Learning Issues

Assumption: Pupils are able to solve (knowledge qns?) discrete routine questions. Higher order qns involving rearrangement of shapes/lines in composite fig requires more scaffolding (be more specific about the challenges)

non routine, multiple solutions to a single problem, non-linear prescription as a pedagogy

Analysis:

Pupils may think that they will have to find the areas of the different figures, rather than rearranging the figures.

Pupils launch directly into the procedural aspect of problem solving without applying (1) reasoning (2) thinking, e.g. identifying patterns.

Pupils do not venture into alternative methods of solving as they are too used to using a single method. (fixation caused by procedural teaching)

Conditions of
Proposed Solution:

Understanding beyond formula (reasoning)

Developing habits in pupils not to jump into procedural approach. (Metacognition - discuss, think aloud & reflect)

Need for pupils to articulate their thinking and how they solve the problems.
Sharing of the process allows them to learn and discover the different ways of solving (Metacognition - discuss, think aloud & reflect).

Spatial visualization skills:

Observation in pupils who are able solve:
1. Ability of pupils to simplify the question.
2. Ability of the pupils to visualize the different shapes

Role of ICT

How shapes interact through the use of ICT, but at the same time allowing them a platform to also pose questions, craft questions and share their reasoning.

Circles

short cut using technology (GSP)

allows you to have various diagrams

students focus on analyzing data rather than calculations

discover relat

Assumption:

Analysis:

Conditions of Proposed Solution:

Pupils view a percentage as a number rather than part of an amount (whole)

conceptualization issue

Conditions of Proposed Solution:

Lower ability: tackle conceptual understanding

Foundation kiids

Subtopic

issue in solving word problems involving discount, GST, interest and respective BASE to use.

conceptualization issue

Conditions of Proposed Solution:

Word problem using conventional base. Transitional activities: tightly contextualized, highly relevant to students' daily lives/experiences (authentic learning)

Assumption: procedural knowledge is not an issue

Going back to fundamentals: concepts (abstract) embedded in contextualized activities linked to students' experiences

Role play: Shopping with the aim of getting the best discount with a designated amount of money

Foundational students

Meaning making of percentages

Repeated practice will potentially lead to improvement

Application of Percentage in real life

Conditions of Proposed Solution:

Understanding terminologies such as discount/GST and relationship to content of maths lesson

Use of multiple situations

Ability to transfer knowledge exhibits understanding

Case base pedagogy

multiple small scenarios/ iteration of knowledge to help students see patterns (easy-->difficult conversion of bases)

Identifying the BASE to use

Contextualizing the question

When the base is non-conventional (eg: 5/7 % of something), students get confused. See numerical quantity of the % at its face value

Percentage is a system for comparison using the baseline of 100

"OF" (indicate multiplication) the "cost price"

"OF" implies a derivation from something

Lesson Implementation (6th Aug)

Lesson Objective

Examine their thought processes as a class

Cyclical approach

Increasing level of difficulty

Word problem. Varying level.
Part whole relationship.

Lesson Progress

Compare and contrast their problem solving process

Platform:Mimio (in house platform)
Restrictive because one page at a time. Varying handwriting size and also challenging to write with stylus

2-3 groups used model to solve. Others used percentage

Deconstruction shown in their working

Feedback

Twiddler

Online whiteboard space

Teacher explored and find it challenging

Computational error

Misread question

Transfer error (from questions)

Conceptual error

Challenges

Unable to recognize the place value of tenths, hundredths etc.

Unable to see decimals relating to fractions

Concept of place values

Concept of tenths and hundredths

Relative value

Proposed solution: audio heuristics

Pupils launch directly into the procedural aspect of problem solving without applying (1) reasoning (2) thinking, e.g. identifying patterns.

Hinders their ability to tackle non routine questions

Application of Decimals in real life

Tools to be used in teaching

IWB flipchart

Applets

SSM resources

screencast for voice up

class blog

different representations

use cubes, coins

Proposed solution

Tapping on prior knowledge

Money

Challenges: syllabus order.

Help them understand place values.

How to link one to tenths?

Linking money to decimals

Challenges: form/structure is similar but underlying concept is different.

Both money and decimal's numeration system includes base 10.

