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, teacher's manual and a homework practice workbook. The book contains 13 chapters with from 7 to 15 lessons.
A quick walk through the teacher's edition reveals remarkably few bells and whistles required for each lesson. In many cases the teacher might present one transparency and the rest is board work. The students do divide into groups from time to time to play a game to work with snap cubes or dot cubes (dice), but these activities relate directly to the lesson. Key concepts and skills, such as number facts and writing number sentences, are given top billing.
In general, the teacher's book gives slightly more aid to teachers in planning and presenting lessons than some of the other programs.
Lessons generally follow the same pattern. Prior to the lesson, the teacher may use one or both of two optional group activities as a getting started activity. Each lesson begins with a short introduction containing review, a story, or some activity lead by the teacher, with student participation, which introduces the topic for the day. This section also directs specific vocabulary development to understand the lesson. The teacher then goes to the first page of the two pages of student work for the day and uses the example at the top of the page as a teaching device to go over the lesson and show students how to do the material for the day. Students finish the exercises on their own, with the teacher checking and helping as necessary. At the end the teacher then closes the lesson with a short discussion question and a verbal skills assessment. There is then a one page homework assignment in the homework book. Each lesson also comes with reteaching, enrichment and problem solving worksheet activities. There is relatively little guidance on when or how to use the optional introductory or follow up activities.
A series of activities related to math are given to fill some of the gaps in the day, such as lining up time. Children do things like skip count, practice number facts, determine how many groups of two there are and so on.
The first chapter has the usual mish-mosh of graphs, data collecting and introductory activities, but later chapters get down to business.
Content Area Evaluations
Addition and Subtraction of Whole Numbers [4.5]
Addition and subtraction are taught in two large sections. Number facts and their mastery are in the first block. Chapters 2-4 move in a smooth progression from addition and subtraction concepts, to actual strategies for learning number facts (doubles, doubles +1, making 10, when adding 9, etc.), to fact families and the relationship between addition and subtraction. At every step students are encouraged to write number sentences.
Chapter 2 stresses the use of stories, drawings, number cubes and the number line to make concrete the relationship between number facts, concrete objects and counting. There is lots of practice in recognizing word stories and related number relationships and in writing word stories to go with an addition or subtraction problem. Although all of the basic addition/subtraction facts are used, with the various aides mentioned above, they are not mastered at this point.
Chapter 3 builds from the concepts of chapter 2 to the teaching of explicit strategies for each group of number facts. This starts with doubles, then moves to doubles plus 1, pairs that make 10, adding 9, and 6, 7, 8 facts. The second half of the chapter then deals with the related subtraction facts. Chapter 4 then builds fact families and starts the drills that should lead to mastery while at the same time teaching the use of addition to check subtraction. 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, nor is it clear that there are enough interspersed activities to practice these to mastery.
There is then a gap in this topic until chapters 8, 9 and 10. Chapters 8, 9 and 10 provide another thorough and systematic approach to addition and subtraction, this time for 2 and 3 digit numbers. Regrouping (carrying, borrowing, renaming, trading) is presented as the tool for adding and subtracting multidigit numbers. (Of course, the children start out with addition and subtraction problems not requiring regrouping). The text clearly deals with how addition can be used to check subtraction. In fact, the relationship between addition and subtraction is stressed from the earliest lessons on addition and subtraction. This is an important topic to stress and master early.
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 8 contains 8 lessons leading to the standard regrouping method for addition. This starts with stressing, using tens sticks and ones cubes, that adding 2 tens plus 3 tens is 5 tens, just as 2 ones plus 3 ones is 5 ones. Then a 100 grid (a table of 1-100 in 10 rows of 10) is used to illustrate adding multiples of 10 to any number. This leads to adding multiples of 10 to any number mentally, by increasing the tens place accordingly. Regrouping is introduced in 8.6 via a game in which two one digit numbers, modeled with blocks, are added. If there are more than 10 blocks in the ones area, then ten are removed and a tens stick is added to the tens area. The next lesson extends this to two digit plus one digit addition with carrying (regrouping, renaming, trading etc.), still using blocks as a modeling tool as well as using numbers. In lesson 8.8 formal carrying is introduced while maintaining the examples with blocks, for two digit plus one digit addition. There is surprisingly little explanation of carrying in the book or scripted in the teacher text, although a little box above the tens place and the examples guide the students to carry the extra 10 ones as a single ten. The next lesson extends this to two digit plus two digit addition, still with blocks as a backup. 8.10 finally has the students add and carry without the manipulative backup, and even eliminates the box to guide placement of the carried number. There are no problems at this point with answers greater than 99, so there is no carrying from the tens place to the hundreds place. The next two sections introduce adding cents (totals below one dollar) and adding 3 two digit numbers (total less than 100).
