Cling to the main vine, not the loose one.
Kei hopu tōu ringa ki te aka tāepa, engari kia mau ki te aka matua

123Thoughts on Teaching and Learning of Mathematics

Lesson #10 Revised 25/1/20

Assessment and Learning or perhaps Learning and Assessment!
know that assessment is for learning
know assessment that works for you

"We do not make a pig heavier by weighing it." (Dave Boardman - Advisor of Mathematics. University for Waikato 2000, 2001). I taught with Dave in Hokitika 1984 to 1987 and we talked a lot about mathematics. We wrote new schemes and assessments. We caused a lot of achievement. He was the first teacher I met who had a big picture grip on curriculum and it became our problem to present the material in an organised cyclical fashion so students met new ideas and returned to them later. We unpacked fractions, decimals, area, geometry, algebra and I focused on Statistics and Dave on Calculus at senior levels. We had both studied both subjects. We were both into problem solving then and Canta Math had a competition that our Year 7 and 8 students creamed in 1986. It was fun. It was at that time Dr Peter Hughes of AU, NZ Numeracy Project, produced his maths books which I bought and saw new insights from him as well. Thanks Peter.
  1. Assessment is for Learning
Assessment is about "forming learning" and "summarising learning". Diagnostic and Summative.

Forming Learning is about finding out what students know and building on that knowledge and those skills. This is strongly connected to "knowing the student (Ch 1)" and "developing a robust professional relationship (Ch 2)" with the student so they know you care and only say things they are ready to hear. Time to develop these precious "AKO" situations is needed. The pedagogical talent of the teacher then takes center stage and learning contexts appropriate to the student, school, environment, what is available and what is needed are constructed. This is "Understanding by Design" (McTigue)

Summarising Learning is about finding out if the learning that was designed happened.
A Word of Warning! We should be very careful not to presume we are measuring what we think we are measuring. There are more questions we do not ask when we test than we do ask.  We do not know what kids were actually learning. Worse, we do not know what in fact or what else we are actually teaching. A test is "a one of event" and the variation in small samples is astonishing as I hope everyone knows. Statistics has known about small sample sizes and variation for a long time. (Read "The Lady Tasting Tea"). In my mind the only way to get some firm estimate of the progression through the curriculum or the effectiveness of some learning to get a new concept across is over time assessment. (See Chapter 13.)
There are many assessments in mathematics including PAT, Question Data Banks, NCEA, asTTle, Numpa, LOMAS, Level tests, Teacher tests, Unit tests, End of term tests, Mid Year exams, End of Year exams, Aussie Maths, Otago Maths and so on and on and on. We do not need a lot of data however, just good data.

Mathematics is traditionally a place of tests and the traditional mathematics teacher is fascinated by marking each question √ or x. By doing this we are actually teaching students that mathematics is a "right-wrong" world. (a hidden part of what else we are teaching and is the real learning that is happening). It is much better to complement the student on the approach and argument, asking about how they did their solution method, and ask more interesting and thinking questions that allow the student to demonstrate understanding and application.  Instead of asking "What is 3x8?" ask a much more revealing question would be "How many questions can you think of that will give the answer 24?" We are now teaching the student that they are an integral part of the learning process and that there are many ways to solve a problem.

Here are a couple of alternative ways to see how much thinking development has happened.

Try these yourself now...

"Ask the Answer"
"How many questions can you think of that will give the answer 24?"

A shape has an area of 64cm^2. Draw some shapes with this area.

Create 10 data sets that have a mean of 5 but contains no "5's".

Three unit fractions (1/x) add to make 1/2. What are they? Are the fractions the same or different?
3,4,5 is a Pythagorian Triple. Find a few more triples. (Squares and Pythagorus Theorem)
Think of two numbers. Add them and cross out one of them. Keep adding and crossing out for about 15 to 20 repetitions. Now divide the last two answers. Did you getr 1.618 or 0.618? What is happening here.
One sixth (1/6) = 0.16 (chopped to 2 dp). Can you find more fractions with this property?
"Ask the Answer"
"How many questions can you think of that will give the answer 24?"

I think of 3x8 and 6x4 immediately, 4! because I know about factorials, 1/2 of 48 and √576, {1,2,3,4,6,8,12,24] which are the factors of 24, 5^2 - 1, 96/4, -48/-2 , 480/10, .048/.002
What do my answers tell you about my numeracy? Now look at your answers and ask the same question.

Learning First
In reality we do not know what students are learning when we are teaching. We might be teaching them the "right-wrong" world of mathematics (see above). They could well be learning about fraction addition or whatever it is I am discussing at the time. However, each student is a universe of self interest and experiences, interactions and personality, prior learning and self generated concepts...among a few million other things. They may have just had a fight or have just fallen in love or have just had a birthday or are just tired. These are some of the questions we never ask, but hopefully notice.

I always ask past students what they remember from my classes and usually they say "we had fun" or "I enjoyed your classes Sir". I ask "Do you remember the mathematics we did?" and very rarely they say "Yes". What they seem to report is the fun and the enjoyment. I know it was not all fun and enjoyment and so do they, after all "they were there". I ask them what they do now and do they use mathematics and I am happy to report that the vast majority have good jobs and actually can describe how they use mathematics or/and statistics. These are not just engineers and doctors but jobs, trades and professions from a wide spectrum in society including building, environmental monitoring, courier owners, courier drivers, painters, nurses, shop owners, lawyers and teachers and also one os also the current fastest 400m Electric Car Record Holder in NZ. Young people are a very capable and talented resource!

