Test Information
Here is the place to find information about tests.
Information will be posted shortly before each test.
General Guidelines and Information
Format
The format of most questions on exams will be open-ended and free-form.
That is, you should not expect to find lots of multiple choice, true/false
or the like.
Rather, you should expect to see questions that require
a written response, often requiring some sort of support for your claims.
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Short answers.
Many types of questions fall into this category, and the questions you
have seen on reading assignments and quizzes are good examples.
You may be asked to make a list, to give a definition, to provide examples,
to compare or contrast two concepts, readings or positions,
to explain, etc.
The most
important thing to remember on short answer questions (and other questions
too, but especially on short answer questions) is that
I am not looking merely for "correct" answers, I am looking
for quality answers.
Your answers will therefore be graded on such criteria as
truth, accuracy, importance,
conciseness, coherence, and supporting evidence.
In particular more is not always better. Don't tell me everything you know
about the topic in question; show me that you can distill the most
important issues or most essential features into a precise, coherent
response. If asked to give an example of something, try to give
the best possible example; one very good example is usually better than
several poorer examples.
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Problems and Exercises.
You will be required to do some mathematics on each test. Hopefully,
some of the tasks on each exam will be exercises for you and some will
be problems, but exactly which are which may vary somewhat from student
to student.
In particular, you should be prepared to do tasks that are similar to
any exercises or problems asigned, worked on in groups, presented in
readings or discussed in class. If you have understood these tasks from
class, this type of task should be an exercise for you on the exam, even
if it was a problem when first encountered in class.
It is also possible that you may be given a task that is a genuine problem
for you. Of course, if you have forgotten how to do something or didn't
completely understand it the first time around, a task of
the type described above may also be a problem for you. But I may also
design some tasks with the intent that they be problems rather than exercises.
These will usually require only background knowledge and problem solving
skills that have been used in other tasks in the course,
but may require you to do some experimenting and use problem solving
heuristics before you will be able to come to a solution.
Solutions to either type of mathematical task should be presented clearly
using words as well as symbols, explaining your reasoning, etc.
This is especially important if your solution is "incomplete". Significant
progress, good ideas, reasonable attempts, realization of errors, etc. as
well as quality of presentation (good communication) will all be considered
when evaluating your work.
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Essays.
More rarely I may give you an essay question.
Essays are just longer version of short answers. Typically the difference
will be in the scope of the question. An essay may cover more or the issues
may be more involved or just more difficult to present, so that a
longer answer is required.
Nevertheless, the same criteria as for short answers apply,
with an additional criterion: organization. Essays should be written using
good writing skills so that the organization and flow of your answer
are readily apparent.
Important Topics -- Big Ideas
Obviously a one-hour exam cannot cover everything we have read
in this class so far. In fact, it cannot even cover every important
topic. Thus the exam will include questions covering
a sample of (some of) the most important topics.
Part of your job in preparing for the exam is to identify the major themes
and learn the big ideas.
One good way to this is to go through the questions on
the readings and mark the questions from each chapter that you think
are most important.
For each chapter or reading assigned, ask yourself questions like:
What are the big ideas here?
What are the key examples or supporting evidence?
Do I understand the key terms used to describe the big ideas?
(Could I give a concise definition or a good example for each?)
Class discussions can give some indication as to what kinds of
topics I think are important,
but do not neglect a topic simply because it was not
discussed at length in class.
Mathematics Content
Remember, one of the goals for this course is that you have the mathematical
competance to teach K-8 mathematics. Thus part of the exam will be devoted
to making sure that you all have the necessary knowledge, skills and
background for this. Note that this includes both procedural knowledge
and conceptual knowledge and the ability to relate the two.
For each exam you should make a list of the content areas covered.
(For example, for the first test, the specific content areas from
the K-8 curriculum that we have covered include attributes, patterns,
sets and set operations, the meaning of the four basic arithmetic operations,
and basic facts.)
Mathematical content may be tested using exercises, problems or short answer questions.
Activities
You should be familiar enough with the activities we have done in class
(like the A-block activity) that a very brief description recalls
the activity to your mind. You may be asked to do portions
of an activity, to answer questions about the methods or goals
of a particular activity, or to use activities as examples in
answers to other types of qeustoins.
Teaching Methods
This will become even more important later on, as we have more examples
from elementary school curriculum, but you should prepared to answer
questions about how to teach mathematics, advantages and disadvantages
of various methods, how to use models to develop mathematical understanding,
typical errors students make and how to help them avoid or unlearn them, etc.
Synthesis
Avoid the temptation to overly compartmentalize your learning. We can
only discuss so much at a time in class, and we must read things in
some order. But once we have covered a number of things, don't forget
to watch for connections. For example, how does what Van de Walle
says about problem solving fit in with the the articles we read
and the material from Musser, Burger, and Peterson?
with your experience solving problems in class?
with the video Double-column addition?
Later in the semester we will repeatedly connect the material from
this early part of the course with specific content areas throughout
the curriculum.
Test 1 Notes
Material Covered: Test 1 covers through Basic Facts. Consult
the course calendar for a list of readings, etc.
Much of the material you have read
falls into the category of foundational work for the rest of the semester.
Chapters 3 and 4 of Van de Walle are especially important in this regard,
and you should be sure you understand the ideas presented there
(and our discussions in class pertaining to those ideas) particularly well.
