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 Math Diagnosis and Remediation

Math Principles

Visuals for study are from a variety of Internet sources.

Polygon

A polygon is a closed plane figure made up of line segments.

Flip = Reflection

Slide = Translation

Turn = Rotation

Vertex is:

The common endpoint of two or more rays or line segments

Ray is a line which starts at a point and goes off in a particular direction to infinity.

Line segment is a straight line which links two points without extending beyond them.

Diagonal of a polygon is a line connecting to non-adjacent vertices of a polygon. When you go from one vertex to another, that's a diagonal.  The diagonal outside a polygon is concave.  The diagonal inside the polygon is convex.

Lines of symmetry:

Pi=3.14=C/D

Mean, Median, Mode, Range Lesson Plan

Brief Description

Students use photos of raisin example to calculate mean, median, mode, and range.

Objectives

The learner will:

• Calculate the mean, median, mode, and range.

• Discuss the concepts with peers in the online discussion thread.

Goal

To assist the student in making links between statistical principles and the journal articles they read.

Graduate students in experimental research, but these concepts correspond to the following:

Data and Probability
2. Select and use appropriate statistical methods to analyze data
A Describe and analyze data

 Grade 6 Grade 7 Grade 8 find the range and measures of center, including median, mode and mean find, use  and interpret measures of center and  spread, including ranges find, use  and interpret measures of center, outliers  and spread, including range and  interquartile range 2 2 2 MA 3  1.10 MA 3  1.10 MA 3  1.10

Onground Class Materials Needed

small packages of raisins (one for each student).  Online class will use photos.

Procedural Steps

Before the Lesson Onground

Provide each student with a package of raisins (onground) or script (online).

Students can work with other students and each other. In this activity, onground students may work on their own, in pairs, in small groups, in larger groups and, finally, as a whole class to figure mean, median, and mode of collections of raisin data.

Activity

Begin this lesson by modeling the process students will use to calculate and record mean, median, and mode for their raisin. Then students work on their own to do the following activities:

• Count and graph the number of pieces of raisins in their packs of raisins.

• Use data to figure the mean, the average number of raisins in each box.

• Use the data to calculate the median, or middle number.

• Use the data to calculate the mode, or the number that occurs most frequently.

Finally, collect data for the entire class and challenge students to use that data to calculate mean, median, and mode for the entire class.

Assessment

Students will post their answers and discuss the process in the online discussion thread (online students) or as a whole (onground class).

Online Script

Quantitative Logic 1

Welcome to my office!  We have been talking about the mean, median, mode, making estimations and inferences, and range.  We will start with some definitions then use a concrete example to illustrate.  You can find definitions in your glossary or online, for example.

Mean: The most widely used measure of location.  The sum of all observations divided by the number of observations.  Sample means are symbolized by  (x(bar)), while population means are generally symbolized as .

Median: The “middle value” if the data are listed in rank order.  If there are two central values (n even) then the median is simply the average of these.  The median is a useful statistic when we are dealing with highly skewed data.

Mode: The most commonly observed value (or set of values) in a data set.  For continuous variates we cite the modal class (or classes).  The mode is a useful characteristic when we wish to quote the most “fashionable” observation.

Range: The difference between the highest and lowest values.  Perhaps the simplest measure of dispersion in data, but by definition, it is strongly influenced by extreme untypical values.

Outlier:  A data point that is an "unusual" observation and likely should be discarded. Note: The median is less affected by outliers than is the mean. A number that is far apart from the rest of the data; an extreme value either much lower or much higher than the rest of the values in the data set. Outliers are known to skew means or averages.  Many researchers will remove outliers to increase the accuracy of their calculations.

I want you to estimate the number of raisins in one of these boxes.

As a researcher, the more information you have, the better chance you can predict or hypothesize correctly.  In this case, begin with an estimate.  Number your paper from 1-7, then record your answers.  When you finish, post your answers and talk with other students on the Discussion Board.

1.  Write down how many raisins you think are in the box.  After you have an estimate, scroll down to the next picture.

Now, based on the number of raisins you seen in the end of the box, do you want to revise your estimate.  You can scroll up, but not down.

2.  What is your estimate of how many raisins in the box?  When you have written down your estimate, scroll down to the next step.

Here are the estimates from several people.