One student who gained a Doctorate of Applied Mathematics explained to me that I never told her the answer to any problem. Another said "When I asked if the problem was right, you just said believe your in your own thinking!" and that helped me gain a Chemistry degree and I am now a teacher!" So what in reality were these students learning? I suspect I knew what I was teaching them besides the mathematics but that was a few years before the New NZC which says it is as important.

For me the lesson is be careful about what you think you are actually measuring, and measure only what you need to measure. If you find data in your mark book that is not analysed,  not pondered and unused, or there is a pile of marking waiting to be done then "you did not need the data, so why did you bother?" Why did you waste good learning time for the students.

I witness leaders in schools justify mid-year examinations with the reason "Students need to learn how to sit an examination, in a large room for 3 hours, silent!" Bunkum! What an unsubstantiated load of nonsense this hypothesis represents. Ask the students if they need practice at sitting examinations! Better, think of a better way to assess the learning. Sometimes an examination is appropriate but not always! It is just a habit that we have developed.

A Brief NCEA Comment
In NZ and in Mathematics, NCEA L1,L2 and L3 is typically measured by an assessment in quiet on your own conditions and is marked against a rubric which tries to identify agreement with the Achievement Standard being assessed. The more I see this happening the more I think it is not actually a very nice way to assess learning. It has the effect of limiting what is taught to what is turning up in the test and narrowing the learning. I am not against the NCEA Standards Based Assessment. It is a very powerful qualification if used in an intelligent manner. I am questioning how we assess for NCEA.

A Numeracy Project Gem.
There are many critics of the NZ Numeracy Project. I think it was a huge and wise investment and had many benefits across a wide spectrum. It pretty much informed the 2007 NZC, the Adult Numeracy Progressions, The LPF Learning Progression Framework, the language Primary and Secondary Math teachers use and helped thousands of mathematics teachers become better mathematics teachers. It made several doctorates and masters qualifications. It built a huge collaboration between many many NZ teachers. It was singly the biggest investment in mathematics in NZ ever. We still benefit today.

Dr Peter Hughes and Dr Grigor Lomas created a written version of what was used for data collection in the Numeracy Project called the NUMPa Inteview. The issue with an interview is it takes time. The value is it allows the interviewer to "see" inside the head of the student and clearly understand what the student is reasoning. More on if you look for it.

What happened (about 2008)was Peter and Grigor crafted some very clever questions and published a research paper  (all available on The result was a series of tests trialed collated as Test A, Test B, Test C, Test D and three more parallel tests made. I call the tests LOMAS and some schools now understand and use them to get a fast and pretty accurate point sample of ability after one test and over time a more reliable time series view of each student and whole class or year group ability. I suggest testing students on Week 3 of each Term and keeping a chart over the Junior Years 6 to 10. There is more about this test here on my website and in a later Chapter.

The LOMAS test measures thinking involved with number and parallels the Number Framework created by researchers in the Numeracy Project. There are 4 tests Part A, B, C and D which align to NZC Levels 2, 3, 4and 5. I contend that what we are actually measuring here is the complexity of thinking and a student takes this thinking into all learning areas and daily life. As this thinking improves so does the thinking in other learning areas. So we are actually measuring a really important indicator which is a deep use for all teachers.

At Level 1 students hold concepts and ideas a independent and unconnected events. This is counting and very little generalisation or patterning is noticed.
At Level 2 students weak connections are seen and as number knowledge and place value ideas develop so do the connections. Basic ideas of sharing and fractions are formed.
At Level 3 these connections strengthen but pretty much are linear or in line or from here to here, in steps. Early ideas of groups and use of groups start forming.
At Level 4 the thinking abruptly changes to thinking of two things at once and having reasons for everything. This because of "this" rather than this and then "this". Fraction ideas and sharing is well connected and early decimal and percentage ideas improve drastically. Measurement, geometry, probability and statistics all become well formed and students read and write quite well also. Here is the first goal for all young students Y 1 to 10. I describe this type of thinking as two dimensional.

At Level 5
all the learning with whole number snow happens with fractions and decimals as well. The thinking become quite complex and rate and ratio make sense. Students automatically choose fraction solutions and use multiplication as a standard procedure. They not only give reasons but question the solution and justifications as well.

The critical thinking improves to adult levels. Now perseverance, learning to work, enduring, creating, participating, contributing and abstracting are developed. A student will understand the reasons for studying and the purpose of new learning. This is a longer term goal and about 50% of the human race actually get here before they die.

(Added 2019)
At Level 6 students become "big picture thinkers". This is Merit and Excellence at NCEA L1. Level 5 is Achievement. A big pictrure thinker will teat a test differently from a Level 5 student by reading all the details, pondering and consider whether the actually have to do the test. Level 6 students are exciting people to meet. They are a young person with an adult brain. They are switched on and in teh right social and financial conditions with some good moral fibre will become our future leaders. In comparison, a Level 5 thinker will still try and obey and do. A level 4 student will struggle with understanding aspects of questions, proportion, but for most problems figure out a way. A Level 3 student will still try and add everything. A Level 2 student will see numbers that just keep getting bigger and more obscure. A Level 1 student will not even see the issue and just be happy. If there was a Level 0 they would be content to breathe, smaile and look for Mummies eyes. Such is brain development and thinking. See

The LOMAS test is so fast that 20 minutes is all a student needs and within a period a teacher can have the class analysed and not long after that can have a picture of the Year level so all can ponder next steps. Much more on how all this is done in a future chapter using real examples.

Hence Lesson #10 and Teacher TASK
Describe an assessment you used recently, its purpose and its effect. Try and explain if it did the job.
List three ways you will use the results from your latest assessment.
Did the assessment improve your future lessons?
Rework the assessment to three tasks that will do the same job.