The instructional materials reviewed for JUMP Math Kindergarten partially meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied; however, there are missed opportunities concerning the variety of problem types called for by the Standards. 

Kindergarten standard K.OA.2 specifically calls for students to apply their mathematical understanding as they “solve addition and subtraction word problems, and add and subtract within 10, e.g. by using objects or drawings to represent the problem.” The K.OA.A cluster heading calls for students to “understand putting together, adding to, taking apart, and taking from situations” (see Table 1, CCSSM page 88). Much of the work in this program includes put together, add to, and take from. All four problem types are not represented equally, there is a missed opportunity for students to work with take apart situations.

Students are given multiple opportunities to practice representing addition and subtraction problems with drawings and objects in routine applications. Take apart situations are underrepresented in the materials. Examples of put together, add to, and take from include:

• Teacher Resource, Part 2, Unit 7, Lesson OAK-6, “Students do ‘Put Together with Total Unknown’ additions within 5 using objects. Acting out ‘put together’ additions. SAY: Let’s do another kind of story. Choose five volunteers from the class, two boys and three girls. Separate the boys from the girls at the front of the class. SAY: In this story, some boys and some girls are playing. ASK: How many boys are playing? Show me on your fingers. (2). How many girls are playing? (3). How many children are playing in all? (5)” (K.OA.1,2)
• Teacher Resource, Part 2, Unit 8, Lesson OAK-19, “Students do ‘Add To with Result Unknown’ additions within 10 using pictures and ten-frames. Six children are sitting on seesaws. Then, two more children come to play on a seesaw. How many children are playing on seesaws in all?” (K.OA.1,2; K.CC.3)
• Teacher Resource, Part 2, Unit 11, Lesson OAK-29, “Students do ‘Take From with Result Unknown’ subtractions within 5 using pictures. Activity, 5 spiders are climbing up a wall. 2 spiders get washed away by the rain. How many spiders are left?” (K.OA.1,2)

The instructional materials have some opportunities for students to engage in non-routine application throughout the grade level. Examples of non-routine applications include:

• Teacher Resource, Part 1, Unit 4, Lesson CCK-27, Extension 6, “Five friends want to eat apples. There are two red apples and two green apples. Can each friend eat an apple? Have students solve the problem using tools of their choice, such as pencil and paper to draw pictures, or blocks. Then have students explain their solution to a partner.” (K.CC.1,3-5)
• Teacher Resource, Part 2, Unit 8, Lesson OAK-17, Extension 4, “Distribute BLM Addition Story Blanks and 10 counters or blocks to each student. Give students a starting number-for example, 4. Ask them to write all the number stories they can think of with up to 10 cats in all that start ‘4 cats and then some more cats come.’ Have students use counters or blocks to help find how many in all for each story.” (K.OA.1,2; K.CC.3)
• Teacher Resource, Part 2, Unit 12, Lesson OAK-35, Extension 4, “I have a big plate and a small plate, and 9 cookies. How can I put the cookies on the plates so that there are always more cookies on the big plate?” (K.OA.1)

The Teacher Resource states, “The deepest work in a JUMP math lesson often happens in the extension questions, which appear at the end of most lesson plans.” It goes on to say, “All students should be given the opportunity to engage with the extension questions.” However, there are instances where the extension questions state that they are “best suited for advanced or very advanced students.” Examples include:

• Teacher Resource, Part 1, Unit 2, Lesson CCK-16 “Note: Extensions 4-6 are best suited for very advanced students”, Extension 4, “Give each student three clear plastic bags, one with 3 identical blocks, one with 10 identical blocks, and one with 25 identical blocks. Challenge students to show ‘more’ in three different ways by pairing two groups at a time.”
• Teacher Resource, Part 1, Unit 6, Lesson MDK-3 “Note: Extensions 2-3 are best suited for very advanced students”, Extension 2, “Provide individuals (or student pairs) with connecting cubes (or other objects) of three different colors. The number of cubes of each color should be different and should not exceed 10 (e.g., 9 blue, 7 red, 4 yellow). Students sort the objects by color and then count how many of each color. Students place the groups in counting order without using a number chart. Bonus: Provide students with an additional 6 cubes in a fourth color and ask them to place all four colors in counting order.”
• Teacher Resource, Part 2, Unit 7, Lesson OAK-9 “Note: Extension 3 is best suited for very advanced students” Extension 3, “a) Give students five counters and have them find all the ways to make 5 by adding three numbers. Have students start without zero, then include zero. Provide blank paper for them to record their work. b) Have students make 5 by adding 4 numbers, with no zero.”

The instructional materials reviewed for JUMP Math Kindergarten partially meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. The instructional materials address specific aspects of rigor, and the materials integrate aspects of rigor; however, not all aspects are addressed equally. Heavy emphasis is placed on conceptual understanding and procedural skill and fluency. While students are given opportunities to engage with application problems throughout the materials, these are often teacher directed.

All three aspects of rigor are present independently throughout the materials. Examples include:

• Conceptual Understanding: Teacher Resource, Part 2, Unit 7, Lesson OAK-6, Extension 2, students represent Put Together with Total Unknown additions within 5. “Distribute BLM Adding Stories (4) or BLM Addition Story Blanks, two yarn circles, and five blocks or counters to each student. a) Have students write all of the number stories that can be shown with their blocks that start with ‘Two cats and some dogs’. b) Have students explain to a partner how they know they’ve written all the stories, and how they know how many pets in all in each story.”
• Procedural Skill and Fluency: Teacher Resource, Part 2, Unit 13, Lesson OAK-47, students use dots as visual representations of the numbers 1 to 5 to develop fluency in addition and subtraction within 5. Students are shown pictures with two different colors of dots. Students find and add the parts, subtract a part, and write addition and subtraction sentences together. Students are given the opportunity to practice this procedural skill and fluency in the accompanying Assessment and Practice Book pages.
• Application: Teacher Resource, Part 2, Unit 11, Lesson OAK-26, Activity, “Students draw or cut out pictures from magazines to create their own subtraction stories. For example, they may draw three ducks on a pond and two ducks flying away to illustrate ‘5 ducks take away 2 ducks equals 3 ducks.’ Students use their pictures to tell their subtraction story to a classmate.”

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials; however, a heavy emphasis is placed on conceptual understanding and procedural skill and fluency. Examples include:

• Teacher Resource, Part 2, Unit 8, Lesson OAK-19. “Six children are on seesaws. Then two more children come to play on a seesaw. How many children are playing on seesaws in all? (8)” This lesson incorporates application and conceptual understanding.
• Teacher Resource, Part 2, Unit 12, Lesson OAK-39, Activity, students do subtractions within ten from BLM Subtraction within 10 using a manipulative of their choice such as blocks, counters, fingers, or drawing a picture. This lesson incorporates conceptual understanding and procedural skill and fluency.

The instructional materials reviewed for JUMP Math Kindergarten partially meet expectations for carefully attending to the full meaning of each practice standard. The materials do not attend to the full meaning of MPs 2 and 4.

Examples of the materials carefully attending to the meaning of some MPs include: 

