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Table of Contents. 2. Title Page ...1 Introduction ...4 Purpose of the Math City Project ..5Comparing Backwards Thinking and Traditional Design Process .6Backwards Thinking -- 6
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1. A Designed-Based Learning Math Curriculum for Middle School Education Part II
2. Table of Contents 2 Title Page
... 1
Introduction
... 4
Purpose of the Math City Project
.. 5
Comparing Backwards Thinking and Traditional Design Process
. 6
Backwards Thinking -- 6 ½ Steps Process in Math City Project ............................................................. 7
Backwards Thinking ................................................................................................................................ 8
Backwards Thinking -- Steps 4, 5, and 6
9
The Story of Math City
.. 10
Long-Range Planning Boards
12
2-D Stylized Portrait of the Challenges Sequence
. 14
3. Table of Contents (Cont.) Pathway 1: Introductions
15
Pathway 2: Body Object
. 19
Pathway 3: NBS Instant City
. 21
Pathway 4: Site
... 23
Pathway 5: Division of Land
.. 26
Pathway 6: Design Parcel
... 28
Pathway 7: Structures
. 30
Pathway 8: Place of Exchange
32
Pathway 9: Movement
34
Pathway 10: Forms of Entertainment
. 36
Conclusions
.. 38 3
4. Introduction I Believe Math Classes Should
Train students to be self-sufficient
Increase higher level thinking skills
Improve communication skills
Train students to become better problem solvers and
problem seekers
Inspire creative thinking and originality
Develop proficient computational skills
Develop an understanding of mathematical concepts
Make connections from classroom work to the real world
Incorporate the mathematics state standards
Develop team work skills
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5. Purpose of the Math City Project To include all the elements that I believe should comprise mathematics classes, I applied the Doreen Nelson Method of Design-Based Learning called Backward Thinking to develop the Math City project. The purpose of the Math City project was to increase transfer of learning, work collaboratively, promote higher order thinking, creative problem solving, reasoning, and proficiency in math through an integrated standards-based curriculum for all students. The Math City project was hands-on, sequential progressions of city building developed teach students the connections from the classroom to the real world. The mathematical concepts taught in class linked to students lives as they learned to identify and solve problems manifested in their Math City. Students discussed, collaborated, evaluated, reflected, and enjoyed working on the Math City project: a designed-based mathematics curriculum for middle school education that was unified through a physical and psychological vehicle, the Story of Math City. 5
6. Comparing Backwards Thinking and Traditional Design Process Backwards Thinking is a 6 ½ step process
Identify the theme, concept , or standard to teach
Identify the problem from the curriculum
½ State a Never-Before-Seen design challenge that results in a 3-D model to resolve the problem
3. Students talk to each other about the challenge, the teacher monitors and asks questions, and students build their 3-D model
4. Set criteria for assessment that correlates to the chosen theme, concept, or standard
5. Students present their model with their results, then teacher presents the guided lessons
6. Students assess and synthesize what they learned, then make necessary revisions
Traditional Design Process: Frontwards is a 7 step process Identify and define problem
Gather and analyze information
Determine performance criteria for successful solutions
Generate alternative solutions and build prototypes
Evaluate and select appropriate solutions
Implement choices
Evaluate outcomes
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7. Backwards Thinking - 6 ½ Steps Process in Math City Project Step 1: Identify the theme, concept, or standard to teach. Example: Students deepened their understanding of the measurement of plane and solid shapes and used this understanding to solve problems.
Step 2: Identify the problem from the curriculum. Example: Structures needed to be built for people to live and for places of exchange.
Step ½: State a Never-Before-Seen design challenge that results in a 3-D model to resolve the problem. Example: Build structures to accommodate the residents, places of exchange, and public spaces on their parcel. 7
8. Backwards Thinking
Dont Want and Needs were the check off lists for the students. The criteria list introduced vocabulary.
For example, ABS means already-before-seen and NBS means Never-Before-Seen. This is to promote originality.
