ASSESSMENT AND EVALUATION

Assessment is no longer viewed and used the way it used to be. A variety of assessments is used to improve student learning and guide teachers in their planning and instructional approaches. Today’s shift is focused on learning and not so much on grades. Evaluation and marks come after providing students with a lot of practice and feedback on their understanding of concept and skills. Assessment is the gathering of information or observable evidence about what a student can do, where as evaluation refers to the process of judging the quality of student work on the basis of established criteria and assigning a value to represent that quality (reference to A Guide to Effective Instruction in Mathematics,Kindergarten to Grade 6 – Volume 4).

Three Types of Assessment

• Assessment for learning (diagnostic)
• Assessment as learning (formative)
• Assessment of learning (summative)

Assessment for learning (Diagnostic)

Is the process of gathering and interpreting evidence in order to provide clear, timely feedback to students to support their ongoing learning and achievement. This type of assessment is descriptive and not evaluative; frequent and ongoing. It is aimed at helping students improve their learning.

To effectively prepare a diagnostic assessment, consider looking at the previous grade level’s expectations and prepare an activity to get a baseline of your students’ mathematical processing skills and levels of understanding of a concept. When preparing a task or question, consider all possible misconceptions and prepare a task that would challenge their understanding of the concept. Incorporating a communication aspect to your assessment will also help you see how deep your students’ knowledge of the concept is and whether they have any misconceptions. All information will guide you in planning and differentiating lessons in a way that best fits your students' zone of proximal development. For an example of a diagnostic assessment of Grade 4 fractions, please click on the link here.)

Diagnostic Assessment Strategies

·        KWL charts

·        Clickers (SMART response, eInstruction)

·        RAN chart

·        Exit cards

·        Checklist

·        Instructional rubrics for students

·        Draw pictures to explain thinking

·        Role plays

·        Wordle

·        Thumbs up/Thumbs down

•       Read a children’s book on a particular Math concept http://childrenspicturebooks.info/articles/picture_books_for_math.htm (This is a website with a list of books for various Math strands)
•      Use the Senteo program on the SMARTboard and create a few questions based on last year's curriculum expectations.
•     Solve a problem from the previous grade (with the availability of various math manipulatives)

Assessment as learning (Formative)

It is based on the conviction that students are capable of becoming adaptable, flexible, and independent in their learning and making-decisions. It provides an opportunity for the student to monitor their own learning and progress. It fosters self-monitoring through meta-cognition and the application of self-regulatory strategies.

Our goal is to determine a child’s progress and improve learning. To effectively prepare formative assessments, careful planning needs to be considered. Consider the design down model of planning. Begin with the end in mind. Begin by asking yourself what is critical for students to know by the end of this unit. Then, determine what skills and concepts need to be learned and in what order. Planning lessons and assessments need to be carefully thought ahead of time and purposeful. Consider both the previous and next grade level’s expectation in the curriculum. Determine what possible misconceptions that we may encounter in your lesson plans, and write up good probing and prompting questions to get students to think about their math ideas and skills. Plan for a variety of lessons that will allow the students to show us what they know and have learned, and what they still need to learn. Be prepared to record your observations and to give students ongoing feedback on both their strengths and areas of improvement. Encourage your students to give each other constructive feedback and provide them with instructional rubrics (with kid-friendly words), where they can monitor or help each other monitor their learning progress. A big resource for teachers is the students’ parents. Use newsletters, a class website or a brochure to inform your students’ parents about the difference between assessment and evaluation. Inform and engage them in supporting you in your Math program. Give them ideas on how to support and extend their child’s mathematical ideas and skills. They are great resources! Here is a sample newsletter for your parents.

Formative Assessment Strategies

·        Assessment for learning tracking sheets (on their knowledge and understanding; thinking and problem-solving skills; communication skills and application skills)

·        Observations

·        Anecdotal notes

·        Math journal

·        Three part lessons

·        Teacher-student conferencing

·        KWL charts

·        Clickers (SMART response, eInstruction)

·        RAN chart

·        Exit cards

·        Checklist

·        Instructional rubrics for students

·        Peers assessments

·        Math portfolios

·        Concept maps http://www.linkslearning.org/Teachers/1_Math/2_Curriculum_Planning/2_Math_Concept_Maps/index.html

·        Draw pictures to explain thinking

·        Role plays

·        Wordle

·        Thumbs up/Thumbs down

•      Write a question related to particular concept

Assessment of learning (Summative)

Includes all summative assessment; it occurs when a teacher deems it necessary to determine the extent of a student’s achievement in relation to an established standard.

