1
Q
What is the purpose of the before lesson phase?
A
To activate prior knowledge, ensure students understand the task and establish clear expectations for the task.
2
Q
What kinds of questions can you ask to make sure students understand the task in the before lesson phase?
A
- What is the problem asking?
- What does that mean?
- Do we have enough information?
- What do you know to get you started?
- Questions about key vocabulary
3
Q
What is the teacher’s role in the during lesson phase?
A
Let go and allow students to spend time with the task, notice students’ mathematical thining, and provide appropriate support and extensions.
4
Q
What questions can you ask to support student thinking without telling them which strategy to use or giving the answer away?
A
- What have your tried?
- Where did you get stuck?
- Have you thought about….
- What if you used…to help you?
5
Q
What questions can you ask to extend the task in an interesting way?
A
- What if….?
- Could you find another way to solve it?
6
Q
In which phase does the most learning occur?
A
the After Phase
7
Q
What is the teacher’s role in the after phase of the lesson?
A
Promote a mathematical community of learners, listen actively without evaluation, and summarize main ideas and identify future tasks.
8
Q
What are some high leverage routines you can use to develop numeracy and active engagement in mathematical thinking.
A
- 3 Act Math Tasks
- Number Talks
- Worked Examples
- Warmups and Short Tasks
- Learning Centers
9
Q
What are the 3 acts in a 3 act math task?
A
- Act 1: The teacher shares visual context for a problem, such as a picture or video, that peaks student interest and curiosity.
- Act 2: Students identify possible variables needed and define a solution path.
- Act 3: The teacher reveals the problem through digital media, and students share and discuss the math behind it.
10
Q
What is a number talk?
A
A - minute discussion about a specific problem and how it might be solved.
11
Q
Worked example
A
Correct, incorrect or partially completed problems that students analyze to develop procedural and conceptual knowledge.
12
Q
What kind of knowledge do worked examples improve?
A
procedural and conceptual
13
Q
What are the 3 ways you can differentiate a lesson?
A
- Content: What you want them to know
- Process: How they will engage in the task.
- Product - What they will show, write or tell to demonstrate learning
14
Q
Open questions
A
broad and invite meaningful responses at many different developmental levels
15
Q
Tiered Lessons
A
provide students with similar problems that focus on the same goals but are adapted for different levels.
16
Q
What are the 4 ways tiered lessons can be differentiated?
A
- Degree of Assistance: provide examples or allow students to work with a partner
- How the task is structured: Students with special needs may need increased structure while gifted students will benefit more from open ended tasks.
- Complexity of task given: Tasks can be more concrete or more abstract and include different levels of difficulty and applications.
- Complexity of process: The pace of the lesson can vary as well as how many instructions are given at one time. The complexity of the process can also be adjusted with the number of high-level thinking questions.
17
Q
Parallel task
A
Students are allowed to choose which to complete. All tasks are focused on the same goal.
18
Q
What are the five process standards from Principles and Standards for School Mathematics?
A
- Problem Solving
- Reasoning and Proof
- Communication
- Connections
- Representation
19
Q
What are the 8 Standards for Mathematical Practice from CCSS
A
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively
- Construct viable arguments and critique the reasoning of others
- Model with Mathematics
- Use appropriate tools strategically
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoining.
20
Q
What does it mean to reason abstractly and quantitatively?
A
- Make sense of quantities and their relationships in problem situations.
- Represent situations using symbols (e.g. writing expressions or equations)
- Create representations that fit the word problem.
- know and use flexibly the different porperties of operations and objects to solve the problem
21
Q
What does it mean to look for and make use of structure?
A
- Identify and explain mathematical patterns or structures
- Shift viewpoints and see things as single objects or as comprised of multiple objects or see expressions in many equivalent forms.
- Explain why and when properties of operations are true in a particular context.
22
Q
What does it mean to look for and express regularity in repeated reasoining?
A
- Notice if patterns in calculations are repeated and use that information to solve other problems.
-Look for and use general methods or shortcuts by identifying generalizations. - Self assess as they work to see whether a strategy makes sense, checking for reasonableness prior to finalizing their answer.
23
Q
Instrumental understanding
A
Students know what to do but don’t know why they do it
24
Q
Relational understanding
A
Students know what to do and why they do it that way. They can make connections to other areas.
25
What does it mean to do math?
Finding and using strategies to solve problems and then checking to see whether the answer makes sense.
26
What are the 5 strands of mathematical proficiency?
Conceptual understanding, procedural fluency, productive disposition, adaptive reasoning, strategic competence.
