An Important Message To Teachers, Parents, and Students
What is number sense?
Number sense is basic insight into the structure and use of numbers. For example, a student without number sense will do long division to find 600 divided by 4. A student with number sense will say that half of 600 is 300, and half of 300 is 150, saving energy and time. Or, a person without number sense will subtract across many places. Someone with number sense will add back rather than subtract to obtain the answer faster. Number sense does not replace or precede basic arithmetic algorithms. However, one who truly understands the basic algorithms knows how and when to alter them. Number sense is to math as common sense is to life.
What is the difference among number sense, mental math, and quick math?
All are related. Mental math implies that students do problems solely after hearing them. Quick math implies that students do problems primarily for the sake of speed. Number sense implies that students may see and/or hear problems; understanding is paramount to tricks. Moreover, with a number sense approach, students may minimally write work.
Why are problems in this book in horizontal rather than vertical form?
Vertical form lends itself to using standard arithmetic algorithms. Horizontal form lends itself to using number sense.
Why should students not show full work for the problems?
Everyone teaching and doing math needs to understand the wide range of math problems. Some, done by professional mathematicians, may require dozens of pages of work. At the other extreme, some should be done completely mentally. Problems commonly seen in textbooks for grades one through twelve require a few digits of writing up to a few inches. Students should not show full work for the problems in this book. Full work counteracts the development of number sense.
How does a student learn to do the problems without full work?
MAVA Math: Number Sense Solutions provides not only answers but also helpful ideas between the two columns of problems. Apply methods from the examples shown to the other problems.
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 number sense may find this book useful.
What are the two numbers at the top left corner of each page following the topic identifier?
The number to the left of the dash indicates the school grade; the number to the right of the dash provides a possible sequencing within each grade.
How firm are the two numbers at the top left corner of each page?
Both numbers are approximations because students’ abilities and schools’ curricula vary. The school grade number corresponds to material in basal texts by national publishers. Still, a given skill may appear in many grades: initially as introduction, the next year as review, and the following year as mastery. Moreover, children at the start of a grade are different students than at the end of the same grade.
Does a child need to develop number sense given the availability of calculators?
Thankfully, for children who do not develop number sense, calculators are available in later life. However, no adult should close a door on a child in any subject. Math is one of the two most basic skills in life (language the other) and is essential for so many careers. Overdependence on a calculator cripples a child’s numeric thinking.
What is the best order of topics to follow?
Math does not have one correct order in which to study topics. Students learn math best by studying arithmetic and geometry in parallel with complementary strands of number theory, probability, and statistics. With topics in alphabetical order, students or teachers may find problems that meet a specific need or supplement school lessons. Still, some topics have an inherent order as LCM before fraction addition with different denominators.
Why do some topics have so many more pages than other topics?
Math for first graders begins with place value, addition, and subtraction. These topics, needing deep and careful study, lend themselves to the growth of number sense. For older elementary children, multiplication, for example, offers vast richness for the development of number sense, while a topic such as probability requires a middle school student to study its subtleties.
Are these topics and questions all that exist for number sense development?
No, these topics and problems are a representative sample within a framework of 298 pages of math. One can always think of additional valuable exercises. Also, at the middle school level, more sophisticated concepts are appropriate for number sense development.
Should a topic without pages for a grade level not be studied at all in that grade?
A void of pages for a topic in a grade does not imply that the topic is inappropriate at that grade. Some topics at a certain grade lend themselves better to a different style of math book. For example, a first grader could draw lines of symmetry through shapes or locate figures in the positive-positive quadrant of the coordinate plane. Check www.mavabooks.com for other style textbooks.
What is the significance of the ordering of pages within a topic?
With addition, for example, pages could be organized by regrouping: all problems with no regrouping followed by all with regrouping. Alternatively, as this book does, pages could be organized by number of digits: all one-digit problems, followed by all 2-digit and 3-digit problems. Arguments could be made for either case. Teachers, students, and parents should use pages in any order that suits their specific needs.
