoughts on Teaching and Learning of Mathematics

Measuring and Monitoring #13

At the start of each chapter [or page] is a couple of key outcomes I see as useful and achievable.

- to increase your understanding of getting good quality datato improve learning

- to learn new efficient ways of looking at data

Introduction.

Data, data, data. My experience of 40 years in education suggests that mathematics teachers, especially, collect test data every few weeks, examination data once or twice a year, e-AssTTLe data, PAT data, spot test data and then pretty much do nothing with it. The mark book, paper or electronic, is full of attendance data and marks out of 10, 20, 100 and 37 or some other strange total. A pig does not get heavier by weighing it more often! (Thanks Dave). There is a lot of data not collected as well. Student voice data, journal data from students and the from the teacher's journal, observation data from another teacher, informal data from early lesson starters and late lesson "try this" or "next time!" data. Less is better when it comes to measuring and monitoring a class full of emotions, attitudes, randomness, choices and events completly out of the teacher's control and usually knowledge.

A 2022 note. The new changes in NCEA and Numeracy expected in 2023/4+ bring attention to this page. It will be essential for all Y7 to 11 teachers, Math HODs to get an infallible measure of where students are in relation to the NZC. The new online Numeracy assessment will credit students with a pass if they are at or above NZC Level 4.5, so quite multiplicative. I have been harping on about this essential measure for 20years! Read this page carefully. Overtime!

Hence Lesson #1

If you do not intend to use the data do not go to all that trouble and waste student's learning time doing a test.

What data do you need?

In my project schools I suggest we collect 4 to 6 items of data each year per student. It is nice to have a math measure from the previous year, one from each term, and an outside check; total 6. From these I can create cohort, class and individual profiles and report accordingly. Learning is hap-hazard and random and any measure has an MOE (Margin of Error) that confuses any sense making meaning from one test. Deep learning takes time. All the little Unit Tests that teachers persist in giving students, the daily tests and the weekly tests are all useless to an HOD who is trying to see the big picture. The variation is wicked in these tests and the reliability pretty close to zero. I only want the over time data using a consistent measure.

Hence Lesson #2

Quality overtime data, one per term, reduces the variation and increases the reliability of sense making.

How is quality data measured?

Quality data is about keeping variation to a minimum and is a hard learned lesson. I suggest using one test or style of test that has multiple NZC Level assessing ability. NZCER has PAT, the Ministry has e-AssTTLe, teachers can use the PACT Tool and the Learning Progression Framework, past tests if you are good at writing such things, and there is always an OTJ or Overall Teacher Judgement where the teacher compares several items of student work with NZC expectations.

I use what I call the LOMAS test. It was developed around 2007 by Dr Peter Hughes and Dr Grigor Lomas. All tested and researched. The questions are carefully constructed and are concerned with the thinking strategies used by a student with the knowledge he or she has in the Number Strand of the NZC Levels 1 to 5. There are four parallel tests and I might have a go at writing another one in the near future. See the link above for more detail and the actual test components. The test is efficient taking only 20minutes or so to complete by a student and less than that to mark the complete class and enter it into a set up spreadsheet. It is possible to have a complete update in one day for a cohort and many hundreds of students. The LOMAS test does not have a reading issue nor a writing issue. No calculators are allowed and the student can get their previous term answer sheet back to check or improve answers. I have used the same test with same group of students on a period by period, day by day, week by week repetition and always found the same measure results. Learning can change over a term so increase is expected.

The data I assist HODs and teachers to collect is

- a measure for the previous year

- a LOMAS test result for each of 4 terms

- an e-AssTTLe result.

How is quality data analysed?

Here is an example of a data record for a Year 9 student.

Y8 LT1 LT2 LT3 LT4 eA4

Surname First Name 3 3 4 4 4 4 LT1 = Lomas Test Term 1. eA4 is the e-AssTTLe test in Term 4. "e" means online.

This student made a strong improvement from NZC L3 to NZC L4. Typical improvement is half a curriculum level in one year but often I am now seeing a 100% acceleration or 1 NZC level and often better.

Here is an example of the gain a class of 26 students made last year. The year 8 data has been removed. The graph shows a gain of 1.38 NZC levels using the Lomas measure. The end of Year e-AssTTLe test was not quite as kind as the LT4 test which I think is probably due to the different style of test and the reading (literacy) required to interpret the questions.

Class data for 26 mixed ability students, showing steady gain over 1 year.

Here is another view showing the spread of the students.

Same data as above showing the Blue Term 1 data to the Green Term 4 data.

A measure that I use to check everyone is one track, learning is happening and we are all focused on the same future is the %Mult or percentage of the class at or above NZC L4. Term 4 is a shorter period of time in NZ schools so the graph would be expected to slope less. The e-AssTTLe check measure pretty much agreed with the final result and that confirms and is reassuring. Note by Term 4 the L2 tally had decreased almost to zero. Typically seen is decreasing L2, a fairly stable L3 as students migrate in and out, an increasing L4 and L5. If this pattern is evidenced then I am confident targeted learning to the needs of the different groups is being enabled, students are engaged, the teacher is being effective and the class is doing what should be happening.

Same class showing the strong trend of become Multiplicative.

The cohort of students individually is also shown. Student #9 had a strong growth year moving form Level 1 through Level 2 to Level 3. Likewise student # 22 gained similarly moving form Level 3 to Level 5. I call this graph a "Skyscraper Graph". This graph shows how messy learning can be. Student #18 for example peaked at L5 and dropped back to L4 in the EA measure. Student voice after this test shows that literacy was indeed an issue for this student who said that "Some of the questions were a bit hard to understand and confused me."

This cohort performed very well. The class almost mirrored this performance. Every class accelerated learning and improved the %Mult measure.