20

29

30, 30

40

70 (Outlier)

Answer the following questions.  If you need a reminder, you can look up the meaning of these words in the course glossary.

3.  What is the range?

4.  What is the mode?

5.  What is the median?

6.  What is the mean?

Now look at the actual number of raisins in the two boxes.

7.  How many raisins are in the box?

As it happens, the outlier of 70 actually brought the estimation to be pretty accurate regarding raisins.

8. Imagine that the data are test results, however.  The problem with the outlier is it can screw the data.  If we remove the score of 70 and recalculate, we have the following:

20

29

30, 30

40

The person who received the high score of 70 points probably came into the test knowing information the other students didn't know.  You would be more accurate in determining the class knowledge by removing the outlier.  Now the results are

Mean = 29.8

Median = 30

Mode = 30

When you look at these descriptive statistics, you have a more accurate view of the students' test performance.

How did your estimate compare to the actual results?

Are you learning what you need to learn to read communication research?

Describe a problem in communication.  How can mathematics help you solve that problem?

Discuss this quote by a math professor:  "Statistics are like a light pole to a drunk.  They just provide support.  Illumination has nothing to do with it.  The research adds the illumination by deciding what number to use and what they mean."

NEED MORE PRACTICE?

Here are some practice problems with answers:  http://lessons.ctaponline.org/~cdenton/pretest1.key.htm

You are already familiar with this kind of information regarding test scores.

·        Remember the pretest you took. Interpret the class results.  What do you predict would happen if students completed the posttest week 1 of the course?

·        This week?

·        Week 8?

·        In this course you have a pretest, treatment (instruction), and a posttest.

Hypothesis:  After using multiple learning strategies, student knowledge of experimental research will increase.

Independent variable:  CA 517 learning strategies.

Dependent variable:  Student knowledge.

Operational definition of learning:  Student knowledge:  Answers on the CA517 pre-post test.

Course  Pretest  Results

Based on the readings so far, you should be able to look at this data a make sense of it.

Highest Score: 9 pts. (90%)

Lowest Score: 4 pts. (40%)

Range: 5 pts.

Mean: 6.23 pts.

Median: 6 pts.

Mode: 6 pts.

Difficulty (Mean P-Value): .62

Standard Deviation: 1.59

Semi-Interquartile Range: 1

Number of Respondents: 13

Number of Questions: 20

Points Possible: 10 pts

Frequency Distribution: 9 pts. 2 (15.38%) 92.31
7 pts. 3 (23.08%) 73.08
6 pts. 4 (30.77%) 46.15
5 pts. 2 (15.38%) 23.08
4 pts. 2 (15.38%) 7.69

Week 4 Quantitative Logic 1 Answers:

3.  What is the range? 70-20=50

4.  What is the mode? 30

5.  What is the median? 30

6.  What is the mean? 36.5

Source

Prof. Eddie Smith (class sessions TE413) and

Hopkins, G. (2008).  Candy colors: Figuring the mean, median, and mode.  Education World.  Retrieved June 16, 2008 from http://www.education-world.com/a_lesson/03/lp293-02.shtml

Review Marilyn Burns article

Important:

• Helping students make connections

• Building on prior learning

• We want justification

Based on what Marilyn Burns. . .

Build In Vocabulary Instruction

Division is repeated subtraction.

Figure difficult problems in their head.

Associative property—the group you hang around with.

Commute—drive back and forth.  Moving location.

Elementary school:  Marilyn burns is an amazing elementary school diagnostician.  Games and things she has published.  Published many books on how to teach elementary school kids.  Especially for teachers who are math phobic.

Classroom products—AngLegs are excellent. Come with activity cards.  They snap together and you can explore things with them.  There’s a protractor that will snap in.  Each color represents a link.  Fun looking, attractive cards.  Can use for modeling shapes.  In geometry, visualization is huge.

convex--kite

Concave—dart--what’s the shape of a cave.  It goes in.

ABCD  The b has to be next to the A and so on.  Can go clockwise or counterclockwise.

Just draw a four sided figure to start.  Once you place the A, there are only two other spots for a B.  You can go clockwise or counterclockwise.

Surprised when some students came up with the right answer, but didn’t understand the process.

Students really didn’t understand what a lot of math concepts represented.

