Exploring the Use of UbD in the Teaching of Mathematics DR. GLADYS C. NIVERA Mathematics Faculty,...
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Transcript of Exploring the Use of UbD in the Teaching of Mathematics DR. GLADYS C. NIVERA Mathematics Faculty,...
Exploring the Use of UbD in the Teaching of
Mathematics
DR. GLADYS C. NIVERAMathematics Faculty, PNU
“Would you tell me please
which way I ought to go from here?”
“That depends a good deal on where you want to get to,” said the Cat.
“I don’t much care where…” said Alice.
“Then it doesn’t matter which way you go,’ said the Cat.
Begin with the end in mind.
Quoted from the 7 Habits of Highly Effective People (Coven, 2005):
By beginning with the end in mind (the habit of vision), you get
A clear definition of desired results A greater sense of meaning and purpose Criteria for deciding what is or what is not
important Improved outcomes
The foremost strength of UbD is the commonsense nature of its principles and strategies.
- John Brown (2005)
Designing a Course by Common Sense
SED 4: Principles and Strategies in Teaching Mathematics
Course Description: Designed to prepare students in the teaching of mathematics to
secondary students Enhances students understanding of the nature of mathematics
and the philosophy of teaching mathematics Familiarizes students with various teaching methods and
strategies Provides students an opportunity to enhance their teaching skills
and to plan and execute a lesson successfully.
Framework of Mathematics
• Reflect on their views on the nature of mathematics, on the theories of learning, and on the mathematics curriculum
Methods and Strategies
• Discusses different methods and strategies and evaluates the effectiveness of each
Teaching Skills
• Prepares a good lesson plan
• Teaches a lesson effectively
What is an effective teacher?
NCBTS 7 Domains: Social regard for learning The Learning Environment Diversity of Learners Curriculum Planning, Assessing and Reporting Community Linkages Personal Growth and Professional Development
How to Assess the Students
Assessment OutputsFramework
• Reflection papers on the nature of math, theories of teaching math, one’s philosophy towards teaching
• Reaction paper on articles
• Film review
Teaching Strategies
• Discussion and reflection on different teaching strategies
• Actual lesson plans from first draft to final form
• Evaluation forms on their peers’ selection and use of strategies
Teaching Skills
• Actual Demonstration Teaching
• Visual aids• Evaluation of
classmates’ demonstration teaching
• Teacher’s and peers’ evaluation of one’s teaching
Common sense tells us to expand assessment tools and repertoires to create a photo album of student achievement rather than a snapshot.
- Wiggins and McTighe
Everything is wrapped up in a portfolio.
Criteria 4 3 2 1
A. Visual Appeal (20%)Cover, Lay-out, Tone/Mood, Creativity,
Resourcefulness, Neatness
B. Organization (20%)Order of entries, coding technique, readability of entries, correctness of form
C. Content (30%)Statement of purpose, completeness of entries, diversity of selections
D. Reflections (30%)Depth of understanding, application of ideas
Final Rating
4 – Outstanding 3- Very Satisfactory2 – Fair 1- Needs Improvement
Analytic Rubric for Student Teaching Portfolio
Summary Sheet
Criteria Self(20%)
Average of Peers (30%)
Mentor(50%)
A. Visual Appeal (20%)Cover, Lay-out, Tone/Mood, Creativity,
Resourcefulness, Neatness
B. Organization (20%)Order of entries, coding technique, readability of entries, correctness of form
C. Content (30%)Statement of purpose, completeness of entries, diversity of selections
D. Reflections (30%)Depth of understanding, extent of reflections, application of ideas
Final Rating
Components of the Grade
Portfolio (40%) Demonstration teaching (40%) Oral report (10%) Participation (10%)
Then we draw up the ‘syllabus’ or the relevant experiences and instruction that the students need to have in order to reach the desired outputs.
Some Activities That We Do
Debate and reflect on issues Analyze the BEC document Critique Films Conduct Group Discussions React to journal articles Discuss and demonstrate different
strategies Perform Micro-teaching
Stages of the Backward Design
1. Identify desired results
2. Determine acceptable evidence
3. Plan learning experience and instruction
Work in Progress..
Development of Math 1 and 2 Syllabi
Use of UbD in the design
Mathematics Unit on Measures of Central Tendency (Wiggins and McTighe, 2005)
Essential Question:
What is fair – and how can mathematics answer the question?
