An Important Message To Teachers, Parents, and Students
What are grade reviews?
The word “grade” refers to elementary school, or grade school as it was called decades ago. The word “review” suggests repetition, although certainly practice may include some extension of knowledge. The word “grade” parallels the word “middle” in MAVA Math: Middle Reviews which is a collection of reviews at the middle school level.
What is the importance of cumulative review?
Children cannot learn without cumulative review. All children, regardless of math ability, have difficulty remembering skills and concepts unless they practice them within a year as well as from year to year.
On which curriculum is this book based?
A grade level curriculum outline corresponding to the problems in this book appears after the fifth grade reviews. As no one national curriculum exists and curricula vary somewhat from school to school and book to book, this book uses a blend of common approaches.
Does this book use a developmental approach?
Yes, this book’s curriculum advances each topic from level to level. For example, Level 1 has addition of two 2-digit numbers without regrouping; then Level 2 has addition of two 2-digit numbers with regrouping. Or, Level 1 asks students to draw one line of symmetry through a figure, while Level 2 asks students to draw up to four lines of symmetry, as well as recognizing when no line of symmetry exists. Careful checking of the problems in this book has produced a logical progression of concepts and skills within each topic.
Are these topics and problems all that exist in a typical curriculum?
No, these topics and problems comprise a representative sample within the framework of the book–namely, ten problems per page in a two-column format. Especially in the higher grades, some valuable exercises, such as statistical graphs, cannot fit in the allotted space. Creating a math book that is totally comprehensive is virtually impossible. While other problems could be created, a curriculum that is too broad does not permit mastery.
Does this book’s curriculum vary dramatically in any way from those typically seen?
A significant difference is this book’s introduction of fraction multiplication in fourth grade instead of fifth. Teaching a child to simplify a fraction in fourth grade is illogical without knowing that simplification means crossing out an unneeded multiplication by one.
How is this book best used?
At one school students may study fractions in the fall, whereas at another school students in the same grade may study fractions in the spring. Thus, to make this book universally available, the review problems cover the entire grade level rather than follow the sequence of one school or another. This book is best used by starting a grade level in January of that grade, completing about one third during the second semester, completing another third during the summer, and then completing the final third during the first semester of the next school year. A typical first grader would do Level 1 Numbers 1–15 (fifteen) from January to May of first grade, Level 1 Numbers 16–29 (fourteen) from June to August, and Level 1 Numbers 30–44 (fifteen) from September to December of second grade. The supervising adult should take care to spread the reviews over the available number of weeks.
Why does this book have so many problems?
People need practice to master any skill–not only in math but also, for example, in music and sports. The MAVA Math series of textbooks provides plentiful problems, so needed by students but often unavailable.
What is the grade level of this book?
While this book aims at grades one through five, some children may begin earlier. Also, children in middle school lacking certain concepts and skills may find this book useful.
What do the headers “Level” and “Number” mean?
“Level,” a synonym for “grade,” is a more flexible word because children may work above or below grade level. “Number” simply counts the review pages. Within a level, a lower number does not imply that the review is easier than one with a higher number.
How does this book interact with MAVA Math: Number Sense?
The day is too short and the subjects are too plentiful to have adequate time in school (or home school) to learn math competently. Essential supplementary work includes practicing number sense and completing cumulative reviews. As the two approaches are different but complementary, MAVA Math: Number Sense and MAVA Math: Grade Reviews should be used concurrently.
Should students use mental math when they are able or always show full work?
Students who can use mental math and get problems consistently correct should do so. Forcing a competent child to show work will only frustrate the child and slow progress. Moreover, teaching rigid, absolute methods to talented children leads to lack of trust and understanding. However, if errors are frequent, then the supervising adult should insist that the child show more work. Evaluate the child before sacrificing flexibility, keeping in mind that more complicated problems require work by all children, especially to practice skills.
Should students complete a page before starting another page?
Because curricula vary among publishers and schools, students can omit occasional problems from a page and then return to them later. Also, students need not do the pages in order within a grade level.
