Introduction
This is part of a series of second, fifth, and seventh grade Mathematics Program Reviews. This review includes a summary of the structure of the program, evaluations of a selected set of content areas, and evaluations of program quality. Ratings in these areas were made on a scale from 1 (poor) to 5 (outstanding). The overall evaluation was made using the traditional system of letter grades. For details of the methods used in this evaluation see Methods for Second Grade Program Reviews.
Structure
This program comes with a student text and teacher's manual, as well as additional materials for homework, reteaching and extension. Programs for particular states may contain exercise books to prepare students for the specific state assessment. The book contains 12 chapters that, according to the well presented pacing charts, take from 11 to 19 days.
The teachers guide contains a clear and detailed pacing guide. If followed, 168 days would need to be dedicated to instruction and activities from the text to cover the material in this program. As most of the topics covered are important at this age, it becomes critical for teachers to keep on track. Math must be taught every day, it cannot be skipped just because there is an assembly or a special activity like the Halloween parade.
Although there are exceptions, the book stays reasonably clear of calculators and computer games. In general, the book is complete, without much reliance on technology. The pages are clean, without a lot of distracting photos. There is a relatively heavy use of manipulatives even into late in the course, especially blocks for ones, tens and hundreds. In some cases this might be viewed as less efficient than stressing the meaning of place value and the number system directly. Teachers might be well advised to tailor the use of these to the needs of individual children, allowing those who have mastered the skills to skip the manipulatives and advance on to the enrichment sheets provided in the teacher's manual.
Within a chapter, in addition to the individual lessons, there are mid-chapter and end chapter reviews, a chapter test, and various extra practice activities including games and non-game activities.
The lessons are generally similar in design. Prior to the lesson, there are possible manipulative warm ups or connections to other subjects. There is then a Daily Review with both problems related to recent lessons and drill questions on number facts, and a Problem of the Day. The lesson itself begins with an introduction in which the teacher explicitly presents issues related to the day's lesson, and then asks the students to do related things or to answer related questions. The teacher then guides the students through the example on the first of the student book pages related to the lesson with discussions of various possibilities. Students then do the problem set and often do an additional activity such as review exercises or additional exploration activities. Finally, the teacher closes the lesson by checking for student understanding, usually by asking for explanations of how students solved various problems. Although not explicitly stated, the practice, reteaching or extension exercises are then given as homework or extra class work. The directions to teachers have above average clarity as to the course scheduling and to presentation of individual lessons. These are not fully scripted, but the descriptions of what a teacher should do and say to present the program as planned are more detailed than in most other programs.
Content Area Evaluations
Addition and Subtraction of Whole Numbers [4.3]
Addition and subtraction are taught in essentially two groups of lessons. Number facts are taught in Chapters 1 and 2. Chapter 1 deals with addition and subtraction up to 12, while Chapter 2 carries on to addition and subtraction up to 18. Strategies used include counting on or counting back (introduced with a number line), finding all pairs that add up to a particular number (10 or less) or all subtraction facts from a particular number, doubles, doubles + 1, the "nine rule" which is discovered, and making ten before adding on the leftover (in say 8 + 3 goes to 10 + 1 = 11). The idea of fact families appears at the end of Chapter 2. The presentation sometimes seems a bit jumpy, and perhaps not as directed at systematic mastery of the facts as it might be. The use of two different manipulative systems (cubes and two color counters) looks as if it may end up being more confusing than informative. This section has a relatively heavy emphasis on "patterns" as a way of doing and learning things. This same emphasis carries over to other sections. At times the emphasis on patterns seems misguided or less efficient than other methods of presentation.
Although there are problems in the book and some extra drill exercises in class, one does not see any assessment that asks children to demonstrate mastery of all the sums to 18, although there are about 4 daily "Fast facts" number facts to be answered at the start of math time.
Place value is introduced in Chapter 3, teaching students to recognize, for example, 30 as 3 tens. Students then divide sets of items into groups of 10, noting how many tens and writing that as a number as well, leading to doing this when there are some ungrouped items, such as 3 tens and 2 ones = 32. Although not directly addition and subtraction, this is key for later multidigit arithmetic.