Rhombus—parallelogram that is equilateral

Give the best name.  Not all the answers.

Necessary and sufficient.

Build the things and letting the kids see it helps immensely.

Diagnosing

MD is concerned with what can’t you do.  An educator is concerned with what can you do.  Figure what already exists and use to help.

Part of what surprised me this summer is how different students suddenly understood and made major advances.

What learnin style are they?  Hone in on.

Simplify as either this or this.

No one is completely good at one and defunct of another.  Which one does the student exhibit more often.

If you were taught with a graphing calculator, you were taught why.

A graphing calculator allows students to do LOTS of graphs.  Student can take ownership in grasping the concept and figuring out himself.  Look at the graphs, does it open up or open down. The student who needs the visual can do it over and over until they get it.  The graphing calculator also has a table in it.  For students who are highly numeric, this is great with automatic tables.

Learning Style I

Prefer “recipe” approaches to mathematical procedures – step-by-step

Need the HOW before the WHY

Seldom estimate

Tend to remember parts rather than wholes

Strong need to talk themselves through the process

Once process is mastered, diligent in carrying it out

Unaware of the underlying principles that give meaning to what they’re doing.

Learning Style II

Uses skills that involve reverse processing (subtraction/division)

Impatient with step by step procedures

Make mistakes when trying to apply step by step procedures

Good at estimating

Resistant to using paper and pencil techniques

Superior at recognizing large-scale patterns

Good at appreciating various spatial relationships

Good at geometric situations involving both 2D and 3 D configurations.

Good test-taking skills because can work backwards from the multiple choice answers by estimating.

Want to be manager who runs through aisles at HYVee and calculate.

What’s the preferred they go to immediately, and are they successful with the preferred.  If not, may want to try another.

Show their work—Is it a concept error or a calculation error or moving things incorrectly.  “Your brain is a dark cave.  You showing me your work lets me see.”  Help them diagnose themselves when they have a problem.

Conservation of number—counting on (4 + 5 + 6 + 7)

How do we help floundering students who lack basic math concepts?

By Marilyn Burns

Nine Ways to Catch Kids Up

Author:      Marilyn Burns

Magazine

Association For Supervision

And Curriculum

November 2007

Vol. 65  No. 3

Grappling with interventions

Three issues that are essential to teaching mathematics

# 1    It’s important to help students make connections among mathematical ideas so they do not see these ideas as disconnected facts.

# 2   It’s important to build students’ new understandings on the foundation of their prior learning.

#3   It’s important to remember that students’ correct answers, without accompanying explanations of how they reason, are not sufficient for judging mathematical understanding.

Essential Strategies

I have found  the following nine strategies to be essential to successful intervention instruction for struggling math learners.  Most of these strategies will need to be applied in a supplementary setting, but teachers can use some of them in large-group instructions.

Marilyn Burns

Strategy # 1

Determine and Scaffold the Essential

Mathematics Content

Determining the essential mathematics content is like peeling an onion.  We must identify those concepts and skills we want students to learn and discard what is extraneous.  Only then can teachers scaffold this content, organizing it into manageable chunks and sequencing these chunks for learning.

Example:  For a child to multiply 683 x 4, they need a collection of certain skills.  They must know the basic multiplication facts.  They need an understanding of place value that allows them to think about 683 as 600 + 80 + 3.  Then students must be able to multiply 4x3 (a basic fact), 4x80 (understanding the power of 10), and 4x 600. To master multidigit multiplication, students must be able to combine these skills with ease.  Lesson planning must ensure that each skill is explicitly taught and practiced.

Strategy # 2

Pace Lessons Carefully

We’ve all seen the look in students’ eyes when they get lost in math class.  When it appears, ideally teachers should stop, deal with the confusion, and move on only when all students are ready.   Yet curriculum demands keep teachers pressing forward, even when some students lag behind.  Students who struggle typically need more time to grapple with new ideas and practice new skills in order to internalize them.  Many  of these students need to unlearn before they relearn.

Strategy # 3

Build in a Routine of Support

Students are quick to reveal when a lesson hasn’t been scaffolded sufficiently or paced slowly enough:  As soon as you give an assignment, hands shoot up for help.  Avoid this scenario by building in a routine of support to reinforce concepts and skills before students are expected to complete independent work. There is a four stage approach.

