1. Worthwhile Tasks

2. Four Fours and Operations Problem
Use four 4s and some symbols +, x, -, ÷,and ( ) to give expressions for the whole numbers from 0 through 9: for example 5 = (4 x 4 + 4) ÷ 4.
Solve the problem.
Sharing solutions

3. Analyzing the Problem
What mathematics did you use to solve the problem?
When would you use this type of problem?
What related problems are usually found in textbooks?
What makes this task a worthwhile task?

4. Worthwhile Tasks

5. The teacher of mathematics should pose tasks that are based on ---
Sound and significant mathematics;
Knowledge of students’ understandings, interests, and experiences;
Knowledge of the range of ways that diverse students learn mathematics;
And that engage students’ intellect;
Develop students’ mathematical understandings and skills;
Stimulate students to make connections and develop a coherent framework for mathematical ideas;

6. Cont’d
Call for problem formulation, problem solving, and mathematical reasoning;
Promote communication about mathematics;
Represent mathematics as an ongoing human activity;
Display sensitivity to, and draw on, students’ diverse background experiences and dispositions;
Promote the development of all students’ dispositions to do mathematics.
Reference: National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM

7. How to Determine Worthwhile Tasks
In selecting, adapting, or generating mathematical tasks, teachers must base their decision on three areas of concern: mathematical content, the students, and the ways in which students learn mathematics.

8. Mathematical Content
Teachers should consider how appropriately the task represents the concepts and procedures entailed.
Teachers must also use a curricular perspective, considering the potential of the task to help students progress in their cumulative understanding of a particular domain and to make connections among ideas they have studied in the past and those they will encounter in the future.
Teachers must also assess what the task conveys about what is entailed in doing mathematics.
Teachers must also consider how well a task helps in the development of appropriate skill and automaticity.

9. Students
Teachers must consider what they know about students in deciding on the appropriateness of a given task.
Teachers must consider what they know about students from psychological, cultural, sociological, and political perspectives.
When selecting tasks, teachers must think about what their students already know and can do, what they need to work on, and how much they seem ready to stretch intellectually.
Teachers must know their students interests, dispositions, and experiences.

10. Knowledge About Ways In Which Students Learn Mathematics
The mode of activity, the kind of thinking required, and the way in which students are led to explore the particular content all contribute to the kind of learning opportunity afforded by a task.
Teachers must be aware of common misconceptions about mathematical concepts.
Teachers should deliberately select tasks that provide them with windows into students’ thinking.
Reference: National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM

11. Problem Solving Situation based on 1 Hunter by Pat Hutchins
A hunter walks past two elephants, three giraffes, four ostriches, …, and ten parrots. How many animals were there including the hunter?

12. Resourceful and Flexible Problem Solving1 2 3 4 5 6 7 8 9 10 11 15 19 26 36
55 animals
= 55 animals
PURPOSE: Show different strategies used by students to make sense of a problem
SPEAKING POINTS
Literature stories can provide an interesting context for problems. This problem from the story, 1 Hunter (Hutchins, 1982) describes a hunter who sees a variety of animals in the jungle. The students are asked to figure the total number of animals in the story.
These second graders used strategies that show different types of organization and efficiency. Students not only use a strategy that reflects their understanding of the situation, they also apply and strengthen their understanding numbers with such tasks. As students discuss their solutions they usually discover any counting or computational errors and correct them. Explaining their solutions gives them a chance to articulate their thinking and make more precise. Looking at other approaches also helps them determine which strategies are more useable and efficient.
Solving these types of problems support the learning of “the basics” and engages students in substantive reasoning that develops higher-order thinking skills and problem solving strategies.
REFERENCES: Principles and Standards: pp
54 animals (student miscounted) 12

13. Kindergarten Students– Number of Pockets
Keith
Lynda
Mark
Nicki
Octavio
Paula
Quinton
Robert
Sam
Wendy
Victor
Yolanda 1 2 3 4 5 6 7 8 9 10
Anthony
Barbara
Christine
Donald
Eleanor
Fred
Gertrude
Hannah
Ian
Pockets in Our Clothing
PURPOSE: Description of a data experience for young students, the increased emphasis on data is intended to span across the grades
SPEAKING POINTS
A primary PURPOSE of collecting data is to answer questions when the answer is not immediately obvious. By the end of second grade, students should be able to organize and display their data through graphical displays and numerical summaries.
This graph shows the data collected by kindergarten students about the number of pockets in their clothing. Here, the data for each student is represented. Kindergarten students many use blocks or other counting devices to figure the total number of pockets in their clothing.
These experiences help students understand different ways to represent the data they have gathered to give information and answer questions.
REFERENCES:
Principles and Standards: pp