Two-digit addition and subtraction are introduced in Chapter 6, then continued in Chapters 7 and 8. At the end of Chapter 11, addition and subtraction are extended to 3 digit numbers to 999. In these sections, regrouping (carrying, borrowing, renaming, trading) is presented as the method to solve these problems. There appear to be only three of the practice or mixed review exercises refreshing addition or subtraction skills between Chapter 8 and the three digit addition exercises in Chapter 11. This is, according to the teacher's lesson planner, 40 lessons (at least 8 weeks) with little obvious practice of these important skills.
Because of the importance at this age of mastering addition and subtraction of multidigit numbers, a detailed description of the scope of adding such numbers follows. Subtraction follows a parallel path, usually a chapter later.
Chapter 6 "explores" the concepts of multidigit addition and subtraction. The first lesson connects addition of two one digit numbers, say 4 ones and 4 ones to get 8 ones, to adding 4 tens and 4 tens to get 8 tens. This is a fairly standard way to start this topic, but the presentation here seems a bit less clear than in other programs. The next set of lessons deals with adding and subtracting groups of tens by counting on or counting back by tens. This is again a standard second lesson on this topic with a less-than-completely-clear presentation. The next sections in this chapter introduce mental math with two digit numbers. Unfortunately, even if by design, these lessons appear before the students understand carrying (regrouping, renaming, trading) in the context of place value. There is a nice problem solving exercise on two step problems as well.
Chapter 7 begins with a regrouping exercise, for example, given 12 ones blocks regroup them into 1 tens stick and 2 ones blocks. Students decide if a number can be regrouped and then do it both with blocks and with numbers. Two digit plus one digit addition is modeled first with blocks ( for example, 16 as one ten stick and 6 ones plus 8 as eight ones blocks, the ones are regrouped into one tens stick and 4 ones blocks, for a total of 2 tens and 4 ones). These are not explicitly stated as addition. A similar strategy is used to introduce two digit plus two digit addition, although this time it is made explicit that this is addition. In this same exercise, formal carrying first appears, with the presence of an empty box above the tens place in the written addition problems. Indeed, the appearance of carrying is accompanied by a note that you can add tens and ones without models. Word problems invoking these skills follow these exercises. The importance of checking addition by adding the other way ( a + b = b + a) is given a full lesson. The chapter ends with adding cent amounts of money (no sum more than 99 cents).
Chapter 11 builds from this on to addition up to 999. Most of the chapter is devoted to place value and ordering numbers in the range of 100-999, but three lessons toward the end of the chapter cover regrouping of tens into hundreds (12 tens is 1 hundred and two tens), addition of three digit numbers modeled as blocks and with numbers, including carrying twice, and additional practice.
This is an acceptable but somewhat mixed coverage of the topic. Basic facts may not be mastered as efficiently as one might hope, but the coverage of adding and subtracting larger numbers is generally decent.
Multiplication of Whole Numbers [3.0]
At this level, multiplication is largely an exposure topic rather than one targeted for final mastery. On the other hand, understanding of the concept should be built and some facts can be learned and applied.
This book contains some, but perhaps not as many as might be hoped, skip counting exercises with 2, 3, 4, 5, and 10 in the first half of the book. Multiplication is directly covered in five days in Chapter 12, the last chapter.
The first lesson takes one day and is an exploration of multiplication. The class starts with a review of skip counting, then divides into groups of 2 for an exercise in understanding multiplication as addition of equal groups. Each group has a five space spinner and some counters. One student's spin determines the number of counters per group and the second student's spin determines the number of groups. Students then figure out the product for each combination by skip counting. In principle, students could see a large fraction of the multiplication pairs up to 5 x 5 in a relatively short time. The homework continues this with groups already given.
