Why Should Teachers Assign Higher Level Tasks?

Why Should Teachers Assign Higher Level Tasks?North Carolina State UniversityStudent TeachersFall 2010

NCTM Problem Solving PrincipalsInstructional programs from prekindergarten through grade 12 should enable all students tobuild new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; monitor and reflect on the process of mathematical problem solving.

NCTM Communication PrincipalInstructional programs from prekindergarten through grade 12 should enable all students to organize and consolidate their mathematical thinking through communication; communicate their mathematical thinking coherently and clearly to peers, teachers, and others; analyze and evaluate the mathematical thinking and strategies of others; use the language of mathematics to express mathematical ideas precisely.

Common Core StandardsMake sense of problems and persevere in solving them.Reason abstractly and quantitatively.Model in mathematics.Look for and make use of structure.

Types of TasksNovice skill/procedural knowledgeApprentice performance assessmentsExpert multi-day/complex/portfolio

Types of Mathematics TasksStein, Smith, Henningsen, & Silver, 2009Low-Level DemandsHigher-Level DemandsMemorization TasksExamples: Recall or Memorizing facts, rules or definitionsTask follows a specified reproduction of workProcedures with Connections TasksExamples:Focused on the use of the procedure to develop the sense of the conceptStudent must engage in the idea to make sense of the problem

Procedures without Connections TasksExamples:AlgorithmsFocused on the procedure/correct answerRequires only limited cognitive demandDoing Mathematics TasksExamples:Requires in-depth, conceptual thinkingRequires students to rely on experiences and previous knowledge to develop an answerDepth of KnowledgeLevel 1-RecallLevel 2-Basic Application of Skill/ConceptLevel 3-Strategic ThinkingLevel 4-Extended Thinking

Mathematical TasksWhat is cognitive demand?Focus is on the sort of student thinking required.Kinds of thinking required:MemorizationProcedures without ConnectionsRequires little or no understanding of concepts or relationships.Procedures with ConnectionsRequires some understanding of the how or why of the procedure.Doing Mathematics

Examples of Mathematical TasksLevel 1MemorizationWhich of these shows the identity property of multiplication?A) a x b = b x aB) a x 1 = aC) a + 0 = aProcedures without ConnectionsWrite and solve a proportion for each of these:A) 17 is what percent of 68?B) 21 is 30% of what number?Too much of a focus on lower level tasks discourages student involvement in learning mathematics.

Examples of Mathematical TasksLevel 2Procedures with ConnectionsSolve by factoring: x2 7x + 12 = 0Explain how the factors of the equation relate to the roots of the equation. Use this information to draw a sketch of the graph of the function f(x) = x2 7x + 12. Doing MathematicsDescribe a situation that could be modeled with the equation y = 2x + 5, then make a graph to represent the model. Explain how the situation, equation, and graph are interrelated.Higher level tasks, when well-implemented, promote involvement in learning mathematics.

Characteristics of Higher Level TasksHigher-level tasks require students to

do more than computation.extend prior knowledge to explore unfamiliar tasks and situations.use a variety of means (models, drawings, graphs, concrete materials, etc) to represent phenomena.look for patterns and relationships and check their results against existing knowledge.make predictions, estimations and/or hypotheses and devise means for testing them.demonstrate and deepen their understanding of mathematical concepts and relationships.

Traditional ProblemNicoles Carpeting Task

Nicole was redecorating her house. She has decided to recarpet her bedroom, which is 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase?

Advanced ProblemNicoles Carpeting Problem

Nicole wants to redecorate her bedroom. She decides to recarpet. If her room is 5 feet longer than it is wide, write an equation to represent the area of her room. If you know her room is 10 feet wide, how many square feet of carpet will she need? If the carpet is sold by the square yard, how many square yards will she need?