• MP5: Teacher Resource, Part 2, Unit 13, Lesson OAK-45, Extension 2, “Make the following tools available for students: number lines from BLM Number lines, blocks, counters, yarn circles, paper, and colored pencils. Have students choose tools to solve the problem and then explain their solution to a partner. a) There were 8 dogs at the park. 3 dogs ran away from the park. 1 dog came back. How many dogs are at the park? b) Rani has 7 red apples and 2 green apples. She gives 4 red apples to Zara. How many apples does Rani have now?” Students choose from various tools to solve a given problem.
• MP6: Teacher Resource, Part 2, Unit 14, Lesson MDK-11, Activity 1, “Students make shapes and designs by putting together two or more pattern blocks. In pairs, tell each other about the shapes they used and their designs.” Students attend to precision and use mathematical language correctly when they tell each other about the shapes and designs they made.
• MP7: Teacher Resource, Part 1, Unit 6, Lesson MDK-2, Extension 6, “Draw 3 dots in a row. SAY: Count the dots. ASK: How many dots are there? (3) Add one more dot to the row. ASK: How many dots are there now? (4) Can you find the answer without counting form 1? (yes, count 1 more from 3) Repeat with six dots and one more, then eight dots and one more. Have students explain to a partner how they can always find the number of dots quickly without counting from 1.” Students make use of structure when they recognize they do not have to recount from 1 when they add 1 more dot to a group that they have already counted.
• MP8: Teacher Resource, Part 2, Unit 8, Lesson OAK-13, Extension 6, “Have students model each situation with blocks. a) There are 3 frogs in the pond. 2 more frogs join them. How many frogs are in the pond now? b) There are 3 children at the park. 2 more children join them. How many children are at the park now? c) There are 3 birds in a tree. 2 more birds join them. How many birds are in the tree now? d) If you have 3 of anything and 2 more join, will you always get 5? How do you know? e) In pairs, explain your answer to part d). Do you agree with each other? Discuss why or why not.” Students look for and express regularity in repeated reasoning when they recognize that when they have three of something and add two more that they will always have a sum of 5.

Examples of the materials not carefully attending to the meaning of MPs 2 and 4 include:

• MP2: Teacher Resource, Part 2, Unit 1, Lesson OAK-4, Activity, problem 1, “The first one says 1 frog plus 2 more frogs. Your job is to find how many frogs there are in all. You will put blocks or counters in the first box to show how many at the start. Next you will put blocks or counters in the second box to show how many at the start. Next, you will put blocks or counters in the second box to show how many more. ASK: How will you find how many in all? (count) SAY: Then you will count all the blocks and write how many in all.” Students do not contextualize or decontextualize the problem as the teacher is directing students how to solve the problem. 
• MP2: Teacher Resource, Part 2, Unit 11, Lesson OAK-27, Modeling subtraction from 5 with frogs. “SAY: In the song, every time a frog jumps off the log, we take away one finger. We can also say that we subtract. Let’s tell some other frog stories. Draw on the board a log long enough for five paper frogs to sit on.” Students do not contextualize or decontextualize the problem as the teacher is directing students how to solve the problem. 
• MP4: Teacher Resource, Part 2, Unit 1, Lesson OAK-3, Telling a number story, “SAY: I am going to tell a story that has a number in it: Two children are playing in the park. One more child comes to play with them. The end. Ask: At the end of the story, how many children are playing in the park? (let students answer--do not respond to the answers) Are you sure? SAY: Let’s act out the story and check.” Students do not model the problem since the teacher is directing students to act out the problem. 
• MP4: Teacher Resource, Part 2, Unit 2, Lesson OAK-16, Activity, “Give each student 10 counters and two five frames. Read each of the following stories. As you read a story, have students find how many in all. Have students use fingers instead of five-frames for some of the stories.” Students do not model with mathematics since they are given the model to use.

The instructional materials reviewed for JUMP Math Kindergarten partially meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards. 

There are few opportunities in the Teacher Resource or the Assessment & Practice for students to construct viable arguments or analyze the arguments or the work of others. MP3 is identified in the margins of the lesson. Examples of where the materials prompt students to construct viable arguments or analyze the arguments of others include, but are not limited to:

• Teacher Resource, Part 1, Unit 4, Lesson CCK-29, Extension 1, “Students compare two ways of counting ones blocks. Give students 9 ones blocks in a bin and Black Line Master Using Blocks to Find How Many? (3). Direct them to Question 9 on the BLM. For the first way, They count the ones blocks as they place them in order on the boxes in Question 9. Then, they remove the blocks and put them in a pile. For the second way, students try to count the pile of blocks by touching or pointing at each one individually, but without moving them. ASK: Can you do it? Is one way easier than the other way? Explain why.” Students construct viable arguments. 
• Teacher Resource, Part 2, Unit 8, Lesson OAK-13, Extension 6, students model situations with blocks. “a) There are 3 frogs in the pond. 2 more frogs join them. How many frogs are in the pond now? b) There are 3 children at the park. 2 more children join them. How many children are at the park now? c) There are 3 birds in a tree. 2 more birds join them. How many children are at the park now? d) If you have 3 of anything and 2 more join, will you always get 5? How do you know? e) In pairs, explain your answer to part d). Do you agree with each other? Discuss why or why not.” Students construct arguments. 
• Teacher Resource, Part 1, Unit 2, Lesson CCK-19, Extension 3, “Give pairs of students two groups of blocks, one with three blocks and the other with five blocks, and have some extra blocks available. a) Have students identify the group with more blocks. b) Have students add one block at a time to the group of five blocks. Each time students decide which of the two groups has more. c) Will adding blocks to the group with five blocks ever change which group has more blocks? d) Have partners explain their answers to part c). Do partners agree with each other’s explanations? Have them discuss why or why not.” Students analyze the arguments of others. 
• Teacher Resource, Part 1, Unit 3, Lesson GK-6, Extension 4, “SAY: Ron puts 4 red blocks on his table, and 5 blue blocks underneath. He says 4 equals 5, since there are 4 red blocks and 5 blue blocks. Ron writes “4 = 5”. Write on the board: Ron writes 4 = 5. ASK: Do you agree with Ron? How would you explain to Ron what the equal sign means? Have students explain to a partner what the equal sign means and why they agree or disagree with Ron.”
• Teacher Resource, Part 2, Unit 12, Lesson OAK-36, Extension 5, “Lisa has 5 toy cars. She gives 2 cars to Raj. Lisa says she has 7 cars now because 5 + 2 = 7. a) Do you agree with Lisa? Use any tools to explain why or why not. b) In pairs, have students explain their answers to part a). Do they agree with each other? Have them discuss why or why not.”

Examples where the materials miss opportunities to prompt students to construct viable arguments or analyze the arguments of others include, but are not limited to:

• Teacher Resource, Part 1, Unit 6, Lesson MDK-5, Extension 8, students are shown a picture of 6 light squares arranged in a line and 8 dark squares arranged in a line directly underneath the light squares. “Without counting, can you tell if the picture shows a number and the next number? How do you know?” Students do not construct an argument or analyze the arguments of others. 
• Teacher Resource, Part 2, Unit 9, Lesson GK-16, “Hold up a rectangular prism that is not a cube, such as a long skinny cardboard box. ASK: Does this shape look just like a cube? (no) Holding a cube in one hand and the long skinny box in the other, ASK: How is this shape different from a cube?” Describing the differences in shapes is not creating a mathematical argument. 
• Teacher Resource, Part 2, Unit 11, Lesson OAK-27, Extension 3, students make up subtraction stories within 5 in which you subtract one and then add one from the starting number. “Example: 4 frogs are on a log. 1 frog hops off. Another frog hops on. How many frogs are on the log now? For the first few stories, have students use blocks and then have them predict the final answer without using blocks. Have students explain to a partner how they know the answer without using blocks.” Students do not construct arguments.

The instructional materials reviewed for JUMP Math Kindergarten partially meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. While students are given opportunities to construct viable arguments and analyze the reasoning of others, the materials provide limited assistance to teachers in engaging students in both constructing viable arguments and analyzing the arguments of others.

Teacher Resource, Part 1 states, “When students work with a partner, many of them will benefit from some guidance, such as displaying question or sentence stems on the board to encourage partners to understand and challenge each other’s thinking, use of vocabulary, or choice of tools or strategies. For example:

• I did ___ the same way but got a different answer. Let’s compare our work.
• What does __ mean?
• Why is __ true?
• Why do you think that__?
• I don’t understand__. Can you explain it a different way?
• Why did you use__? (for example, a particular strategy or tool)
• How did you come up with__? (for example, an idea or strategy)

Once all students have answered the ASK question, have volunteers articulate their thinking to the whole class so other students can benefit from hearing their strategies.” While these generic question and sentence stems are provided, there is no further guidance or examples for how or when they should be used.