Assessments of learning are only done after lots of opportunities to explore and represent a concept (through oral activities, small group activities, class debates, class discussions, many real-life problem-solving exercises, etc.) and after many feedback from the teacher, peers, parents and self-assessments. Summative assessments will assess the students’ knowledge and understanding, thinking and problem-solving skills, communication skills and application skills. They can be done orally, written and/or through a performance task. Summatives should reflect what was explored and learned in class. There should be no surprises to the students or the parents.

Effective Ideas to Support Learning for All Students (reference to Damian Copper)

1.      Classroom assessment serves different purposes at different times (assessments are not always evaluations).

2.      Classroom assessment must be planned and purposeful (Practice the design down model).

3.      Classroom assessment must be balanced (assess oral, written and performance responses).

4.      Classroom assessment and instruction are inseparable because effective assessment informs learning.

5.      Feedback delivered in words is helpful to students.

6.      Classroom assessment is a collaborative process that is most effective when it involves self-, peer- and teacher-assessment.

7.      Performance standards are an essential component of effective classroom assessment (Have success criteria readily available and kid-friendly).

8.      Grading and reporting is a caring process that requires teachers’ professional judgment.

9.      Inform and engage parents in your Math program.

References

1.      OCDSB’s Educators’ Resource Guide: Assessment, Evaluation and Reporting of Student Achievement, 2008

2.      Assessment for learning http://www.etfo.ca/Resources/ForTeachers/Documents/Assessment%20for%20Learning.pdf

3.      Assessment as learning http://www.edu.gov.mb.ca/k12/assess/wncp/ch4.pdf

4.      Assessment FOR, AS, OF learning http://www.crcs.bc.ca/teacherlinks/for-as-of.html

5.      A Guide to Effective Instruction in Mathematics,Kindergarten to Grade 6 – Volume 4

6.      The Key to Good Assessments: Eight Ideas to Support Learning for All Students http://www.eqao.com/eMagazine/2009/02/eMagArticle.aspx?Lang=E&ArticleID=04&ItemID=34

Learning Environment and Inclusion

PRINCIPLES FOR TEACHING THE FACTS
A Guide to Effective Instruction in Mathematics, Volume Five, Teaching Mulitdigit Computations.

• Most students can learn the basic facts accurately, although their speed may vary considerably
• Students should learn the facts in a problem-solving context
• Students should have many experiences modeling the facts using concrete and pictorial representations
• Students should be encouraged to look for pattern and relationships between the operations and number in the facts
• Students need strategies that help them reason their way to the solutions for the facts, rather than strategies for memorizing the facts.
• Student who learn basic facts without understanding them do not know when or how to use what they know. Such learning is often transitory
• Students should not be compelled to memorize facts if they have limited strategies for solving facts. Students who have a repertoire of strategies will be able to find an accurate answer, and over time their speed will naturally increase"

MAKING MATHEMATICS ACCESSIBLE TO ALL

Differentiation and celebration of diversity in key in making Mathematics worthwhile. Differentiation is a teaching skill that is essential in providing an effective Mathematics' program, one that would reach many students with various backgrounds and exceptionalities. Factors to consider are listed below.

4 FACTORS THAT AFFECT LEARNING IN MATHEMATICS

Based on the booklet Teaching and Learning Mathematics: The Report of the Expert Panel on Mathematics in Grades 4-6 in Ontario

