Download my thoughts and possibilities for discussion

NZ Curriculum Development • 21 June 08
The following is mostly my own thoughts on where we are going with the NZ
Curriculum due for implementation in 2010. The HOD Days in Tauranga 22 May and
Hamilton 12 June have contributed. My SSS colleagues at primary and secondary
level, and fellow advisors have also contributed. An NCEA “re-alignment” will also be
undertaken beginning this year with the achievement standards and the unit
standards. Some thoughts follow on that process and outcome as well.
NZQA has released a document to principals outlining the six principles by the
review will be guided. This review applies to all levels of achievement standards and
unit standards. It will concern itself with the issues of duplication of standards,
credit parity, consistency, fairness and coherence. There will be an expert group to
check accountability of subject advisory groups involved in the review.
The principles say that a standard:must come from the curriculum,
have a clear purpose,
be valid and reliable,
have A, M, E grades based on higher level thinking
have credit parity
and will not duplicate others.
The conditions for assessment will be attached to the standard. Note I have
interpreted these principles for the secondary mathematics situation and assumed
A,M,E grading and better means higher order thinking rather than more of the same.
Together these appear to be a reasonable directive. It will be interesting to see what
develops in the next few weeks.
Jim Hogan, Seccondary Mathematics Advisor, WaiBOP 2008
Deep Understandings in Mathematics
What is it we want our students to know?
I am trying to establish the concepts, ideas or and/or
notions that will remain with a student after they have
departed school. These are the real understandings we
are targeting and hence describe the real assessment
events as well. Read “Understanding by Design,
Wiggins and McTighe” if you want an excellent
description of deep understandings.
The NZC has 3 strands,, Number and Algebra,
Geometry and Measurement, Statistics (and
Probability). One deep understanding for each of these
would inform classroom business on a daily basis. One
deep understanding covering all would be the purpose
for studying mathematics. All of these understandings
must be in harmony with the school vision and the
departmental vision.
The following are developmental and will change as I learn more and am informed
by working with other mathematics teachers.
Assessment grades for achievement standards; general note.
Achievement will be a numerical solution. Merit will be the clear explanation of the
solution and or other solutions. Excellence will be the generalisation of the problem
clearly expressed. These reflect increasing ability of the key competencies.
Overall Deep and Enduring Understanding of Mathematics, Year 1 to 11.
Students will understand that mathematics is a connected complex relationship of
quantity, space, time and data. We can make sense of this world by learning the
language of mathematics, exploring these relationships and expressing our thinking
and findings for others. [Mathematics is the thinking and a language. It is logical;
deductive; skills; concepts; but above all…connected. It can be abstract and applied.
Mathematical literacy involves interpreting graphs, diagrams and other
representations. In mathematics the words add and multiply.]
Overall Deep and Enduring Understanding of Strands, Year 1 to 11. Unit
Standards, not described here will come from Curriculum Levels 4, 5 and 6.
Achievement standards will come from Curriculum Levels 5 and 6. This is then
consistent with TEC Adult numeracy and literacy expectation of Curriculum
Level 4 or multiplicative number thinking as a minimum.
Attempt 1, Number : Students will demonstrate numbers can be taken apart,
operated upon, and re-assembled to solve problems in new and familiar situations. [
This is using place value and part-whole notions with N, W, I and R. Number
connects with all other strands. At first this is a counting world, then additive and
multiplicative to complex applications and proportionality.]
Attempt 2, Number : Students will solve, and explain and generalize solutions of
number problems in new and familiar situations.
[ This is using place value and part-whole notions with N, W, I and R. Number connects
with all other strands. At first this is a counting world, then additive and multiplicative
to complex applications and proportionality. A is a numerical solution, M is the clear
explanation of that solution and E is the generalisation.]
Algebra: Students will demonstrate generalisation of patterns and number
properties and be able to model situations to solve problems in new and familiar
situations. [ This is using the variable, developing the syntax of algebra and using
abstract reasoning with increasing complexity and sophistication. Algebra connects
with all other strands.]
Geometry: Students will demonstrate knowledge and application of shapes and
their properties to solve problems in new and familiar situations. [This is the
physical world, angle, manipulation and symmetry. Proof.]
Measurement: Students will demonstrate understanding and use of common
measurement systems to solve problems in new and familiar situations. [This is the
interaction of number and geometry.]
Statistics: Students will demonstrate understanding the inquiry cycle of posing a
problem, planning getting data, analysing the data and making conclusions. [ This is
the world of data, variability, graphical literacy, deduction and inference.]
Probability: Students will demonstrate knowledge of uncertain events and apply
this to model and solve problems in new and familiar situations. [This is the random
world and links to variability and modelling.]