Students were given points for each item that was done correctly. For example: In the Structure Challenge the type of structure was a tall rectangular prism because there was not much land to support the number of people in the city.
The items on the list led to the guided lessons from the curriculum. 8
9. Backwards Thinking Steps 4, 5, and 6 9 Step 4: Students talked to each other about the challenge, the teacher monitored and asked questions, and students built their 3-D model.
Step 5: Students presented their 3-D model with their results, then the teacher presented the guided lesson and the students wrote about their discoveries.
Step 6: Students assessed and synthesized what they learned, then made the necessary revisions.
10. The Story of Math City At the beginning of the school year the students did not know each other or Design-Based Learning (DBL). My Math City project for the year sequenced ten pathways which became progressively more complex to provide an integrated curriculum by combining strands for the mathematics standards. Each pathway had a challenge followed by guided lessons. First, students made objects to introduce themselves to each other and to classify objects by finding common factors. Then, they dressed up as the objects to learn another way of knowing about each other and to learn scale.
After they got to know each other, they needed a place to live. They built an Instant City to learn that buildings should be in proportion to the population and their needs. Then, they chose a parcel of land from a real location. They assessed the way the land is used and ways to improve it. They supported their decisions based on their mathematical computations.
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11. The Story of Math City (Cont.) Next, they built structures for shelter, work, and social or cultural functions, using measurement and proportion. They needed to work in order to support themselves, so they used percentages and algebraic equations to design a Place of Exchange for commerce.
They discovered they needed a way to go from one place to another. Studying the use of rates and distances they made a new way of Movement. Once commerce, living areas, and Places of Exchange were made , they had time to relax with family and friends so they made NBS Entertainment using probability and statistics to determine if the game was fair.
To commemorate their achievements for prosperity, they built a NBS Time Capsule that contained a review and assessment of the years mathematics concepts learned within the context of the Math Story. 11
12. The Long-Range Planning Boards I made the Long-Range Planning Boards to display my integrated standards-based mathematics curriculum for a school year and sequenced each month as a pathway. Each pathway presented a problem for students to solve by completing a challenge, using set criteria as a guideline. The pathways were connected or held together by The Story of Math City. Each pathway had guided lessons.
The ten boards provided a visual representation of the required guided lessons and standards that were taught throughout the year. I showed this to parents, administrators, teachers, and the students. I found that the boards served as a reference for all to perceive the standards, lessons, and increasing level of difficulty as the project progressed. 12
13. The Long-Range Planning Boards (Cont.) 13
14. 14
15. Pathway 1 Introductions Innovation is a specialty of our species. Students used innovation to transform items from their backpack into meaningful objects.
The problem was to introduce themselves to each other and to classify the objects by finding common factors.
When students made a 3-D object as an introduction of themselves, they learned to identify their character traits. Then they classified the objects and interpreted the data in a bar graph. 15
16. Pathway 1 (Cont.) The NBS Intro Challenge in both years, 2008 and 2009, was met with enthusiasm by the Math 7 and Basic Math students. A few students did not believe it related to math and thought they were getting some free time. I learned from 2008 the importance of making a clear deadline to finish building the object. In 2008, a whole period was needed to build the object. In 2009, the time limit was 30 minutes. Although I worried that their objects would not be as well done or thought through since they had less time to build, they tied in their objects to the criteria list better in 2009 than 2008. 16
17. Pathway 1 (Cont.) Her group completed all the tasks successfully as she proudly displayed their final task, the bar graph of the categorized objects made for the introductions challenge.. 17 In 2009, the criteria list was better than in 2008. The Needs list required specific answers without struggling to choose just one memory or a single event, please see 2008/2009 criteria list on page 18. Two minutes was too long, especially since some students favorite memory exceeded the time limit. They were graded on the accurate completion of the bar graphs and the table with the correctly ordered fractions and decimals.