Teacher’s	-   teaching and learning experiences (background) -   teaching and learning style -   sensitivity to the students’ needs, learning styles and backgrounds -   approaches to teaching (balance between traditional and 3 part lesson) -    balanced program in Math- guided, shared and independent Math -    organisation and skills in preparing sequential and well scaffolded lessons -    attitude towards and knowledge of Math -    positive and constructive modeling (as well as modeling and teaching students that it is okay to make mistakes, get messy) -    ongoing assessments -    collaboration with other teachers
Student’s	-    background (socio-economic situation, gender, language and culture, special needs, limited Math foundation) and prior knowledge -    learning style -    family circumstances and met basic needs -    attitude towards Math -    choice to be an active learner versus a passive learner -    preconceived notions of Math (i.e., Math is for Math nerds, Math is suppose to be solved quickly, etc.) -    peer group influences
Learning environment	-    Safe environment to try new things, make mistakes and not be ridiculed -    Seating plans -    Flexible groupings -    Availabilty of manipulatives to work with -    Rich environment filled with Math resources and literature -    Math corner with challenges and games, as well as a Math word wall and/or anchor charts -    Post metacognitive questions or write them down while they are working. -    Technology as a resource -    Opportunities to problem-solve and share new ideas/strategies -    Differentiated tasks with various entry points -    Lessons: cross-curricular, progressive/sequential/cumulative, 3-part structure ensuring students have an opportunity to consolidate -    Fun, meaningful problems to solve and to chose from (differentiated and parallel tasks) -    Structured and well thought out lesson plan -    Clear expectations and goals -    Opportunities for Guided, Shared and Independent Math -    Celebrate diversity
Parents’	-    Attitude and knowledge of Math -    Involvement in helping his/her child understand Math Big Ideas -    Preconceived notions of Math -    Positive modeling of perseverance in problem-solving

EFFECTIVE ACCOMODATIONS BENEFICIAL TO ALL

·      Use a problem-based approach with a problem that allows a range of entry points (while considering their prior knowledge and backgrounds)

·      Give students a problem that is relevant or that they can relate to

·      Form heterogenous groups or groups based on a goal you set out for a specific group of students (flexible grouping)

·      Allow students to work by themselves

·      Model communication of ideas. Write ideas down on chart paper or the blackboard.

·      Tell students that they can use various manipulatives to show their work.

·      Tell students what the goal is and what the expectations are

·      For ESL students, read the question with the class, and ensure a complete understanding of the question by examining the “math information”

·      Provide differentiated and parallel tasks

·      Provide a choice activities

·      Provide a choice of extended activity for the stronger and quicker students

·      Acknowledge all strategies and ah ha moments during the consolidation stage of learning

·      Use anchor charts and lots of visuals!

·      Accomodate the different learning styles (visual, auditory and kinesthetic learner)

·      Use mini whiteboards when in large group activation activities - to allow students to test drive their theory before sharing with the class

·      Provide time to think and solve a problem

·      Use technology to reinforce concepts learned and increase procedural efficiency

·      Get parents involved

Assessment

·     Use different ways to assess knowledge and skills (oral interviews, learning logs, more time for completion of tasks, portfolios, etc.)

·     Check for students understanding regularly and frequently

·     Allow time for students to process the question you asked

·     Provide “cloze” activities or “complete the sentence” activities to help the student communicate his/her  ideas and thoughts

BARRIERS THAT COULD POTENTIALLY KEEP A CHILD FROM LEARNING

·    Teacher’s attitude towards math

·    Teacher’s understanding of the big ideas in Math

·    Teacher’s willingness and commitment to work with a multidisciplinary team to differentiate task

·    Teacher’s ability to provide a rich environment for learning

·    Teacher’s ability to provide differentiated tasks

·    Teacher’s ability to provide a balanced program

·    Teacher’s ability to challenge students metacognitively

·    Teacher’s ability to stay organised, assess students regularly, and communicate his/her observations with the parents, as well as to get them involved

·    Students’ socio and economic status

·    Students’ race

·    Students’ needs are not met

·    Students’ preconceived notion of Mathematics

·    Students’ fears

·    Students’ understanding of the problem

·    Students’ attitude towards Mathematics

·    Parents’ involvement

·    Parents’ attitude towards Math

·    Parents’ preconceived notion of Math

·    The learning environment: safe atmosphere, clear goals and expectations, richness of the classroom resources and literature, visuals, anchor chart, choices, differentiated and meaningful problems, availability of technology, school funding.

Research have indicated that one's race and income affects one's achievement (http://media.curriculum.org/curriculum/CSC256OERP/EN/10GRashtiWMartino.wmv ). This is also witnessed in my classroom. Many of my students' families are from low socio-economic backgrounds; many are ESL or ELL students who live in a family who struggles financially.


