Teaching As Inquiry in Mathematics
NZAMT 2011
Louise Addison
laddison@mcauleyhigh.school.nz
Ideas
Background
- My class
My plan for teaching
- Key Ideas in Maths vs Essence Statement
Teaching as Inquiry
- School goal “To promote student ownership
of learning through the Teaching as Inquiry
cycle”
Focusing Inquiry
- Development of Teaching as Inquiry goal
Teaching Inquiry
- It’s not what you do it’s the way that you do it
- Three obligations
- Cause, be and do
- Teaching for tomorrow
Learning Inquiry
Resources
Flip Video
Key ideas in Maths Posters
Plan for the year
Appraisal system at McAuley High School
Focusing Inquiry Sheet
Teaching Inquiry Sheet
Teaching considerations
Observation 1 (Argumentation and Proof)
- EGG technique
- Number Proofs
- Power Point
Learning to learn
- Think Board
- Poster
- Wait time - What do good learners do
‘Revision’
- Algebra assessment
Making connections (Structure and Generalisation)
- Relationships tasks
- Geogebra files
Observation form 2
- Quadratic modelling
Work in progress
Teaching as Inquiry
1
KEY IDEAS IN MATHEMATICS
2
1
5
7
2
4
6
3
5
4
1-2
Change and variation
Students uncover stories in which variation is
omnipresent.
6
SSG
Structure and generalisation
Students unlock stories using models,
abstractions, and representations.
Weeks
Argumentation and proof
Students tell stories using evidence and
reasoning.
Term
Theme
Statistical Reasoning
Involves identifying problems
that can be explored by the use
of appropriate data, designing
investigations, collecting data,
exploring and using patterns and
relationships in data, solving
problems and communicating
findings. Statistics also involves
interpreting statistical
information, evaluating databased arguments, and dealing
with uncertainty and variation.
Algebraic Reasoning
Number involves calculating and
estimating, using appropriate
mental, written, or machine
calculation methods in flexible
ways. It also involves knowing
when it is appropriate to use
estimation and being able to
discern whether results are
reasonable. Algebra involves
generalising and representing
the patterns and relationships
found in numbers, shapes, and
measures.
Geometrical Reasoning
Geometry involves recognising
and using the properties and
symmetries of shapes and
describing position and
movement. Measurement
involves quantifying the
attributes of objects, using
appropriate units and
instruments. It also involves
predicting and calculating rates
of change.
AOs
Big Ideas
Contexts
Learning to learn
Assessment
- plan and conduct surveys and experiments using the
statistical enquiry cycle
determining appropriate variables, cleaning data, using
multiple displays, and re-categorising data to find
patterns, variations, in multivariate data sets, comparing
sample distributions visually, using measures of centre,
spread, and proportion, presenting a report of findings.
- plan and conduct investigations using the statistical
enquiry cycle
justifying the variables used, identifying and
communicating features in context (differences within
and between distributions), using multiple displays,
making informal inferences about populations from
sample data, justifying findings, using displays and
measures.
PPDAC
Variation
Within and between
distributions
Sampling effects
Variables and
measures
Informal inference
Us and them
Census @ School –
finger length
Birth weight data
Baby names
Titanic
Kiwi Kapers
The eyes have it
Using data to verify
Worry questions
Use of representations
Design process
Justify findings
1.10 Investigate a given
multivariate data set
using the statistical
enquiry cycle (4)
- Compare and describe the variation between
theoretical and experimental distributions in situations
that involve elements of chance.
- Investigate situations that involve elements of chance:
comparing discrete theoretical distributions and
experimental distributions, appreciating the role of
sample size, calculating probabilities in discrete
situations.
Distribution
Sample size
Compare expected and
experimental
Simulation
Hospital Problem
Games
Birth numbers
Baby names
Questioning
Using data
Compare and contrast
Theoretical reasoning
1.13 Investigate a
situation involving
elements of chance (3)
- find optimal solutions, using numerical approaches
- solve linear equations and inequations, quadratic and
simple exponential equations, and simultaneous
equations with two unknowns
- relate graphs, tables, and equations to linear,
quadratic, and simple exponential relationships found in
number and spatial patterns
- relate rate of change to the gradient of a graph.
Connecting
representations
Parameters
Starting points
Island Paradise
Geogebra tasks
Create the picture
Growth and motion
modelling
Use of models
Use of technology
Visualisation
Making connections
Abstraction
1.3 Investigate
relationships between
tables, equations and
graphs (4)
Generalise properties
of operations
Algebraic identities
Expressions and
Equations
EGG technique
Always, sometimes,
never
Squares and
derivations
26 x 24
Generalisation
Turn it into something I
can solve
Using a concrete
example
Using models
1.2 Apply algebraic
methods in solving
problems (4)
Vitruvian Man
Ohakune Carrot
Body
measurements
Accuracy
Using prior knowledge
1.5 Apply
measurement in
solving problems (3)
(1.1 Apply numeric
reasoning in solving
problems (4))
3-D
Triangles
Proofs
Deduction
Proof
Use of formula (2 out
of 3)
1.6 Apply geometric
reasoning in solving
problems (4)
- generalise the properties of operations with fractional
numbers and integers
- generalise the properties of operations with rational
numbers including the properties of exponents
- form and solve linear equations and inequations,
quadratic and simple exponential equations, and
simultaneous equations with two unknowns.
- convert between metric units, using decimals
- deduce and use formulae to find the perimeters and
areas of polygons, and volumes of prisms
- find the perimeters and areas of circles and composite
shapes and the volumes of prisms, including cylinders
- apply the relationships between units in the metric
system, including the units for measuring different
attributes and derived measures
- calculate volumes, including prisms, pyramids, cones,
and spheres, using formulae.
- deduce the angle properties of intersecting and parallel
lines and the angle properties of polygons and apply
these properties
- recognise when shapes are similar and use
proportional reasoning to find an unknown length
- use trigonometric ratios and Pythagoras’ Theorem in
two dimensions
- deduce and apply the angle properties related to
circles.
Level of precision
Metric system
Area / Volume
Derivation of formula
Optimisation
Modelling
Similarity
Angle rules
Geometric Proofs
Loci
Proportional reasoning
Trig ratios and PT
Revision
3
4
5
Focusing Inquiry
What is important (and therefore worth spending time on),
given where my students are at?
The focus for this teaching as inquiry cycle is going to be developing students understanding of different algebraic
representations and the links between them. This is traditionally a very challenging area for students in mathematics and I
would like to explore how the use of technological and other visual representations can best support the development of this
understanding.
Baseline data:
Curriculum Level
Algebra Topic Test
Graphs Topic Test
Algebra Exam
Graphs Exam
3
0
0
3
3
4
3
10
11
8
5
3
7
5
5
6
13
2
0
3
The above table is my class’ results from Year 10. From this it is clear most students are about levels 4/5 of the curriculum. They
also struggled to replicate class test results in the exam – this is important as both the standards for this topic are externals.
McAuley High School
National results
Standard
Title
N
A
M
E
N
A
M
E
90147
Use straightforward algebraic methods
and solve equations
19.6
45.7
34.8
0
33.9
31.5
28.3
6.2
90148
Sketch and interpret graphs
23.9
56.5
17.4
2.2
23.9
43.2
23.6
9.3
Highlighted areas show where McAuley students do better than the national average, however this is mainly because only our top
third of students sit the exam whereas nationally about the top two thirds would be sitting.
Students were also be given the practice examination at the start of the unit next term so that a baseline can be established – the
results were Not Achieved for all students in both papers. The highest number of questions students had correct was 2 out of the
16 in the procedures paper and 3 out of the 18 questions in the graphs paper. The majority of the class (15 out of 18) had no
questions correct in either paper.
This shows students have already learned:
From the Year 10 data it is clear that students have a good procedural understanding of basic Algebra skills, however they are not as
proficient at demonstrating this understanding in an exam situation. The graphs understanding of the majority of the class is at
Level 4 of the curriculum currently which suggests they have good understanding of linear relationships only. However this
understanding was not demonstrated in the pre-test and thus the teaching of linear understandings will also be necessary within
this topic.
These students need to learn:
A key understanding for the new standard in algebra is linking the three different representations of graphs, equations and tables.
In order to move to Level 6 of the curriculum work on quadratic and exponential relationships and equations is needed.
My goal for this inquiry cycle is:
To investigate the use of teaching strategies that support the development of students’ understanding of quadratic and exponential
algebraic representations, in order for them to meet the requirements of Merit and Excellence in the external examination.
6
Teaching Inquiry
What strategies (evidence-based) are most likely to help my students learn this?
In this teaching inquiry, the teacher uses evidence from research and from their
own past practice and that of colleagues to plan teaching and learning opportunities
aimed at achieving the outcomes prioritised in the focusing inquiry.
Observation 1:
I was very happy with this lesson in the way it engaged all students to work towards a conceptual understanding of
algebra. This process was very new to the students who are more used to a textbook approach to algebra. Use of the
EGG technique enabled all students to access the excellence requirements of this topic and enabled a future focus on
what mathematicians actually do. This was quite a teacher directed task and whilst most students were very engaged, a
couple did need reminding to participate fully. Over this topic I would like to develop student initiated reflection on
ownership of their learning by using a greater variety of assessment for learning strategies.
Key information / research about this topic:
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Mathematics BES
E-learning research from at Auckland University
Paper written for e-learning maths Masters level course “how does your technological knowledge effect how you
use technology as a pedagogical tool”
Te Kuaka – being the agent to cause learning
Use of instrumentation and instrumentalisation theory
Feedback article from PD
Teaching and learning opportunities I intend to trial:




