Irish Principal Blog

This chapter discussed tactics that are most appropriate for more advanced mathematics in Grades 5 through 8.  Math instruction should be more grounded in real-world experiences and applied applications of mathematics to solve problems.  In anchored instruction applied mathematics from Grade 5 through Grade 8 is “made concrete” by using real-world story problems coupled with video vignettes.  These video anchors are highly motivating and provide some “emotional intensity,” leading students into increased involvement with the problem and toward problem solution.  Researchers have also coupled the use of these video anchors with subsequent applied tasks based on the video story.  A web site that has information about an anchored instruction curriculum is available at:

http://peabody.vanderbilt.edu/projects/funded/jasper/jasperhome.html

www.erlbaum.com/jasper.html

There are other ways to teach mathematics in an “applied” fashion.  Pgs 113 and 114 discuss a Teacher’s Bank and checkbook.

Pgs. 115 through 120 discuss the PASS strategy.  This strategy helps students sort through and discover strategies that work best for them in a series of thirty minute lessons.

This chapter covered strategies to improve a student’s metacognition which means “thinking about thinking”; it is also defined as planning and monitoring how one performs a task.

In metacognitive instruction, students are taught to specifically plan their thinking and subsequently monitor their own performance of those steps.  If math teachers can teach the steps that a student must engage in to complete a math problem, that student will be better able to plan the stops in the problem and to monitor his or her progress in that problem.  This will lead to higher math achievement.

Bender explained once again that students with learning disabilities will struggle in math.  They will struggle with metacognition because they have difficulty with organization and planning tasks.  Thus, specification of these steps for the student allows the student to “think about his or her thinking” in solving the problem and in monitoring his or her performance.

Bender presented a few strategies that could be used:

1.        RIDD

Read the Problem

Imagine the Problem

Decide What to Do

Do the Work

2.       STAR Learning Tactic

Search the Word Problem

Translate the words into an equation in picture form

Answer the problem

Review the solution

3.       SQRQCQ Tactic:  A student’s reading skills can directly hinder his or her achievement in mathematics.  Therefore, teachers of mathematics must also teach reading, and one way to do this is to provide students with a graphic organizer that will assist them in seeing concepts and operations in the problem.

a.       Survey: Read the problem quickly to get a general understanding of it

b.      Question:            Ask what information the problem requires.

c.       Read:                    Reread the problem to identify relevant information, facts, and details needed to solve it.

d.      Question:            Ask what must be done to solve the problem:  “What operations must be performed and in what order?”

e.      Compute:            Do the computations and compute a solution.

f.        Question:            Ask whether the solution process seems correct and the answer reasonable.

4.        Schema-Based Instruction:  Pages 101-106 discuss the use of schemas.  Students are provided schematic diagrams, and their task is to “fill them in” during their reading and solving of a word problem.

Page 107 gave a very good idea for an activity that could be planned for a tear out group.  Students act out word problems in a short play.

Students need to develop a deep understanding of mathematics and they can do this as long as they have mastered the prerequisite skills for a particular problem and are supported by the teacher and curriculum as they “ construct” further understanding of the math problems.  Teachers are facilitators who provide appropriate supports and should withdraw those supports and allow students to work more independently as they mature in their cognitive understandings of math concepts.

One way to accomplish this is to use a cognitive guided visualization strategy.  First, working in a small tear out group, the teacher would provide a problem for the students to consider.    Next, the teacher should allow students to solve the problem in their own way.  Then, the teacher will have the students share they strategies they used.  Finally, using focused questions with students who got an incorrect answer, the teacher should guide them through the visualization process to arrive at the correct answer.

Another strategy is to use Scaffolded Instruction.  The teacher assists individual students by providing prompts and guidance that is tailored to their specific needs.   On page 75, Bender suggests guidelines to be used:

1.        Identify what students know

2.       Begin with what students can do.

3.       Help students achieve success quickly

4.       Help students to ‘be’ like everyone else.

5.       Know when it’s time to stop

6.       Help students be independent when they have command of the activity.

Bender presented ways to scaffold such as word problem mapping and graphic representations of word problems on pages 77-83

Another teaching strategy that was presented was Process Mnemonics to Teach Computation.  On pages 87-90 Bender gives an example for subtraction, addition, multiplication, and division.  At Washington Elementary I have seen teachers use a mnemonic for rounding.  That mnemonic is called “Bullies and Wimps.”  I was very impressed when it was presented and it seemed to be extremely effective.

This chapter covered 3 teaching strategies that could be used by teachers to help struggling students and to differentiate instruction. 