The Standards of Qualification
Having broadly described the deep and enduring understandings it appears obvious
that they also describe the standards that we should construct, with appropriate
levels of difficulty, to assess the learning for progression and qualification.
At junior levels we are more interested in learning and progression. Here the NZ
Numeracy Project clearly directs and leads the way forward in the world of number.
Becoming multiplicative is a major achievement and essential for mathematical
competency. At senior levels this multiplicative understanding is developed and
applied to a wide spectrum of situations and perspectives. At Years 11, 12 and 13
these abilities are summatively assessed for qualification.
I have taken the liberty of renaming the current three NCEA awards as follows to
highlight their individual and different purpose.
Year 11: NZCEA: First qualification and assesses learning for employment.
Number, Algebra, Geometry Achievement Standard: External 3 hr: Curriculum Level
5 and 6. Three sections with three results.
Measurement Achievement Standard: Internal 1 hr: Curriculum Level 5 and 6.
Statistics Achievement Standard: Internal 1 hr: Curriculum Level 5 and 6.
Probability Achievement Standard: Internal 1 hr: Curriculum Level 5 and 6.
Year 12: NZCMA: Second qualification option to assess more advanced
learning for technical trades. Algebra, Geometry, Trigonometry, Calculus,
Statistics and Modelling. I have not developed this yet but explaining and
generalising would appear important. Development of algebraic manipulation and
modelling, applications of coordinate geometry and trigonometry, formal proof,
calculus fundamentals and the PPDAC inquiry cycle in modern statistics an
probability.
Year 13: NZCEU: Third qualification option to assess advanced understandings
for higher level trades and university study. Algebra, Calculus, Statistics,
Mechanics, Programming. This might be developed with university involvement.
Mathematics at this level is specialised with an algebraic, complex number and
calculus theme running along side applications of statistics and probability. Applied
mechanics, programming, numerical analysis and internet applications could be at
this level.
Year 11 Achievement Standards: CL 5 and 6
Value of standards is important and as most time in Years 7 to 11 is spent on
number this standard should be more valued. The following table shows my
weightings which would reflect the time involved.
Standard
Year 11
Credit
Number
8
External
Algebra
3
3hr paper
Geometry
3
no calculators
Measurement
3
Internal
Statistics
4
Internal
Probability
2
Internal
Number and Algebra and Geometry: Students will solve, explain solutions and
generalize solutions of number problems in new and familiar situations.
Students will demonstrate generalisation of patterns and number properties and be
able to model situations to solve problems in new and familiar situations.
Students will demonstrate knowledge and application of shapes and their properties
to solve problems in new and familiar situations.
Calculators will not be allowed.
Measurement: Students will demonstrate understanding and use of common
measurement systems to solve problems in new and familiar situations.
Statistics: Students will demonstrate understanding the inquiry cycle of posing a
problem, planning getting data, analysing the data and making conclusions.
Probability: Students will demonstrate knowledge of uncertain events and apply
this to model and solve problems in new and familiar situations.
ASSESSMENT QUESTIONS, Year 11
This is an attempt at writing some questions that could assess the appropriate
standard expected at Year 11. The NA, A, M, E grading has been retained with the
general expectation that:A, Achievement is solving a numerical problem or demonstrating straightforward
understanding appropriate to Curriculum level 5 with supporting evidence.
M, Merit is to demonstrate an elegant solution, or alternative solutions showing
deeper understanding or knowledge appropriate to Curriculum Level 6 with
supporting evidence.
E, Excellence is to demonstrate the abstract or general form of the problem or
evaluative thinking appropriate to Curriculum Level 6 with supporting evidence.
The implication here is that Merit and Excellent grades are a prerequisite to study at
Curriculum Level 7. This may or may not always be appropriate and would require
the discretion of the HOD.
Number Sample Question
Question 1
Part A: Scotty takes 60 minutes to drive 72 km to his work every morning.
He usually leaves home at 7:00am but today he was late and left at 7:10am.
If he arrived at the correct time describe what his average speed must have
been.
Part B: Show your working to Question 1 to clearly describe your solution.
You may also describe different solutions.
Part C: For any journey that normally takes T minutes and we are t minutes
late, what is the mathematical relationship that describes the factor by
which the speed must change?
Question 2
Part A: Twelve men can build a house in 36 days. How long will it take 9 men
Part B: Show your working to Question 1 to clearly describe your thinking.
You may describe extra or alternative solutions.
Part C: If x men can build a house in z days, how long will it take y men to
build a house?