Use of geogebra to build representational thinking
Use of algebraic proofs of number properties
Links to learning to learn for mathematics with student evaluations of their development in these
Use of teaching model for linear relationships to build a structure for the quadratic and exponential understandings
needed
7
1. Effective Teaching is characterised by a commitment to three
obligations:
- the obligation to cause successful learning
- the obligation to cause greater interest in the subject,
- the obligation to cause greater confidence, feelings of self-efficacy and
intellectual direction in learners
Wiggins, Grant. (2010). What’s my job? In Robert Marzano (Ed.). On Excellence in Teaching.
Bloomington, IN: Solution Tree Press. p 11.
2. The Impact of Cause on Teacher Qualities and Actions
Aitken, G. (2011). Excellence in Teaching in a Faculty of Education. IN: Te Kuaka. pp 4-6.
3. Teaching for tomorrow by Ted McCain
1. Resist the temptation to “tell”
2. Stop teaching decontextualised content
3. Stop giving students the final product of our thinking
4. Make a shift in our thinking – problems first, teaching second
5. Progressively withdraw from helping students (Be less helpful).
6. Re-evaluate evaluation
8
Lesson plan 1 – Algebraic generalisation
This lesson explores the use of algebraic generalisation to generate algebraic identities.
Key
Competencies
Number and Algebra Level 5