27
Strategic competence
the ability to formulate, represent, and solve math problems.
28
Adaptive reasoning
the capacity students have for logical thought, reflection, explanation, and justification.
29
Productive disposition
The tendency to view mathematics as sensible, useful and worthwile.
30
What strategies can math teachers use to improve student learning, considering prior knowledge, communication, reflection, and diversity?
- Build new knowledge from prior knowledge
- Provide opportunities to communicate about mathematics
- Create Opportunities for Reflective thought
- Encourage Multiple Strategies
- Engage students in productive struggle
- Treat errors as opportunities for learning
- Scaffold new content
- Honor diversity
31
What is the problem with teaching for problem solving?
Students learn a concept and then apply it to story problems, they learn that they problems they complete ask them to use the skill they just learned. They are not developing problem solving skills, only the ability to pick out the numbers and apply the skill.
32
What is the difference between teaching for problem solving and teaching about problem solving?
Teaching for problem solving is teaching a skill to have students apply it to a problem, but teaching about problem solving is teaching students how to solve problems as well as the strategies that can help them.
33
What is the 4 step problem solving process?
1. Understand the problem
2. Devide a plan
3. Carry out the plan
4. Check your work
34
list the 7 problem solving strategies
1. Visualize
2. Look for patterns
3. Predict and check for reasonableness
4. Formulate conjectures and justify claims
5. Create a list, table or chart
6. Simplify or change the problem
7. Write an equation
35
What is teaching through problem solving
Students learn mathematics through inquiry and exploring different concepts, problems, and situations.
36
What are some of the things students do when they engage in problem solving and inquiry of mathematics?
- Ask questions
- determine solution paths
- Use mathematical tools
- Make conjectures
- Seek out patterns
- Communicate findings
- Make connections to other content
- Make generalizations
- Reflect on results
37
What does teaching through problem solving do for students?
- Focuses student attention on ideas and sense-making
- Develops mathematical practices and concepts
- Devleops student confidence and identity
- Builds on student strengths
- Allows for extension and elaboration
- Engagees students so that there are fewer discipline problems
- Provides formative assessment data
- Invites creativity
38
Tasks that promote problem solving
Tasks that promote problem solving create experiences for students where they can develop their mathematical skills and practices.
39
What are the 3 requirements of a task that promotes problem solving?
1. High level cognitive demand
2. Multiple entry and exit points
3. Relevant contexts
40
Multiple entry and exit points
A problem with a variety of different ways it can be approached with varying levels of difficulty as well as multiple ways of expressing the solution.
41
What are 2 ways you can make relevant contexts for tasks that promote problem solving?
Using literature and connect to other disciplines
42
What kinds of questions can the teacher ask to promote mathematical discourse?
1. How did you decide what to do? Did you use more than one strategy?
2. What did you do that helped you make sense of the problem?
3. Did you find any numbers or information you didn't need? How did you know that the information was not important?
4. Did you try something that didn't work? How did you figure out it was not going to work?
43
Wait time
Give students time to think before responding
44
What are the 5 talk moves for understanding ideas?
1. Wait time
2. Partner talk
3. Revoicing
4. Say more
5. Who can repeat?
45
What are the 3 talk moves for deepening student reasoning and understanding?
1. Why and When
2. What do you think?
3. Tell me more
46
List examples of why and when talk moves
Why do you think that is true?
When will that strategy work?
47
What are the three things teachers should tell students in the after phase?
mathematical conventions, alternative methods (if they aren't getting it on their own), and clarification or formalization of students' methods.
48
What is the I-Think framework for
it encourages students to use metacognitive thinking and guide their discourse as they solve problems cooperatively.
49
What does I-THINK stand for?
I - Individually think about the task
T- Talk about the problem
H- How can it be solved
I- Identify a strategy to solve the problem
N- Notice how your strategy helped you solve the problem
K- Keep thinking about the problem. Does it make sense? Is there another way to solve it?
50
Subitizing
the ability to instantly recognize the number of objects without actually counting them.
51
Number Sense
Number sense is an intuition about numbers and their relationships. It is the ability to think flexibly about numbers and the various ways to represent and use them.
52
What two skills are involved in verbal counting?
The ability to say the counting words in order and connect the sequence in a one to one correspondence with the objects being counted.
53
How can you help a child develop an understanding of cardinality?
1. After they count, ask them how many there are to help them make the connection.
2. Give them counters, and ask them to give you the same number of counters as there are dots.
54
What are some ways to help students develop numeral writing and recognition?
