21st Century Competencies in the Mathematics Classroom
Restricted – for Reference by Staff of MOE Schools 1
Professional Development Guide
for
21st Century Competencies in the Mathematics Classroom Video
This video outlines an example of how the 21st Century Competencies, specifically
Critical and Inventive Thinking (CIT), enhanced by Self-Directed Learning (SDL)
processes, are deliberately developed in a Primary 3 Mathematics classroom. (A
brief description of 21CC, as well as SDL and Collaborative Learning (CoL), is found
in Annex A. The unit plan can be found in Annex B.) As you watch the video, you will
notice the following strategies being used by the teacher:
1. Use of thinking routine: Think-Puzzle-Explore
Generates students’ interest
Allows students to take ownership of their learning
Develops students’ information literacy skills
2. Use of 5Ws and 1H
Guides students to formulate questions
3. Use of ICT
Facilitates the gathering of different viewpoints
4. Use of thinking routine: “What makes you say that?”
Prompts students to:
i. Describe what they see or know
ii. Construct explanations
iii. Exercise evidence-based reasoning
5. Use of thinking routine: “I used to think… Now I think…”
Facilitates reflection on students’ thinking process
Helps students recognise cause-and-effect relationships
Encourages students to learn from each other
6. Use of Project Work
Facilitates transfer of learning in a new and authentic situation
7. Use of Presentation
Develops communication skills
Assesses students’ understanding
21st Century Competencies in the Mathematics Classroom
Restricted – for Reference by Staff of MOE Schools 2
As you watch the video, take note of…
How the strategies presented in the
video apply to your classroom
Ideas with which you agree or
disagree
Share your thoughts and ideas with your discussion group.
After watching the video once or a few times…
What connections do you draw
between the teaching & learning
practices seen in the video and your
own teaching experience or what
you already know?
What new ideas did you get that
extended or broadened your
thinking in new directions?
What challenges or puzzles have
come up in your mind, which were
triggered by the ideas and
strategies presented?
Adapted from Thinking Routine “Connect-extend-challenge” Source: Ritchhart, R., Church, M., Morrison, K. (2011). Making Thinking Visible. San Francisco, CA:
Jossey-Bass.
Questions for discussion within your group:
Strategies to Develop Emerging 21CC through Mathematics Lessons
1. The emerging 21CC that the teacher aims to develop through the lesson is
Critical and Inventive Thinking (CIT).
a. What are some strategies you have used to develop CIT in your
current classroom teaching & learning practice?
21st Century Competencies in the Mathematics Classroom
Restricted – for Reference by Staff of MOE Schools 3
b. How do / can you create a classroom environment that is rich in
thinking?
c. How do / can you as a teacher model critical thinking when facilitating
classroom discussions?
Use of ICT
2. ICT was used in this lesson to collect data from the class to generate bar
graphs.
a. (for Mathematics teachers) Which ICT tool/s would you choose to
teach this topic and why?
3. ICT was also used to capture students’ thinking and reflections, thus making
students’ thinking visible.
a. What ICT tool(s) would you use to document students’ thinking? What
are the advantages of doing so?
b. How would you use the reflections written by students to inform you
about their learning?
Discipline-Specific Skills
4. One of the objectives of this lesson was for students to learn the following
discipline-specific skills:
Read scales Read and interpret bar graphs in both horizontal and vertical forms Complete a bar graph from given data Solve problems using information presented in bar graphs Read, interpret and present information in written, graphical, diagrammatic
and tabular forms (in the first lesson, students will only be able to interpret information)
Use mathematical language to communicate mathematical statements accurately and logically
a. Do you have an alternative way of teaching these skills while
developing 21CC in your students?
b. To what extent, in your view, are the featured lesson activities effective
in achieving the intended objective of helping students develop the
abovementioned skills?
21st Century Competencies in the Mathematics Classroom
Restricted – for Reference by Staff of MOE Schools 4
Next Steps
You may wish to use the following learning plan to plan your next course of action
after this PD session.
Questions Responses Next Course of Action (If Applicable)
1. What is the immediate next step you need to take?
2. What further information do you need?
3. What specific actions will you take to get there?
4. What specific support will you need?
5. What evidence will you look for to demonstrate growth and progress in your teaching?
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and CHIJ St. Joseph’s Convent (2012) A-1
Annex A
Introduction to 21st Century Competencies (21CC), Self- Directed Learning (SDL) and Collaborative Learning (CoL) To help our students thrive in a fast-changing world, MOE has identified
competencies that have become increasingly important in the globalised world of the
21st century. These competencies will enable students to tap into rich opportunities
present in the emerging digital age.