Chapter 10 introduces numbers up to 1000, with addition beginning in section 10.8. Development of the topic extends over 4 sections which basically recapitulate the presentation for adding two digit numbers and finishes with students able to add two three digit numbers with sums less than 1000.
Multiplication of Whole Numbers [2.5]
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 topic is covered in 6 lessons of Chapter 13, the last chapter in the book. Some skip counting exercises with 2, 3, 4, 5, 6 and 10 is done earlier in the book.
The first lesson deals with the ideas of equal groups, say groups of 3 or groups of 4. Unfortunately, the particular presentation is cumbersome and confusing. Each group isn't just three things, like 3 blocks, it is 3 apples in a bag, or two wheels on a bike and so on. This may be a good problem solving strategy later on, but at this point it confuses two different ideas. The second lesson contains a review of skip counting and then builds from this to add repeated addition and introduce the "x" for multiplication. Again, the presentation makes a good idea less than fully clear.
After two bad presentations, the third lesson has a very good presentation of arrays, with students noting the number of rows, the number per row, and the total. The next lesson builds on this system with the introduction of the term product and instruction on the fact that 3 x 5 = 5 x 3.
The fifth lesson on this topic introduces vertical multiplication notation, in addition to the horizontal notation used before, and students now model with pictures of objects rather than just with blocks. Section 6 is a problem solving activity applying the ideas presented in 1-5 to solve problems and section 9 deals with when to use a particular operation such as multiplication.
In the meat of the lessons, there is no obvious introduction of any specific group of key facts (say the 2's table or the 5's table). There is one Gifted and Talented activity that involves making the full multiplication table up to 5 x 5, but that is it.
This book covers the concept of multiplication in multiple ways, but some of the presentation is less than clear. Some of the basic terms are presented and some sense of when to multiply is gained, although students do not come to own any set of multiplication facts.
Time [3.5]
Time is taught in Chapter 7. The chapter begins with the concept of a minute. The children are asked to think about activities taking more or less than one minute. The clock face is presented and detailed, starting with the statement that there are 60 minutes in an hour. Students are taught to write time using "o'clock", and the colon method. The chapter continues with sections on telling time to 5 minutes, half hour, and quarter hour. Alternative expressions for times (half past, 15 minutes before) are taught. A two page section on reading a schedule provides an introduction to elapsed time (in hours).
The calendar section is brief, covering two pages. The first page has students fill in a calendar for the current month and answer questions about the number of days of the week, days in this month, one date that is on a Thursday, and so on. The second page presents a full year of calendar pages, having the children circle certain favorite dates and make a table of months having 28, 30 or 31 days.
Little emphasis is placed on writing the date in this chapter. The teacher's edition presents a daily calendar routine that includes stating the date and finding it on a calendar as well as stating the time.
Money [3.5]
Money is presented in Chapter 6. The chapter is divided into two sections, A, dealing with counting, and B, dealing with applications. The first 4 sections concern counting coins with emphasis placed on skip counting by 25's, 10's, 5's, etc. The children are asked to use different combinations of coins to make the same amount, and, in section 4, are actually asked to make charts in which they find all possible combinations of 3 kinds of coins to make a given value. This requires a higher level of organization and thought than seen in other texts.
Section 5 gives two pages of practice finding the fewest coins to make an amount and section 6 introduces the dollar bill. The structure of the dollar amount is explained, pointing out that the decimal point separates the dollars from the cents. The students precede to find combinations of coins equal to $1.00. The last section of chapter six involves making change with amounts under $1.00.
Chapters 8 and 9, which present two digit addition and subtraction with regrouping, each contain a 2 page section on computing with coin amounts. The students are reminded to use the cents sign in their answers. Addition and subtraction with dollar amounts is not represented in this text.