The second section actually takes 3 days. The first day starts with a warm-up exercise repeating the spinner activity of the day before and a quick review of a few of the "group" problems. It then introduces the idea of multiplication as repeated division, the term multiplication, the "x" notation for times, and the horizontal representation of a multiplication sentence. Finally, there is an introduction of some of the 1 x, 2 x and 3 x facts. An exercise for early finishers suggests that they make flash cards for multiplication facts. The second day of this lesson is similar to the first, with introduction of the term "product," and exercises with some 4's and 5's facts in the lesson. The third day in this section introduces the commutative property (2 x 3 = 6 and 3 x 2 = 6) and the term "factor," as well as mixture of problems using facts up to 5 x 5. An explore exercise at the end of the practice sheet contains multiplication by 0 up to 5 x 0, and sums it up with the zero property of multiplication.
Lesson 3 is multiplication problem solving, with making tables equivalent to a portion of the multiplication table. In a game after the midchapter review, students make a table of facts up to 5 x 5.
For what it does this presentation is quite clear about multiplication concepts and, through various review problems and activities, does systematically introduce students to a reasonable portion of the multiplication table. It is too bad that the area/array representation of multiplication was not also presented since it fits with other topics and makes some concepts clearer for some students. In five days it is unlikely that any but a few number facts will be committed to memory. This is not a complete coverage of the material that this review would hope to see, but students will be moderately prepared to learn more in the next year.
Time [4.0]
This book is very thorough in its presentation of time using analog and digital clocks. Telling time is presented in three sections, starting with hour and half hour, moving to quarter hour and 5 minutes. The alternative expressions for telling time (half past, quarter to) are clearly presented. The problem sets are fairly short but there are many accompanying practice pages offered in the teacher's edition. Once the basics are taught, emphasis is placed on finding the time a certain number of minutes later, or a quarter of an hour later, than the given starting time. The second half of the chapter has the children write daily schedules and solve time problems by working backward (A sale starts at 12 o'clock, the bakers must arrive two hours earlier. When should they arrive?) .
A six page section on reading a calendar follows. It includes writing the date correctly, finding the number of days in a particular month and locating such dates as the "fourth Tuesday in May." Ordinal numbers are defined and used in stating the date.
The chapter concludes with a section on reading schedules, sadly, using a TV guide as the example. The students use the schedule to compute elapsed time and to recognize overlapping times.
Money [3.8]
Money is introduced in Chapter 4. Section 1 involves counting nickels, dimes and pennies. Quarters and half dollars are introduced in the second section. A 2 page problem solving section follows, along with some games which reinforce counting skills. Lesson 4 introduces the dollar bill and provides counting activities with dollar amounts up to $3.00 and change. Section 5 compares money amounts and includes amounts over $1.00. The counting section is reviewed with a money game in which children add and subtract coins from a piggy bank. The final section deals with making change. Although 2 digit addition and subtraction have not yet been taught, the children make change from $1.00 by counting up (the classic method of making change).
Money is added and subtracted in Chapter 8, following instruction in regrouping. Both coins up to $1.00 and 2 digit dollar amounts are added and subtracted (example, $79 + $69). Lesson 9 of Chapter 11 presents adding and subtracting of dollar amounts up to $9.99. While decimal notation is not noted in the student text, the introduction section of the teacher's edition suggests looking at such problems as 975 - 384 and $9.75 - $3.84 and discussing how they are similar and how they are different. Presumably, although not explicitly stated, the role of the decimal point will come up and, possibly, the concept of whole dollars vs. hundredths of dollars will be explained.
Measurement of Length, Weight, Volume and Temperature [3.5]
Measurement is taught over 8 lessons in Chapter 10. Lesson 1 introduces the concept of measurement with non-standard units, followed by inches, feet, yards, centimeters and meters. The exercises involve looking at various classroom objects, estimating the measures of these in the specified units, and then using a ruler, yard stick, or tape measure to find the measurement. No fractional units are considered.
Prior to lesson 1, there is a short optional activity (math center) that involves working with a partner and measuring distances on a tracing of the hands of the students. Generally each student measures his or her own hand and compares distances among his/her own measures. One worries about exercises in which students start comparing aspects of their own bodies relative to other students. As set up, this may not be a problem, but certain presentations or extensions might set up unwanted negative social interactions.
Lesson 2 introduces perimeter, having students find perimeter for classroom objects and simple polygons. An "explore" activity, briefly introduces the concept of area. The square unit is presented, and paper squares used to cover various objects and record the area. A second short problem set has the students color in a certain number of square units on a grid. No attention is given to the way perimeter might vary for different shapes having the same area, even though this exercise seems to lead directly to that discussion.