Generalise the properties of
operations with fractional
numbers and integers.
Number and Algebra Level 6
 Generalise the properties of operations with
rational numbers, including the properties of
exponents.
To explore algebraic generalisations of number strategies
Task
Introducing


Exploring

Assessment
Extending






Working like a
mathematician –
Andrew Wiles
example
Use of example to
explain EGG
technique of explain,
give other examples
and generalise
x + x + x + x = 4x
Working in teams of 4
to explore different
situations of Task One
Summarising key
ideas from Task One
– complete Task Two
for homework (link to
consecutive numbers)
Proof “Show that the
sum of two
consecutive numbers
is always odd” “Show
that the sum of three
consecutive numbers
is always divisible by
three”
Success
Criteria
Generalising and
representing the patterns
and relationships found in
numbers.
Aim
Essence
Statement
Generalise ideas (GI)
Use generalisations (WS)
Use words and symbols to describe patterns and generalisations (WS)
Use appropriate vocabulary to explain ideas (V)
Compare and contrast ideas (CC)
Critically reflect (CR)
Participate actively in a collaborative team or community (CT)
Curriculum
Objectives




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

Mathematics Level 7
 Manipulate rational,
exponential and logarithmic
algebraic expressions.
I can generalise from a number strategy
I can explain why an algebraic identity is always true
I can use identities to manipulate algebraic expressions
I know key algebra vocabulary and recording conventions
Pedagogy
Making connections to
prior learning and
experience:
Sometimes true / always
true / never true
Focus on why as well as
how
Accept all answers, lead
academic discussion
Learning
Use of language to
support explanation:
variable, co-efficient, LHS,
RHS
Use of algebraic recording
conventions: = sign,
identity symbol , no times
sign, coefficient first,
letters of alphabet
Learning to learn
Use of examples and
linking strategies
Use of mathematical
language
Linking number and
algebra
Facilitating shared
learning:
Observation of groups to
explore current
understandings and look
for opportunities to
develop further
Linking of factorised
expression to expanded
version.
Development of ‘rules’ in
mathematics
Working in a group
Encouraging reflective
thought and action /
Enhancing relevance of
new learning:
Supporting students to
make links, sharing of
light bulb moments
How is the variable being
used?
Link to other
representations – what
would a graph of each
‘side’ look like?
Nature of proof “Show”
Use of key ideas
The difference between
proof and example
Students can work independently to derive the common algebraic identities
Use of correct vocabulary
Simplifying, expanding and factorising tasks
Link to 1.2 assessment, expanding, factorising and proofs
9