THINKING WITH NUMBERS
Suggestions For Using Lessons
Suggestions For Using The Lessons

At The K-2 Grade Level

Background Information

Before you start the lessons, it is important to know you will not start with lesson 1 and continue until you have completed all the lessons. Altogether there are about 850 5-minute lessons, …far more than what you will have time to use and far more than what you will need.

By listening to the students explain their solutions, you will know when they have developed understanding and the confidence to move on to the next group of lessons. You can always revisit these lessons as a review or as needed. Each group of lessons has a 5-item assessment to inform and or confirm your decisions.

Representations

Chapters 1 and 9 introduce the ten frame and the number line. These representations, together with the animations used with them are used to help students visualize and make sense of addition and subtraction. Familiarity with these representations will help students prepare for the addition and subtraction lessons.

Addition And Subtraction

Chapters 2-8 and 10 are explore lessons and previews for what students will master later in their regular school mathematics curriculum. Daily use of Thinking with Numbers provides extra time for students to explore and preview addition and subtraction concepts in an informal, low-stakes environment. There is not one lesson in chapters 2-8 and 10 that needs to be mastered before moving on. But collectively over several months, they will help students make sense of addition and subtraction.

Reasoning Strategies

Chapters 11 and 12 provide structured lessons to help students make sense of reasoning strategies for addition and subtraction. After students can spontaneously use each strategy, there are “practice the strategy” lessons to help develop reasonable fluency in using that strategy.

Why Should These Lessons Be Used In September When You Don’t Teach That Topic Until February?

Exploring a topic for just a few minutes a day over an extended period of time has proven to be much more effective than spending more time per day on fewer lessons or even a unit of instruction. The research on this has been definitive for at least 75 years. Originally, the focus was on massed versus distributed practice with the focus on skills. But recent evidence supports that meaningful distributed instruction also works extremely well for conceptual learning. The focus of these lessons is on conceptual learning.

Making sense of addition and subtraction involves far more than “getting the answer.” By exploring these concepts over time, they will become familiar and comfortable with the following:

ways to represent these concepts,
actions on models such as ten frames and number lines,
using appropriate language and symbols
different ways of reasoning, and
knowing when to apply these ideas in everyday life.

Daily use of these lessons will provide the experiences and time students need to deeply understand all of these aspects of addition and subtraction and begin to develop flexibility in their use. Previewing these topics (chapter 2-8 and 10) provides that opportunity.


Outline Of The Lessons

Getting Ready For Addition
Chapter 1 Ten Frame (30 lessons and 3 assessments)
Chapter 2 Making Sense of Addition
Part 1  Joining: How Many In All? (36 lessons and 6 assessments)
Part 2  Describing: How Many In All? (36 lessons and 6 assessments)
Chapter 3 Equals With Addition (10 lessons and 1 assessment)
Chapter 4 Parts Of Numbers (30 lessons and 3 assessments)
Getting Ready For Subtraction
Chapter 5 Making Sense Of Subtraction
Part 1  Separating: How Many Are Hidden? (36 lessons and 6 assessments)
Part 2  Describing: What Is The Other Part? (36 lessons and 6 assessments)
Chapter 6 Equals With Subtraction (10 lessons and 3 assessments)
Chapter 7 Equalizing (10 lessons and assessments)
Extending Addition And Subtraction Concepts
Chapter 8 Relating Addition And Subtraction
Part 1  Joining: How Many More? (36 lessons and 6 assessments)
Part 2  Joining: What Did You Start With? (36 lessons and 6 assessments)
Part 3  Separating: What Did You Start With? (36 lessons and 6 assessments)
Part 4  Separating: How Many Moved? (36 lessons and 6 assessments)
Chapter 9 Number Line (12 lessons and 2 assessments)
Chapter 10 Making Sense Of Comparison
Part 1  How Many More Or Less? (36 lessons and 6 assessments)
Part 2  How Many In The Big Set? (36 lessons and 6 assessments)
Part 3  How Many In The Small Set? (36 lessons and 6 assessments)
Developing Fluency
Chapter 11 Reasoning With Addition
Counting On To Add (38 lessons and an on-line assessment with reports)
Using Ten To Add (38 lessons and an on-line assessment with reports)
Using Known Facts To Add (43 lessons and an on-line assessment with reports)
Flexible Use Of Strategies
Tree Diagram (25 lessons)
Open Number Line (25 lessons)
Chapter 12 Reasoning With Subtraction
Counting Back To Subtract (38 lessons and an on-line assessment with reports)
Counting Up To Subtract (38 lessons and an on-line assessment with reports)
Using Ten To Subtract (38 lessons and an on-line assessment with reports)
Using Known Facts To Subtract (43 lessons and an on-line assessment with reports)
Flexible Use Of Strategies
Use Nice Numbers, Then Adjust (15 lessons)
Change The Problem (10 lessons)
Chapter 13 Choose A Strategy (10 lessons)












