2nd Grade - Gateway 2

The materials reviewed for Math Mammoth Grade 2, Light Blue Series, partially meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. The materials do not provide opportunities to develop conceptual understanding throughout the grade level as students are provided the procedure to solve problems during the introduction of conceptual understanding.

The materials do not provide opportunities for students to develop conceptual understanding throughout the grade level. Examples include, but are not limited to:

• Worktext 2-A, Chapter 4: Regrouping in Addition, Going Over to the Next Ten, Question 1, students write the answer for problems that require them to add within 100. “Circle ten little cubes to make a ten. Count the tens and ones. Write the answer. a. 13 + 9 = _____”  Students are shown a picture of one tens stick, with three ones block next to it and a group of nine ones box together. Before the question students are provided with a box that says, “Sums that go over to the next ten”, and then provide students with two different procedures of how to add a sum over ten.  This question does not provide an opportunity for students to build a conceptual understanding of 2.NBT.5 (Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.), as students are provided with the procedure to solve the problems.

• Worktext 2-B, Chapter 6: Three-Digit Numbers, Adding Whole Tens, Question 1, students write the answer for problems that require them to add within 1000.  “Add whole tens. a. 160 + 30 = _____”  A picture is shown of one hundreds block, 6 ten sticks and 3 ten sticks. Three similar questions b - d are provided. Students are then shown a box that states, “651 + 20 = ??  Add 50 and 20, or the tens. That is 70.  The answer is then 671.  Notice: Only the TENS digit will change!  The ‘6’ of the hundreds does not change, nor the “1” of the ones.” Students then work on Question 2 questions a - f to show their understanding. An example: Question 2, “Add whole tens. You can draw illustrations to help. Underline the tens digits to help. b. 412 + 70 = _______”  This question does not provide an opportunity for students to build a conceptual understanding of 2.NBT.7 (Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.), as students are provided with the procedure to solve the problems. 

• Worktext 2-B, Chapter 8: Regrouping in Addition and Subtraction, Regrouping in Subtraction, Part 1, a box is provided with the following information, “We will now study regrouping (‘borrowing’) in subtraction. As a first step, we study breaking a ten-pillar into ten little cubes. This is called regrouping, because one ten ‘changes groups’ from the tens group into the ones.” Question 1, “Break apart a ten into 10 ones. What do you get? Draw or use manipulatives to help.” d. Five tens and five ones are pictured with an arrow indicating the space for the student to draw the broken apart ten. “____tens____ones$$\rarr$$ ____tens____ones.” This question does not provide an opportunity for students to build a conceptual understanding of 2.NBT.7 (Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds), as students are given a procedure to solve the problems. 

There is a cluster and/or standards that specifically relate to conceptual understanding that is/are not developed. For example, there is no evidence of how conceptual understanding is developed for 2.MD.6 (Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.)

The materials reviewed for Math Mammoth Grade 2, Light Blue Series, partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. The materials include multiple opportunities for students to engage with and independently demonstrate routine applications of the mathematics throughout the grade level. However, the materials include few opportunities for students to engage with and independently demonstrate non-routine applications of the mathematics throughout the grade level.

Examples of students engaging in routine application of grade-level skills and knowledge, within instruction and independently, include:

• Worktext 2-A, Chapter 1: Some Old, Some New,  Subtracting Whole Tens, Question 5, students add the weight of items together to solve for total weight. “Solve a. Three suitcases weigh 30kg, 18kg, and 20kg. How much is their total weight?” This problem allows students to apply the mathematics of 2.OA.A (Represent and solve problems involving addition and subtraction) in a routine application problem independently.

• Worktext 2-A, Chapter 3: Addition and Subtraction Facts Within 0-18, Review: Going Over Ten, Question 6, students solve world problems and write an addition or subtraction sentence depending on the situation. “Solve the word problems. ALSO, write an addition & subtraction sentence for them! a. You have $$\$8$$ and you buy a toy for $$\$5$$ and candy for $$\$2$$. How much money do you have now?” This problem allows students to apply the mathematics of 2.OA.A (Represent and solve problems involving addition and subtraction) in a routine application problem independently.

• Worktext 2-B, Chapter 9: Money, Review Chapter 9, Question 6, student  “Lily has $$\$1.26$$. Alex has two dimes, two quarters, and seven pennies in his piggy bank. How much money does Alex have? How much money do the children have together?” This problem allows students to apply the mathematics of 2.MD.8 (Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using \$ and ¢ symbols appropriately) in a routine application problem independently.