Test 2 Notes
Material Covered: Test 2 covers through the algorithms for
the four arithmetic operators (addition, subtraction, multiplication, and
division) including other bases. This builds upon a solid understanding of
place value (again including place value in other bases). You should also
be familiar with models for place value and how to use them to model the operations.
You should be able to draw pictures of these models on the test. Base 5 pieces will
be available during the test, but you will not be required to use them.
You may be asked to work in bases other than 5 and 10, too.
A good source of example questions are the recent homework problems and the
activities from class. You should know by name the algorithms mentioned on
activity sheets
25 and 26 as well as "long division" and the "big 7" method
for division. (The big seven is also called "low stress division" on one of
our handouts.) For any other algorithms, you will either be given an example
to follow or be allowed to choose any agorithm you like so that the names are
not important.
The test is cummulative, but will emphsize the newer material
and old material that is directly connected to more recent material.
In the case of test 2, this clearly includes (but is not limited to) things like:
the "big triangle of representations" (models-symbols-words),
meanings of the operations and categories of word problems,
basic facts and other prerequisites for the pencil-and-paper algorithms,
and
conceptual vs. procedural understanding.
Test 3 Notes
Material Covered: The primary new topics for Test 3
are factors and multiples, fractions, and fraction operations (addition,
subtraction, multiplication, and division). Here is a list of
things you should be sure to know about. (It is not intended to be exhaustive.)
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Factors and Multiples:
- definitions of factor, divisor, multiple, prime, composite,
prime factorization.
- GCD (greatest common divisor), LCM (least common multiple), Euclid's algorithm.
- divisibility tests and why they work. (We discussed tests for
divisibility by 2, 3, 4, 5, 6, 9, and 10.)
- how to determine if a number is prime.
- how to get prime factorization.
- how to use a prime factorization to determine
greatest common divisor, least common multiple, number of factors
- relationship between GCD and LCM of two numbers
- Fractions
- meaning of fractions
- models for fractions (3 types with examples of each)
- conceptually based fraction comparison
- Fraction operations
- meaning/models of operations with fractions
- algorithms for addition, subtraction, multiplication, division
and why they work
- common denominators and equivalent fractions (how, why, and when)
A good source of example questions are the recent homework problems and the
activities from class.
The test is cummulative, but will emphsize the newer material
and old material that is directly connected to more recent material.
In the case of test 3, this clearly includes (but is not limited to) things like:
the "big triangle of representations" (models-symbols-words),
meanings of the operations and categories of word problems,
and
conceptual vs. procedural understanding.
Test 4 Notes
Material Covered: The primary new topics for Test 4
are decimals and percents, ratios and proportions, and integers (positive
and negative numbers, including operations with them.
subtraction, multiplication, and division).
Of course, there is a lot of connection of some of these topics to old topics,
epsecially fractions, place value, algorithms for the arithmetic operations,
etc.
Her is a list of specific things you should be sure to know about.
(It is not intended to be exhaustive.)
- Conceptual teaching of new material
- the big triangle (models, symbols, words);
big star (models, symbols, words, real world, pictures)
- the proper role of models
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making connections (to other mathematics topics, to "real world",
to other disciplines, etc)
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attention to procedural/conceptual knowledge involved
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Decimals and percents
- meaning, connections between fractions, decimals and percents
models and place value
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algorithms for basic operations (how and why)
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word problems, especially percentage change problems
- converting between fractions, decimals and percents,
including repeating decimals
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Ratio and Proportion
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difference between ratio and fraction
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word problems
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three solution strategies (unit rate, scaling, cross-multiply)
and why they work
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Integers (positive and negative numbers)
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models, situations
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operations
We'll hold off on estimation and mental math until the final exam.
Final Exam Notes
The final exam will cover material since the beginning of the course.
To prepare, look over your old tests and quizzes and the review materials
for each test. As you do so, in addition to making sure you can do all
the problems listed there, ask yourself questions like:
- What other types of questions could be asked to get at the same
understanding?
- How can this type of question be modified to get at something a
little different? (Examples: If I had you do a taks with a certain
type of model, can you do it with other models as well? If I had you
answer a question about a certain operation, can it be asked about other
operations as well?)
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Was there anything that was "on the list" but didn't get on thet test?
Was this an important topic that just "didn't make it" or was this a
less important topic?
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What themes and topics come up repeatedly throughout the course?
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What topics that came later in the course are related to this topic?
Should I have a deeper understanding of this topic now than
I had at the time because of things we did later in the course?
Note: There has also been some new material since fourth test:
- Estimation and Mental Math
- difference between mental math and estimation
(as we used those terms)
- strategies and explaining how you arrived at your results
- why teach these topics?
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Number Systems: Counting Numbers, Whole Numbers, Integers, Rationals, Reals
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closure properties
- density
- how to represent rationals
(as "fractions", certain kinds of decimals)
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how to represent reals
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examples of reals that are not rationals
(and why they are not rational)
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Pythagorean Theorem (careful statement, uses)
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representing irrationals on a geoboard
- Algebra
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Using variables and equations to represent number relationships
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Solving algebraic equations for unknown quantities (2 rules)
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Models for algebra
- Graphs
- situations -> graphs; graphs -> situations
- role of axes (independent and dependent variables)
- continuous vs. discrete
This page maintained by:
Randall Pruim
Department of Mathematics and Statistics
Calvin College
rpruim@calvin.edu
Last Modified:
Saturday, 01-Dec-2007 11:14:44 EST