The majority of opportunities for students to engage in  MP3 occur in the extension problems. These include sample answers and often refer teachers back to the prompts listed on page A-33, but give no further guidance on how to build students ability to construct an argument around their thinking or how to critique the reasoning of others. Teachers are often prompted, “In pairs, have students explain their thinking. Do they agree with each other? Discuss why or why not.” however, no guidance is given as to which questions to ask in regards to that specific problem, how to help the students defend their answer, or why an answer makes sense. Additionally, materials include some sample explanations relating to the correct answer being given, but do not always give guidance for teachers on how to effectively guide the conversation if an incorrect answer is being defended. Examples include:

• Teacher Resource, Part 1, Unit 3, Lesson GK-10, Extension 7, “Provide each pair of students with cards from BLM Open Shape Cards and BLM Closed Shape Cards. Partner 1 chooses any three cards, and Partner 2 explains why the shape on each card is or is not a triangle. Partners switch roles and repeat for circles squares and rectangles. NOTE: Encourage partners to ask questions to understand and challenge each other’s thinking (MP.3) and use of math words (MP.6)-see page A-33 for sample sentence and question stems to guide students. Look for students to notice the number of sides or corners, and whether the shapes are open or closed.” No suggestions are given for questions students could ask their partners, which questions from page A-33 are relevant to this problem, or how students could defend their thinking. 
• Teacher Resource, Part 1, Unit 5, Lesson CCK-41, Extension 2, “Arrange 6 blocks in a circle. Touch each block once as you SAY: Ava counts like this the first time: 1, 2, 3, 4, 5, 6. Touch each block once, and the first block again as you SAY: Ava counts again like this: 1, 2, 3, 4, 5, 6, 7. Ava says there are 6 or 7 blocks, it depends how you count. a) Do you agree with Ava? b) In pairs, have students explain their thinking. Do they agree with each other? Have them discuss why or why not. Sample answer: b) I do not agree with Ava, because there cannot be 6 or 7 blocks. It can only be one number. When Ava counted the second time, she counted the first block twice, once at the beginning and once at the end. When you count blocks, you only count each block once. NOTE: For part b) encourage partners to ask questions to understand and challenge each other’s thinking (MP.3) and use of math words (MP.6)-see page A-33 for sample sentence and question stems to guide students.” No guidance is given for guiding the conversation when students have the misconception that Ava is correct or specifically how to challenge their partner’s thinking.
• Teacher Resource, Part 2, Unit 10, Lesson NBTK-11, Extension 5, “Eddy says that spheres are better than cubes for sliding, because they are round. Do you agree with Eddy? Explain why or why not. Have students explain to a partner orally or in writing. Sample answer: I disagree with Eddy. A sphere is round, but that’s why it is good for rolling, not sliding. A cube would be better than a sphere for sliding because a cube has flat faces. Redirecting students: Encourage students to think about what sliding means and what rolling means, and how they are different.” No further guidance is given to help students critique Eddy’s reasoning or defend their answers.

The instructional materials reviewed for JUMP Math Kindergarten partially meet expectations for explicitly attending to the specialized language of mathematics.

Accurate mathematics vocabulary is present in the materials; however, while vocabulary is identified throughout the materials, there is no explicit directions for instruction of the vocabulary in the teacher materials of the lesson. Examples include, but are not limited to:

• Vocabulary is identified in the “Terminology” section at the beginning of each unit.
• “Vocabulary” is identified at the beginning of each lesson.
• The vocabulary words and definitions are bold within the lesson.
• There is not a glossary.
• There is not a place for the students to practice the new vocabulary in the lessons.