# Final Exam Question Pool

ED 411/518 Teaching Children Mathematics

Fall 2003

**FINAL EXAM QUESTION POOL**

**AND DETAILS ABOUT PREPARING FOR AND TAKING THE FINAL EXAM**

Teaching Children Mathematics (ED 411/518) is a professional course. The standards for your performance in this course have been tied to those you will be expected to meet as a teacher: meticulous preparation, appropriate use of professional knowledge, careful consideration of alternatives, genuine curiosity about ideas and about learners, exercise of professional judgment, collegial work on teaching, analysis and reflectiveness, skills of ongoing professional learning, clear expression, organization, and timeliness.

The final exam questions have been designed to assess your progress in the core areas of the course, and your performance on these tasks will be evaluated on these same standards. For example, good responses to the questions will demonstrate appropriate use of professional knowledge drawn from a variety of sources, including records of your own work this term; your classroom experience; your study of records of practice such as videotapes or student work; professional readings; and discussions with colleagues. Good responses will also be clearly expressed, grounded in evidence and particulars, and with clear explanations of professional judgment and reasoning. The core domains are:

- your knowledge of mathematics for teaching and your ability to use that knowledge in teaching;
- your sensitivity and skill at attending to children’s mathematical ideas and interpreting what they think and know;
- your knowledge and skills with respect to designing and enacting instruction for students in classrooms, as well as your ability to observe, attend to, and analyze teaching and learning;
- your ability to reason and to reflect on your own deliberations and decisions, to assess your own progress as a professional. Reflection includes both looking backward and looking ahead to future learning and practice.

The questions and tasks below comprise the pool of possible final exam questions. Eight questions are listed below. The actual final exam will be made up of a subset of 4 – 5 of these questions, or parts of these questions, although the numerical examples and details of the questions will be different. For example, if you are asked to analyze the meaning of operations in story problems, the examples on the exam will not be the ones included here. If you are asked to examine a student error, the numbers in the example and some specific features of the student’s error, will be different from the examples offered in this document.

You are encouraged to study and prepare these questions, and to do so with one another. If you do this, you will be well prepared for the final exam. You may bring all notes and resources with you to the final exam, including your course notebook. You will be provided with grid, plain, and lined paper as well as an assortment of writing tools, although you may of course bring your own if you wish.

The classroom in 4212 (School of Education Building) will be available as a study site if you want to meet and work there, and have access to the manipulatives and the videos. Dates and times for study are:

Thursday, December 47:00 – 10:00 p.m.

Sunday, December 74:00 – 6:00 p.m.*

(*The building is locked on weekends, so it would help if you could come at 4. We will also post a cell phone number that you can call if you come later.)

Tuesday, December 97:00 – 9:00 p.m.

Two or three of the course instructors will be there, and can help clarify questions or help you find things. We can answer questions on topics or ideas that you are reviewing. However, we will not go over possible responses to the questions, or other kinds of detailed help specific to the final.

The times for the exams are as follows:

518-001: Monday, December 8; 10:00 a.m. – 12:00 p.m.

411-001:Thursday, December 11, 10:00 a.m. – 12:00 p.m.

518-002: Thursday, December 11; 1:00 – 3:00 p.m.

411-002:Thursday, December 11; 4:00 – 6:00 p.m.

411-003:Monday, December 15; 1:30 – 3:30 p.m.

During the exam, you will work independently on the questions, and will be called up for an individual session at which one or more of the oral questions will be posed.

* * *

- (Oral question) Model a computation with manipulatives (addition, subtraction, or division).