Requirements for the Estimation Center

Engaging and creative.Students can work independently.Makes a connection to real-world or practical applications.Encourages thoughtful classroom discussion.Uses digital cameras and other multimedia tools.Presented in a power point presentation.Includes the title of the estimation center, mathematical concepts and connections addressed, and materials and set-up neededAll sources are citedWorksheet for students to useGrading rubric for students submissions

Implementing the Estimation Center

Students must make use of a wide variety of problem solving skills

Students are required to write a thorough description/explanation of the techniques used while attempting to solve the problem

These explanations form a basis for classroom discussion, with the main focus being on process and strategies, not on the final answer

Discussion

Results of the estimations are discussed, not to determine who got the answer right, but as an examination of effective strategies

The thoroughness of the various approaches and the clarity of the written summaries are also discussed. Although a definite answer may not be possible, some strategies may yield more accurate results than others.

The class data can be reviewed to determine what generalizations and assessments can be made about the problem. The class helps answer the question, What did we learn from the activity?

Estimation CenterMr. Wonkas Dilemma An Estimation CenterMathematical Concepts addressed:EstimationExponentsAreaVolumeMathematical Connections addressed:Connections between area and volumeConnection between area formula, volume formula and exponentsMaterials and equipment needed:Estimation Center Power PointStudent worksheetCalculatorPencilLCD ProjectorWhite BoardGobstoppersSet up needed:Students can either work independently or in pairsComputer, LCD projector and board at front of room so that all students can see the estimation centerAccess to a computer and internetAccess to the library

Mr. Wonkas DilemmaAn Estimation CenterMegan CoatesCherelle Cole

The BackgroundWilly Wonka and his candy factory have been doing quite well ever since the invention of the everlasting gobstopper. Kids from all over the world come to the factory just to see if they can get a glimpse of the everlasting gobstopper machine!

There is such a demand for the everlasting gobstoppers that Mr. Wonka has to put the gobstopper producing machine on overdrive for days!

The Dilemma But oh no!!!! The machine cant keep up with the demand and goes haywire! The machine explodes and there are everlasting gobstoppers spewing everywhere! The entire floor of the everlasting gobstopper room is covered by gobstoppers! The area of the floor of the room is the size of two and a half football fields.

The ChallengeMr. Wonka has to clean up the mess and he needs your help to determine how many gobstoppers are littered all over the floor! He also needs your help to determine how big of a candy dispensing machine he needs to store the displaced gobstoppers.

Can you take on this honorary task? The rewards are great!

Remember..Be sure to explain your process and show all work. Mr. Wonka will not be able to reward you if no work is shown. Be prepared to discuss your findings and explain how you came to your conclusions since Mr. Wonka may need to employ your method if something like this ever happens again.

Sourceshttp://www.grubbyhalogallery.com/mklacy/gallery/theatre_sets/images/P1010216.large.jpghttp://picsdigger.com/image/176d7fdf/http://steelkaleidoscopes.typepad.com/steel_kaleidoscopes/2007/09/the-everlasting.htmlhttp://rubistar.4teachers.org

Possible SolutionArea of floor: Size of 2 and a half foot ball fields The American football field is approximately 120 yards long by 53.3 yards wide.The area of one foot ball field is approximately The area of the floor of the Gobstopper room is approximately Figure out how many gobstoppers fit in one square inch, foot, etc. Approximately 225 gobstoppers fit in one square foot (15 gobstoppers times 15 gobstoppers).Convert units to find out how many gobstoppers are in one square yard. (This is not the only conversion possible).I will convert feet to yards. There are 3 feet in one yard. There are 9 square feet in one square yard. So 9 square feet is equivalent to one square yard.Multiply the number of Gobstoppers in one square foot by 9.

So approximately 2025 Gobstoppers are in one square yard.Find the number of Gobstoppers that are covering the floor in the Gobstopper room.

So 32,379,750 gobstoppers are covering the floor in the Gobstopper room.Now find the size of the container that will be needed to store the gobstoppers (there is more than one way to do this, one could find the volume of a gobstopper and go from there).So find the amount of gobstoppers in one cubic foot : gobstoppers/ft3 Now divide 32, 379,750 gobstoppers covering the floor by the 3,375 gobstoppers in one cubic foot to find the size of the container needed to store the gobstoppers.