The materials provide few opportunities for students to engage with and independently demonstrate non-routine applications throughout the grade level. Examples include: 

• Worktext 2-A, Chapter 5: Geometry and Fractions, Making Shapes, Question 1, students combine different shapes to create new shapes that match given pictures. “Cut out the shapes on the next page. What shapes can you use to make the given shapes? There may be several possible solutions.”  Students are tasked to cut out triangles, squares, and parallelograms and combine them to match the given shapes a - f. This problem allows students to apply the mathematics of 2.G.A (Reason with shapes and their attributes) in a non-routine application problem.

• Worktext 2-B, Chapter 7: Measuring, Mixed Review Chapter 7, Puzzle Corner, students find different numbers that fit a given criteria. “The DIGITS of the number 467 are 4, 6, and 7. The sum of its digits is 4 + 6 + 7 = 17 (just add its digits). Find a number that… 

  • is more than 100 but less than 200;

  • the sum of its digits is 11.

  There are actually 9 different numbers like that. Can you find all of them?” This problem allows students to apply the mathematics of 2.NBT.B (Use place value understanding and properties of operations to add and subtract) in a non-routine application problem.

The materials reviewed for Math Mammoth Grade 2, Light Blue Series, partially meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. The materials do not balance all three aspects of rigor, as there is an over-emphasis on procedural skill and fluency. 

The materials provide some opportunities for students to develop conceptual understanding, procedural skills and fluency, and application separately throughout the grade and some opportunities where multiple aspects of rigor are used to support student learning and mastery of the standards. However, there are multiple lessons where one aspect of rigor is emphasized.

Examples of conceptual understanding, procedural skill and fluency, and application presented separately in the materials include:

• Worktext 2-A, Chapter 3: Addition and Subtraction Facts Within 0-18, Adding with 9, Question 2, students add doubles of numbers within 20. “It is good to memorize the doubles, also. Fill in.  2 + 2 = _____  3 + 3 = _____  4 + 4 = ____”  This activity provides students with an opportunity to develop procedural skills and fluency with 2.OA.2 (Fluently add and subtract within 20 using mental strategies).

• Worktext 2-A, Chapter 5: Geometry and Fractions, Some Fractions, Question 2, students divide a rectangle into fourth to answer questions about the number of squares.  “Complete. d. Divide this into fourths.  Color \frac{1}{4}____ little squares in one fourth.  ____little squares in the whole rectangle.” Students are shown an outline of a rectangle on a grid. This activity provides students with an opportunity to develop a conceptual understanding with 2.G.3 (Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape).

• Worktext 2-B, Chapter 7: Measuring, Meters and Kilometers, Question 5, students solve a word problem about the distance traveled between bases in Finnish baseball. “The picture shows the field for Finnish baseball game (‘pesäpallo’). How many meters do you run with these ‘routes’? a. You run from the home base to the 1st base and then return to the home base. b. You run from the home base to the 1st base and on to the 2nd base, plus one meter over, because you cannot stop in time. c. (Challenge) You run all the way around the field.” Students are provided with a diagram of the field with “Home base”, “1st base”, “2nd base”, and “3rd base” labeled and the distance between the paths in meters. This problem allows students to apply the mathematics of 2.MD.5 (Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units) in a routine application problem independently. 

Examples of students having opportunities to engage in problems that use two or more aspects of rigor include:

• Worktext 2-A, Chapter 1: Some Old, Some New, Adding with Whole Tens, Question 1, students write an addition problem to solve using ten-sticks and one-dots. “The numbers are shown with ten-sticks and one-dots. Write the sums. d. ___ + ___ = ___” The problem shows two ten-sticks plus one ten-stick and three one-dots. Students develop conceptual understanding and build procedural skills and fluency for 2.NBT.5 (Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.) as they use the visual images of ten-sticks and one-dots to write the equation and find the sum.

• Worktext 2-B, Chapter 8: Regrouping in Addition and Subtraction, Add in Columns: Regrouping Twice, Question 6, students solve addition word problems that require various strategies. “Solve the word problems. a. From Flowertown to Princetown is 148 miles. You travel from Flowertown to Princetown and back to Flowertown. How many miles is that?”  Students are given a 3-column grid to write the equation in.  Students develop conceptual understanding and application of 2.NBT.7 (Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.) as they solve word problems.