Measurement of Length, Weight, Volume and Temperature [3.8]
Measurement is presented in 13 lessons of Chapter 11. After an introductory, "explore" section, in which non-standard units are used to measure common objects, the text presents inches, feet, and yards and the relationship among these units. The next section has the children measure items with centimeter rulers and meter sticks, and gives them some examples of common items that measure about a centimeter or a meter. The relationship between these units is directed for explicit statement in the teacher's guide, but does not appear in the student text, nor does it appear to be practiced. For neither the metric nor standards systems are fractional units discussed, even if this is implicit in the conversion of feet to inches or yards.
Note that this unit has a number of exercises involving measurement of students' bodies (e.g. unit 11.2). This involves students in measurement and makes things real to them, but there may be unwanted social interactions if these become public exercises. For example, do we really want to establish a situation in which we can break up into the long arms and the short arms, or the big feet and the little feet? Is the teacher prepared for the unintended and probably unwanted diversity and tolerance lesson on the differences between students' bodies?
Following these lessons, perimeter is defined and used in several exercises which test visual thinking as well as ability to measure. Area is presented in a hands on section in which various geometric shapes are covered in unit cubes (subbing for square unit areas). Students do some interesting exercises in which they use 12 cubes to create various rectangles and then measure the perimeter of each. Through this exercise they learn the fairly sophisticated concept that shapes can have the same area, but different perimeters.
The following lessons in the chapter introduce students to pounds, kilograms, cups, pints, quarts and liters. Most of these are done by showing what object might have a given unit weight or volume and then asking whether other items have measurements greater or less than this. The relationship between cups, pints and quarts is presented arithmetically with unit conversions clearly stated. A section on choosing the correct measuring tool (ruler, scale, measuring cup) follows. A single lesson deals with temperature measurement in Fahrenheit and Celsius. Both scales are presented on a single thermometer, so that students can gain a feel for the relationship between the two scales. The teacher's book asks for explicit discussion of the degree symbol and of the two temperature scales. Teachers are also directed to mention key benchmark temperatures including room temperature, water freezing and water boiling. Temperatures on each scale are related to activities and weather consistent with that temperature. Unfortunately, student exercises ask students to color in a thermometer to match a particular temperature, but not to read a thermometer (although one alternative teaching strategy suggests reading thermometers and writing the temperature).
Altogether, this is a thorough coverage of the topic, with the exception of temperature which has some strong points and one notable weakness.
Perimeter [4.5]
As noted above, perimeter is covered in the measurement chapter, with perimeter introduced before area. The first lesson begins with the teacher reading the definition of perimeter from the top of the work sheet. This leaves no doubt as to the meaning of perimeter. Both the work sheet and the teacher stress that one measures all sides and adds their measures to get perimeter. Exercises practice on classroom objects and on shapes on worksheets. None of the problems strain the previously mastered addition skills of the students. After area is introduced, students calculate and compare the perimeters of a series of multisided objects with the same area (as defined by a set of unit area markers). There are no exercises with curved shapes or drill with more difficult numbers or complicated shapes not generated with unit area blocks. In summary, the key points are covered with some drill, but some extensions of the idea are not covered.
Program Quality Evaluations
Mathematical Depth [4.0]
This program is particularly strong in addition/subtraction and perimeter, moderately good in time, money and measurement, and relatively weak in multiplication. The level of depth is sufficient that a student who masters the material will be well prepared to move on at the next level and in appropriate real world situations.
Quality of Presentation [3.0]
Presentation is variable within this text. Some topics, such as addition/subtraction, are reasonably well presented, with the exception that the discussion of carrying is perhaps not as thorough as it should be. Others, such as multiplication, are disappointingly poor in presentation and level. As a general problem, the teacher's text seems a bit thin in terms of aiding a teacher in actually teaching the lesson at hand. For the experienced teacher with well developed teaching strategies for each key topic, this may not be a problem, but for a teacher who is less knowledgeable about the underlying mathematics, or who is new to teaching, more guidance in terms of presentation of each and every topic may be important.
Quality of Student Work [3.9]
The amount of student work is well matched with other programs. The level of student work generally follows the mathematical depth as described above. This means that there is good coverage of perhaps the key second grade topic (addition/subtraction) and weaker coverage of some other topics such as multiplication.
Overall Program Evaluation
This book has much to recommend it, although the level is low in a few topics. Some areas are very well taught and at an excellent level. At the top level of students, and for the eager mid-level, some topics should be augmented (multiplication, time, money). For the lower level student needing to catch up, this is a reasonable level of content. The real question becomes, will the presentation in the individual classroom allow all students to succeed at or close to the level of the program.
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