1. Trace over the numerals
2. Make Numerals from clay
3. Trace numerals in shaving cream
4. Press on a calculator
5. Write them
55
What are ways students can develop an understanding of relationships between numbers 1 through 10?
1. One and two more/one and two less
2. Benchmark numbers of 5 and 10
3. Part Part Whole Relationship
4. Missing Part Activities
56
How can you develop student understanding of the relationships of numners 1 through 10 by using benchmark numbers of 5 and 10?
Use 5 frames and 10 frames to model and allow children to explore these numbers using them and discuss the relationships.
57
How can you help students move past counting every counter b y ones when using 5 and 10 frames?
Ask questions like
1. What are you looking at in the ten frame to help you find how many?
2. How does knowing you have a full row help you find how many you have?
58
How can you encourage students to reflect on the part part whole relationships within a number?
Have them say or read (or both) the parts out loud and then write them down on a recording sheet to help them connect part part whole concepts with addition and subtraction.
59
What are some ways sudents can represent part part whole relationships?
1. Drawings
2. Fill in the blank (a group of ____ cubes and a group if ____ cubes.
3. Addition Equations
60
Missing part activity
Students are given the whole and a part and use their knowledge of the part part whole relationship to identify the missing part.
61
What are ways students can develop an understanding for relationships between numbers 10 through 20 and beyond?
Pre-Place Value Concepts, Extending More than and less than relationships, and early introduction to numbers to 100
62
What kinds of questions can you ask to help students develop an understanding of estimate, or the use of the word about in relation to associating numbers with measures of length, weight and time.
1. Is it more or less than _____? - Will the rug be more or less than 10 footprints long?
2. Closer to ____ or to ____? - Will the apple weigh closer to 10 blocks or closer to 30 blocks?
3. About____? - About how many cubes are in this bar? (you can suggest possible numbers as options)
63
How can you help a child who does not know the counting sequence?
1. Use a puppet that makes a variety of counting errors having students correct the mistakes.
2. Practice counting together out loud (forward and backward)
3. Use counting books
4. Prepare cards with numerals and have children place them in order on a number line.
5. Match the written numeral with the written number word.
64
How can you help a child who counts without using a one to one correspondence?
1. As the child counts objects, have him place one object in each space of an egg carton or ten frame.
Make a plan for counting: arrange objects in a row, count objects from left to right, touch one object and say each number word out loud, move each object as it is counted across a line on a work mat or place into a bag or box.
2. If a child splits the count across two syllable counting words, have the child work on matching the written nuymeral with the written nymber word.
3. Have children use a pointer to toufh each object as they count.
65
How can you help a child who struggles with counting on?
1. After a child counts out one set and states how many, cover the collection with a sheet of paper or put the collection in a cup. The idea is to remove the objects from sight, forcing the child to create a mental image of the objects. Let the child peek at the hidden collection if needed, but encourage the child to think about how many before peeking.
2. Use quick images to work on child's subitizing skills
66
How can you help a child who is confused be perceptual cues such as spacing or size of counters?
1. Students lack the ability to conserve, which develops through more experiences and activities with counting. Pose situations that ask students to use one-to-one correspondence to prove the two amounts are equal.
2. Use matching to compare sets. For example, stack counters on top of images to match the sets.
67
How can you help a child who does not understand the cardinality principle?
1. Play board games that have a linear path and instead of moving one marker along the path, leave one counter in each space. For example, if the child rolls a 4 on the die, she places 4 counters, one in each space on the board. Ask, "how many spaces did you travel?"
2. Provide lots of counting opportunities, followed by asking "How Many?"
3. When counting collections together, say "1,2,3,4,5. We have 5 pencils."
68
How can you help a child who has difficulty counting the teen numbers or decade numbers?
1. Play numerous games with counting or count as children line up to leave the room. Consistent practice will provide familiarity with the teen numbers.
2. Play a game that "crosses the decade" where one child in a pair has the numbers ending in 9 up to 100 such as 29, the other has the decade number, such as 30. One student holds up a card the other must find in their set of cards, either the number that comes before or after.
69
How can you help a child who writes the numeral backwards or reverses the digits in the teen numbers?
1. Use a vertical nymber line to show the pattern in writing the teen number.
2. Give children a sheet of numerals and ask them to circle the numerals that are not written correctly. It is important for students to see counterexamples of numerals written backwards to sort between the visual appearance of correctly written numerals and incorrectly written ones.
70
How can you help a child who is not sure of the magnitude of numbers from 1-20?