1. Introduce and discuss the essential question: What is “fair” and what is “unfair”? (M)
Very good, trigger question.
Students have very strong opinions about the words “fair” and “unfair”.
2. Introduce the 7th Grade race problem. Which 7th Grade class section won the race? What is a fair way to decide? Small-group inquiry, followed by class discussion of answers. (M)
Presents a real-life situation
Promotes discussion and mathematical reasoning and communication
3. Teacher informs students about the mathematical connections derived from the problem analysis, and lays out the unit and its culminating transfer task.
Provides an overview of the unit and its expected outcomes
4. In small group jigsaw, students share their answers to the inquiry sheet, then return to their team to generalize from all the small-group work. Discuss other examples related to the concept of “fairness” such as the following:
Utilizes various strategies
Promotes cooperative and investigative work
a. What is a “fair” way to rank many teams when they do not play each other?
b. What is a fair way to split up limited food among hungry people of very different sizes?
c. When is it “fair” to use majority vote and when is it not “fair”? What might be fairer?
d. What are fair and unfair ways of representing how much money the “average” worker earns, for purpose of making government policy?
Applies the concept of “fairness” to different real-life situations
Promotes transfer of learning
5. Teacher connects the discussion to the next section in the textbook – measures of central tendency (mean, median, mode, range, and standard deviation). (A)
6. Students practice calculating each type of measurement.
Presents the concepts and formulas but connects them to previous discussions
Practice computations
8. Teachers leads a review and a discussion of the quiz results. (A) (M)
9. Group task worked on in class: What is the fairest possible grading system for schools to use?(M) (T)
Check and consolidation of concepts and skills learned
Provides more transfer tasks which while require a new perspective
10. Individuals and small teams present their grading policy recommendations and reasons. (M) (T)
Adopts performance assessment to show multiple dimensions of students’ understanding
11. Culminating transfer task: Each student determines which measures (mean, median or mode) should be used to calculate their grade for the marking period and write a note to the teacher showing their calculations and explaining their choice.
Authentic Assessment
Promotes application, transfer, empathy, and self-awareness as well as team work
12. Students write a reflection on the essential question.
Deepens and consolidates understanding
UbD’s Strengths (Brown, 2004)
It follows common sense.
It could curb the tendency in public education to teach to the test and to emphasize knowledge-recall learning.
Workshop Mechanics
Form groups of 5
Decide which of the two units you would like to design
Geometry (Volume and Surface Area)
Statistics (Measures of Variation)
Workshop Mechanics Fill up the template.
Modify the activities in the initial template as you see fit
You may use extra papers.
Be ready to share your design.
Geometry Unit – Stage I
Established Goals:
II. Math7C3b, 4b: Use models and formulas to find surface areas and volumes.
II. Math9A: Construct models in 2D/3D; make perspective drawings.
Understandings
The adaptation of mathematical models and ideas to human problems requires careful judgment and sensitivity to impact.
Mapping three dimensions onto two (or two onto three) may introduce distortions.
Sometimes the best mathematical answer in not the best solution to real-world problems.
Essential Questions
How well can pure mathematics model messy, real-world situations?
When is the best mathematical answer not the best solution to a problem?
Students will know…
Formulas for calculating surface area and volume Cavalieri’s Principle
Students will be able to…
Students will be able to… Calculate surface area and volume for various 3-dimensional figures
Use Cavalieri’s Principle to compare volume
Stage 2 - Assessment Evidence
Performance Tasks 1
Packaging problem: What is the ideal container for shipping bulk quantities of M&M packages cost effectively to stores?
(Note: The best mathematical answer is not the best solution to the problem.)
Performance Task 2
As a consultant to the United Nations, propose the least controversial 2-dimensional map of the world. Explain your mathematical reasoning.
Other Evidences
Odd-numbered problems in full
Chapter Review, pp. 516-519
Progress on self-test, p. 515
Homework each third question in subchapter reviews and all explorations
Stage 3 Learning Plan
Learning Activities:
Investigate the relationship of surface areas and volume of various containers (e.g. tuna fish cans, cereal boxes, Pringles, candy packages)
Exploration 25, p. 509
Investigate different map projections to determine their mathematical accuracy (i.e. degree of distortion)
Read Chapter 10 in UCSMP Geometry Exploration 22, p. 504 Exploration 22, p. 482
Statistics – Stage I
Established Goals:
Develop statistical literacy by analyzing, comparing, and solving for the measures of variability accurately.