Figure 1 – Framework for 21st Century Competencies and student outcomes
As reflected in figure 1, the framework is anchored in core values that shape the
beliefs, attitudes and actions of a person. The middle ring signifies the Social and
Emotional (SE) competencies which are necessary for students to recognise and
manage their emotions, develop care and concern for others, make responsible
decisions, establish positive relationships, as well as to handle challenging situations
effectively. The outer ring of the framework represents the emerging 21CC
necessary for the globalised world we live in. These are: Civic Literacy, Global
Awareness and Cross-Cultural Skills; Critical and Inventive Thinking; and Information
and Communication Skills.
SDL and CoL can be seen as learning processes that students engage in, through
which they acquire essential 21st Century Competencies that lead to the
realisation of the desired outcomes of education. In SDL, the student exerts effort to
gain knowledge and skills, both for personal development and to explore ways to
contribute to his community. Students engaged in self-directed learning will exercise
ownership of, as well as manage, monitor and extend their learning. In CoL, students
are engaged in social interactions that help them achieve better understanding about
a concept or phenomenon, or to co-create a piece of knowledge or solution to a
problem. Students who learn collaboratively will engage in effective group processes
and be committed to both individual and group accountability for their learning.
Figures 2 and 3 provide detailed explanations of the constructs of SDL and CoL.
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and CHIJ St. Joseph’s Convent (2012) A-2
Figure 2 – Constructs of SDL
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and CHIJ St. Joseph’s Convent (2012) A-3
Figure 3 – Constructs of CoL
Both SDL and CoL enable students to acquire skills and competencies to learn
and work effectively in the 21st century knowledge-based environment. ICT provides
many new possibilities to facilitate SDL/CoL, e.g. students could systematically
search for information and expert help, and use various online tools to synthesise
and represent the data (SDL), or engage in a shared task that is constructed and
maintained by and for the group (CoL).
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-1
Annex B
Math Lesson Plan
Subject: Mathematics Level: Primary 3 Topic: Bar Graphs Duration: 60 minutes per lesson
Desired Results
(for the entire unit)
Established Goals (from syllabus documents):
Acquire the necessary mathematical concepts and skills for everyday life, and for continuous learning in mathematics and related disciplines.
Develop the necessary process skills for the acquisition and application of mathematical concepts and skills.
Enduring understanding
Students will understand that…
Bar graphs are used to represent data and information in everyday life.
Essential Questions:
Why do we use graphs to represent a set of data? How does a set data get represented in graphs?
Knowledge
Students will know that….
Graphs are used to represent a set of data or information.
Graphs can be presented in both horizontal and vertical forms.
Skills
At the end of the lesson unit, students will be able to…
(Subject-specific skills)
Read scales Read and interpret bar graphs in both horizontal and vertical forms. Create a representation of a bar graph from given data Solve problems using information presented in bar graphs
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-2
Read, interpret and present information in written, graphical, diagrammatic and tabular forms (in the first lesson, students will only be able to interpret information)
Use mathematical language to communicate mathematical statements accurately and logically
(Critical and Inventive Thinking Skills) in particular to this lesson
Explain his or her reasoning and decisions explicitly and precisely using mathematical language to demonstrate understanding of the concept of representation using bar graph (CIT 2.1a)
Sequence Resources /
Materials
Evidence of 21CC
(CIT)
Evidence of SDL
Lesson 1
(15 min)
Teacher begins lesson by introducing students to the concept of bar
graphs by priming students’ prior concept of graphs when they learned
picture graphs
Discuss the concept of ‘graphs’ as a class
- Link picture graph (learnt in P2) to bar graph - Real world examples (ACE pedagogy) - “Why are we studying graphs?” - “Why are graphs important?” “How are graphs useful?” “What are
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-3
Sequence Resources /
Materials
Evidence of 21CC
(CIT)
Evidence of SDL
the advantages of graphs?” - “Where are you likely to see graphs?” (this could be something for
students to find out over the week)
SDL: Ownership of Learning
(10 – 15 min)
Teacher asks students to do a T-P-E (Think-Puzzle-Explore) (SDL:
Ownership of Learning) on Google Form and tabulates the students’
responses using word cloud. Students are to fill in the ‘T’, ‘P’ and ‘E’
parts.