ACCOMODATIONS FOR SUPPORTING LOW INCOME AND/OR ESL/ELL STUDENTS
(With reference to the book “Many Roots Many Voices: Supporting English language learners in every classroom; A practical guide for Ontario educators)

FOR LOW INCOME STUDENTS

·    Provide food through a school breakfast program

·    Ask students to bring in school donations of clothes for the less fortunate

·    Have a snack bin where students can exchange their junkfoods for healthier foods

·    Have them involved as much as possible in their school community

·    Provide homework clubs

·    Get them hooked up with the Big Brothers and Sisters program within their community

·    Form support groups for high risk students

·    Praise all accomplishments to build self-esteem

·    Provide a safe and positive learning environment with many resources, manipulatives to work with

·    Provide differentiated tasks and choices in activities

·    Set realistic goals and hold high expectations

·    Provide parents with tips on how to help their children at home (i.e., homework – literacy and numeracy)

 

FOR ESL/ELL STUDENTS AND/OR LOW INCOME STUDENTS

·   Be familiar with the ESL/ELL stage of development

·   Assess their prior knowledge (Give the student a variety of Math concepts and strategies and ask him/her which ones they are familiar with)

·   Use their background, knowledge and skills to enrich our lessons

·   Engage learners in activities that appeal to their interests and build on their existing knowledge, skills and backgrounds

·   Relate mathematics to real-life experiences

·   Use mathematics as a tools for developing the learning community

·   Modify some or all curriculum expectations

·   Assign tasks that are appropriate to the student’s level of proficiency in English

·   Teach and model basic vocabulary (avoid idioms and jargon)

·   Use questions to help students develop language and concepts of mathematics

·   Explicitly teach the vocabulary of mathematics using questions to help students develop the language and concepts of mathematics.

·   Ensure that your instructions and expectations are clear (explain homonyms when necessary)

·   Simplify language

·   Speak clearly- articulating your words and pause often

·   Repeat instructions

·   Use a variety of visuals, manipulatives, charts, pictures, diagrams and graphic organizers

·   Use gestures

·   Print key words and instructions

·   Write key ideas down or have a scribe for ESL/ELL students

·   Encourage peer tutoring and class discussion

·   Integrate as many Math words as possible in other subject areas or any community/school event

·   Pair the student up with a another student who speaks the same language

·   Teach and model to English-speaking students ways they can help English language learners

·   Communicate positive feedback about language learning

·   Communicate positive attitudes towards newcomers and their cultures

·   Provide positive feedback on student effort

·   Establish a safe, respectful and supportive environment, where errors are accepted as a normal part of the learning process.

·   Work with a multidisciplinary team ( ESL/ELD teacher; board resource staff; MLO; school administration, and all who are involved in this child’s education) - set regular meetings where the homeroom teachers and special education teachers gather together and learn how to examine student work, looking at strengths and possible misunderstanding and misconceptions, examining the key goals and ideas we want students to master and generate questions that could challenge students to deepen their understanding of a specific concept - while ensuring that the student needs match the learning Math goals].

 

Curriculum Planning With The Big Ideas In Mind

What Are Big Ideas in Math?

Big Ideas consist of a comprehensive list of mathematical knowledge and concepts that provide a basis for developing more complex thinking and acquiring a deeper understanding of big mathematical ideas. The linkages and connections between math concepts are made explicit by linking previously learned big ideas to new concepts and problem solving situations. By emphasizing the big ideas in each lesson, teachers can build students' acquisition and use of key conceptual knowledge across lesson plans.


The Bigs Ideas are (Marian Small 2005)

1. Number Sense
2. Operations
3. Measurement
4. Geometry
5. Patterning and Algebra
6. Data Management
7. Probability

5 Mathematical Processes To Consider When Planning (Based on the Ontario Curriculum of Mathematics (revised 2005)

1. Problem solving
2. Reasoning and proving
3. Reflecting
4. Selecting tools and computational strategies
5. Connecting
6. Representing
7. Communicating

Benefits of Planning With The Big Ideas in Mind (Marian Small 2010)