Since every student turned in the work with logical reasons for their categorization choices, I considered this a successful design challenge. Despite the improved criteria list, next time I do this challenge a time limit will be placed in the needs column of less than one minute.
18. Pathway 1: Criteria Lists 18
19. Pathway 2 Body Object We construct our own realities, and we can reshape them. Students enlarged an object that represented them.
The problem was to learn scale.
When students made the Body Object they learned to enlarge their small object, then they calculated scale by using proportions. 19
20. Pathway 2 (Cont.) In 2008, the Basic Math students got into the spirit of this one. Someone delivering papers to the classroom wanted to know why he was not invited to the costume party and a student was quick to reply, Hey, this is no costume, but who we are. The students instant response was significant since it indicated the students role-playing involvement. There was a lot of good dialogue. For instance a student applied estimation and then measured to get the actual measurement, Well, if Im this tall and minus my head and below the knees, its probably 3 feet.
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21. Pathway 3 NBS Instant City What we want is often not what we need. The students had to restructure their city to accommodate the population by making their community center or parks smaller so more living areas could be built.
The problem was to identify places and spaces in the built environment, the Instant City.
When students built an Instant City, they learned to identify the various solids that made up their model, then they categorized these solids and calculated the area of the solids that represented different types of structures for commerce, living area, and public space.
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22. Pathway 3 (Cont.) The specific dimensions in the needs column were important in order to calculate the number of people per building and to determine the size of the entire city. Also, the homes were given in metric units whereas the cardboard the city was built on was given in customary units, so conversions were taught in their guided lessons. The places the students listed were considered in their evaluation determining if the needs of the people were met. For instance, based upon the number of people living in the city, they decided if there were enough work places.
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23. Pathway 4 Site
The world is not always as it seems. A site in the local community of Walnut that was familiar to the students was chosen. The created an enlarged version that enabled them to assess the reality (the way the land was currently utilized), then to visualize changes to the site in order to improve it.
The problem was to choose a site from a map of a region familiar to the students, then they enlarged this area to take a closer look at the selected site.
When the students selected a landsite, they learned to identify differing parts of a map, then they reconstructed the chosen site into a larger size in order to see the details.
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24. Pathway 4 (Cont.) Since protractors were not used before this project, reading angles was taught. I drew angles on the overhead projector and placed the protractor on the vertex of one angle and lined it up correctly. By determining whether an angle was acute or not, they could decide whether to use the inside measurements or the outside measurements on the protractor. We did the next two angle measure as a class. Then, I asked them to draw a 65Ί angle and a 140Ί angle. I put up the two angles and they did the neighbor check. Then they worked collaboratively in their groups to complete their challenge.
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25. Pathway 4 (Cont.) Students discussed the best site to choose. Should we include the water tank? What about the train? City Hall would be good to have and the Teen Center
Some students had hushed discussions, while others listed reasons then compared them. There was not one group that did not take this very seriously.
Once they chose their site, they used a piece of yarn to find the scale factor.
They drew the enlarged map and checked for proportional sides and congruent angles.
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26. Pathway 5 Division of Land Nature can be understood and controlled. The uses of each land parcel determined the amount of land an owner needed.
The problem was to divide the land that represented a real location into not necessarily equal parts, but into equitable parts. Each group received one parcel.
When students divided the land into equal parts or enough parts for each group of students to own a section of land, they learned to recognize equitable parcels. Then they justified the division of the land of the site into parcels to be distributed.
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27. Pathway 5 (Cont.) Criteria List for Division of Land
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28. Pathway 6 Design Parcel We construct our own realities and we can reshape them. Students reshaped their own land parcel to improve the value of their land.
The problem was to determine how to restructure the land development to increase its value.
When students built on a land parcel in a real location, they learned to select the amount of area needed for buildings and open spaces. Then, they supported their decisions based on their mathematical computations. 28
29. Pathway 6 (Cont.) Unlike the 2008 students, the Advanced Math class in 2009 completed the Instant City and thought about modifications to improve their city since every student in the class volunteered an opinion while the criteria list was made.