Broaden the decimals, approx

How to anchoring of the topic--

How is it connected to decimals?

Linking to Fraction

Use concept of fraction as bridge to representation in decimals (parts vs whole; equivalence)

Number line

to give them a sense of a quantity

Using a benchmark to trigger students' reasoning (more/less)

making use of the prior knowledge of the mathematical words- concrete, picttorial,e.g. 7 tenths, 7/10, place values

Which number is bigger, 7/10 or 0.75? in the number line/class blog

Students' understanding of mathematical thinking/reasoning/depth in conception more than mathematical representation

Learning reveals itself through actions. Dependent on experiential learning (prior knowledge).

leverage on students' prior knowledge (eg: fraction) to help students best relate to the topic of interest

Help students understanding the importance of decimals in their lives

Rationale for variation within the numerical system: we need smaller denominations when we use money

Direct instructions

teacher-directed.
room for student practice in differentiated instructions. teacher-modelling, teacher as a facilitator.
give cyclical practice opportunities (repeated practice of sense making) to help them understand the concept.
modelling and telling-->student practice; teachers as facilitator, teacher gives feedback--> consolidate as a class

Problem based learning

Deconstructive activities

Rationale: Able to tap and build on prior knowledge

Teacher scaffolding

Inquiry based learning

Process and thinking- focused. Provide a platform for students to represent their thinking and discourse. After which, work with students through constructive feedback.

Duplicate CoP (among teachers) in the classroom with students

Experiential learning

Multiple representations

Cubes

Anchored by understanding of fundamentals such as number lines etc to diminish fear of progressing forward

Number Sense of the size of the quantity is very important

use of the number line

Then we lead to the 4 operations

To hear the articulation in the class blog about the no sense

What is common about this numbers? 0.7 or 0.75 ( pictorial/ % ) to hear the differnet representation.

To be able to tell us the next learning objective

pupils must know place values

pupils must know what are the jargons

pupils must know the literacy part- hundred/hundredths

pupils must know the comparison concept

PCK-Case base learning- pupils articulate how to work out and then come out with the generalisation- reason using the modality

PCK- Pedagogy

PCK-multiple representation

TPACK- no line in excel

1. no line ( manipulate on online) , 2. base 10 blocks ( pictorial or concrete=the same)

role of the technology for the idea formation i.e. it has to be articulated

need to document in the blogs- inquiry and no sensing based on the multiple cases/ qnsmaths rationalisation- to help in the deconstruction,

reciprocal teaching

PCK-maths rationalization with use of blogs

ZPD

Note: 2/3 periods-Teachers must explore the base 10 block and no line online-or to construct in excel. confirmation of blog design ( if we do reciprocal teaching)

no sense-place value

mulitple cases/ qns e.g. case 1: craft qns common cases and misconceptions then come up with generalisation

pupils must show no line / base 10- they must see and the growth of two number lines, no blocks

Pupils must be able to articulate 0.17 + 3 = 17 tenths + 3 ones i.e. 3 ones is bigger than 17tenths.

Pupil must do a print screen when they do the no line when they do the comparison during the comparison a+b, c-d etc--> have 2 no line

Suggested Lesson (Introduction to P4 Decimals)

Prior knowledge: Money sense.
Aim: How to bridge the gap linking it with
prior topic on fractions, tenth place value for
decimals.
Strategy: daily experiences relavant to students
Share on class blog.

Challenges (content knowledge): place value, not able to convert fraction to decimal. use prior knowledge on fractions to transit to decimals. confusion with relative size due to expression of fractions in words (eg: thousandths etc)

Pedagogical knowledge (existing):

Radical method-Teacher posed on the qns

another PCK

Design for failure to happen

after which, give students experience to play with different ideas to lead towards the correct conception

Suggested lesson plan

phases

conducted in 3 classes

HA, MA, LA

similar problem sums across the 3 classes

3 questions of increasing difficulty

first question to boost confidence

Concerns: time consuming

Each group 1 question

groups of 3

randomly assigned

use of ipad app for explanation

problem solving process will be recorded

IT platform (software) allows recording of students' scribbles, audio and thinking process

Concerns: pen and paper first? Every group given ipad with question embedded. write on ipad and explain. recorded on ipad simultaneously. later link to projector for presentation.

can import it to youtube/email back. look at platform
via askNlearn/myCloud so that they can submit their
work to us

recording can be edited at any point. dont have to start from
the beginning

student presentation to class

to surface misconceptions

What is the anticipated misconceptions?