T: “What do you think you know about graphs?”
P: The teacher may need to give an example of a question first – e.g. “I
wonder what graphs are used for?” Then from there, the students would
know what ‘puzzling’ questions are and can generate their own. The
teacher also needs to clarify what ‘puzzle’ means, to ensure everyone is
on the same page
“What questions or puzzles do you have?” Push the thinking further by
inviting the students to wonder more about the topic with additional
questions, such as “What would be interesting to investigate and learn
more about? What are you wondering about? Are there things about this
topic about which you are curious?”
Thinking routine: T-P-
E (Think Puzzle
Explore): Setting the
stage for deeper
inquiry
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-4
Sequence Resources /
Materials
Evidence of 21CC
(CIT)
Evidence of SDL
E: “How can we explore these puzzles?” Ask students to identify a puzzle
from the list, or at times teacher may want to highlight a couple of puzzles
for the class, and ask students how the class or an individual might
explore those puzzles further. “Whom might you ask? Where could you
get further information? How would you frame your search key words?
What sources would be worth tapping? How could you find ways to
answer your own puzzles?”
Teacher gathers a few responses from students and shares them with
class. Teacher takes notes of students’ questions and refers to them and
addresses them as he teaches the lesson on bar graph.
Lesson 2
(30 min)
“How many of you have made a decision today?” or “How many of
you have made a decision in the past week?”
Teacher to ask students this question to get them thinking about what
decision making is, and help them realise that they make decisions every
day. This will set the stage for the lesson as they will be engaged in skilful
decision making.
Task:
From 21CC Standards
and Benchmarks for
P3:
The student is able to
explain his/her
reasoning and
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-5
Sequence Resources /
Materials
Evidence of 21CC
(CIT)
Evidence of SDL
Teacher to prompt students to recall previous lesson on graphs and
set the context for the task to be completed on this lesson on bar
graph
“Your class is going to organize a games day for all primary 3 students in
the school. However, we can only play 3 games on that day. How can
you, as organisers, decide which 3 games to play? How can you find out?
You will need to explain your decision to your teacher and convince him
that the 3 games you chose should be played on games day, explaining
the criteria (CIT2.1a) you used to decide on the choice of game ”
Class discussion( teacher facilitates this)
Understanding the task
Teacher asks the class what they understand of the task. “What do you
think we are supposed to do here?” Teacher guides the class in the
understanding of the task - students have to make a decision on the
choice of games for games day.
Teacher facilitates a class discussion for coming up with a few criteria for
helping to choose the games to be organised. “What kinds of criteria
1:1 computing
Show the challenge
statement on
powerpoint slide or
write it on the board.
decisions. (CIT2.1a)
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-6
Sequence Resources /
Materials
Evidence of 21CC
(CIT)
Evidence of SDL
should we use to decide on the choice of games?”
(Some examples: most students play this game (popularity), availability
of facilities/ equipment for the game, safety, time, team/individual games
etc.)
Deciding on the method for collecting information/ data
Teacher asks the class what kinds of data the class would need to gather.
“What kind of information do we need to help us decide?”
Teacher asks for suggestions on the methods for collecting the data
needed.
Students to submit their suggestions on Google Form which will be
collated and presented on Wordle.
Teacher acknowledges all the suggestions in a word cloud (“We have
many ideas from the class such as…”) and discusses the feasibility of the
more popular suggestions (words in bigger font) with the class.
“Do you think this is a good way of collecting the information we need?”
Google Form
Wordle
(www.wordle.net)
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-7
Sequence Resources /
Materials
Evidence of 21CC
(CIT)
Evidence of SDL
“How else could we have done it?”
The teacher looks at the ideas on the interactive whiteboard and
facilitates the class in evaluating the effectiveness of each idea with the
help of a graphic organiser (Appendix B1). Teacher may wish to
explain the concept of pro/con to the class at this point. Teacher
proceeds to participate in weighing the effectiveness of the ideas and
coming to a decision whether to accept or reject the ideas. For each
pro/con, teacher puts a +/- sign against the idea. (CIT2.1a)
“What do you think of idea A? Why? Why not?” (Listing pros and cons)
“Student X, what do you think of idea B?”
“Student Y, how about idea C?” and etc.
“Looking at all the + and – signs next to the ideas, which do you think is
the best way for collecting the information we need?”