Enlarging the land parcel and checking the measurements to be sure the two figures were similar, proportional sides and congruent angles, was second nature to them now.
Discussions regarding size and location of the places took an entire 50 minute period. Students were already talking about the shapes of the structures. 29 Criteria List for Design Parcel
30. Pathway 7 Structures There are many paths to the good stuff. Structures did not need to have conventional designs. There were many ways (or paths) to make structures become more functional and aesthetically pleasing for a future lifestyle.
The problem was to make structures for the building types on their land parcels.
When students built structures that became a building type such as a cylinder with a half sphere dome for a government building, they learned to identify solids. Then they calculated surface area and volume to justify the size of the structure for its function. 30
31. Pathway 7 (Cont.) 31
32. Pathway 8 Place of Exchange We shape intermediaries and then they reshape us. Students will determine NBS goods or services that were valuable to consumers.
The problem was to construct a place to exchange goods or services.
When students built a place of exchange, they learned to identify the value of goods or services and the costs involved. Then, they justified the success of the place based on their mathematical computation. 32
33. Pathway 8 (Cont.) Place of Exchange was completed once in 2009 with an Advanced Math class. Retail costs of products and services were discussed before students worked in their groups. The class was told that for convenience sake the wholesale price was 30% less than the price cited on the list of retail prices of products. For services, $8.00 per hour was used for minimum wage.
After the groups decided on products and services needed for the city, the cost and profit sheets were drawn up. Then the students designed their structures.
Students enjoyed designing the place of exchange, but formulas were used to calculate cost and profit.
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34. Pathway 9 Movement We shape intermediaries and then they reshape us. Humans invented the motor vehicle and the motor vehicle shaped the way we live today. It also shaped our lives environmentally (pollution), economically (gas and manufacturing jobs), and politically (control of oil). Students devised ways of going from place to place to think about reshaping our lives for the better. Students became environmentalists who envisioned a future of movement that would be more healthy, in other words more biking and walking, and alternative power sources would be used.
The problem came from the need to move people or goods from one place to another in the newly designed city. This fit into the math curriculum because a movement object was made from combining solids and analyzing how weight and angles affected the performance of the movement object.
When students built a movement container, they learned to recognize how the size and the shape of the load affected movement, then they evaluated if the movement container worked based on their mathematical computations. 34
35. Pathway 9 (Cont.) 35
36. Pathway 10 Forms of Entertainment As Mary Poppins says, In every job that must be done there is an element of fun. Everyone has different views of what was entertaining, but it took ones creativity to make work into a fun form of entertainment.
All the necessary components for daily living were completed so now they had time to relax, so the problem was to create NBS Entertainment using probability and statistics.
When students made a form of entertainment, they learned to identify if the form of entertainment was fair. Then, they created a marketable form of entertainment and assessed its profitability by using mathematical computations. 36
37. Pathway 10 (Cont.) 37
38. Conclusions Storing supplies and organizing the classroom was a difficult task when I first began the Math City project. But as I progressed with each project I was able to address these matters and expand my teaching practice.
I was able to add to my 2009/2010 NBS Introductions pathway from my 2008/2009 school year. The students were better organized and reached higher levels of thinking. They tied the criteria list to their object and were able to categorize their objects without the teachers assistance.
When surveyed in 2010, 80% of the students said that lecture and taking notes was a better way of learning for them. In elementary school the teachers lectured and did note taking with the students. This was their first year in the Math City project. The students may have chosen what they were familiar with instead of the DBL method.
In the Math City project the students simultaneously learned creative problem solving and had fun integrating math into the DBL projects. In the survey, 82% of the students marked they learned from classmates and discovered different ways to solve problems.
The Math City project allowed the students to apply math to their lives. Overall they enhanced their problem solving skills and understanding of mathematics. 38