Stage 1 – The teacher models what students are expected to learn and  records the appropriate mathematical representation on the board.

Example of Stage 1:

The students are expected to multiply 2x3x4.

The teacher models the possibilities.

2 x 3 x 4                    2 x 3 x 4                        2 x 3 x 4

2x3=6x4=24            2x4=8x3=24                   3x4=12x2=24

It is important to understand that solving a problem in more than one way is a good strategy for checking your answer.

Stage- 2:   The teacher models again with a similar problem such as 2 x 4 x 5.  This time the teacher elicits responses from students. The teacher asks which two numbers might the students start with and what should be that answer.  The teacher then asks what number should be multiplied next, always using correct mathematics vocabulary such as factors and product instead of numbers and answer.

Stage – 3:  The teacher presents a similar problem.

Example:  2x3x5; After thinking about the problem, the students work in pairs to solve the problem in three different ways, recording their work.   As the students report back to the class, the teacher writes on the board and discusses their problem solving choices with the group.

Stage – 4:  Students work independently, referring to the work recorded on the board if needed.  This routine both sets an expectation for student involvement and gives learners the direction and support they need to be successful.

Strategy # 4

Foster Student Interaction

Giving students opportunities to voice their ideas and explain them to others helps extend and cement their learning.

To strengthen the math understandings of students who lag behind, make student interaction an integral part of instruction.  Using the think-pair-share or the turn and talk strategies give students the opportunity to think on their own, discuss the problem with a partner, and finally share their ideas with the whole group.  This gives students opportunities to express their math knowledge verbally. This is particularly valuable for students who are developing English Language Skills.

Strategy # 5

Make Connections Explicit

Students who need intervention instruction typically fail to look for relationships or make connections among mathematical ideas on their own. They need help building new learning on what they already know.

Example:  6x8; How is this related to 6x9?

Students need to be able to connect the meaning of multiplication to what they already know about addition. That is to think of combining 6 groups of 8. They need time and practice to cement this understanding for all multiplication problems. In addition to 6x8, they should be able to associate other factors such as 6x2, 6x3, and so on, looking at numerical patterns of these products. There should be as many of these experiences as possible to enable students to grasp the relationships of how these equations are connected.

Strategy # 6

Encourage Mental Calculation

Calculating mentally builds students’ ability to reason and fosters their number sense.  Once students have a foundational understanding of multiplication, it’s key for them to learn the basic multiplication facts.  Students’ experience with multiplying mentally should expand beyond these basics.  Students should investigate patterns that help them mentally multiply any number by a power of 10.

When students calculate mentally, they can estimate before they solve problems so that they can judge whether the answer they arrive at makes sense.

Strategy # 7

Help Students Use Written Calculations

to Track Thinking

As students learn and practice procedures for calculating, their calculating with paper and pencil should be clearly rooted in an understanding of math concepts.  Help students see paper and pencil as a tool for keeping track of how they think.

Example:  14x6

Students can write 6x10=60 and 6x4=24 to reach partial products. Then they can add the two sums together.  This also can be used for multiplying three digit numbers by single digit numbers.

Strategy # 8

Provide Practice

Struggling math students typically need a great deal of practice. Its essential that practice be directly connected to students’ immediate learning experiences.  Choose practice problems that support the elements of your scaffolded instruction, always promoting understanding as well as skills.

• Give assignments through the four stage support routine allowing for a gradual release to independent work

• Games can be another effective way to stimulate student practice. Games like Pathways that make use of multiplication skills by marking boxes on the board that share a common side and that each contain a product of two designated factors.

Strategy # 9

Build In Vocabulary Instruction

The meanings of words in math such as even, odd, product, and factor often differ in meaning than when used in a common language.  Many students that need math intervention have weak mathematic vocabularies.

It’s key that students develop a firm understanding of mathematical concepts before learning new vocabulary, so that they can anchor terminology in their understanding.  We should explicitly teach vocabulary in the context of a learning activity and then use it consistently.  A math vocabulary chart can help keep both teacher and students focused on the importance of accurately using math terms.

When Should We Offer Intervention

Three Scenarios

While the class is studying the topic -  We must also provide comprehensive instruction geared to repairing the student’s shaky foundation of understanding.