So Mr. Wonka needs a container that is 9594ft3 to store the gobstoppers.

Digital Scavenger HuntChelsea LewisMatt HovisMary Katherine Miller

On NC State Universitys campus there are two benches for every one disposal. If Home Depot donates 320 benches, how many disposals will NC State need to buy?

The school wants to build another right triangle to support the air conditioning unit. If the base and height of the triangle are 2 ft and 3 ft respectively, find the hypotenuse.

What is the difference between intersecting lines and perpendicular lines?

Dunkin Donuts wants to cover the donut with powdered sugar. Find the surface area of the donut if the diameter is 8 cm.

Define an ellipse. How many ellipses are in this picture?

Why can you tessellate a hexagon and not a pentagon?

What equations graph would look most like this picture?

Gavin and the Giants ButtonStephanie WoodJenny Randall

One day while Gavin and his grandma were playing in a park, they came across a huge red button. Gavin wondered aloud, How big is the giant who lost that button?

ActivityQuestions for Gavin to considerHow can we determine the size of the giant button? What attributes of the giant and the button are important in deciding the giants height based on the size of the button? What attributes of Gavin and his buttons are important in deciding the giants height?

More QuestionsWhat else do we need to know in order to determine the height of the giant? What if we also want to know the amount of fabric needed to make the giant a coat? If the giant needed a drink of water, how much water would be equivalent to that in terms of a humans glass(es) of water? How large would the giants pack of gum be?

Use the student page Gavin and the Giants Button to explore relationships between enlargements and reductions called size changes, and the measurements of length, area, and volume using cubes.

ExploratoryGavin and the Giants Worksheet1. Use cubes to investigate what happens to the surface area and volume of a prism when each dimension is magnified by 200%, 300%, or other factors. Use the table to organize and record workMagnification factorDimensions of prism (units)Surface area of prism (sq. units)Volume of prism (cube units)100%1 x 2 x 3226200%300%50%150%100n%2. Conjecture a rule telling how area and volume change when length is changed by a magnification factor.

3. Note that some of the magnification factors in the table are not integer multiples of 100%. On a copy machine, you are also able to reduce the size of a copy or make non-integer magnifications. Do area and volume change in the same way when the magnification factor is not an integer multiple of 100%? Why or why not?

More Exploratory QuestionsAssuming giants have pencils, what are possible dimensions of a giants pencil? Compare the giants pencil to some human-sized object.What are possible dimensions of a pair of eyeglasses for the giant?What are the dimensions of the giants footprint? Could the giant step inside our classroom? Why or why not? What human sized object is approximately the size of the giants footprint?Using a humans paper cup as a model, determine the dimensions of a giant-sized paper cup. How much fluid will the paper cup hold? Compare the size of the cup to some human-sized object.

Share and SummarizeWhat assumptions were made about the giant? What assumptions were made about the items used to decide on the sizes of the giants items?Keeping all other factors constant, how do your results change if:

A. the giant is another gender?B. The giant is from another generation?C. The button is a jacket button?D. The button is a shirt button?If you were the giant, how tall would a human be?Suppose your button is the giants button. How big is a humans button when compared to your giant button?Look back at the table you completed in Gavin and the Giants Button. How do proportions arise from the table? How can we use proportions to help solve problems like these?

BibliographyRubenstein, Rheta N., Charlene E. Beckmann, and Denisse R. Thompson. Teaching and Learning: Middle Grades Mathematics. Hoboken, NJ: John Wiley & Sons, Inc., 2004. Print.

http://www.made-in-china.com/showroom/lishun-button/product-detailbQmEUTRMtnrp/China-Shirt-Buttons-C08031111-.html

http://gofifo.com/playground.htmThank you so much for listening to our presentation! Are there any further questions?