Should students complete a page before starting another page?
Students do not need to complete a page before starting another page. In fact, spiraling among pages introduces a cumulative review effect.
What is the meaning of the instruction “find”?
Some problems have a specific operational instruction: add, subtract, multiply, or divide. However, many problems require the student to obtain a valueąfor example, a volume or a factor. While instructions of this general type such as “give,” “express,” “complete,” or “name” are often synonymous, this book consistently uses the instruction “find” to avoid confusion. Students should practice the skill and not ponder the wording of the instruction.
Why does this book have so many problems?
People need practice to master any skilląnot 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.
Can a child really operate from left to right and not get confused?
By third grade, some children can notice quickly that a problem has no regrouping. Then writing its answer from left to right is faster and more natural. Reading and writing occur from left to right, as does division. Teaching rigid, absolute methods to talented children leads to lack of understanding and trust. Evaluate the child before sacrificing flexibility.
Why do the “Contents” pages have colons and semi-colons?
Pages with a single theme have an entry in the “Contents” without a colon or semi-colon. However, pages with two themes have their two entries separated by a semi-colon on one line. If a descriptor pertains to both themes, it appears first, followed by a colon.
What if the page descriptions in the “Contents” are difficult to understand?
The page descriptions in the “Contents” are confined to one line each. They identify exact skills and are best read while looking at the actual corresponding pages of problems.
What happens if students start doing problems mechanically, without thinking, by seeing a pattern in the answers?
Always consider if a student can do the problem on a cumulative review sheet quickly and accurately. If yes, then the mechanical approach has a foundation. Otherwise, stop and return to the page later.
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.
Why do the answers abbreviate words such as “Mon” for Monday or “E” for even?
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.
Why are the answers to 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. The questions in this book either do not require labels or, in the case of unit conversions, provide labels.
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.
Why do some pages have different font sizes?
Often problems have a smaller font size to provide multiple answer columns or to fit within the framework of the book. Overall, a large font size is best for elementary school children.
What do the words “increment” and “decrement” mean in MAVA Math: Number Sense Solutions?
The comments use the computer programming meanings of these wordsąincrement, to add one or increase by one and decrement, to subtract one or decrease by one.
Why do some pages have multiple answer columns?
Just as a music student will practice scales regularly, so may a math student practice certain problems multiple times. When space permits, this book provides several answer columns so that students may do a page more than once. First, use the outermost answer column. On the second use, cover that column with a clean sheet of paper, revealing the next answer column to the left. Continue in this manner until all answer columns are filled. Redoing a page after a period of time or the following year acts as a good review. Triple columns may be used in consecutive years for introduction, review, and mastery.
Why do some skills have more than one technique discussed in the answer key?
Math has inherent richness and beauty, yielding multiple approaches to many problems. Some students prefer way one of thinking and some another.
What happens if a student has trouble with a page in this book?
The fundamental guideline for this book is “break it down.” For any skill, do it with smaller numbers, or do it with whole instead of fractional answers. Break down the skill until it is as simple as possible. Then, after a student is successful with the most basic task, build it back up gradually to the more advanced skill. All skills cannot be broken down totally in 298 pages. Also, a student does not need to attempt every problem or every page.
What is the best way to memorize basic facts?
Old fashioned flash cards are very helpful when used daily at bedtime for 5 minutes or less. Short, daily repetition over a long period of time is important. 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.
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 or forty-one hundred in another problem.
What happens if MAVA Math: Number Sense Solutions contains an error?
MAVA Math: Number Sense 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 firstname.lastname@example.org.
Should math be fun?
Of course, math should be fun. However, teaching math solely as a game does not lead to growth. Students who develop number sense gain insight into the beauty of math, in turn leading to natural enjoyment. Competence is pleasurable.
© 2007 Marla Weiss