This shows a class of 26 students and each has a story of the year of learning. Every student gained in this class.

The cohort view is also available.

Here we see the cohort of 130 students gained over 1 NZC level, or were accelerated by 100%. Better was the multiplicative level moved form near 20% to nearly 80% making most of these students ready for proportional thinking development. The cohort spread shows the decreeasing NZC Level 2 and a decreasing NZC L3 pattern, the holding pattern is more about NZC L4 as students move in and out as the year progresses, and increasing NZC L5 and NZC L6 student numbers.

Total test time = 4x20 minutes and a 1hr eAssTTLe experience. The rich data that resulted took 5 hrs to compile into a class by class breakdown and a cohort summary with comments. The report is presented to the teachers who are asked to comment and explain any flat topped "Skyscrapers", missing test data, insufficient mean gain, poor %Mult gains and so on. It is easy to run but very hard to hide from this quality data and effective analysis. Overloaded Markbooks can not produce this sort of clear sense making so that more time is available for planning and targetting.

Sensible and Informed Target Setting

All to often I see in BOT plans targets plucked out of thin air for new and existing cohorts. With data as above ensible targets can now be set for the following year for the cohort and each class. I would expect the mean for the cohort to approach NZC L5 and the %Mult to improve further to nearly 90%+. Experience shows that a Year 10 student at NZC L5 will gain a good selection of Merit and Excellent credits, enjoy mathematics and open doors across pathways. This cohort will have many choices.

Hence Lesson #3

Less is better. Analyzing and reporting quality data means it will be used. Maximize time spent planning and targeting time and energy.

I once had a computer business and a question I would ask a keen buyer of a new computer was "How is this computer going to save you time?" If a new computer did not save time, or make more money then buying a new computer was going to be a waste of money. I realised I was in the business of helping people save time and make more money from the beginning. It was not just about selling a new computer. The measuring and monitoring above is another example of better use of time. I read an article once where a business was struggling selling electric drills. Once the focus had been moved to "selling holes" the company starting performing. So what is the business of a school and every teacher?

BUT IT IS MEASURING ONLY NUMBER!

In the background on this page is a cartoon that presents two perspectives.

Background cartoon on this page.

There are two perspectives on teaching and learning mathematics as well. One is that we are teaching MATHEMATICS (and statistics) and the other is that we are teaching THINKING. As the man with the wooden leg said, "It's all a matter of a pinion!".

Mathematics teachers use the context of MATHEMATICS to teach the deeper learning and longer lasting learning called THINKING in all its different guises. I explain this all in earlier chapters but briefly as we move from NZC L1 to NZC L5 our thinking moves from not connected or organised, to simply connected, well connected, organised, logical, creative, critical and reflective. Maths has been taught in schools for so long teachers no longer recognise this stepping organised procedure and what is really important. They just teach maths, hopefully form a text or source that is orgainsed, and co-incidentally also teach thinking. By knowing you are actually teaching thinking we are now able to focus and put our energy where it will be most productive.

Hence Lesson #4

Know what your business is about!

But you are only measuring NUMBER!

If you look thoughtfully at the LOMAS test, at mathematical concepts introduced at each NZC Level, the types of language and questions asked, the projects and tasks expected in texts I quite firmly assert that you are actually looking at an increasing complex structure of thinking. Number allows us to look at that thinking and to measure that thinking. Number is quite intuitive to most people and even animals have rudimentary counting ability. The complexity with which someone uses NUMBER reflects the complexity of their thinking. What we are actually doing when we use the LOMAS test (or most other tests NZC referenced) is to measure THINKING.

The cohort graphs above could well be re-labelled as

Mean NZC = Increasing complexity of thinking, %MULT = % of those students who can think of two things at once, Spread graph showing how diverse the thinking actually is.

Lastly

For twenty years I was in a classroom teaching. I wondered from time to time what I was actually teaching. I became an advisor of mathematics and have been one for twenty years now. That gave me time to rub shoulders with Math ED and talk with many people about the Teaching and learning of Maths. In the last 5 years I have been more about "Making Dreams Come True" when schools have asked me for help. It has been a busy time and as you can see form the data and analysis above some astonishing results can be realised. There are other projects in NZ that claim acceleration rates of 100% or so for groups of needy kids. My work has created 100%+ acceleration for whole cohorts including those needy kids. Knowing that we are in the business fo teaching thinking (read the NZC blurb about mathematics - strategic, logical, creative and critical thinking takes center stage!) means a purposeful team with teh same shared goal.

Somewhere in the last 40 years of being a teacher I learned that I was actually teaching thinking. Somewhere in the last 10 years I became aware of this and have shared it with the schools I have worked in to produce positive outcomes. Much of this chapter is in other places in this book.

Last Lastly

Data is just data. It is a blurry image of what is actually happening. Students learn while they are doing a test. Triangulating data with OTJ, student voice, what you see and hear, all contibutes to a making sense of what you are doing.

I am not a fan of 1st in Maths Awards. I am a fan of most improved and high achievers. See Teacher Task below.

Some questions...

• Do you teach Thinking or Mathematics?

• Is it better to ask "How did you get that answer?" or say "Well done, that answer is correct!"

• Is it more challenging to say "The answer is 24, what was the question?" or "What is 2x12?"

• Do you have a test that you know inside and out that provides NZC Levels of where your students are now placed.

Teacher TASK

• Find the LOMAS resources on this site, pretty easy, and trial Test B or Test C with a group of Year 9 students. While they are sitting the test write your considered estimate of the NZC Level for each student.

• Design a spreadsheet that will produce the graphs displayed above. You could always email and ask me to send a version or check yours.

• Which students in the class of 26 above would you give awards to for High Achievers and Most Improved?

• Which students in the Cohort would you target for High Achievement?