Data collection
Teacher suggests gathering students’ favourite sports by collecting data
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-8
Sequence Resources /
Materials
Evidence of 21CC
(CIT)
Evidence of SDL
using an online tool (Google form) as opposed to pen/paper survey.
(Teacher may elicit some advantages of using online survey from the
class. Some reasons are: online survey saves time, students can do this
at their own time, online survey platforms also collates the results for you)
-> possible teachable moment for CIT 2.1.
At this point, the class would have decided on conducting an online
survey. The teacher will suggest to the class to conduct the survey with
their class first.
Teacher will ask the class to brainstorm on the survey question. E.g.
“What is your favourite sport?” Teacher suggests using Google Form for
gathering of data.
Teacher will set up the survey on his computer that is projected to the
class screen. Since the class makes use of 1:1 computing, once the
Google Form is completed, the students will go in and enter their
responses. The class will come together and watch the teacher collate the
responses into the table (this is auto-generated by Google).
Google Form
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-9
Sequence Resources /
Materials
Evidence of 21CC
(CIT)
Evidence of SDL
Teacher pauses at this stage (table should be shown on the screen)
and asks a few questions. Showing the table on the screen provides the
advantage of enabling the students to see the process of collating
responses and the teacher can easily point to the students the top three
choices.
“What do you think of this table?”
“Is it easy to read?”
“Is there another way of presenting this information such that it is easier to
read?”
Students respond to the questions. Teacher probes to make visible the
thinking behind students’ responses by asking “Why?” or “Can you
explain why presenting the information this way is better than the other
way?”] (CIT2.1a)
Teacher may want to point out to the students that while there are many
ways to represent the data, they need to take into account time efficiency
in representing the data at hand.
At this point, the class would have agreed to present the information in
the form of a graph or a visual diagram. (Note: It is important at this
point to point out that while graphs and visual diagrams serve the function
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-10
Sequence Resources /
Materials
Evidence of 21CC
(CIT)
Evidence of SDL
well here, other forms of visual representation will serve the function
better for other contexts. This is so as not to discount the generation of
alternative ideas by other students.)
Teacher clicks on the option to churn a bar graph from the table of results.
The screen should show the bar graph now. Teacher asks questions.
“How is this different from a table?” (visual diagram as opposed to
numbers)
“How do you read this graph?”
“We will be learning about bar graphs today”
Sound reasoning CIT
2.1a
Students are
encouraged to reason
and explore the best
way to represent the
information on the
screen
(15 min)
Students discuss the graph with their group members
Task:
From the graph, focus on the information you need to determine the three
sports to organise for sports day. (You may look at the most popular
sport, least popular sport, comparison between 2 sports, etc.) Decide, as
Sound reasoning CIT
2.1a
Thinking routine:
“What makes you say
that” (Interpretation
with justification)
ICT-facilitated CoL,
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-11
Sequence Resources /
Materials
Evidence of 21CC
(CIT)
Evidence of SDL
a group, which 3 sports to organise for games day and explain your
decision. Type out your discussion in the online forum or blog.
Teacher gives out a graphic organiser (Appendix B2) for students to fill in
based on the criteria they choose to decide on the sports to organise.
Question for students to ask one another in their groups:
“What makes you say that?”
Questions have to be clearer so as to frame the context for students to
interpret the graph
10 minutes
Teacher to call one or two groups to share their discussion and decision
with the class.
- Some questions for teacher to keep in mind. Did anyone not respond to the survey? (concept of total number)
- Which is the least popular sport? How about the most popular sport? What makes you say that?
- How many more students prefer Sport A to Sport B? What makes you say that?
- Which three sports should we organize for games day? - What makes you say that? (Asking questions so as to make
students’ thinking visible) (CIT2.1a) Teacher may wish to pose
SDL.
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-12
Sequence Resources /
Materials
Evidence of 21CC
(CIT)
Evidence of SDL
criteria to help students in their decision making process (e.g. availability of infrastructure, time constraint and popularity of the sport), in so doing capitalising on the possible teaching moment that the most popular decision may not always be the best decision for everyone
- What do you notice about the bars itself? (equal width) If students ask why they are equal, we’ll leave that as an ‘E (Explore)’ question at the end of the lesson.
- Is there another way of presenting the bar graph? (horizontal and vertical or in 3D)
(10 min)
Reflection
Teacher asks students to pair up (think-pair-share) and do some
reflection (thinking about thinking) questions:
“Was this a good way of gathering the information (data) we needed to
organise games day?”