Qn2: conveniently think of a concept that can
best solve the problem. will not check for accuracy based on reasoning

group discussion precedes presentation by one group
member to the class

within that group, students may teach
each other and get the problem right.

share ideas and learn from each others' mistakes

Feedback

peer assessment

more vocal students can give constructive feedback

learning as a class

TPACK:
use youtube/LMS to exploit mobility for further analysis

Concerns: platform for students to showcase thought processes?

Free app

Screen Cast Omatic.
Allows teachers to select any screen of choice on the
laptop. Play, record, over ride. Can publish it out on the
website. Similar function to what was suggested

Lesson Reflection (6th Aug)

Lesson Objective

Verbalise problem solving process in proper
mathematical language

App to capture their discourse

Use specific structure to justify their reasoning

Why agree/disagree

Understand their problemsolving process

Improve confidence. Enhance logical thinking process.
Improve the way they structure their thought processes
on paper

Learning through failure

Progress

Implemented with 2 classes in CCK

1 Ha, 1 Ma

Lesson flow

Presentation videos

P5 HA group: 50% found the answer.
P5 LA group: made assumptions that led to obstacles, brainstormed and eventually was able so solve

Feedback

Using comparisons as a form of scaffolding

Proportional reasoning

Questioning: seeing the part whole relationship

Moving forward

Developing a platform to discuss each others' thinking process/reflection

Parking the cognitive artifact

especially useful for problems with
multiple solutions

Understanding the various properties of angles & lines

Analysis:

Conditions of Proposed Solution:

Estimating Angles

Analysis:

Conditions of Proposed Solution:

Measuring Angles

Analysis:

Conditions of Proposed Solution:

Drawing Angles

Analysis:

Conditions of Proposed Solution:

Drawing Triangles

Learning Issue 1: Using a protractor

Analysis:

Conditions of Proposed Solution:

Learning Issue 2: Drawing interior or exterior angle

Analysis:

Conditions of Proposed Solution:

Geometrical Construction

Drawing Triangles

Drawing 4-sided Figures

Lesson Suggestion

Lesson Objective

Construct what they intend to construct

Practise their construction skills

Use of geogebra platform

Measurement specific

exposure to various shapes

Use of screencast-o-matric to record the process

Use of Edmodo

Upload to youtube

Use of manipulation

Feedback

Schoology

24th Sept Lesson Suggestion

padlet on composite figure, address learning issues

Thinking aloud via ICT, constructing parallelogram

sketch and geogebra reasoning

Learning Issue 1: Identifying Fractions

Learning Issue 2: Simplification

Learning Issue 3: Parts of a set

Word Problems

Learning Issue 1: Students are unable to relate ratios to real life applications

Analysis:

relationship with other topics. Fractions, percentages. Craft task to relate all 3 topics. Multi representation of ratios

At P4 level, able to leverage on decimal to see the link between ratio and fraction

Relevance to real life situations

Fraction is a pertinent issue among students

Ratio->fraction->percentages

Related representations

Big idea behind Fraction/thinking skills involved/mathematical reasoning

What is the underlying concept of fraction?

Equal parts of a whole

Derived from equal division/sharing

Comparison of 2 or more quantities

quantities are of equal part

equal parts of a whole (fraction/ratio)

Subtopic

Action needed to help students understand the concept of fraction

Technology

Manipulative methods

SSM resources

every fraction is anchored by a whole

students get hands-on experience, kinaesthetic learning

Challenges

students find it challenging to grasp abstract concept

solution: use more concrete and physical examples (concrete manipulatives). Eg: mass, money

Inability to relate to real life scenarios/Dissociation with real life

Eg: exchange rate. Students are not familiar

Solution: ratio of syrup and water needed to make drinks.

Solution: inter level connectedness between topics.

How to help students understand the link?

