4th Grade - Gateway 2

The instructional materials for Grade 4 partially meet the expectation for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics without losing focus on the major work of each grade. Overall, although word problems are included in the instructional materials, the problems are often routine. Many problems are single-step, and problems that are multi-step are often scaffolded. However, there are ten Problem Solving Lessons designed to help students "isolate and focus on [problem solving] strategies."

In Grade 4 there are several standards that call for application. Standard 4.OA.3 requires students to solve word problems posed with whole numbers and having whole number answers using the four operations, including problems in which remainders must be interpreted. Standard 4.MD.3 requires students to apply the area and perimeter formulas for rectangles in real world and mathematical problems. Standard 4.MD.7 requires that students solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems. Standard 4.NF.3d requires that students solve word problems involving addition and subtraction of fraction referring to the same whole and having like denominators, and Standard 4.NF.4c requires students solve word problems involving multiplication of fraction by a whole number. The instructional materials include some problems that allow students to engage in applications of the mathematics.

• Assessment and Practice Part 1 Unit 2 Lesson 23 Problem 3 consists of three multi-step word problems. Problem 2a states: “There are 7 dogs at the shelter. There are three more cats than dogs at the shelter. How many cats and dogs are at the shelter?” Problem 2b states: “There are apples and pears on the table. There are 8 apples and 3 fewer pears. How many apples and pears are on the table?” Problem 2c states: “Darya invited 17 friends to a birthday party. Seven of them are boys. How many more girls than boys did Darya invite to her party?”
• Assessment and Practice Part 1 Unit 3 Lesson 17 Problem 6 asks: “Class 4A collected 287 books and class 4B collected 476 books for charity. A) About how many books did 4A and 4B collect together? B) Is your estimate higher or lower than the actual answer? How do you know?” This problem requires students to make estimates and use their estimates to reason about the actual answer.
• Assessment and Practice Part 2 Unit 3 Lesson 38 includes multi-step word problems. For example, Problem 2 asks “Diana is two years older than Farhad. Farhad is 10 years old. Farhad is 7 years older than Chen. How old are Diana and Chen? Diana is ______ years old and Chen is ______ years old.” However, not all problems in this lesson are multi-step word problems. For example, Problem 11 asks “An elephant weighs 13,000 pounds and is 13 feet tall. Is this elephant 1,000 times heavier than it is tall? Explain.”

Word problems can be found in many lessons throughout the instructional materials; however, they are mostly routine, similar to problems previously encountered by students, and/or encourage the use of strategies modeled in the Teacher Resource. As a result, the instructional materials do not present sufficient opportunity for students to engage in non-routine application problems.

• Teacher Resources Part 2 Unit 3 Lesson 32 includes scaffolded problems such as "There are 32 children in a class. 13 of them are boys. a) How many girls are there? b) How many more girls are there than boys?"
• Teacher Resources Part 2 Unit 3 Lesson 32 includes one-step problems such as "A book costs $10. A poster is $4 cheaper than the book. How much does the poster cost?"
• Teacher Resources Part 2 Unit 3 Lesson 35 is aligned to 4.OA.3, but the problems are one-step multiplication word problems. For example, "Alex is 8 years old. Jo is three times as old as Alex. How old is Jo?"
• Teacher Resources Part 2 Unit 4 Lesson 11 includes routine word problems. Students answer questions about pizzas and pies.
• Teacher Resources Part 2 Unit 8 Lesson 26 lists 4.NF.3 and 4.NF.4. This lesson focuses on measuring and drawing line segments and objects of a given length in inches, to the quarter of an inch. There is not a clear focus on solving word problems.
• Teacher Resources Part 2 Unit 8 Lesson 27 lists 4.NF.3 and 4.NF.4. This lesson focuses on measuring and drawing line segments and objects of a given length in inches, to the closest eighth of an inch. There is not a clear focus on solving word problems.
• Assessment and Practice Part 2 Unit 3 Lesson 32 focuses on Addition and Subtraction Word Problems. In Problem 1, students complete a chart that scaffolds the problem into parts. Problems 3-6 are word problems that follow the same routine without the chart.
• Assessment and Practice Part 2 Unit 3 Lesson 33 includes word problems using diagrams with multiple steps that are scaffolded for students. In Lessons 4-34 through 4-36, students are provided with either a sample response, a chart to complete, or some other form of scaffolding for each problem set rendering each problem routine.

Problem Solving Lessons include word problems but are often heavily scaffolded and focused on the use of a particular problem-solving strategy. However, there are some instances where application problems are found. For example, Problem Solving Lesson 6 includes problems such as "A school fundraiser has a bake sale that sells muffins and cake. A muffin costs $2 and a piece of cake costs $3. The bake sale sold 30 items altogether and made $71. How many muffins and how many pieces of cake were sold?"

