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**Exploring the Use of UbD in the Teaching of Mathematics**

DR. GLADYS C. NIVERA Mathematics Faculty, PNU

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**“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.

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**Begin with the end in mind.**

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**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

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**The foremost strength of UbD is the commonsense nature of its principles and strategies.**

- John Brown (2005)

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Designing a Course by Common Sense

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**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.

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**Prepares a good lesson plan Teaches a lesson effectively**

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

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**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

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**How to Assess the Students**

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**Assessment Outputs Framework**

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

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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

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**Everything is wrapped up in a portfolio.**

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**Analytic Rubric for Student Teaching Portfolio**

4 – Outstanding Very Satisfactory 2 – Fair Needs Improvement Criteria 4 3 2 1 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

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**Summary Sheet Criteria Visual Appeal (20%) B. Organization (20%)**

Self (20%) Average of Peers (30%) Mentor (50%) 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

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**Components of the Grade**

Portfolio (40%) Demonstration teaching (40%) Oral report (10%) Participation (10%)

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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.

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**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

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**Stages of the Backward Design**

1. Identify desired results 2. Determine acceptable evidence 3. Plan learning experience and instruction

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**Work in Progress.. Development of Math 1 and 2 Syllabi**

Use of UbD in the design

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**Mathematics Unit on Measures of Central Tendency (Wiggins and McTighe, 2005)**

Essential Question: What is fair – and how can mathematics answer the question?

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**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”.

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**Presents a real-life situation**

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

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**Provides an overview of the unit and its expected outcomes**

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

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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

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**Applies the concept of “fairness” to different real-life situations**

What is a “fair” way to rank many teams when they do not play each other? What is a fair way to split up limited food among hungry people of very different sizes? When is it “fair” to use majority vote and when is it not “fair”? What might be fairer? 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

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**6. Students practice calculating each type of measurement.**

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

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**Check and consolidation of concepts and skills learned**

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

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**10. Individuals and small teams present their grading policy recommendations and reasons. (M) (T)**

Adopts performance assessment to show multiple dimensions of students’ understanding

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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

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**12. Students write a reflection on the essential question.**

Deepens and consolidates understanding

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**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.

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**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)

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**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.

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**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.

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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.

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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?

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**Students will know… Formulas for calculating surface area and volume**

Cavalieri’s Principle

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**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

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**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.)

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Performance Task 2 As a consultant to the United Nations, propose the least controversial 2-dimensional map of the world. Explain your mathematical reasoning.

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**Other Evidences Odd-numbered problems in full**

Chapter Review, pp Progress on self-test, p. 515 Homework each third question in subchapter reviews and all explorations

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**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

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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

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**Statistics – Stage I Established Goals:**

Develop statistical literacy by analyzing, comparing, and solving for the measures of variability accurately.

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**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

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**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?

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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

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**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.

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**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.

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Performance Task 2 Critique real-world statistical analyses and misleading graphs that you can find in magazines, newspapers or internet.

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**Other Evidences Odd-numbered problems in full**

Chapter Review, pp Progress on self-test, p. 515 Homework each third question in subchapter reviews and all explorations

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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?”

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**2. Present the points scored by several athletes across several games**

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?

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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.

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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?

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**Which provides a more stable measure of variability**

Which provides a more stable measure of variability? Support your answers. How will you interpret the computed variability of your scores in math quizzes?

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**5. Graph your scores. Ask the students to compare the graphs obtained**

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.

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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.

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**Is it fair to consider the highest and lowest scores only**

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.

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**Ask the students to graph the grouped data**

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?

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**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?

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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?

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**Some Comments About UbD**

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**UbD’s Strengths (Brown, 2004)**

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.

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**UbD’s Strengths (Brown, 2004)**

It could help all students develop a deep conceptual understanding of what they are studying.

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**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?

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**Essential Questions (Brown, 2004)**

2. How can we expand our ability to access models, benchmarks, and exemplars of UbD units and related curriculum products?

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**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?

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**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)?

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**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?

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**Apprehension should not lead to inaction.**

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Think BIG. Start SMALL.

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Thank you!

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“To begin with the end in mind means to start with a clear understanding of your destination. It means to know where you’re going so that you better understand.

“To begin with the end in mind means to start with a clear understanding of your destination. It means to know where you’re going so that you better understand.

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