Understandings…
Student will understand that… Statistical analysis often reveals patterns
that prove useful or meaningful. Statistics can conceal as well as reveal Abstract ideas, such as individual
differences and consistency, can be modeled statistically
Essential Questions
How do people and events differ? What is a consistent performance - and
how can mathematics help us answer the questions?
How well can statistics reveal patterns in usually messy, real-world data?
Students will know…
Formulas for calculating variability of a given set of data (e.g. range, standard deviation)
Interpretation of the results obtained by the measures of variability
Students will be able to…
Calculate the different measures of variability relative to a given set of data, grouped or ungrouped (e.g., range, S.D.)
Give the characteristics of a set of data using the measures of variability.
Make a valid interpretation regarding the variability of graphs and data.
Stage 2 – Assessment Evidence
Three students must be selected to represent the school in a quiz bee. The principal asks the teacher to randomly select three students from any of the classes since all classes have the same mean. Is this fair? If the policy of the school is to really randomly select the 3 representatives from any of the classes, in which class should the teacher get the representatives - the one with high or low variability? Explain your answer.
Performance Task 2
Critique real-world statistical analyses and misleading graphs that you can find in magazines, newspapers or internet.
Other Evidences
Odd-numbered problems in full Chapter Review, pp. 516-519 Progress on self-test, p. 515 Homework each third question in
subchapter reviews and all explorations
Stage 3
Learning Activities
1. Introduce and discuss the essential questions: “How do people and events differ?” “What is a consistent performance - and how can mathematics help us answer the questions?”
2. Present the points scored by several athletes across several games. Which athlete gave a more consistent performance? What is a good measure of consistency in performance? Did two athletes have the same mean? If yes, were their scores exactly the same? Who performed more consistently?
3. Connect the discussion to the section in the textbook on measures of variation. Give exercises on computing the range and standard deviation using ungrouped data.
4. Ask the students to list their grades in mathematics quizzes for the past quarter and let them compute the mean.
How can you compare the data with the same average but different range? What do the measures of variability reveal?
Which provides a more stable measure of variability? Support your answers.
How will you interpret the computed variability of your scores in math quizzes?
5. Graph your scores. Ask the students to compare the graphs obtained. How does a graph reveal the variability of the scores?
6. Give a quiz on finding the range and standard deviation of ungrouped data.
7. Cooperative Work:
The teacher will show the grades in mathematics of the students from two classes (without reflecting the students’ names). Ask the students to group the data and compute for the mean and range.
Is it fair to consider the highest and lowest scores only? Support your answer.
Based on the results of the mean, speculate on which group of data is more variable.
Ask the students to graph the grouped data. Based on the graphs, speculate which set of data is more variable?
Ask the students to compute for the standard deviation of both sets of data. Were your speculations correct?
8. Let the students investigate the following:
Suppose you remove the highest and lowest scores in a set of data, what happens to the variability of the scores?
If each score in a set of data is increased by the same number, what happens to the variability of the scores?
When the scores are the same, what is the variability of the scores? How does the graph look like?
When no two scores are the same, what is the variability of the scores? How does the graph look like?
Some Comments About UbD
It could encourage teachers to discuss and work together to identify what is essential for students to understand.
It could guide and inform the process of school renewal and educational reform.
UbD’s Strengths (Brown, 2004)
UbD’s Strengths (Brown, 2004)
It could help all students develop a deep conceptual understanding of what they are studying.
Essential Questions (Brown, 2004)
1. How do we overcome educator’s anxiety and tension associated with the changes in mind-sets and practices required by UbD?
Essential Questions (Brown, 2004)
2. How can we expand our ability to access models, benchmarks, and exemplars of UbD units and related curriculum products?
Essential Questions (Brown, 2004)
3. How can we overcome the misconception that UbD is just for the best and brightest, and not for all students and teachers?
Essential Questions (Brown, 2004)
4. How can we acquire and ensure the long-term availability of resources, required to sustain successful UbD implementation (e.g., time, materials, curriculum development)?
Essential Questions (Brown, 2004)
5. How can we ensure that UbD is a clear and natural part of instruction and learning for all students, including those in primary grades, those enrolled in special education or ESL instruction, and those who are socioeconomically disadvantaged?
Apprehension should not lead to inaction.
Think BIG.
Start SMALL.
Thank you!