Comments About The Flow Chart

There is not one correct way to select which lessons to use the explore lessons in Chapters 2-8 and 10. They can be used any time as long as the students do not get frustrated with the level of difficulty. Further more, most students will not need to do every lesson in each chapter. By listening to student explanations and using the paper-and-pencil formative assessments at the end of each section of the lessons, you will be able to identify an appropriate level of difficulty and know when to move on.

The lessons on the ten frame, chapter 1-parts 1 & 2 need to be used prior to using Getting Ready For Addition-Chapters 2-4. After students can comfortably and confidently answer the questions in chapters 2-4, Developing Fluency-Chapter 11-Counting On-part 1 can be used. The remainder of Chapter 11-Counting On can be used after Chapter 9-Number Line-part 1 has been introduced. Note that fluency is never mentioned and not expected until Chapter 11-Counting On-Practice Counting On-part 4, is being used. It is crucial that students understand before trying to develop fluency. Delay expectations for fluency until they do understand.

Similarly, the lessons on the ten frame, Chapter 1-parts 1 & 2 are prerequisites for using Getting Ready For Subtraction-Chapters 5-7., Developing Fluency-Chapter 12-Counting Back-part 1, can be used. The remainder of Chapter 12-Counting Back can be used after Chapter 9, Number Line, part 1, has been introduced. Note that fluency is not expected until Chapter 12-Counting Back-Practice Counting Back-part 4 is being used.

Then Developing Fluency, Chapter 12, Counting Up can be used. Again, fluency is not expected until Chapter 12-Counting Up-Practice Counting Up-part 4 is being used.

At this point, students should be reasonably fluent and confident in using counting on to add and counting back and counting up to subtract. By practicing the strategies, the students gradually will be able to use conceptual thinking more quickly. There is no absolute standard for response time expected. Students are measured against their own prior performance in the on-line assessments.

Next, the lessons in Extending Addition And Subtraction-chapters 8-10 can be used. The structure and language of these problems is often more difficult for students, so they may need more experiences and time to understand.

After students are comfortable and confident in answering the problems in Chapter 8, they may be introduced to the strategy lessons Using Ten and Using Know Facts. Since there are four strategies, two for addition and two for subtraction, it makes sense to use each addition strategy before using the corresponding strategy for subtraction. After students are capable of confidently using each of these strategies, part 3 of the lessons for each strategy, can be used to gradually develop fluency.

Comparison lessons, Chapter 12, can then be used to further extend addition and subtraction concepts. Students may need more experiences because comparison problem structures are more complex and the language is more difficult for students to understand (How many more? How many fewer? How much more? How much less?).

Chapter 13 provides students opportunities to compare the use and efficiency of different strategies. There is no right or wrong way to solve these problems. Students will each have their own idiosyncratic way of reasoning.

Making A Plan

To organize the lessons, schools should identify when each reasoning strategy will be taught in their curriculum and when they expect fluency with that strategy. Note that fluency does not mean an instantaneous response. It means a student is able to start the thinking process for a strategy and the ideas flow smoothly as the student uses that strategy. There should be no hesitation while the student needs to recreate what to do next. The time it takes a student to complete this reasoning process is highly individual, so no absolute time demands are appropriate. After practice using each strategy, most students are able to think through these reasoning processes in 2 or 3 seconds.

The following is one possibility. Remember it depends on when your school curriculum teaches the strategies and when the school expects fluency
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