• Facilitates long range/unit and day planning, and assessment when we understand how various mathematical concepts are interconnected and which concepts are more broader and encompassing than others
• Helps teacher set better learning objectives
• Allows you to create more opportunities for more meaningful teaching and learning experiences by continuously using the same key concepts to teach a variety of math skills and processes
• Builds students' acquisition and use of key conceptual knowledge across lesson plans
• Helps students gain a better understanding of the connections between all mathematical big ideas
• Increases the joy of learning Math
• Develops problem-solving and critical thinking skills
• Assesses more accurately the consolidation of knowledge of various Math concepts

A New Way of Teaching: The Three-Part Lesson Approach
Reference to The Magazine Of The Ontario College of Teachers
http://professionallyspeaking.oct.ca/march_2010/features/lesson_study/three-part.aspx

Before: Getting Started: 10-15minutes

The purpose of this lesson is to activate prior knowledge. Using an activation questions, get students are cognitively preparing themselves for the lesson problem by thinking about ideas and strategies they have learned and used before. The teacher reviews concepts and/or strategies learned through large group class discussion or think/pair/share activities. Then, expectations will be communicated to the students.

During: Working on It: 30–40 minutes

At this time, students are actively solving the problem. During this part of the lesson, students work in pairs or individually. Students are encouraged to use manipulatives. Students are encouraged to persevere and find a solution to the problem.

During this part of the lesson, the teacher is walking around the classroom, observing, listening to students' train of thoughts and assessing their reasoning. The teacher may also use prompts to help weaker students get started. Nearing the end of this part of the lesson, the teacher will carefully choose several Math work to show various key concepts and clarify misconceptions if any.

After: Consolidation and Practice: 10–15 minutes

 In this phase, the teacher strategically places student solutions, using a mathematical instructional strategy like bansho (http://dl.dropbox.com/u/22184571/Bansho%20with%20annotations.pdf) or math congress or a gallery walk. The teacher will then draw out the math from the students' work and facilitate a whole-class discussion that will provoke students to make connections between their math ideas and those of their peers. This is where students consolidate their understanding of the learning goal of the lesson. The teacher can also make an anchor chart with all the connections made, as well as the strategies explored by the class. What the teacher learns from students about their understanding is directly related to the types of questions asked. What the teacher learns from this discussion will guide the direction of future lessons or activities.

Great videos outlining the 3 part math lesson:
Before:
http://dl.dropbox.com/u/22184571/The%20Three%20Part%20Lesson-Part%201-Before%20copy.wmv

During:
http://dl.dropbox.com/u/22184571/The%20Three%20Part%20Lesson-Part%202-During%20copy.wmv

After:
http://dl.dropbox.com/u/22184571/The%20Three%20Part%20Lesson-Part%203-After%20copy.wmv


Effective Planning Tips

• Use the Ontario Curriculum to guide your planning
• Use and order the big ideas based on which expectations come first and/or encompass others
• Ensure a clear progression of difficulty in your lessons
• Encourage the use of manipulatives
• Have a clear learning goal
• Begin planning with the end goal in mind
• Assess prior knowledge
• Plan engaging and meaningful lessons that students can relate to
• Choose good problem-solving questions that are conducive to good math discussions
• Choose various assessment strategies to check for all mathematical processes
• Choose a culminating task that relates back to the expectation that you are assessing
• Share your plans with a colleague
• Team up with an experienced colleague when planning
• Always look for ways to integrate other subject areas in your planning
• Use technology to reinforce your lessons

Helpful Resources

Ministry Documents

1. Ontario Curriculum
2. Early Math Strategy (2003) http://www.eworkshop.on.ca/edu/resources/guides/ExpPanel_K-3_Math.pdf
3. Teaching and Learning Math Expert Panel (2004) http://www.eworkshop.on.ca/edu/resources/guides/ExpPanel_456_Numeracy.pdf
4. Guides to Effective Instruction in Mathematics (2006)

ð       Vol. 1 - Foundations of Mathematics Instruction

ð       Vol. 2 - Problem Solving and Communication

ð       Vol. 3 - Classroom Resources and Management

ð       Vol. 4 - Assessment and Home Connections

ð       Vol. 5 - Teaching Basic Facts and Multidigit Computations

5. Guide to Effective Instruction in Mathematics, 4 to 6

ð       Number Sense and Numeration (2006)