Was this book field tested?
Yes, this book in draft form was field tested at an independent school with about eighteen students in each of three classes per grade. Virtually all students raised their standardized test scores, both in the national and independent school norms. Some students had very significant increases. Equally importantly, the children showed happiness toward math as they gained familiarity, comfort, and understanding.
May parents help with the reviews?
During field testing, the students who received help from their parents showed significantly less growth in math than the students who learned to work independently. Parents should only monitor a child’s work, determining weak areas.
Why are occasional problems quick or easy?
An occasional quick or easy problem on a problem set is a welcome respite for a student and keeps both the pacing and progress moving forward.
Does this book prepare for future math instruction?
Yes, this book looks ahead to skills needed in classic algebra word problems such as age problems needing charts. Moreover, it provides a solid foundation for middle school math.
Should children use a calculator with this book?
Children should not use a calculator with this book. They are still learning basic facts as well as developing number sense.
Why are the decimal points bold?
Some children do not see decimal points in normal font. Similarly, some students do not write decimal points darkly enough. A happy medium exists between a light dot and a wart.
How are answers distinguished from work in MAVA Math: Grade Reviews Solutions?
Work is shown in the same font as the problem. Answers appear on the answer lines or in the charts. Otherwise, the answers are circled to distinguish them from the text. Students need not always circle their answers as their pencil writing will appear different from the printed book. Additional comments appear in the smaller “chalkboard” font.
Why are some answers in bold font?
In charts, number lines, and the like, some numbers come with the problem and some numbers are answers. To distinguish between the two situations, the bold numbers are answers. The numbers in regular font appear in the student book as part of the problem.
Why do some answers abbreviate words?
Math time should not be treated as an opportunity to teach language arts, whether spelling or handwriting. Too many students have difficulty with math or learn to dislike the subject. Attaching verbal skills to the study of math handicaps many students otherwise talented in math. Ideally, students will learn to correctly spell and neatly print math vocabulary words. However, requiring these skills while learning math is self-defeating. The abbreviations used in this book are found on page vi.
Why are the answers to some measurement problems a number without a label following?
Consider the question: How far do you live from school? The answer could be 2 miles or 2 turtle steps. A label is needed for accuracy. Now, consider the question: How many miles do you live from school? The answer may be 2 without ambiguity because a label, namely miles, is built into the question. Requiring a label at all times is incorrect.
What does the word “unit” mean?
Unit is a general term. Regarding distance, “unit” may mean many different measurements such as inches, feet, miles, or centimeters. Understanding that the label must be square units for area and cubic units for volume is more important than what the actual unit is. By removing the unit, a student may focus on the specific skill.
Should all improper fractions be converted to mixed numbers?
No, the term “improper fraction” is a misnomer. Converting from a fraction greater than one to a mixed number is a valuable skill, but it need not always be done. For example, as a solution to an equation, the fraction is better because it may be plugged in directly to check its validity. However, measurements are best as mixed numbers. For example, one and three fourths cups flour is more helpful than seven fourths cups.
Why do some 4-digit numbers have commas and some do not?
The comma in a 4-digit number is optional. Its use best depends on whether the emphasis is thousands or hundreds. For example, 4100 may be more helpful as four thousand one hundred in one problem but as forty-one hundred in another problem.
Which are more valuable–fractions or decimals?
Decimals are useful in working with money. However, decimal math may be easily done on a calculator. Fractions are more important in higher math. For example, a student who does not understand how to add 1/2 + 1/3 cannot possibly add 1/x + 1/y. Furthermore, fractions may yield an exact answer whereas decimals would yield an approximation if the numbers in the problem convert to repeating decimals.
May students and teachers write diagonal fraction lines?
Never! For correct fraction work, students must clearly see numerators and denominators, only accomplished by writing horizontal fraction lines. This book uses diagonal fraction lines only in paragraph form (see above answer) or in the hints due to limited space.