1.  The heart of this chapter is the use of a specific teaching strategy labeled CSA.  CSA stands for Concrete, Semi-Concrete, and Abstract.  This is one way to offer varied instruction and meet the needs of all students.  CSA is referred to as the “mental tool” view of math instruction, which suggests that mental images of mathematic equation or problems may best be formed using physical materials or concrete objects.  Students need to “visualize” math problems in order to understand them.  Clearly, concrete objects or pictorial representations of them can greatly assist in this visualization process. 

Learning to use concrete objects and semi-concrete representations is a prerequisite to abstract learning from many students; and for students who struggle in math, use of CSA instructional approaches can be critical.  Many students with learning disabilities exhibit visual and auditory perception problems that negatively affect mastery of math.  The use of CSA can foster efficient learning of math concepts and provide these learners with strategies to solve problems in every math area.

The concrete level is the lowest level of comprehension.  This involves the use manipulative objects during instruction.  Some examples would be using tally marks for numbers, manipulatives that can readily be configured to portray the concept , circles divided into fractional parts, or even edibles such a an orange divided into segments.  These manipulatives should be accompanies with a worksheet for students to record results of problem solving.

Semi-concrete level uses representation of concrete object is instruction, such a s drawings of objects.  This level uses no hands-on support materials.  Worksheets for activities at this level should have graphic representations on them, as well as spaces for students to draw their own.

Bender points out that whenever a teacher gives a concrete or semi-concrete demonstration they should stress the relationship to the math concept under discussion.

Abstract thinking should be the teacher’s overall goal.  In the abstract phase, mathematics problems without any representations are used, but such worksheets also can be constructed to include space on the sheet for students to develop their own representation of the problem.   Bender gives many ideas and examples for teachers on pages 51-59.

2.  Another teaching strategy covered in this chapter was the Errorless Learning Procedures.  This strategy emphasizes that children require high levels of success in order to be motivated to continue their work in any curriculum area.  Psychologists indicate that students need to succeed at least 85 percent of the time in order for learning to take place.  An errorless learning procedure is an instructional procedure that precludes students from performing an incorrect response.  Thus the students experience much higher success in learning.

These procedures seem most viable in cooperative learning activities and one-on –one teaching situations.  Three procedures that were presented were:  prompting, time delay, and cover, copy, compare.  The last tactic seemed most practical.  Copy, cover, compare follows a process where students copy the solution to a math problem such as a math fact or equation, the student then covers all examples and tries to write the fact or equation from memory.  Finally the student compares what they wrote with the original fact or equation to make sure it was correct. 

3.  A third teaching strategy that was presented was called Classwide Peer Tutoring For Differentiated Instruction.  This strategy will only be effective if a teacher designates a certain amount of time each day for the tutoring experience.  During this time students are put into pairs and one at a time they serve as a “tutor.”  The tutor will present math facts, equations, or problems on flash cards and record the responses.  Student progress should be charted each day.  The teacher is a facilitator and helps students when needed.  Obviously, teachers could have each pair working at a different level or on a different concept or skill.  Bender points out that this tactic allows teachers to truly individualize their lessons for an entire classroom.

Below is my summary of the latter half of Chapter One which focused on Number Sense.  I’ve already had good conversations with teachers about the points that Bender made in these pages.  He makes a lot of sense and our Math Committee is considering changing or adding to our school improvement goal.

The concept of number sense may be as fundamental in learning mathematics as phoneme instruction has become in learning to read.

Without number sense, the child may never succeed in math at even the lowest levels since concepts such as numeration, addition, or subtraction would have no substantive meaning.  Clearly, development of number sense is a critically important first step in math instruction.

On pgs 15 and 16 you can find the 5 levels of Number Sense development and a short description for each level.

Pgs 16-22 have teaching tactics for developing number sense.

Many of the tactics can be used throughout the day in other subjects.

One tactic that I really liked would help teach fractions by playing the game, ‘musical chairs.’

Other tactics that I would like to try are:

Giving students opportunities to move since brain research has shown that movement will enhance memory.

When teaching various bar graphs, stand the students in the shape of each bar. Use color because it results in increased attention from the learning brain.

Use edible foods as counters.

Personal learning timelines.

Use songs to help teach concepts such as, telling time.

There are printable worksheets on http://math.about.com

This year I read the book, “Differentiating Math Instruction“, by William N. Bender.  I have invited my teachers at Washington Elementary to participate in this book study.  Anyone else who is interested in joining us in this steady is welcome to add their comments.  The purpose of this book study is two-fold. 1.  One of our school improvement goals is to have every 3-5 grade student reach grade level proficiency on Math Computation.  Hopefully, this book study will help us reach our goal. 2.  Professional learning communities collaborate about best instructional practices.  It is my goal to create a strong collaborative blog site.    Please, join me on this maiden voyage.