“Is the information gathered good enough for us to determine which 3
sports we should organise for games day?”
“What else may we need to find out?
“Was the information accurate?”
Students are to enter their reflections as well as what they have learned
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-13
Sequence Resources /
Materials
Evidence of 21CC
(CIT)
Evidence of SDL
about graphs today in the online platform, Wall Wisher.
During this time, teacher should look through all the Ps (T-P-E) and group
them into a few categories and ask the class to choose which of these
areas they are interested to find out more about on their own at the end of
the class.
Conclusion
Teacher concludes the lesson by going through students’ reflections on
Wall Wisher to help students distil learning points. Teacher also highlights
some of the ‘P’ and ‘E’ from students’ T-P-E and poses thinking questions
for students think further. Teacher also directs students to resources to
allow them to explore the topic further.
Lesson 3: Project work (Lesson Extension) (SDL: Self-managed
learning)
(10 min)
Teacher to divide students into groups of 4. In their groups, the students
are to design a survey form using Google Form, to better understand their
classmates’ opinions on the groups’ chosen issues. Teacher to present
some examples of possible questions that students can work on, such as:
How do students travel from home to school every day?
What subjects do students like the best?
What are the popular hobbies of students?
SDL: Self-managed
learning.
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-14
Sequence Resources /
Materials
Evidence of 21CC
(CIT)
Evidence of SDL
(10 min)
Students are to design survey forms in their group using Google Form to
try out this data collection process with their classmates. Teacher to guide
them by providing the benchmark of good survey questions (encapsulated
by the “CFC” guidelines – Clear, Focused, Concise). The whole class
should be surveyed.
(10min)
Students to complete the respective surveys in 10 minutes before
resuming group discussion.
(30 min)
After the 10 minutes have lapsed, in their groups, they are supposed to
present the information they have collected in the form of bar graphs and
present their findings to the whole class.
Teacher and classmates to provide feedback on the group presentations.
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-15
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-16
APPENDIX B1
DECISION MAKING GRAPHIC ORGANISER 1
CONS OPTIONS
PROS
- - - + + +
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-17
APPENDIX B2
DECISION MAKING GRAPHIC ORGANISER 2
OPTIONS
CRITERIA FOR CHOICE OF SPORTS
SAFETY POPULARITY AVAILABILITY OF
EQUIPMENT
STUDENTS’
READINESS LEVEL
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-18
Suggested Assessment Tasks
Suggested Performance Task(s):
Authentic task
(Individual work) Students present the food items they have in their
home refrigerator in the forms of a table and a bar graph. They
should include a short write-up of their work as well as, either:
Give suggestions for adding or removing food items to ensure healthy diet for their family. or
Describe a dish that can be prepared with the food item in the refrigerator and explain the cultural significance (e.g. Chinese, Malay, Indian, Hokkien, Teochew, Thai, etc.) of the dish. (CGC 3 – Displays socio-cultural awareness and sensitivity)
Teacher may ask students to present their work to the class as an
oral presentation, or put up their work on an online platform where
Other Evidence of Learning:
(E.g. quizzes, tests, academic prompts, observations, homework,
journals, etc.)
Collaborative group work: Every student is to think of a question that they would like to know about their classmates. In the next lesson, they will form groups of 4-5 and share their questions with one another and think of presenting their information as a bar graph. They should consult their teacher with their ideas. They will then gather the data from the rest of the class and present their findings as a bar graph. Some examples of the questions are, “Do you have a pet? Y/N” and “How many siblings do you have?” There can be either Yes/No answers or answers with varied responses. The groups will have to discuss with their teacher how best to present the information.
Homework (formative) “How are graphs relevant to us?” Share some examples with the class next week, with interpretation of the graphs. (ICS 2.1a)
21st Century Competencies in the Mathematics Classroom
Jointly developed by CPO, CPDD, ETD and St. Andrew’s Junior School (2012) B-19
classmates can also view and enter their comments.
Rubrics: Teacher should design appropriate set of rubrics to assess
the task. Rubrics should be given and explained to the students
before they begin their work. If presentation is part of the work, oral
presentation skills could be one of the assessment criteria.
Homework (formative) “Where are we more likely to find graphs in everyday life?” Share some examples with the class next week and explain the use of the graph. (ICS 2.1a)
Journals (formative) Students write down what they have learned about graphs (K-W-L) in the online journal
Self-assessment
Worksheets