Why do the pages for levels 4 and 5 have a smaller font sizes?
Younger elementary school children are more comfortable with a larger font. Older children may handle a smaller font needed to accommodate longer problems in the allotted space.
Why do some problems or skills have more than one method shown in the solutions book?
To find the perimeter of a rectangle, should one add the length and width and then double, or should one double each measurement and then add? Students should understand both methods and decide based on the numbers in the problems, always aiming for easier arithmetic. Similar questions may be asked of other techniques. The inherent richness and beauty of math yield multiple approaches to many problems. Some students prefer one way of thinking and some another.
What is the best way to memorize basic facts?
Flash cards are very helpful when used regularly at bedtime for 5 minutes or less. Short, oral, daily repetition over a long period of time builds retention. One operation each night is fine. Doing flash cards with a family member is supportive and nurturing. Over time with number sense work, other facts such as 85 + 15 or 12 x 5 become second nature.
What happens if MAVA Math: Grade Reviews Solutions contains an error?
MAVA Math: Grade Reviews Solutions was thoroughly proofed. However, any needed corrections will be posted on www.mavabooks.com. If a correction does not address your concern, then send a concise and precise e-mail to info@mavabooks.com.
What happens if a student has trouble with a problem type in this book?
Cumulative reviews help to find topics that children have not truly learned. Remediation, backing up in grade level as much as necessary, should occur.
How can one learn more about various problem types and solution methods such as GCF, LCM, prime factorization, cross-tabulation charts, and multiplication principle?
While MAVA Math books do not have adequate space to fully teach concepts and skills, www.mavabooks.com does offer public service chalkboard slide shows giving instruction on various topics. Visit the website often because new material appears continually.
This book uses the term “function table.” Are elementary school students supposed to know the correct definition of a function?
No, function is not an elementary level concept. These tables could be called “operation tables.” However, this book uses the term “function table” to match most national textbooks. In general, teaching a child correct terminology initially is preferable to changing vocabulary in later years. However, one always runs the risk of children misunderstanding or teachers misinforming.
Should a student in first grade know the meaning of an ordered pair?
Again, problems may use the correct notation without students totally understanding the meaning of the parentheses. With time, students will see that, for example, (3,4) is not the same ordered pair as (4,3). The concept of ordering develops over time.
Why are equilateral triangles also labeled isosceles?
The correct definition of isosceles triangle is a three-sided polygon with at least two congruent sides. Therefore, an equilateral triangle is isosceles, but the converse is not true.
How rigid is the vocabulary list at the back of the book?
The vocabulary list, similar to the curriculum guide, is approximate. One must have guidelines to chart the progress of students and to have finite work for each school year. However, students learn at different paces across math in general and within specific topics. Moreover, as just one example, a two-year-old may be capable of identifying hexagons. By the end of seventh grade, students should know over five hundred math vocabulary words. This list appears in MAVA Math: Middle Reviews.
Why do some of the word problems have labels next to the work?
Many children have difficulty with word problems. While the labels are not necessary, if they help children focus on how their work matches the problem, then the labels are beneficial. Again, full words are not necessary as long as the students understand the abbreviations they select.
Should a student assume that lines are perpendicular and angles are right even if the symbol is not present?
Yes, at the elementary level, for example, one assumes that a picture of a square is a square, whereas at the high school level students may assume nothing. While some right angles in this book are detailed, marking all of them would introduce too much clutter for young children.
Why does this book have fictitious operations?
Fictitious operations are found in the math sections of the College Board SAT. Many high school students skip these problems because they find them unfamiliar and strange. In fact, these problems are quite logical and serve as another way to practice substitution. As with other topics, by starting simple and progressing in difficulty, understanding builds.
Should math be fun?
Of course, math should be fun. However, teaching math solely as a game does not lead to growth. Students who complete weekly cumulative reviews gain enough practice to truly learn math. Understanding in turn leads to natural enjoyment. Competence is pleasurable.
© 2008 Marla Weiss