The materials reviewed for Math Mammoth Grade 2, Light Blue Series, partially meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Although the materials attend to the specialized language of mathematics, there is limited evidence of the intentional development of MP6 to meet its full intent in connection to the grade-level content.

The materials has limited evidence of intentional development of MP6 throughout the grade level. Examples include, but are not limited to:

• Worktext 2-B, Chapter 6: Three-Digit Numbers, Which Number is Greater?, Instructional box, states “Remember, the open end (open mouth) of the symbol < and > ALWAYS opens towards the bigger number.” Question 4, students compare if a number is greater than or less than another number. “Write either < or > in between the numbers. e. 200 ___ 190“ Although the symbols < (less than) and (greater than)  > are frequently used in comparison problems throughout the materials, the materials do not call them by those names, and students do not have to explain the meaning of the symbols.

• Worktext 2-B, Chapter 6: Three-Digit Numbers, Bar Graphs and Pictographs, Question 2, students fill out a chart and then use a bar graph created by that chart to answer questions. “Below, you see page counts for 14 different second grade math books.” The book's page counts are the following numbers: 217, 388,  365, 290, 304, 315, 243, 352, 289, 392, 346, 308, 329, and 323. “Count how many books have between 200 and 249 pages. Count how many books have between 250 and 299 pages. Continue. Write the total pages in the chart. After that, draw a bar graph using the numbers in the above chart. a. How many books have a page count between 350 and 399 pages? b. How many books had 300 pages or more? c. How many books had less than 250 pages? d. What was the lowest page count?” A chart is provided with the labels “Page count” and “Number of books”. A bar graph is provided with the title “Page count of 2nd grade math books” and the labels “number of books” and “pages”. The scales on the bar graphs are also filled in. This problem does not allow students to attend to precision as they do not have the opportunity to label the chart and graph appropriately.

• Worktext 2-B, Chapter 7: Measuring, students are tasked with answering questions dealing with different units of measurement (centimeters, inches, feet, miles, meters, kilometers, pounds, and kilograms). Throughout the chapter, there are examples of the units of measurement being specified for the students by the problem. For example: Some More Measuring, Question 5, students are tasked with measuring some items and recording the results in a table. “Measure some things in your classroom or at home two times. First measure them in inches, to the nearest half-inch. Them measure them in centimeters, to the nearest whole centimeter. Remember to write “about” if the thing is not exactly so many inches or centimeters. Write your results in the table below.” A table is provided with three columns labeled “Item”, “in inches”, and “in centimeters”. This problem does not allow students to attend to precision as they do not have the opportunity to measure items and then specify the unit of measure.

The materials attend to the specialized language of mathematics. Examples include:

• Worktext 2-A, Chapter 5: Geometry and Fractions, Shapes Review, Question 4, students draw shapes to answer questions about vertices and sides. “Draw FIVE dots on the right.  Connect the dots with straight lines.  You have drawn a pentagon (penta means five).  It has ______ vertices and ____ sides.  Draw one more pentagon in the space.” This problem attends to the specialized language of mathematics as students use the name and drawing of the shape to find the number of vertices and sides. 

• Worktext 2-A, Chapter 5: Geometry and Fractions, Solids, Question 2, students answer questions about the face of various objects. “a. A face is any of the flat sides of a solid. Count how many faces a cube has. ____ faces. What shapes are they? b. Count how many faces a box has. ____ faces What shapes are they?  c. Count how many faces a pyramid has. ____ faces What shapes are they?”A picture of a pyramid is provided. This problem attends to the specialized language of mathematics as students apply the definition of the geometric term “face” to answer questions.  

• Worktext 2-B, Chapter 7: Measuring, Some More Measuring, Question 6, students draw a shape and then find the perimeter of that shape. “Draw three dots on a blank paper so you can join them and make a triangle. Then, measure its sides BOTH in inches (to the nearest half-inch) and in centimeters (to the nearest centimeter). Write your results in the table. How many centimeters is the perimeter (all the way around the shape)? _____ cm. How many inches is the perimeter (all the way around the shape)? _____ in.” This problem attends to the specialized language of mathematics as students apply the definition of the geometric term “perimeter” to answer questions.