Use bundling sticks, beansticks, or base ten blocks to model the conventional algorithm for each type of computation listed below. Be sure to show the correspondence between the materials and the written form of the algorithm; your goal is to show the meaning of the conventional algorithm in ways that would make plain to a learner the meaning of the procedure, and attend to key “cautious” areas that are crucial to learning this algorithm. Refer to the “Peer evaluation checklist for modeling computational algorithms” that we used in class for guidance about what to pay attention to. Be sure to consider what students might have trouble with and anticipate those difficulties in the care you take with the explanation.

a) addition with and without regrouping

b) subtraction with and without regrouping

c) long division with a 1-digit divisor

Try different numbers to study and practice these; try numerical examples that require different kinds of caution and attention with each algorithm.

- (Oral or written question) Draw an area model representation of a multiplication computation and link it to the full partial product written form.

Make an area model representation on paper for a two-digit whole number multiplication computation or a decimal multiplication computation involving tenths or hundredths. Label the factors in the drawing and label the product. Draw your rectangle clearly so that each of the partial products is clearly visible. Make an explicit link between the written form of the multiplication algorithm and the drawing.

Again, try different numbers to study and practice these; try numerical examples that require different kinds of caution and attention.

- (Written question)
**Distinguish and identify the meaning of the operation in a story problem; calculate the answer; and give the meaning of the answer with respect to the problem.**

*These are the interpretations on which we are focusing:*

Take-away model of subtraction

Comparison model of subtraction

Repeated addition model of multiplication

Rectangular array model of multiplication

Cartesian product model of multiplication

Area model for multiplication

Partitive model of division

Measurement model of division

(a) Write story problems for the following calculations. For each calculation, be sure to write (if possible) one story problem for each of the above interpretations for the operation; and also give and interpret the correct answer for the story context you create.

Examples:

OPERATIONS / EXAMPLESSubtraction / 52 – 19 / 4.5 – 2.3

Multiplication / 6 x 4 / 5 x 0.7

Division / 20 ÷ 5 / 51 ÷ 8

(b) Evaluate story problems. Decide which operation (subtraction, multiplication, division) would best model the problem, and identify the particular interpretation of subtraction, multiplication or division involved in the specific story.

Examples:

- I want to buy enough rope to be able to cut 4 lengths, each 6 feet long. How much rope do I need?
- A terrace is 4 feet wide and 6 feet long. How many one-square-foot tiles do we need to cover the terrace?
- We are having cheese sandwiches for lunch. There are four kinds of rolls and six kinds of cheese. How many different sandwiches can we make, if each sandwich has only one kind of cheese?
- Nicole is planting a flower garden. She wants to plant four rows of flowers with six flowers in each row. How many flower bulbs does she need?
- For a 4th grade festival, a school had 9 teams of 14 students each. How many students participated in the festival?
- 182 students want to play soccer this season. They can make 14 teams. How many players are on each team?
- 182 students want to play soccer this season. How many teams of 14 players can be created?
- There are 52 students in the 3rd grade. The students sit at tables at lunch. Each table seats exactly six students. What is the fewest number of tables needed to seat all the third graders in the lunchroom?
- Melissa has 52 delicious malted milk balls. She wants to give bags of candy to six of her friends, and wants to make sure that she puts the same number of candies in each bag so that no one is jealous. How many malted milk balls can she put in each bag?
- For a summer job, Jacob delivers newspapers. His first week, he makes $13. His second week, he makes $24. How much more money does he make the second week?
- Gabriel and Andreas have $45 to spend on a birthday party. They have already spent $16 on balloons. How much do they have left to spend on the cake?

4. (Written question) **Compare the assets and pitfalls of particular concrete (manipulative) materials used in teaching mathematics.**

Make a table to compare, side-by-side, features and uses of bundling sticks, beansticks, base ten blocks, and money.