While others are learning multidigit multiplication, floundering students may need experiences to help them learn basic underlying concepts such as single digit numbers.

Plus – Intervening at this time may give students the support they need to keep up with the class.

Before the class studies the topic -  Gives struggling students a new concept prior to the rest of the class so that at-risk students will experience a new concept on an individualized basis and then experience the concept again with the rest of the class.

Plus – We are preparing students so they can learn with their classmates

Minus- Struggling students are studying two different and unrelated mathematics topics at the same time.

When Should We Offer Intervention

Three Scenarios

After the class has studied the topic – Offers learners a repeat experience, such as during summer school in a math area that initially challenged them.

Plus – Students get a fresh start in a new situation.

Minus – Waiting until after the rest of the class has studied a topic to intervene can compound a student’s confusion and failure during regular class instruction.

Continued.........

When do you offer

intervention strategies?

……..instruction for all students and especially for at-risk students must emphasize understanding ,sense making, and skills.  I am now much more intentional about creating and teaching lessons that help intervention students catch up and keep up, particularly scaffolding the mathematical content to introduce concepts and skills through a routine of support………..

Marilyn Burns

Math Solutions Professional Development

800-868-9092

Week 3

Experiences to help children learn to count on.  Finding math in everything.   Between 4 and 7 years old, gain ability to correspond symbols with actual number of items.  By relating materials to the real world, students can catch on without realizing it.  You can have students help develop a game.

Early Childhood Corner: Experiences to Help Children Learn to Count On
Linnea Weiland
October 2007, Volume 14, Issue 3, Page 188
Abstract:
Edited by Andrew M. Tyminski. Early Childhood Corner is a regular department of the journal. It features articles related to the teaching and learning of young children. Counting on is a landmark strategy that young children discover through interacting with the world around them. Although early childhood teachers should not try to teach this strategy as a rote procedure, they can be purposeful in creating experiences that encourage counting on. Classroom routines as well as games and activities can be used to create meaningful mathematical situations that help children construct understanding of why counting on works. Let kids play with calculators.  Need simplistic ones without overwhelming buttons.  The conservation of quantity is the essence of the article.

Touch point --count on points of each number.

Chisum bop--add and multiply on fingers--mid 80s

Tims report--mid 90s The Third International Mathematics Science Symposium.  How we compare to industrialized nations, we're near the bottom.  When they talk about this, they're referring to the TIMS report.

CUMON Math  like Sylvan, but specifically for math.  Use Asian methods.

Chip exchange--carry

Until a kid is developmentally ready to count on, they cannot do it.  They should be able to count on between age 4 and 7.

Call to action from 1980s about what's wrong with out math program.  Dr. Shirley Hill--we need national standards.

Play with dice or Yatzee to see if high school students truly don't know how to count on.

Conservation of number means that you should be able to look at it and say "that's five."  That's different from knowing addition facts.

Tally marks in groups of five means they have conservation of numbers.

Use monopoly--money, doing percentages, is it better to pay luxury or 20%.  Make mathematically correct choices.

Card games can help.

Hundred chart--can help the kid learn how to add.  Use with card that has a window.  You can help kids learn how to add and subtract.

borrowing --chip exchange

chip exchange chart, ones tens hundreds columns with different colors in each category.  Use colored chips. Replace ten yellows with a blue.

Can use hundred chart with highlighter, sequencing, skip counting, multiplying, prime numbers.  Will help with estimating and rounding.  Helps take away idea they have to count tally marks or number ine.  Helps them rewrite distributive properties.

rename=chip exchange=borrow

71=7tens plus 1 ones

dyphantine problem--there are multiple correct answers.

Something rotationally symmetric

Conservation of length, conservation of volume.

Number world--sra program

Scott Foresman - Addospm Wesley Math by grade.  Another Look Reteaching workbook

.Cuisinare rods are color coded manipulative.

Use paper folding to show multiplying  of fractions.

Week 5

Processing strategies:  Logical step by step versus gestalt.  See page 91 of original handout.

Gestalt kid will often work backwards.  Or they will estimate and change their guess.

There are always other ways to solve something.

Algebra tiles are a way to have have a physical representation of x squared.  Home made ones, click here.

Punnett Squares in biology gives a grid that would be better than algorythmthly.