The next two sections explore pounds and kilograms. In both cases children are asked to estimate what objects weigh in the given units (more or less than 1 kg or 1 pound objects) and then to measure. The actual measurements are recorded in a table with the words, "about ___ lbs" or "about ____ kgs" provided. Fractional units are obviously not introduced.
Cups, pints, quarts and liters are all presented in brief "discovery" sections. The text provides a table with blanks to fill in for the relationship between cups and quarts. The students use measuring cups and colored water to discover the relationship. The correct numbers are never presented in the text, which is worrisome. Brief mention is made of the fractional units 1/2 cup, 1/4 cup, 1/3 cup, but only to compare briefly.
Lesson 6 introduces Fahrenheit and Celsius temperatures, coupling particular temperatures on each scale to relative conditions of hot, moderate or cool weather, but there is no direct comparison between the two scales. Lesson 7 discusses choosing the appropriate tool (ruler, scale, thermometer. Measuring cup) for making a particular measurement and lesson 8 has them think about reasonable answers to questions such as "on a warm sunny day. About what temperature is it? 8oF, 48oF, 84oF".
Overall, this is an adequate treatment of measurement, although the individual lessons and problem sets are quite brief. Fractional units are not considered and the majority of problems involve estimation and not calculation.
Perimeter [4.4]
As noted in the previous section, this topic is covered in Chapter 10, the measurement chapter. The class presentation involves measuring the distance around the room in "footprints", with explicit discussion that perimeter means distance around. The text follows with an "explore" activity that begins with the direction to "Measure each side and record. Add the sides to find the perimeter." Students then measure classroom objects and some items in the workbook. All measurements are in centimeters. A thought problem and class discussion in the next unit ask the students to consider the differences between perimeter and area. The teacher's book explicitly suggests that the teacher also ask students "Can you tell the perimeter of a shape if you know its area?" There is also discussion suggested in the teacher's book of measuring curved objects, but no actual measurement is done. An "extension" work sheet has students measure and calculate the perimeters of a series of odd shaped or higher sided polygons (octagon). In most cases the multiple addend addition problems should not come close to the top level arithmetic skills learned earlier in the book. In summary, the key points are covered with some drill, but some extensions of the idea are not covered.
Program Quality Evaluations
Mathematical Depth [3.9]
This program reaches a fair level of depth. Some key topics, such as addition/subtraction, go to a moderately good level (but not as good as some other programs), while others, such as multiplication are at a lower level (even if better than quite a few programs). In general, the level of depth is sufficient to allow students to succeed at the next level of their mathematics education and in appropriate real world applications.
Quality of Presentation [3.3]
As with the content level, the presentation level varies from topic to topic. Some key topics (e.g. addition/subtraction) are not as strong in presentation quality as other programs while other topics (e.g. multiplication and time) are presented much better. In terms of aid to the teacher, this program seems to give the teacher far more detailed instruction in the actual teaching of the lessons. This is a two edged sword. If the presentation is well tested and proven to work, then this will be a boon to new teachers and to those who are less mathematically sophisticated. If the presentation is poorly designed but detailed, then both teachers and students will be mislead. Luckily, the guides for teacher presentation seem clear and reasonable ways to teach the lesson are likely to remove negative variation between teachers in the presentation of the material.
Quality of Student Work [3.9]
The amount of student work is completely standard relative to other programs at this grade. Depth of student work generally mirrors the reviews for mathematical depth and teaching, as depth of and teaching style are both reflected in the quality of the student work.
Overall Program Evaluation
Students using this book have a reasonable chance to be well prepared to succeed at the next level of mathematics education and in appropriate real world situations. On the other hand, this book has some very strong points and some notable weaknesses. There is variability in depth from topic to topic and variability in presentation. There is the strong point that teachers are well guided in appropriate teaching of each lesson, but poorly designed lessons remain so even if well taught. This array of strengths and weaknesses forces those deciding which books to use to make a choice about both presentation and content, as well as relative worth or particular topics when choosing a book.
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