Use a walk-on number line to have the students count the number of units (placing down cardstock pieces of the length as the unit) from the start (or zero) of the number line. Show how the lengths of the two rows of units compare.
71
What are the 4 addition and subtraction problem types?
1. Change: Join and Add to
2. Change: Separate and Take from
3. Part Part Whole
4. Comparision
72
Where can the unknown numbers be in change problems?
Result unknown, change unknown, and start unknown
73
What are the three unknowns in part part whole problems?
Whole unknown, one part unknown and both parts unknown
74
What are the three unknowns for compare problems?
difference unknown, larger quantity unknown, smaller quantity unknown.
75
Semantic equation
an equation where numbers are written in the order that follows the meaning of the story
76
Computational equation
an equation where the unknown is isolated on the right side.
77
How can you help a student who treats the equal sign as an operation symbol or as a signal to compute
1. Use a number balance to illustrate the relational meaning of the equal sign.
2. Read the equal sign as "is the same as" and "equals"
3. Avoid reading 5+3 "makes" 8 because the word "makes" sounds like an operation or that you have to carry out an action.
4. Pose true/false number sentences in a cariety of equation formats
78
How can you help a student who overgeneralizes the commutative property of addition to subtraction?
Have students first model the problems with single digit numbers and show how you cannot subtract 7 cubes from 4 cubes.
79
How can you help a student who thinks that adding 0 makes a number bigger and subtracting 0 makes a number smaller?
Use story problems that introduce adding and subtracting zero in a meaningful context that students can act out.
80
How can you help a student that counts tick marks or numbers on the numberlines instead of the units or intervals between numbers?
1. Use the sequence of number line models to emphasize that the unit is what is counted on a number line.
2. Use physical hops and setps as well as arrow and arcs to show what is being counted.
81
How can you help a student who does not relate addition and subtraction and/or multiplication and division. (They do not see the inverse relationship)
1. Avoid the rote use of fact families to teach the inverse relationship as it emphasizes procedures without having students see the relationship.
2. Use concrete materials and have students act out a serise of problems with the same three numbers - showing how the part part whole or the number in each group, the number of groups, and the product relate to each other.
82
How can you help a student who believe that they should always get a larger answer in an addition or multliplication problem.
1. When students point out these partterns say that these ideas are only true for some numbers.
2. Point out counterexamples that are within their reach.
83
How can you help a student who relies solely on a key word strategy to determine which operation to use in a word problem?
1. Cover up the numbers in the word problem, and have them analyze the structure.
2. Have the student use bar diagrams to represent the amounts to make more explicit the relationship between the quantities.
3. Do not teach key words to solve word problems.
4. Make sure to pose story problems and contextual probelms that include all four problem types with the unknown quantity in different locations so children gain experience thinking about and solving a variety of situations.
5. Have available and encourage the use of physical materials for students to model story/contextual problems. Discuss what they have done to determine the answer.
6. Suggest to the student some of the recommendations for analyzing contextual problems (think about the answer before solving the problem and solve a simpler problem)
84
How can you help students who choose the wrong word operation in word problems?
Focus on the structure of the problems using graphic organizers or the story problem sorting activities.
85
How can you help students who think the remainder is left over and not part of the answer?
1. Give students the multiple problems shared in the chapter that show the various ways remainders need to be interpreted and how that interpretation affects the answer.
2. Give students problems from anonymous students that have the remainder interpreted improperly. Have students "grade" the problems and share the findings in a class discussion.
86
How can you help students who are unsure about division by zero?
1. Students must focus on division as having an inverse relationship with multiplication. What number when you multiply it bt zero will equal 5?
2. Ask students to take out 5 counters. How many sets of 0 can you make? Or put 5 blocks in 0 equal groups.
87
What are the 3 developmental phases for learning basic math facts?
Phase 1: Counting Strategies - students use objects or verbal counting to determine the answer.
Phase 2: Reasoning Strategies - Using strategies and known information to derive facts.
Phase 3: Mastery - Students are able to produce answers quickly and accurately.
88
List the 3 ways to teach basic facts effectively
1. Use story problems
2. Use quick images
3. Explicitly teach reasoning strategies
89
What are the 9 reasoning strategies for Addition facts?
1. One more than and two more than.
2. Adding Zero
3. Doubles
4. Combinations of 10
5. 10 + ____
6. Making 10
7. Use 10
8. Using 5 as an anchor
9. Near Doubles
90
How can you effectively teach students the concept of adding zero
1. use story problems involving zero
2. Use drawings that show two parts with one part empty.
3. Explore a set of zero facts, some with the zero first, and some with the zero second. Ask students what they notice.
4. Allow students to create their own stories or illustrate the problems.
91
Making 10
Students use known facts that equal 10 and then add the rest of the number to 10. (also called break apart to make ten)
ex. 28+7 = 30+5
92
How can you help students learn the making 10 strategy?
