What is Thinking With Numbers?

There are 200 five-minute lessons, 50 concrete activities, 10 previews for each thinking strategy, lots of strategy practice pages for each strategy, and interview questions and recording sheets to assess students' thinking.

The lessons focus on problem solving and different thinking strategies that will help children learn basic facts and perform mental computations with larger numbers.

The lessons are designed to supplement the mathematics curriculum.  They can be conducted regularly, if not every day.  They should require only about five minutes that can be used in the daily schedule at any convenient time. 

This distributed curriculum approach is particularly helpful to students who need more time to understand and make sense of mathematics.

What are the goals?

The primary learning goals are to help children develop better number sense through

• fluency with basic facts
• flexibility in thinking with numbers, and
• confidence with mental computation using larger numbers.

Another goal is for the program to be

• easy for teachers to use and implement as a supplement to their mathematics curriculum. 

Are all students supposed to think alike?

There is no intent that every child should solve a given basic fact problem in some predetermined way.  Some children will find one efficient way, while others find a different efficient way to solve the same problem.  While the numbers may lend themselves to one strategy, students may very well use a different approach--and that's okay.

Flexibility of thinking should be encouraged.  A major goal is to ensure that every child has at least one efficient way to solve each basic fact problem.

How is the curriculum organized?

Each set of materials is organized in color-coded lessons.  They include lessons with a focus on:

number sense

• partitions and number relationships

fact strategies

• addition—count on, use doubles, make ten
• subtraction--count back, count up, use, ten, use addition
• multiplication—repeated addition, patterns, split into parts
• division—repeated subtraction, patterns, use multiplication

mental mathematics

• extending the fact strategies and using tens

How are the lessons designed?

There are five distinct parts to each lesson.

• Pose a problem.
• Allow time for students to think.
• Encourage students to share solutions.
• Highlight an efficient strategy.
• Ask all students to try that thinking with a second problem.

Highlight an efficient strategy and extend student thinking by:

• asking students to re-explain that thinking,
• summarizing the thinking, and/or
• demonstrating the thinking concretely. 

How can the lessons be managed?

Because the lessons are designed to be very brief, students do not typically use manipulatives or work with partners in cooperative groups.  They are encouraged to solve the problems by thinking about them mentally.  As the students share different ways they think to solve or verify these facts, the teacher should often repeat, highlight and sometimes concretely illustrate a solution.  Manipulatives on an overhead projector can be helpful.  Drawing a diagram on the board can also be helpful.

Most lessons can be oral.  However, sometimes asking the children to write answers or related number sentences on a scratchpad may help the students see possible connections among facts.  This also is a technique to ensure all students are engaged.

How can you decide which lessons to use?

There is no intent that every lesson in this packet be used one after the other.  Teachers should use the probes to help them decide when to encourage new ways of thinking.  Any remaining lessons may be used later as review lessons.  Every strategy should be revisited for regular review and practice. 

Subtraction lessons can be used at the same time as addition lessons, but do not expect students to memorize subtraction facts until after they have memorized corresponding addition facts.  For example, 8 - 5 is difficult to memorize until students know 3 + 5 and 5 + 3.  Consequently, do not use the subtraction strategy practice activities until after students are fluent with related addition facts.

How is assessment built into this program?

Every fifth lesson has an informal assessment called a probe. As students share their thinking, simply ask how many of them used each strategy.  Continually try to extend students' thinking to new strategies.

Optional Assessments include:

• oral presentation of facts, one each five seconds,
• individual interviews, and
• occasional sixty second timed survey tests.
• computer assessment

When is drill and practice effective?

With this program, drill may not be necessary for some students.  Delaying drill, until after the other students have developed efficient thinking strategies, decreases the amount of drill and practice that is needed.  Three guidelines should be followed during drill. 

• make it quick--one minute is enough
• focus on a few facts at a time
• use drill to speed up or practice a specific thinking strategy
• delay drill until after students have efficient way to think about those facts 

Drill can be provided one line of problems at a time.  Let students complete the first line. Allow less time for each new line.  Recognition should be given for accuracy.  There is no need to test all of the facts at once.

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