14. How Many Pockets Number of Students 1 10 2 3 4 5 6 7 8 9 X
PURPOSE: Show how students in second grade represent the same data that is similar to what was shown in the previous slide
SPEAKING POINTS
In second grade, students might decide to count the number of classmates with different numbers of pockets in their clothing. This method of gathering data is different because they are grouping the data by categories-three students have two pockets, five students have four pockets and so on. They will have to think carefully about the meaning of all the numbers and the information represented on the line-plot. This model gives them an opportunity to write number sentences to determine the total number of pockets represented on this line plot graph.
Another skill students learn through these experiences is how to refine questions, to consider alternative methods for collecting information and to choose the most appropriate way to organize and display the data. These skills are acquired through experiences, class discussions with teacher guidance and carefully planned instruction.
REFERENCES
Principles and Standards: pp
Number of Pockets

15. Find several ways to determine the number of dots on the boundary of the square and then represent your solutions as equations.

16. Equivalence 4 x 8 + 4 = 36 4 x 10 – 4 = 36 10 + 8 + 10 + 8 = 36
PURPOSE: An example of building new mathematical knowledge through problem solving
SPEAKING POINTS
This problem offers a natural way to introduce the concept and term equivalent expressions. Students are likely to see different patterns and this gives an opportunity to discuss different equations that describe the same situation. Students can discuss how the solutions are similar and different.
This example also provides the chance to ask questions that extend the problem. If there were 76 dots, how many would be on each side of the square? Could you form a square with a total of 75 dots? This type of experience encourages students to become problem posers as well as problem solvers.
REFERENCE
Principles and Standards: pp
4 x = 36
4 x 10 – 4 = 36
= 36 16

17. Geometric Reasoning B A
Starting with two identical rectangular regions, cut each of the two rectangles in half as shown. Is region A equal in size to region B. Explain your answer.
PURPOSE: At this grade level, many students are just beginning to develop an idea of a convincing argument. The three solution strategies below show a wide range of abilities to logically analyze the relationships between the two shapes.
SPEAKING POINTS
Compare one of the smaller rectangles to one of the right triangles. Do they have the same area or is one area larger than the other? Justify or explain your answer.
Some students were convinced that they could decide if the areas were equal (or not) by cutting the triangle into a set of haphazard pieces and fitting them on the rectangle, covering the space. Others organized the cutting and pasting, for example, cutting the triangle into two pieces to make it into a rectangle that matches other rectangles. Still others developed ways to reason about the relationships in the figure without cutting and pasting. (Note: Use cutouts of the shapes for the audience to manipulate during the presentation or a dynamic geometry applet for doing the problem.
Students in grades 3-5 should make and investigate conjectures about mathematical relationships and develop mathematical arguments based on their work. They need to learn that posing conjectures and justifying them is expected and central to the development of reasoning.
REFERENCES Adapted from Tierney and Berle-Carman (1998, p. 10). B A
Adapted from Tierney and Berle-Carman (1998, p.10)

18. Flexible and Resourceful Problem Solvers
Roll two number cubes and subtract the smaller number from the larger number. Is one particular difference more likely than any other differences? If you do this twenty times, what will be the results?
Difference
Frequency
PURPOSE: An example of how problem solving can permeate the study of mathematics and engage students’ thinking about mathematics.
[Additional condition for the problem: If both cubes show the same number, subtract one number from the other.]
SPEAKING POINTS: The questions posed in this situation were “problems” for the students in that the answers were not immediately obvious. They had to generate and organize information and then evaluate and explain the results.
The teacher was able to introduce notions of probability such as predicting and describing the likelihood of an event, and the problem was accessible and engaging for every student. It also provided a context for encouraging students to formulate a new set of questions.
For example: Could we create a table that would make it easy to compute the probabilities of each value? Suppose we use a set of number cubes with the numbers 4-9 on the faces. How will the results be similar? How will they be different? What if we change the rules to allow for negative numbers?
The goal of school mathematics should be for all students to become increasingly able and willing to engage with and solve problems.
REFERENCES:
Principles and Standards: pp

19. Tasks that foster skill development even as students engage in problem solving and reasoning
Rolling pairs of dice as part of an investigation of probability can simultaneously provide students with practice in addition.
Trying to figure out how many ways 36 desks can be arranged in equal-sized groups--and whether there are more or fewer possible groupings with 36, 37, 38, 39, or 40 desks--presses students to produce each number’s factors quickly.

20. Where can we find worthwhile tasks?

21. Where can we find worthwhile tasks?
Good ideas can be found in articles in the Arithmetic Teacher, Mathematics Teaching in the Middle School, Mathematics Teacher, Teaching Children Mathematics in the PSSM, the Addenda Series, and Navigations.
Change from products to explanations
Use models
Collect and use a variety of contexts