Quick images or manipulating double-ten frames
ex. cover two ten frames with a problem, like 6+8. Ask students to visualize moving counters from one frame to fill the other ten frame and explain their thinking.
93
Use 10
Start with 10+ and adjust the answer
ex. 9+6 = 10+6 and 9 is one less than 10, so the sum is also one less, 15.
94
Using 5 as an anchor
Looking for fives in the numbers
ex. 7+6 = (5+2) + (5+1) = (5+5)+(2+1)=13
95
How can you help students see numbers as 5 and some more?
ten frames
96
What are the 3 reasoning strategies for subtraction?
1. Think addition
2. Down under 10
3. Take from 10
97
Think addition
Students use the inverse relationship between addition and subtraction to solve problems.
"What plus 8 equals 13?"
98
How can you help students learn the think addition reasoning strategy?
1. Story problems that sound like addition but have a missing addend and model the structure of addition problems. (Join:Initial part unknown, change unknown, and part part whole with part unknown)
2. Triangle or t cards
99
Down under 10
The reverse of making 10.
1. Separate - 14-8, jump down 4 to get to 10, then jump down the remaining 4 for the answer.
2. Compare: Jump down 4 to get to 10 and then 2 more to get to 8. 14 and 8 are 6 apart.
100
How can you help students develop the down under 10 strategy?
write pairs of facts in which the difference for the first fact is 10 and the second is either 8 or 9. Have students solve the problem and discuss strategies.
story problems using separate and comparison will help provide context
101
Take from 10
Use knowledge of combinations that make 10 by decomposing the initial value apart into 10+ _____.
ex. 15-8 = 10+5-8. 10-8=2+5=7
102
What are the foundational facts for multiplication?
2,5,10,0, and 1
103
What is a strategy that can be used for 5s?
Clock facts
104
How can you help students develop the concepts of multiplying by zero and one?
1. Story problems
2. Illustrate ones using arrays to show commutativity
3. Help students notice patterns through exploring stories and using concrete tools
105
What are the reasoning facts for division?
Think of multiplication, and then apply a multiplication reasoning fact as needed. Missing factor stories can help students make this connection.
106
Why do students need to practice near division?
They are more likely to happen in real life, are required for estimating, and are required for solving larger division problems.
107
How can you help students who do not recognize the inverse relationship between basic facts?
1. Use the triangle and t cards
2. Explicitly practice the language used here
108
How can you help who struggle to understand and apply the commutative property?
1. Use ten frames for each addend and reverse them.
2. Add contexts that can be reversed (part part whole stories)
3. For multiplication, use areas and arrays that can be rotated.ow can
109
you help students who lose track of the group size when applying the adding or subtracting a group.
1. Provide examples of the problems using an array.
2. Show how the known fact is used to add or subtract a row (or column) from the array.
3. Emphasize the group/row
110
Commutative Property for Addition
you can change the order of the addends and not change the answer
111
Associative property for Addition
when adding three or more numbers, it doesn't matter the order they group numbers to work with them.
112
What is another name for the zero property?
identity property
113
What are the 4 multiplication and division problem structures?
1. Equal groups
2. Comparison
3. Array and Area
4. Combination
114
The commutative property of multipication
the order of the numbers makes no difference
115
The associative property of multiplication
when multiplying three numbers, you can change the order in which you multiply them.
116
Distributive Property of Multiplication
You can split either one of the factors into two or more parts and multiply each of the parts by the other factor, and add the results.
117
How can you help a student understand why numbers cannot be divided by zero?
1. Pose problems to be modeled that involve dividing by zero.
2. Move toward reasoned explanations that consider the inverse relationship of multiplication and division - put the answer into a multiplication problem as a check "What, when multiplied by 0 produces an answer of 5?"
118
Open Middle Problem
A problem in which there is a single correct answer but has multiple ways to approach and solve it.
119
How should you respond if a student asks, "Is this right?"
- "Why do you think that might be right?"
- "How can you tell?"
- "Can you check that somehow?"
120
What questions can you ask to understand a student's thinking and help them navigate the problem?
- What is the problem asking you to do?
- How have you organized the information?
- What about this problem is challenging?
- Is there a strategy or manipulative you could try?
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