Cling to the main vine, not the loose one.

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

Thoughts on Teaching and Learning of Mathematics

Assessment and Learning and Lesson #10

• being aware that assessment is for learning

• knowing about 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.

Assessment is about progression and is basically "forming learning" or "summarising learning". Diagnostic and Summative.

There are many assessments in mathematics including PAT, Data Banks, NCEA, asTTle, Numpa, LOMAS, Level tests, teacher tests, unit tests, end of term tests, mid year exams, end of year exams, and so on and on and on. We do not need a lot of data, just good data.

Mathematics is traditionally a place of tests and the teacher is fascinated by marking each question √ or x. We are actually teaching students that mathematics is a "right-wrong" world whenever we mark in that way. It is much better to complement the student on the approach and argument, asking about how they did their solution method, and ask more interestign 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?"

"Ask the Answer"

Try this yourself now

"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

What do my answers tell you about my numeracy?

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.

I always ask past students what they remember from my classes and pretty often they say "we had fun" or "I enjoyed your classes Sir". One student who gained a Doctorate of Applied Mathematics explained to me I never told her the answer to a 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!"

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 and pondered or has not been used, or there is a pile of marking waiting to be done then you did not need the data so why did you bother and why did you waste good learning time for the students.

I hear 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 hypothesis that statement 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.

LOMAS

This is a series of tests trialed and researched reported and then extended during the NZ Numeracy Project by Dr Peter Hughes and Dr Grigor Lomas. 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.

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.

Broadly...

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 a student 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. Studnets 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.

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.

Hence Lesson #10 andTeacher TASK

Do you understand the impact of the assessment you are using?

List three ways you will use the results from your latest assessment.

Does assessment improve your lessons?

CHAPTER NAVIGATOR1. Intro and Relationships, L#1

This is to help look around my pages. I have tried to make it consistent in all chapters. The Planned chapters are only ideas at the moment.

2. The Main Vine, L#2

3. Beginning a Year, L#3

4. Number and Algebra, L#4

5. Geometry and Measurement L#5

6. Probability and Statistics L#6

7. Problem Solving L#7

8. Investigations L#8

9. Visual Mathematics L#9

10. Assessment and Learning L#10

11. Team and Problem Based Learning L#11

Planned12. Engagement L#12

13. The Classroom L#13

14. Being a Teacher L#14

15. Being a Leader L#15

16. Managing the Principal L#16

17. The Importance of Whanau (Family)L#17

18. The Importance of the Student L#18

19. Math Topic A - Squares

20. Teacher Tools

21. LOMAS

22. Math Phobia