Advanced Lessons often contain application problems; however, they are optional and not assigned to all students and therefore not included in this Report. For example, Teacher Resources Part 2 Unit 3 Lesson 36 (Advanced) includes problems such as "Barret reads 7 books over the holidays. Henry reads three times as many books as Barret. How many more books did Henry read?" Teacher Resources Part 2 Unit 3 Lesson 37 (Advanced) includes problems such as "In a class library, there are four times as many chapter books as non-fiction books. There are three times as many nonfiction books as comics. There are 160 books in total. How many books of each kind are in the library?" Students may not complete these word problems.

The instructional materials partially meet the expectation that the materials balance all three aspects of rigor with the three aspects almost always treated separately within the curriculum including within and during lessons and practice. Overall, many of the lessons focus on procedural skills and fluency with few opportunities for students to apply procedures for themselves. There is a not a balance of the three aspects of rigor within the grade.

• The three aspects of rigor are not pursued with equal intensity in this program.
• Conceptual knowledge and procedural skill and fluency are evident in the instructional materials. There are multiple lessons where conceptual development is the clear focus.
• The instructional materials lack opportunities for students to engage in application and deep problem solving in real world situations.
• There are very few lessons that treat all three aspects together due to the relative weakness in application. However, there are several lessons that include conceptual development leading to procedural practice and fluency.
• There are minimal opportunities for students to engage in cognitively demanding tasks and applications that would call for them to use the math they know to solve problems and integrate their understanding into real-world applications.

The instructional materials reviewed for Grade 4 partially meet expectations that the materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

Materials occasionally prompt students to construct viable arguments or analyze the arguments of others concerning key grade-level mathematics detailed in the content standards; however, there are very few opportunities for students to both construct arguments and analyze the arguments of others together.

In the lessons provided in the Teacher Resources Part 1 and 2, examples identified as MP3 are almost always in a whole group discussion, though there are occasional suggestions for students to work in groups. Students rarely have the opportunity to either construct viable arguments or to critique the reasoning of others in a meaningful way because of the heavy scaffolding of the program. For example, in the Teacher Resources Part 2 Unit 8 Lesson 30, students are converting mixed measurements to measurements in inches. The teacher is prompted to "have students think of how to convert a mixed measurement, such as 3 ft 4 in, into a measurement in inches only. They can discuss the ideas in pairs, then in groups of four. To prompt students to think of the method shown below, ask them to recall what they did with other units of measurement, such as centimeters and meters, or liters and milliliters." Although students are talking in small groups, the discussion is centered around solving a problem in the same manner of previous problem and does not address MP3 by having students construct their own arguments and/or critiquing the reasoning of others. Another example is found in Teacher Resources Part 1 Unit 1 Lesson 12 page 40: "There are counters of two colors. Try to find a way to have 12 counters so that there are four times as many of one color as the other. Then explain why it is not possible." The students are told from the beginning that there is no possible solution. In Teacher Resources Part 2 Lesson 44 page 19, students are asked to "(f)ind and correct the mistake in the long divisions." Students are told from the beginning that the original reasoning is incorrect.

In the Assessment and Practice Books, students are sometimes prompted to construct an argument. For example, in Assessment and Practice Book Part 1 page 1 question 2 asks “ Are all multiple of 8 even? Explain.” Another example is Assessment and Practice Book Part 2 page 23 question 58: “Kyle has 6 books. Ron has three times as many books. How many books does Ron have? Explain how you know.” Although students are prompted to provide written arguments, often using the word “explain,” students are not provided with formal opportunities to share these written arguments with classmates.

In the instructional materials, Assessment and Practice Books, students are rarely provided opportunities to analyze the arguments of others. When items are included that ask for students to critique the reasoning of others, often they are often told up front if the student is correct or incorrect or are provided hints. For example, in Assessment and Practice Book 2 page 8 question 9 provides a pattern. Three rules are provided to describe the pattern. Students are prompted to say whose rule is correct, and students are asked what mistakes did the others make. The problem is suggesting that one is correct and two are incorrect.

The materials reviewed for Grade 4 partially meet the expectation of assisting teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

In the Teacher's Guide, p. A-23, MP 3 is discussed in detail. Examples are provided to clarify the reasoning behind the labeling of MP3 in the materials although examples are not necessarily on grade level.

Within lessons, the teacher materials are not always clear about how teachers will engage and support students in constructing viable arguments or critiquing the reasoning of others. Materials identified with the MP3 standard often direct teachers to "chose a student to answer" or "have a volunteer fill in the blank." Questions are provided but often do not encourage students to deeply engage in MP3. In addition, although answers are provided, there are no follow up questions to help re-direct students who didn’t understand.