ð       Vol. 1- The Big Ideas

ð       Vol. 2 - Addition and Subtraction

ð       Vol. 3 - Multiplication

ð       Vol. 4 - Division

ð       Vol. 5 - Fractions

ð       Vol. 6 - Decimal Numbers

ð       Geometry and Spatial Sense (2008)

ð       Data Management and Probability (2008)

ð       Measurement (2008)

ð       Patterning and Algebra (2008)

Other resources/websites

6. Math Make Sense textbook
7. Nelson textbook (check out the various websites in the textbook)
8. Various resources from Scholar’s choice that are “concept” based
9. Teaching Student-Centered Mathematics by Van De Walle (Grade 3-5)
10. Math Grade 4 Ontario Curriculum exercise books
11. Math Question Puddle Question books (Grade 4) or for a sample question http://www.mcgrawhill.ca/school/imprints/wright+group/mathematics/puddle+questions/index.php
12. Ontario Educational Resource Bank http://resources.elearningontario.ca/
13. Online Teaching Resources http://www.eworkshop.on.ca/edu/core.cfm
14. Ontario Curriculum Web Resources http://schools.tdsb.on.ca/asit/standards/4start/OntWeb.pdf
15. Ontario Curriculum Unit Planner http://ocup.org/units/units55.php
16. Smart Board Resources http://www.ldcsb.on.ca/board/cirt/training/SmartBoards.htm
17. Illumination: Resource for Teaching Math http://illuminations.nctm.org/ActivitySearch.aspx
18. Link of activities that support the Ontario curriculum http://linktolearning.com/

Mathematics Sites Categorized To The Ontario Curriculum

• Mathematics WebQuests - From the WebQuest Locator
• Mathematics Links - From Doug Peterson's Diigo Site
• GECDSB 100th Day of School Resources
  Links to Internet Resources to help you celebrate the 100th day of school.
• The Math Forum
  Resources, Lesson Plans, Issues, all related to math education
• Mathville Homepage
  The homepage for the Mathville products. Includes excellent offline mathematics activities.
• Mrs. Glosser's Math Goodies
  Interesting site with lots of online manipulative mathematics activities for all grades.
• Web Math
  Your own personal online Mathematics tutor.
• Mr. Pitonyak's Pyramid Puzzle
  What would an Egyptian Pyramid cost to build today?
• Tangrams
  The Chinese called this centuries-old game the "Seven-Board of Cunning" because only an exceptionally determined player would attempt its amazing and challenging puzzles.
• Puzzles
  The name says it all!
• Puzzles with polyhedra and numbers
  In this site one can print copies of polyhedron puzzles (for non-commercial purposes only) and one can read several mathematical articles on the subject.
• Currency Converter
  Use this site to get up to date information on currency exchange rates and foreign investment advice!
• Computing Technology for Math Excellence
  Computing Technology for Math Excellence is devoted to resources for teaching and learning mathematics (K-12 and calculus) and the standards movement in education.
• ABCs to Excel
  Resource for spreadsheet modelling.
• Fido
  Can you determine how Fido reads your mind?
• Virtual Pythagorean Theorem
  Interactively work with Java applets to discover the theorem and more.
• Hubbin'
  Check out the various educational hubs at this site
• Geometry Step by Step from the Land of the Incas
  This interactive web site uses a variety of techniques to introduce geometry to the user. Mathematics, music, visual  displays, quizzes, puzzles, science, history, and geography are integrated in this web site. A very visual and engaging site which brings geometry to life.
• Currency Conversion
  From the Bank of Canada, convert Canadian currency to and from others. Also provides great data for Fathom activities.
• Mathematics Learning Objects
  Manipulate objects on the screen to learn various mathematical concepts.
• Squigly's Playhouse
  If you're a primary teacher and looking for some art ideas, activity sheets, or online activities, you've got to pay this site a visit.
• mathFROG
  Collection of mathematics activities for students in Grades 4, 5, and 6.
• Virtual Manipulatives
  Go online and solve mathematics problems with web-based manipulatives.
• Math and Reading Help for Kids
  Math and Reading Help for Kids is a directory of hundreds of original articles, tips and resources centered on the topic of children's learning.
• Internet Math Walk
  Take a wander around the internet and see Mathematics in action.
• Kindersite
  Searchable database of activities for language, music, mathematics, and age appropriate games.
• Mathville 1
  Now, for the youngest of grades, there's Mathville 1.