Your comparison should attend to the following five features of each material analyzed:

- Are there any issues related to equity relevant to the use of this material? If so, what are they?
- Is this material suited best for a
*trading model or a grouping/ungrouping*model? - What particular language or terms for the materials (all the components) and transformations go with this manipulative? (e.g., “bundling,” “unbundling,” “superbundles,” etc. with the bundling sticks)
- What range of numbers can be readily represented with this material? (e.g., whole numbers, decimal numbers, how large or small)
- What are some key considerations related to using this material effectively in class? (e.g., how easy they are to obtain; how readily and effectively they can be displayed publicly on an overhead; how easy/difficult they are to manage in class; how well they work for particular age levels)

5. (Written question) **Analyze a common student error with place value, addition, subtraction, multiplication, or division.**

For the given error:

- Explain what is wrong and what the student probably did to produce that error (i.e., what mistake, confusion, or idea might lead to this error?). Identify a second (different) error that is also likely with the same problem, and explain what would cause it. Consider how the student’s previous experience or knowledge might be shaping each error.

- What would you do to remediate the given error, so that the student would not be likely to make this error in the future? Design a strategy for working with this student on this error. Develop a series of questions or explanations, decide on appropriate materials to use, and anticipate what the student might do. What, specifically, would you be hoping to accomplish in terms of the student’s understanding so that the student will not make the error again?

Examples:

(a)/ (b)

Child counts total as 7. / (c)

(d)

/ (e)

/ (f)

(g)

/ (h)

Child writes 150 to represent number of beans. / (i)

6. (Written question) **Choose numerical examples for a particular mathematical topic at a certain level. **

Make three examples –– one easy, one medium, and one difficult –– and justify what makes each one easy, medium, or difficult.

Modeling numbers with bundling sticks (K – grade 1)

Addition computation (2-digit) (grades 2 – 3)

Subtraction computation (3-digit) (grade 3)

Ordering numbers (whole numbers, decimals, or both) (grades 1 – 8)

Multiplication of decimals (grades 5 – 8)

7. (Written question) **Analyze a segment from a classroom lesson.**

It is a clip from May in the third grade class you have seen several times. You have not seen this episode, however. The students are working on the problem, “Which is more, 4/4 or 4/8?” The transcript is available on the course website.

The video segment will be available for study at < The password will be f03final and will be available starting Tuesday, November 25. You will need Quicktime player to view the videos. If you don't have it you can get it for free from < To view the video on a computer, you must be somewhere where you have a high-speed connection (e.g. DSL, cable modem, or a computer hooked directly into the campus network via ethernet or wifi).

***PLEASE NOTE: Due to agreements made with the families of the children depicted on this video, we ask that you do not give out the site address and password to anyone not in our class.*

a) Identify two specific teacher questions that foster the mathematical development of the lesson and/or engagement of the students with the mathematics. Explain how the question serves to accomplish this.

b) Identify and describe two elements of the classroom culture –– e.g., how students treat one another’s ideas; what sort of student participation is valued and encouraged; how errors are treated; how right answers are determined. Provide clear evidence of the identified elements, and briefly explain what the teacher is doing that may support or promote each element of the classroom culture.

*8. (Written question) **Document and analyze your own learning, and consider your own professional growth.**

For this question, you will compare two examples of your professional work to reflect on your own growth this term.

a) Select two relatively small examples of your work this term that, when compared, show growth in your ability to do professional work at a high standard. These can be specific entries or experiments in your notebook, parts of homework assignments, a section of one of your main projects, or a detailed record of something you did interactively in class or in your field placement.

b) Make photocopies of the work samples to submit with your exam, and label clearly where each one is from. This part of the question must be prepared in advance of the final exam.

c) Explain (in writing) what the two samples, together, represent in terms of your learning as a professional, both in terms of your progress since the summer, and the direction of next steps in your development. Explain how and why the second sample represents significant progress toward professional work. (See the list at the beginning of the final exam for what we mean by “professional work.”)

**This question will be part of the final exam. You must bring copies of your samples to the exam. You may prepare your entire response to this question in advance of the final exam and simply bring it with you to turn in. This question is meant to be responded to in the form of a serious notebook entry. An appropriate length (not including the samples) is not longer than 1 – 2 pages. *

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