Relate to real life applications

Fraction(P5): added dimension- part of the same and different whole. Students find it challenging to grasp this concept

Instruction design: guide student through conversation and action to see the link via multiple representation through ICT tools. Importance of teachers' questioning.

Reasoning

Consistency of familiar words

Connection gets disrupted if discourse/instructional words between topics changes. Students may then find it challenging to draw the link moving from topic to topic

Importance of communication

Visualization

See equal parts

How to link fraction to ratio?

SSM resources

Ratio

Relative quantity

Conditions of Proposed Solution:

differentiated tasks

Potential ICT tools

Exchange rate

Excel folder using formulas

Encourage student articulation/discussion, reasoning

Authentic learning

From the point of view of the learner

Connecting communication to concepts

Use multiple representation to encourage student discussion

Help students make overt their reasoning

Leverage on ICT tools

Use of questioning

address miconception through questions

Encourage students to justify their thought processes

Learning Issue 2: Solving word problems
Students are confused to syntheise the information given i.e. could not see the relationships between concepts.

Analysis:

Conditions of Proposed Solution:

Lesson Implementation (6th Aug)

Lesson objective

Concept of relationship between decimal and
percentage. Vice versa

Use the mathematical conversion platform, guided by Cynthia.

PCK reasoning: by establishing the r/s and see for themselves that base of percentage is 100, and r/s to decimal, their knowledge and then be transferrable

Instil skill ofgenerating patterns

Assess their level of understanding. Sift out their
misconceptions and addressit accordingly

exploratory, inductive

Able to convert percentage to decimal and vice versa,
given any context

Relation of a smaller circle within a larger circle

Inability to relate ratio of the parts to the whole, hence making calculations inefficient

Suggested lesson

big semi circle with two small identical semi circle inside.
Question: find perimeter of shaded parts

Proposed solution:
1) part by part
2) short cut

Use GSP to derive shortcut

Concerns: time consuming to draw so many figures
and manually calculate. GSP enables various diagram
and auto calculate. Pupils focus on analysing the data
instead of the calculation. Gives more examples in a
shorter period of time.

Feedback

CK: Content knowledge not an issue. Concern is students not adopting the most efficient solution.

PK: Current pedagogy of teaching?

Using formula and prior knowledge

Concerns: time consuming esp
in exam context

Discover relationship and pattern through GSP

cultivate alternative way of thinking
about such a problem

cut short alot of time on calculation and help
with visualization

PK: How teachers teach this sort of problems?

Relationship between the radius and the diameter

Using TPACK to see relationships more efficiently

Without GSP, currently help them see basic shapes/patterns.
For more challenging problems, break down the question into smaller portions.

GSP

Still in midst of developing the platform

Aim is for students to discover the relationship

create a context for students to use that tool to justify
certain things.

Concerns

Make assessment. To discard existing practices?

Not the case.

Increasing the alternative problem solving methods?

Yes

Has to emerge in eventual lesson. Giving them a range
of problem solving methods. Draw on affordances of visualization, supported by technology to see the mathematical relationship. Through GSP applet, designed activity to elicit what is/is not the relationship

Cut and paste, shifting of shapes

which is more suitable?

Based on last year's PSLE circle question,
surprised teachers

cut and paste is more challenging in general

Suggested Lesson (6th Aug)

Lesson Objective

Using GSP, understanding value of pie.

Generate multiple cases to see a pattern

relationship between circumference of small circle
within big circle

Feedback

Scaffolding strategies

guide them to specific columns

Ask them to seek trends between values in
the columns

Use a platform to discuss their exploration
strategy. Hide the column and find out what
they come up with

Metacognition is important

Aim of activity is to see the relationship
between radius and diameter.

Next activity is to see the relationship
between big and small diameter for circles
within circles of varying sizes.

Learners' Difficulty

What challenges do students face?

Task Design

ICT for Maths

Finding relationships of circles within circles

Traditionally, hands on paper-cutting

Versus: ICT (more accurate values)

Use ICT to vocalize students' thoughts, hands on learning, enhance visualization skills, ritualize thinking routine

Use of GSP and Padlet

Lesson 1: Pi

Lesson 2: Circumference Investigation

Easy to replicate figures via GSP

Inability to use the most efficient method for word problem questions

Analysis:

Conditions of Proposed Solution: