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Frequently
Asked Questions
The following is list of FAQ's. Simply click on the question
you are inquiring about and it will take you to a detailed explanation.
FAQ ANSWERS
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For
what organizational strategies is this course material designed?
- This course is designed for use with a wide variety of strategies.
At times, you will use a whole-class discussion, while at other times
students will work in pairs, small groups, or individually. Embedded
in the Teacher's Resource lesson walk-through you will find suggestions
that will help you deal with effective groupings for your class. To
address the learning styles of all students, a variety of approaches
should be used throughout the year.
- If you feel uncomfortable with any of the suggested organizational
strategies, carefully read the discussion of management issues in
the mathematics classroom. Planning, organization, and clear communication
are key to its successful implementation.
- "Students benefit, both socially and intellectually, from a variety
of learning experiences, both independent and in collaboration with
others. Instructional settings should include varied learning configurations-whole
class, small group, pairs, and individual-as well as varied activities,
including investigations, discussions, oral presentations and projects."
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For
what types of activities are small groups most effective?
The Investigations often lend themselves to group work.
There are alternative teaching suggestions for the Investigations.
- The Chapter Project and Case Studies will often be most effectively
handled when students work in groups.
- Some of the questions in Check Your Understanding are most effectively
done when students are working in small groups. (This will be indicated
in the teaching notes.)
- You may have students work through a Focus activity in small groups.
However, you should consolidate the concepts after students complete
the Focus.
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When
is it expected that students work individually?
- Students may work through a Focus activity individually.
- Some Check Your Understanding questions are best handled individually.
(This will be indicated in the teaching notes.) You may want to assign
some of these questions as homework.
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When
are whole-class lessons the recommended method of instruction?
- The Suggested Introduction will often be handled as a whole class.
- The Review and consolidation may be handled best as a whole class.
- Discussion questions following an Investigation will be most effective
when pooling information as a whole class.
- You may choose to have students work through a Focus activity as
a whole group.
- Suggestions are made for alternatives that may involve either a
demonstration for the whole class or a whole-class discussion for
some Investigations.
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What
do I do about the student who doesn't want to work as part of a
group?
The group experience is an integral and important part of
this course. Students often learn more effectively when working with their
peers and this allows them to take ownership of their learning, thus reducing
the likelihood of forgetting information. It also offers the opportunity
for communication of mathematics, sharing thoughts of many students at
the same time, sharing ideas of strategies for solving problems, and sharing
ideas. These ideas should be explained to students so that they understand
the need for group work.
If a student is reluctant to work as a member of a group,
talk privately with him or her to determine the reason. Several possibilities
are:
- Uncomfortable in a group setting (shy, insecure, quiet student)
Suggestion: Place this student in a group with a compassionate,
gentle, and outgoing student or start with a group of two.
- Concerned that he or she will end up doing most of the work, and/or
having a lower mark than he or she is used to (strong/highly motivated
student)
Suggestion: Place this student in a group with students who
are not likely to be dominated by him or her. Give him or her a specific
role in the group. Also remind students that while they will work
many times in group situations, they will be assessed individually.
The others in the group will have little influence on an individual's
mark. Also let the students know that the groups will often change
during the year.
Give this student a specific role in the group. Make sure that students
realize that everyone in the group has to work TOGETHER to obtain
maximum benefit from the work. Show students a group work assessment
rubric prior to begining the activity. There is a sample in the Independent
Study chapter in this Teacher's Resource.
- When a student persists in working alone even when he or
she is placed in a group, or when there are not enough students
to make groups effectively, it is likely that you will need to
make sure that a specific task is assigned to that student. Other
members of the group need to be encouraged to draw each other out.
Every so often, asking questions of specific students will involve
even the most reluctant participant.
- "What do you think, Sam?"
- "Any ideas, Martha?"
- "Do you think we need to collect some more data, René?"
What makes an effective group?
Every member of the group:
- contributes to the group activities/projects
- encourages others in the group to contribute ideas
- shows a willingness to question and respond to the ideas of others
in the group and to have his or her own ideas questioned
- is willing to modify his or her ideas when provided with new information
- shows respect for the other members in the group
- supports his or her ideas with observations and facts
- listens to others without interrupting Go over these points with
the students. Don't assume that they know this.
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How
do I organize effective groups?
Decide upon efficient procedures for communicating and effecting
changes in the class format.
- For example, students could be instructed to always look at the
corner of the board for a symbol that will indicate the configuration
of desks for the class activity. A "4" might mean that they move into
their groups of four. (Names of the students in each group could be
posted on the bulletin board or on an overhead.)
- It is worth the initial expenditure of time to discuss seating
arrangements for different groups, so students will know exactly how
to reorganize the classroom.
- This type of organization would mean that class time would not
be used each period to organize and set up the learning situation.
Explain the activity with a clearly defined task and a
specific objective in mind. Make sure that you communicate each to the
students.
- Specify directions and the task before students begin. Task sheets
might be useful.
- The instructions in the Student Book are designed to provide this
direction.
Monitor groups' activities, providing guidance as needed
without undermining the groups' responsibilities.
- Model active listening techniques. Some students will want to ask
you questions and will be reluctant to listen to each other. You can
model appropriate listening skills by asking a question, listening
to the answer, and then directing other students in the group; for
example to "Think about what Jason just said. He has stated an idea
that could be very helpful."
- Provide feedback on effective group process and model constructive
responses to students' ideas. This could include the results of a
checklist that consisted of check marks for engaged/not engaged that
are placed on a seating plan for the class.
Groups of four or five seem to be most effective for many
activities.
During each group work session, roles could be assigned:
- Monitor: ensures that the overall discussion runs smoothly,
and students stay on task
- Encourager: ensures that all members participate actively
in the discussion
- Recorder(s): takes notes on what is being said during the
discussion
- Resource: coordinates distribution and collection of material,
e.g., manipulatives
- Reporter: presents the group's conclusion.
Note: All students must answer questions in their
own notebooks.
Establish a basic set of rules for learning groups. Post
them and make sure that everyone understands them. Examples of such rules
might be:
- be willing to help any group member
- ensure that everyone understands before going on o listen carefully
to each other
- ask for clarification for any point that you don't understand
- make sure that everyone participates and that no one dominates
- treat all people in your group with respect
Teachers are encouraged to spend some time researching how
to effectively implement groups. A key phrase for searching on the Internet
is cooperative learning. One site that may be of some interest
is http://www.ideal.upm.edu.my/~gansl/cl.html
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How
do I decide which students to place in which group?
Begin with an awareness of interpersonal dynamics within
the classroom. At the beginning of the year, you will need to solicit
student input to help you out. (You may want to ask students to fill out
an information form, telling you a little about themselves, their interests,
and their skills.)
- You could assign students to their groups. This avoids some individuals
"not being chosen."
- You may want to create groups randomly.
- Allow groups to form naturally.
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How
can I prepare students to work effectively as a team?
If students have not experienced working in groups in the
mathematics classroom, it is reasonable to take time to talk about the
expectations that are part of this course.
- Why they will be working in groups
- How those groups will be formed
- How they, and their groups, will be evaluated as a functioning
group and as individuals against the outcomes (observation of specific
group work as well as group work in general)
- The roles within a learning group (Make sure all students understand
the different roles. Posting role descriptions would be a good idea.)
- A signal you will use to get their attention (switching the light
on and off works well)
- The physical setting for their group work (specifically, how the
desks need to be set up for different instructional settings)
- The attributes that make an effective group (go over the basic
rules and discuss their importance)
Set up a situation in which students work in groups, then
circulate and discuss with them the observations you would be making.
This is a dry run. This will also help you see if the group is working
effectively.
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How
often should student groups be shuffled?
- As you observe the learning groups in your classroom, you will
note certain individuals who work very well together and others who
are less effective with each other. Changing groups is important for
both. It maximizes the benefit of learning by helping students work
with varying group dynamics.
- Shuffling groups too often is not beneficial to students' learning.
Unless a particular group dynamic is a major problem, it is suggested
that you allow a group time to learn how to work together effectively.
It is not recommended that you shuffle groups for each group activity.
- Partial shuffles may be most effective.
- Make it clear that these groups are not static. How often they
will be changed will depend upon their success. It is reasonable to
shuffle groups at the end of each chapter. The specific situation
may be such that you would like to shuffle groups more frequently.
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How
can I assess how effectively students are learning math in groups?
Tips on using observation as a method of assessment:
- Develop a realistic plan to concentrate on a small number of students
for a given time.
- Have specific criteria in mind.
- Focus on one or two criteria and on a small number of students
for a specific period of time.
- Use a tape recorder or video camera to record group interaction.
- Make notes or keep informal records of what you observe.
- Regularly discuss with students those aspects of their performance
that you will be observing and that will help students to develop
or improve their performance.
- Develop a checklist to help you focus on the criteria to observe.
Guidelines for the construction of observational checklists:
- There are various types of observational checklists. These include:
Mathematical Content
It is critical to remember the following when constructing
observational checklists:
- The checklists should be based on the outcomes for the course,
chapter, section, or task.
- Each item on the checklist should be clear and precise. It should
focus on specifics and be readily understood by, and shared with,
students.
- Design a short checklist with the intention of focusing on a few
specific areas, or a longer checklist with only a few items selected
for attention during a particular observational period.
- Checklists can be designed for either a single observation or a
cumulative record.
- Observational checklists should be kept and used as a vital information
source during student and/or parent conferencing.
Blackline Master A is a checklist for assessing students'
participation in groups.
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How
do I assign marks for a group assignment?
This depends on the outcome and the nature of the group
task. There are tasks for which each group member might receive the same
mark if the group achieves the desired outcome. Individuals might also
receive a mark reflecting their individual contribution and understanding.
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How
can I set up my classroom so that various groupings (whole class,
individual, or small groups) will work effectively?
Organizing your classroom so that the transition between
small group, whole class, and individual situations can be accomplished
smoothly takes some time and planning. After you have developed an organizational
plan, review that plan with students. The following tips might be helpful.
- Number the desks.
- Make it clear to students how they can determine upon arriving in
class their groups for the day. For example:
- put students' names on the desks
- give each student a card and group number
- give each student an equation to solve; the solution reveals
a group number.
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What
is my role in the classroom?
As outlined in Contexts for Learning and Teaching
in Foundation for the Atlantic Canada Mathematics Curriculum (page
31), "[the] change in mathematics instruction is characterized by not
only what mathematics is taught but also the manner in which it is taught."
The teacher has a key role in developing and maintaining an environment
in which students can develop themselves as mathematical thinkers. "They
must learn in a climate in which there is:
– a genuine respect for others' ideas
– a valuing of reason and sense-making
– pacing and timing that allow students to puzzle and to think
– the forging of a social and intellectual community."
The role of the teacher has changed dramatically. Rather
than being the authority figure who taught content and was the source
of answers, the teacher's role is one of coordinator and facilitator.
The educational goals for our students are far broader than just a finite
set of content items. Teaching students a formula and how to apply it
is no longer sufficient for meeting their needs. We are educating students
for their future, not our past. This shift from authority figure to educational
coordinator and facilitator does not mean that you will never lecture
a whole class, lead a discussion, or help students summarize an idea.
It does mean that your role has become much broader in scope.
The mathematics curriculum is more than content. The relevance
of particular content is changing daily; the larger goals of this curriculum,
such as data management, are much more likely to serve our students well.
We need to ask ourselves which of the General Curriculum Outcomes and
Key-Stage Curriculum Outcomes are being addressed by any activity and
by our roles within the activity. The development of reasoning skills,
problem solving skills, and communication skills are more important than
any particular content item.
For example, the fastest way to cover volume of a cylinder
may be to lecture and "teach" students the formula for the volume of a
cylinder, show them several examples, and then assign similar questions.
Your new role is to choose activities that encourage students to generalize
how to determine the volume of any prism or cylinder. This may take longer
but will meet more of the broader and more important goals of mathematics
education, such as inquiry and communication.
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What
management tips will help with the effective implementation of this
course?
Taking time at the beginning of the year to clearly outline
expectations and to discuss these with students is worth every minute
it takes. It is very important that you discuss with students what is
needed in order for various classroom configurations to be effective.
This will include items such as noise level, being on task, staying on
task, task being clearly defined, and so on. Apply these items consistently
throughout the year.
Decide what things need to be done by a professional educator
and take responsibility for those tasks. Delegate responsibilities such
as handing out papers, getting manipulatives, moving desks, putting scissors
away, and so on, to one person in each group/row/area of the classroom.
(For more detail on small group work see Question 2.)
Encourage students to try to answer their own questions
first, and then ask other students, before turning to you for help. This
gives students an opportunity to practise responsibility as well as to
communicate and share mathematical ideas. This is important for students
who are used to seeing the teacher as the source of all knowledge. You
may want to make a chart for your wall stating a series of questions that
students should ask themselves before they seek your help. Such a list
might read:
- Have I reread the question/directions several times?
- Have I checked my calculations carefully?
- Have I asked other members of my group/the class for clarification?
- Have I looked back in my book/notes for clarification?
Write instructions for the day on the board. If you have
the agenda for the day posted on the board when students enter the classroom,
you won't have to wait until they are all seated to explain the day's
activity and organize the room.
Organization is the key. Planning is critical. Throughout
this Teacher's Resource, there are suggestions that will make managing
a multi-faceted classroom go more smoothly.
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What
is the difference between assessment and evaluation?
Assessment is the systematic process of gathering information
on student learning.
Evaluation is the process of analyzing, reflecting upon,
and summarizing assessment information and making judgments or decisions
based upon the information gathered.
"Testing to assign grades is one of the most common forms
of evaluation. But assessment is a much broader and basic task, one designed
to determine what students know and how they think about mathematics."
(NCTM Curriculum and Evaluation Standards,
page 203)
"Assessment and evaluation are essential components of
teaching and learning in mathematics. Without effective assessment and
evaluation it is impossible to know whether students have learned, whether
teaching has been effective or how best to address student learning needs.
The quality of the assessment and evaluation in the educational process
has a profound and well-established link to student performance. Research
consistently shows that regular monitoring and feedback are essential
to improving student learning. What is assessed and evaluated, how it
is assessed and evaluated and how results are communicated send clear
messages to students and others about what is really valued-what is worth
learning, how it should be learned, what elements of quality are considered
most important, and how well students are expected to perform."
(Contexts for Learning and Teaching in Foundation
for the Atlantic Canada Mathematics Curriculum, page 35)
For example, if it is important to evaluate communication,
then the ways in which students are expected to communicate with you and
their peers must be explained. If students are to communicate through
a presentation, then a rubric for how the presentation would be evaluated
could be given to students or be developed collaboratively by teachers
and students.
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What
are some suggested assessment tools for this course?
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Assessment
component
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Description |
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Assignments
(hand-in)
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Relatively
short in length. These can be used to assess process, clarity of
thought, and organization of ideas. Students are expected to do
their very best work. Some assignments (Check Your Understanding
questions, Investigations, etc.) might be submitted by a working
group, and others by individuals. |
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Quizzes
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Relatively
short in length (10–15 min to complete). These can be used
to assess specific outcomes. |
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Tests
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Generally
taking one class period. These are major evaluations and can be
used to measure specific student achievement with respect to specific
outcomes. Typically, a test is given at the end of a unit of study.
Ideally, tests should be cumulative in nature. |
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Projects
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These
are learning activities that take place over a prolonged period
of time (usually several weeks). They are often most effectively
handled when students work in groups and the group submits a single
report or completes a presentation. Students must still be held
individually accountable. |
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Portfolio
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This
provides a student with the opportunity to show a selection of work
for teacher assessment and to self-assess growth, effort, etc. (see
Questions 24 and 25). |
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Class mark/
notebook
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This
mark is a combination of the information you collect using various
observation checklists. Checking notebooks on an ongoing basis allows
you to see that homework is done and all missed work has been completed. |
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Examination
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A
final evaluation for students' achievement across all the outcomes
in the course. |
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Reflection
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Students
reflect on learning through small group discussion and written reflection. |
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The assessment environment should reflect the instructional
environment; consequently, manipulative materials and graphing technology
should generally be available during assessments.
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How
often should I assess student progress?
- Assessment should be ongoing as a method of determining individual
progress and the need for support/enrichment or modifications to the
program. Using questioning, checklists, and observation as formative
assessment tools should allow you to assess student progress on an
ongoing basis.
- The frequency of quizzes, tests, and hand-in assignments will vary
from teacher to teacher and will serve both formative and summative
purposes. The plan for this type of assessment should be carefully
considered and discussed with students at the beginning of the course.
- The student projects will also provide ongoing formative assessment
as you go from section to section in the chapter.
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How
is writing used as an assessment technique in math?
- Written or oral responses to questions can provide a great deal
of insight into students' understanding of ideas or concepts. This
also gives them an opportunity to clarify and consolidate their understanding
of a concept and to reflect on what they have learned and how they
have learned it.
- Writing can be used at the end of a class to provide immediate feedback
regarding what the students learned during a class period.
- Writing can be used at the beginning of a class to determine how
much students recall about what they were doing during the last class(es).
- Students could place writing samples in their portfolios.
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What
is a rubric?
- A rubric is a scoring guide that differentiates, using an articulated
scale, among different levels of student performance.
- The scale on the rubric may use quantitative or qualitative scores.
- Students should be shown the rubric before starting an acitvity.
- A rubric will make your assessment process fairer for assignments,
projects, quizzes, tests, and any other techniques used. In this age
of increased accountability for teachers, using rubrics for your marking
will also provide you with a clearly articulated evaluation strategy.
It is also a useful means of making students aware of your expectations.
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How
can I design an effective rubric for marking?
There is no such thing as a perfect rubric, but you are
in the best position to construct one that meets your needs and reflects
the goals that you have for an activity. You could look at sample rubrics
before creating your own. You begin by thinking about the purpose of the
activity, since the rubric will be built around what you expect students
to accomplish.
There are two types of rubrics:
- holistic—evaluates overall impression
- analytic—evaluates elements of performance
A rubric is usually constructed with three parts:
- scoring criteria or performance categories—the points or categories
for evaluation
- criteria descriptors—these describe expectations for each
criteria
- scoring levels—the range of evaluation choices
Rubrics can assist students in self-assessment tasks. Students
often learn the most about the quality of their work when they do their
own assessment. Students can be involved in the design of the rubric.
To be most effective, students should see the rubric for
any hand-in assignment, project, or presentation when the task is assigned.
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What
role do projects and presentations play in the mathematics classroom?
Daily Presentations
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Advantages
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- Allow you
and your students to observe various approaches to a situation
- Provide
validity to all approaches
- Allow students
to construct their own understanding
- Develop
and reinforce oral communication skills
- Develop
students' confidence in their own ability to learn
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Projects
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Advantages
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- Provide
students with an opportunity to demonstrate individual or group
initiative
- Provide
students with an opportunity to use concrete materials to express
their ideas and talents
- Make it
possible for you to assign projects to reflect student differences
in ability and interests
- Can be
highly motivating for students
- Can be
a powerful learning experience for students
- May increase
self-esteem for some students, broaden their understanding of
mathematics, and broad their mathematical experiences
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It is critical that students understand what your expectations
are for a project. The Chapter Projects suggested in the Student Book
are developed throughout the corresponding chapter, and support is provided
for students (and teachers).
Below is a marking scheme for a project. In some cases,
each larger section should be broken down into smaller units.
Originality (15%)
- Are the diagrams, comments, written work, and design of the project
the student's own work?
- Does the project explore an imaginative/creative idea?
Mathematical Thought and Application (30%)
- Is there a clear indication of the student's understanding of the
mathematics involved in this project?
Accuracy (20%)
- Are solutions, answers, diagrams, and conclusions accurate?
Appearance (15%)
- Is the student's presentation attractive and neat?
- Does the presentation display an imaginative/ creative idea?
Clarity (20%)
- Would the student's explanation of concepts be understood by a reader?
- Are the diagrams, pictures, and notes orderly and clear?
Alternatively, the following analytic rubric can be used.
Score of 5
- The student is extremely knowledgeable about the topic.
- The student demonstrates in-depth understanding of the relevant
and important ideas.
- The student includes the important ideas related to the topic and
shows a depth of understanding of important relationships.
- The project is fully developed and includes specific facts or examples.
- The project is organized around big ideas and major concepts and
principles.
- The presentation is detailed and clear.
Score of 4
- The student is knowledgeable about the topic.
- The student has a good understanding of the topic.
- The student includes some of the important ideas related to the
topic.
- The student shows a good understanding of the important relationships.
- The project includes a good development of ideas and includes adequate
supporting facts and/or examples.
- The project may include some organization around important big
ideas and major concepts and principles.
- The presentation is effective, has some detail, and is clear.
Score of 3
- The student demonstrates some knowledge or understanding of the
topic. The overall answer is reasonable but may show some gaps in
understanding and knowledge.
- The student includes some of the important ideas related to the
topic.
- The student demonstrates some, but limited, understanding of the
important relationships.
- The project includes some satisfactory development of ideas and
includes some supporting facts or examples.
- The presentation is satisfactory and contains some detail, but
may be vague and contain some inaccurate or illogical information.
Score of 2
- The student demonstrates little knowledge or understanding of the
topic.
- The student may include part of an important idea, or even a few
facts, but the connections needed to form a relationship are not present.
- The project contains some misconceptions, inaccurate, or even irrelevant
information.
- The response is limited and lacks clarity.
Score of 1
- The student shows no knowledge of the topic.
- The student writes about the topic using irrelevant or inaccurate
information.
- The student recites the steps gone through by the group, providing
only summaries without comment or explanation.
Score of 0
- The student does not provide a project or a presentation.
- The student writes about something completely irrelevant to the
situation.
- Answers: "I don't know!"
If the students are presenting their projects for evaluation,
the same basic assessment criteria can be used.
- How
often is it reasonable to expect students to prepare projects?
- Detailed projects that may be presented to the class are worthwhile.
Not all projects need to be presented orally. One or two per term
would be reasonable. To provide opportunities for enrichment, you
may want to consider an additional project.
- The Chapter Projects outlined in the Student Book lend themselves
to various approaches. Encourage the use of multimedia. Different
groups may select different projects. Allowing for this flexibility
can be highly motivating for students, as well as maximizing the learning
experience for the collective classroom group.
- Once students have had an introduction to math projects, the course
will be further enriched by assigning different projects to different
groups (or even better, allowing groups to choose projects that most
interest them).
- How
can I prepare students to prepare projects and presentations effectively?
- Being clear about your expectations for a project is essential.
This Teacher's Resource contains examples of rubrics that may help
you to think about how you will evaluate projects.
- It is highly recommended that you spend time up front to design
a rubric for the assessment of a project, and then give students a
copy of that assessment rubric when the project is assigned. You might
develop the rubric with them.
- It is reasonable to allow students some in-class time to work on
their projects. One suggestion would be for 20 to 30 min shortly after
the project is assigned, followed by an additional 20 to 30 min about
a week after it is assigned. Depending on your class, you may wish
to allow a third 20- to 30-min period of project work.
- Provide support for students in planning a timeline for the various
components/steps of the projects. The Chapter Projects are set up
to encourage this approach. It is recommended that, if a project is
assigned to be handed in after four weeks, students be required to
submit certain parts of the project at the end of the first, second,
and third weeks.
- Make yourself available for students who wish to clarify their
understanding of your expectations.
- Be flexible. Expect a lot . prepare to be amazed!
- Give students a self-assessment rubric.
- What
is a mathematics portfolio?
A portfolio is a collection of student work most often selected
by the student to reflect his or her achievement over some period of time,
usually a term.
The form of a portfolio is determined by your resources
and preference:
- file folder
- three-ring binder
- electronic file
The contents of the portfolio will be determined by the
purpose of the portfolio. The purpose needs to be communicated to the
student. The purpose can be quite broad or fairly narrow. For example,
a narrow purpose of "problem solving" could include examples from the
group, individuals, or tests. Other portfolios might be quite broad in
purpose and could include a writing sample, assignments, tests, quizzes,
or a self-evaluation.
Portfolio uses include the following:
- a place where students can "showcase" their learning and their work
o evaluation of student strengths and weaknesses
- an opportunity for students to review, assess, and learn from their
term's work
- an opportunity for parent/teacher/student communication
- goal setting and self-assessment
More information can be found by searching the Internet
using the key phrase student portfolios. An excellent introduction
to portfolios can be found at http://www.adprima.com/student_portfolios.htm
- How
can I assess a mathematics portfolio?
Your evaluation plan should be clear to students at the
beginning of the course; you should prepare, distribute, and discuss a
handout describing your expectations for their portfolios.
A critical objective for the student is to evaluate his
or her progress through the compilation of the portfolio, and to write
about why certain items were selected for inclusion and how they show
growth. Self-evaluation is important for this reason.
As with many evaluation tools, a rubric is often used to
clarify expectations for compilation and assessment of the portfolio.
Each rubric needs to be designed to reflect your priorities and the specifics
of the course design.
In addition to informing students clearly what you expect
them to include in their portfolios, you need to let them know how you
will mark the compilation of their work. Blackline Master F is one rubric
that can be used to assist in your marking of student portfolios.
- How
can I emphasize thinking over memorization?
Emphasize looking at a variety of solutions to any one problem.
- Model this process whenever the opportunity arises.
- Encourage the creative process by highlighting various problem
approaches developed by different students.
- Ask open-ended questions for which a variety of successful responses
are possible.
- Recognize and value all successful first attempts.
Mark for process as well as product. Students will see what
you value by how you assess their work.
Model problem-solving skills as needed when students seem
to be struggling. Make sure students have had time to solve the problem
for themselves first before you solve the problem. It is valuable for
students to see you solving problems. Tell students that you are going
to think out loud. When you are working on a problem, say things such
as:
- "When I look at this problem, I don't know for sure just how I
am going to tackle it."
- "I wonder if a diagram would help."
- "No, that approach doesn't seem to be going anywhere . let's look
back at the problem and see if anything else comes to mind."
Value problem solving and creativity by including it as
part of your assessment scheme. Include at least one non-routine problem-solving
question on each test. Assigning regular problem-solving assignments or
adding a collaborative problem-solving component to the course examination
would serve to illustrate your commitment to the importance of problem
solving.
- Observe problem-solving skills using a checklist (Blackline Master
E). Share the results of your observations and expect to see improvement
as students get more experience and feedback.
- The Student Book provides many problem- solving opportunities. Suggestions
are made throughout this Teacher's Resource that will help you to
implement this important component of education for the twenty-first
century.
- How
can I assess problem solving?
As with all assessment, you need to explain clearly what
you are expecting and allow for trial runs to clarify your expectations.
Sharing a checklist, like Blackline Master E, that you will be using to
mark will help to clarify your expectations, and will serve as a focus
for student interaction on problem-solving tasks. You could also share
the rubric that will be used to assess student progress.
Remember that your inclusion of problem-solving tasks as
part of the assessment for this course will validate their importance.
Telling students how important problem-solving skills are for their lives
while marking only content recall will only tell students that you don't
mean what you say. Make sure that you have students present various ways
to solve problems so they can see and appreciate different ways to solve
the same problem. Remember that all strategies have validity if they work
and should receive equal value when presented.
- What
role does technology play in this course?
"Modern computing technologies have changed the way everyone
learns and works. For many years, they have served as a catalyst for reform
efforts in mathematics education. In today's elementary and secondary
schools, they can and will continue to influence how mathematics is learned
and taught, not only by making, for example, calculations and graphing
easier and more manageable, but also by altering the very nature of what
mathematics is important to learn. New problems as well as innovative
ways of investigating them now become possible.
"The introduction of computers, graphics calculators, video
technology and other technologies into the mathematics classroom allows
students to:
- explore situations with complicated numbers which previously would
have been beyond their capabilities;
- explore individual or groups of related computations or functions
quickly and easily;
- create and explore numeric and geometric situations for the purpose
of developing conjectures;
- perform simulations of situations which would otherwise be impossible
to examine;
- easily link different representations of the same information;
- model situations mathematically;
- observe the effects of simple changes in parameters or coefficients;
and
- analyze, organize and display data."
(Contexts for Learning and Teaching
in Foundation for the Atlantic Canada Mathematics Curriculum, page
39)
This program provides many opportunities to incorporate
technology into your mathematics class. Throughout this Teacher's Resource
are suggestions for the use of technology, and there is a graphing calculator
manual in this binder to help with calculator integration in your class.
There are Internet sites as well as print resources to
support the use of technology in the classroom. One such site can be found
at http://www.ex.ac.uk/telematics/maths/gchmpag.htm
- What
if I don't have ready access to graphing technology?
The use of technology is recommended for some activities.
If this is not possible in some situations, alternative approaches are
suggested for handling all but a few of the curriculum outcomes that require
the use of graphing technology.
- How
can I use the Internet as a resource?
Students will have a wide range of experience with the Internet;
many students have Internet access at home or through other sources. It
can be an invaluable resource tool. If your school doesn't have access
to the Internet, you might provide students with a list of Web sites and
suggest that they visit a library or other access point to the "Net."
It is recommended that, rather than just having students "surf the Net,"
you provide direction by providing some Web addresses.
For students who have access, you might suggest:
- a project researching a topic of interest
- tutorial sites
- sources of help with questions (for example, DR MATH)
Invite students to explore, and be open to the fact that
some students may have more experience with this medium than many teachers.
Sometimes, the Web site is not as useful as it purports to be and students
should be warned of this in advance.
- How
has the use of technology been embedded in the course design and expectations?