Question 3
When Little Suzie walks to school she goes a different way every day. Thwe
diagram shows one way she goes to school.
Home
School
Part A: Work out how many ways there are of going to school.
Part B: Show clearly how you would work out the total ways for your answer
above.
Part C: In a grid with n rows and m columns how many ways are there of
getting from one corner to the other. (Only going left to right, top to
bottom.)
An hour long paper might require four or five of these questions to assure
reliable and valid result. There could be a choice of questions.
Algebra Sample Question
Question 1
You are to investigate the number of intersections or crossroads that are
made when a number of roads are made to intersect. Use the metre rulers
or sticks as your roads. Lay them across each other so that each new road
crosses all others and record the number of crossroads that are made each
time you add a new road. The diagram shows three cross roads when three
sticks are laid across each other.
Build your crossroads and record the results in a table like this.
Roads
Intersections
1
0
2
3
4
3
Part A:
(a) How many crossroads are made when 9 roads intersect?
(b) How many crossroads are made when 99 roads intersect?
Part B:
(c) Show how to solve this problem for 1000 intersecting roads.
Part C:
(c) If there are n roads that all intersect each other how many are there all
together?
Question 2
Part A: Fifteen people in a room all shake hands with one another once.
Altogether how many handshakes happen?
Part B: Show clearly how to solve this problem in at least one way.
Part C: If there are n people and they all shake hands with one another, how
many handshakes are there all together?
Geometry Sample Question
Question 1 Diagram showing an regular nonagon (or 9-sided polygon) with one
external and one internal angle marked.
Part A: Calculate the size of the marked angles.
Part B: Explain how you calculated these angles.
Part C: What is the size of the interior angles of a regular n-gon. (A polygon with n
sides).
Question 2 Diagram showing a triangle with an extended side.
Part A: Measure the size of the exterior angle and each of the interior angles of the
triangle.
Part B: Explain any relationships you know between an exterior angle and the
interior angles of the triangle.
Part C: Prove the Exterior Angle of a Triangle Theorem.
Measurement Sample Question
Question 1: A box, a cone and a cylinder all have the same base area and the same
height. The base of the box measures 30mm by 40mm and it is 70mm high.
Part A: What is the volume of the cylinder?
Part B: Explain how you calculated your answer.
Part C: Which group of shapes have the same volume if they are same height?
Statistics Sample Question
Question 1: Maddie the mathematics teacher thinks more students arrive late to her
class after they have been to PE than any other class. (Data cards of lateness to
classes available).
Part A: What would Maddie need to do to see if she is correct?
Part B: Explain what you would do with any data or information she collected. Data
cards?
Part C: Write a letter to the PE teacher about the problem.
Probability Sample Question
Question 1: Loopy makes up a game for mathematics. He says you can choose
between tossing one die choosing a side number or tossing two dice and calculating
the difference of the top numbers. The winner is the one with the smallest score.
Part A: Play this game and determine which is the better option to take.
Part B: What is the sample space for each option?
Part C: What could be done to rules to make the game fair?
Standard Checking and Question Reliability
It is all very well setting standards but without student trial assessment data to
check we are actually on task this whole system is all nonsense. (I have a good
article on “nonsense” from “The Tao of Physics” by Fritjov Capra.]
So please trial these assessments with a range of students and report results. Write,
publish, share, trial and assess your own questions. We must set and describe an
appropriate standard.
The above information and descriptions are entirely a developmental process and in
no way involve any final decisions. They are for teachers dialogue only at this stage. As
more information evolves and national agreement concentrates I am sure some
definite direction and purpose will meaningfully develop.
Jim Hogan, 12/6/08, [email protected]
Sample Student Answers to Number Question 1.
Question 1
Part A: Scotty takes 60 minutes to drive 72 km to his work every morning. He
usually leaves home at 7:00am but today he was late and left at 7:10am. If he arrived
at the correct time describe what his average speed must have been.
Part B: Show your working to Question 1 to clearly describe your thinking. You may
describe extra or alternative solutions.
Part C: For any journey that normally takes T minutes and we are n minutes late,
what is the mathematical relationship that describes the factor by which the speed
must change?
Student A
The new speed is 87km per hour average speed. (A)
This is best described by using a double number line.
0
? = 12.5 old speed
72
0
10
60
0
?= 12 new speed
72
so ? = 75 +12 = 87
0
10
50
60
Alternatively we could increase the 72 by a factor of 60/50. What was done in 60
minutes is now done in 50. Hence x 1.2. (M)
If we reduce the time by n minutes then we must increase the speed by a factor of
T/(T-n) of the speed.
For example, in the above case the new speed is 72x60/(60-10). (E)