Assignment.  Minimum biographical stuff.  Condense to 2 pages.  Why did you decide this kid is this kind of learner.  UsePowerPoint handout.  Analyze one kids.  Figure out what kind of learner he is.  Due Monday before spring break. Are they visual?  What is your justification for picking one learning style.  5 or more characteristics typed out.  1st name only.  If even distribution, then which do they chose.  Age group.  Learning type I or learner type 2.  Logical or gestalt?

BD kids 9 times out of 10 are kinesthetic and visual learners. Do a table with checkmarks. Due March 17.

D Young

You can give examples.  Two page analysis on our part.  This is the midterm.

Manipulatives

Menu Math is a good book.

Adding It Up is not in color, but an excellent book for teaching math.

Remediation Strategies

1.  Use technology.  Can be many forms.  Can provide individualized instruction through pretests. NCTM.org http://nctm.org/ Strategies Illuminations.

Logical Journal of Zoominis Scholastic Warehouse Sale Zoombinis Software

Another good program
http://www.teenbiz3000.com/

## Hot Dog Stand

Quarter Mile Final  Through algebra 1 and 2 the_quarter_mile_math_bundle_k_6

Prentice Hall support system is math xl  http://mathxl.com/login.htm

## The Geometer's Sketchpad  geometry, middle school, algebra

2.  Make it make sense.  Break it down into pieces the student needs to have to complete the task.  A recipe approach.  Try an analogy.

3.  Tie to previous knowledge.

4.  Write/copy the rules.  5 and up round up; four and down round down.

5.  Use a journal/use writing.  Teach something and ask them to summarize it.

6.  Use estimation.  Good for style 2 kids.

7.  Make it simpler. Break the problem down into components.

7

x + 2 = 5

Subtract 2 from both sides or SIMPLIFY.  The problem is the fraction, so work around the fraction.

7

x + 2 = 5

8.  Look for a pattern.

9.  Mnemonic Device.

10.  Use manipulatives.  Cuisenaire rods.  Paper-folding.  Blocks.

11.  Matching card games.  Concentration.

12.  Other games.  Bingo game.

13.  Connect with a picture

14.  Work backwards.

15.  Use a calculator.

16.  Acting out, especially with word problems.

17.  Worksheet

18.  Alternative rules.

19.  Discuss meaning of.

Not all are feasible.

Sketchpad is a dynamic package.  http://www.dynamicgeometry.com/

The students get accustomed to looking for those things that stay the same and those things that change.

Responsible for 5 remediation activities.

We need to be able to diagnose.  What does the kid know?

See example.  What does the kid absolutely know?

Both of those are correct answers, but they are not simplified.

Kid knows multiplication facts.

Can do equivalent fractions.

Knows least common multiple.

can multiply across the top and bottom.

The child may demonstrate some knowledge.

What doesn't the kid know?

Appropriate use of common denominator.

Hasn't mastered reducing.

"Here's what I was expecting to see."  "The extra steps aren't needed."

Doesn't impose one way on the student.

Tie into previous knowledge.

Write the rules:  Multiply the numerators, multiply the denominators, and reduce if possible.

Use a journal:  Write the rules.  Doing practice.  Showing the difference between multiplying and adding and subtracting.

Decide what's in the kid's best interest.  If the kid is doing it correctly--maybe not what we want--may decide to leave it alone.

Technology could be the calculator.

T x T over B x B

Draw a picture of paperfolding.

Fraction bars

Pie

Using estimation.  How close is the fraction to a whole number.  Tie to a percentage or decimal.

Card game:  Make One  http://mathforum.org/paths/fractions/one.game.html

Working backwards:  What two parts make up an eighth?  What 3 parts make up 3/5?  Visualize the fraction bar.

TI 30s calculator helps the student reduce.

Hershey's chocolate

Counters in egg cartons

Alternative rule:  Get a common denominator on all of them.

Recipe--need half of all the measurements.

Divide a burrito

## www.softschools.com/math/worksheets/  Free worksheets

What does it mean to multiply by a fraction:  Discuss meaning of

Think, pair, share.

Use to get the kids to remediate themselves to determine problems on a quiz.

 Objective to be learned Right Wrong Guess Arithmetic Error Didn't simplify Concept Error Need more study

Final Project Description