This course has been designed with an awareness of the increased
availability of technology and the impact that technology is having on
how mathematics is taught and what mathematics is important to learn.
Technology is treated as a tool for the instruction of mathematics. It
is a powerful tool, providing students with the opportunity to explore
situations that might otherwise be inaccessible, such as linear and non-linear
regression modeling.
Students are encouraged to make judgments throughout this
course about when the use of technology is appropriate and when it is
not.
"As more sophisticated technology becomes accessible in
classrooms, its productive use in support of the mathematics curriculum
will need to be considered. In making such decisions, the opportunity
for improved instruction and learning should be the guiding principle."
(Contexts for Learning and Teaching in Foundation
for the Atlantic Canada Mathematics Curriculum, page 39)
- What
is the role of homework in this course?
"Homework is an essential component of the mathematics program,
as it extends an opportunity for students to think mathematically and
to reflect on ideas explored during class time. The provision of this
additional time for reflection and practice plays a valuable role in helping
students consolidate their learning."
"Traditionally, homework has meant completing ten to twenty
drill and practice questions relating to the procedure taught in a given
day. With the increased emphasis on problem solving, conceptual understanding
and mathematical reasoning, however, it is important that homework assignments
change accordingly. More assignments involving problem solving, mathematical
investigations, written explanations and reflections, and data collection
should reduce some of the basic practice exercises given in isolation."
(Contexts for Learning and Teaching in Foundation
for the Atlantic Canada Mathematics Curriculum, page 30)
The Student Book has many questions designed to serve as
homework, extra practice, or review. It is assumed that some Check Your
Understanding questions will be assigned as homework.
- What
do I do to ensure adequate time to cover all outcomes?
Have a Long-Term Plan
It is important to manage well the time that you have. Planning
for the implementation of the curriculum needs to be done before the course
begins and timelines should be set at that point. These timelines will
help you track progress at regular intervals so that you will not run
short of time as the course draws to an end. There will be sufficient
time to cover the course material.
Have a Short-Term Plan
As you start each chapter, you should make a plan to cover
the outcomes presented in each part of the chapter. If you find yourself
short on time, this plan can be modified to omit material that may not
be helping students to achieve the outcomes of the course. Remember: It
is important that your yearly plan stays on track.
- How
does this material help support the student who needs extra support
or enrichment?
- There are suggestions for additional questions for challenge in
the Teacher's Resource.
- There are suggestions in the Teacher's Resource for alternative
approaches or strategies to use when students are experiencing difficulty.
- Case Studies and Extensions can be used as enrichment.
- Allowing students to work in groups will allow weaker students
greater access to the mathematics at hand and will allow them to hear
other perspectives on the problem.
- What
is the role of manipulatives in this course?
"The use of manipulative materials in the mathe-matics classroom
supports the development of understanding in students of various ages
and developmental levels. Manipulatives may be used to introduce new concepts,
verify results and/or provide remedial help.
"Every student should have the opportunity to explore with
manipulative materials. Manipulatives should clearly represent the concept
being taught."
(Contexts for Learning and Teaching in Foundation
for the Atlantic Canada Mathematics Curriculum, page 38)
Many of the Investigations in this course make use of manipulatives.
There are also many other materials students can use to experience concepts
hands on. The Teacher's Resource gives material lists.
- What
is authentic assessment?
Authentic assessment is done in a real-life context. Students
participate in an active display of their learning and are assessed on
their knowledge, skills, and attitudes. These assessments show whether
students can relate what they have learned in a real-world context.
- What
are learning styles?
Learning styles are divided into three main categories:
visual, auditory, and kinesthetic. All students learn best in one of these
styles. You should try to use all three learning styles in their lessons
to meet the needs of all students.
Visual learners further their understanding using charts,
graphs, mind maps, and symbols. Auditory learners can learn best from
listening and then repeating the information to themselves. Kinesthetic
learners prefer "hands-on" work. They also learn well after a type of
movement is associated with a concept.
Students should be encouraged to determine their own preferred
learning style and to alter their study habits to match their style.
- What
is constructivist teaching?
Constructivism involves students being given the opportunity
to construct their own meaning for concepts. This gives students ownership
of their learning and creates more interest in and understanding of material.
The students play a more prominent role in all aspects of the class, especially
class discussions. Retention, understanding, and transfer of concepts
are also improved.
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