• Teacher Resources Part 1 Unit 1 Lesson 4 page 15: “Find the missing number in each pattern. Explain the strategy you used to find the number.” These are fill-in-the-blank problems that require very brief explanations.
• Teacher Resources Part 1 Unit 2 Lesson 1 page 2: "Ask students to write a few numbers the Egyptian way and to translate those Egyptian numbers into regular numbers (using Arabic numerals). Have students write a number that is really long to write the Egyptian way (Example: 798). ASK: How is our system more convenient? What is it helpful to have a place value system (i.e., to have the ones, tens, and so on always in the same place)?" The focus of this activity is on writing and translating numbers, and the teacher is not given guidance on how to engage students in MP3 other than to have students notice that our number system is more convenient.
• Teacher Resources Part 1 Unit 2 Lesson 4 page 15: "You have one set of blocks that makes the number 13 and one set of blocks that makes the number 22. Can you have the same number of blocks in both sets?" This question can be answer with a yes or no, and the teacher is not given support to help students engage in MP3.
• Teacher Resources Part 1 Unit 2 Lesson 14 page 43: "Write on the board: 83 hundreds + 7 tens + 5 ones = ___ thousands + ____ hundreds + ____ tens + ____ ones. Have a volunteer fill in the blanks. Point out that now we can write the number by writing the digits from left to right. ASK: What would happen if we ddid that with the original representation, 83 hundreds + 7 tens + 5 ones. Would we still get the same answer? (yes!) Discuss why that is the case. Emphasize that because hundreds are the largest place value, regrouping the hundreds won't affect how we write the number." The original question is a fill-in-the-blank problem completed by one volunteer. One of the questions is a yes or no question, and the teacher guidance about the discussion downplays regrouping, the focus of the lesson.
• Teacher Resources Part 2 Unit 3 Lesson 34 page 28: "Draw the tape diagrams below and ask which of them would fit the situation and which would not. Have students explain why the diagrams that do not fit the situation do not work." For the tape diagrams that do fit the situation, no explanation of student reasoning is required, and for the tape diagrams that do not fit the situation, the explanation provided to the teacher is very brief. No guidance for the teacher to support students struggling with interpreting tape diagrams.
• Teacher Resources Part 2 Unit 3 Lesson 3 page 12: "Five eights of a pizza was eaten. What fraction is left? (3/8) How do you know?" Although students are asked to explain how they knew their answer, this question could simply be answered by showing work. The teacher is not provided any guidance around how to get students deeply engaged in MP3 with this question.
• Teacher Resources Part 2 Unit 3 Lesson 6 page 22: "Is there a fraction equivalent to 3/8 with an odd denominator? Explain. Answer: No. The denominator will always be a multiple of 8, so it will always be even." Teachers are not given guidance or examples of student work to help them support students in developing their answers.
• Teacher Resources Part 2 Unit 6 Lesson 41 page 9: "Can 13 be a factor of 12? (no) How do you know? (because 13 is greater than 12; no whole number times 13 can equal 12)." No follow-up discussion or support for the teacher is provided with these questions.

Overall, some questions are provided for teachers to assist their students in engaging students in constructing viable arguments and analyzing the arguments of others; however, additional follow-up questions and direct support for teachers are needed.

The materials reviewed for Jump Math Grade 4 partially meet the expectation for attending to the specialized language of mathematics. Overall, there are several examples of the mathematical language being introduced and appropriately reinforced throughout the unit, but there are times the materials do not attend to the specialized language of mathematics.

Although no glossary is provided in the materials, each lesson includes a list of vocabulary that will be used in that lesson. The teacher is provided with explanations of the meanings of some words.

• Unit introductions sometimes include vocabulary. For example, in Teacher Resources Part 2 Unit 9 page T-1, the definition of an angle is provided.
• Vocabulary words are listed at the beginning of each lesson plan in the Teacher’s Guide, but definitions, if any, are within the lesson.
• Teacher Resources part 1 page A-30 states that for vocabulary words listed for each lesson teachers should “explain the meaning of these terms and write them on the board as they appear in the lesson.”

While the materials attend to the specialized language of mathematics most of the time, there are instances where this is not the case.

• Often students are not required to provide explanations and justifications, especially in writing, which would allow them to attend to the specialized language of mathematics. For example, in Teacher Resources Part 2 Unit 9 Lesson 14, vocabulary includes the terms endpoint, intersect, intersection point, line, line segment, point, and ray. Each time, however, that these words are used in the lesson, they are used by the teacher. The student is not required to provide an explanation or justification for their answers that would allow them to use the words in this lesson.
• Many of the discussion prompts provided are guided by the teacher so that the student is merely repeating the teacher's language. This limits student ability to actively use mathematical language.
• Some activities include words that do not attend to the specialized language of mathematics. For example, in the Assessment and Practice Part 2 page 5, students are prompted to write fill a blank to indicate how many dots are remaining even though the lesson itself included vocabulary like remainder.
• At times the words themselves are the focus, not the language of mathematics. For example, Teacher Resources part 1 page A-17 states that “In some areas of math (e.g., geometry), the greatest difficulty that students face is in learning the terminology. If you include mathematical terms in your spelling lessons, students will find it easier to remember the terms and to communicate about their work.”