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Building Academic Skills in Context Enhancing

Mathematics Achievement through CTE Instruction

- Leslie Carson
- leslie.carson_at_sreb.org

Welcome

Group Norms and Housekeeping

- Group Norms
- Participate
- Ask questions of each other
- Work toward solutions

- Housekeeping
- Restrooms
- Breaks
- Lunch
- Punctuality

Communities of Practice

- Offers the more powerful conceptual model for

transforming schools - (R. DuFour)
- Collaborative Teams
- Collective Inquiry
- Action, Orientation and Experimentation
- Continuous Improvement
- Results Orientation
- Lets form communities.

The Power of Team Dynamics

- Instructions in planner
- 20 minutes to plan, practice
- I will act as timekeeper and give the start

signal - After the challenge, there are reflection

questions to ponder - GOAL Build the tallest free-standing structure

Essential Questions

- Why is helping students understand math

everyones job? - What does math look like in the non-math

classroom? - How do math and CTE teachers support student

understanding of math?

At the turn of the 20th century, we were an

experience rich, information poor society.

Today we are an information rich experience poor

society. Dale Parnell

Preparing our Students to be Successful

Mathematically

- Previously it was enough for our students to

just be able to solve a given math problem such

as - What is 45 divided by 7?
- Reading off of a calculator, the answer is

6.428571429

The Good News Is..

Did you know? 2005 Skills Gap Report-National

Association of Manufacturers

- 84 of employers surveyed believe public schools

are failing to prepare students for the

workplace. - The biggest deficiency is in areas of science and

mathematics, noting that - word problems seldom resemble real-world

experiences - mathematics teachers teach standard approaches

(e.g. algebraic symbol manipulation) to the

detriment of mathematical reasoning. - too often students' mathematics experiences are

characterized by repetition learning rather than

problem solving

Did you know?

- 1/3 of NASA employees were born on the Indian

subcontinent - The Visionarys Handbook 2000 Watts Wacker

Jim Taylor w/Howard Means

Did You Know?

- American Industry is spending nearly as much

each year to educate their employees

mathematically as is spent on mathematics

education in public schools - A selection from Numeracy by Lynn Arthur Steen

Lynn Authur Steen, St. Olaf College

- Children learn to read and write not solely

because of their language arts instruction in

school, but equally because of the reinforcement

provided by other school subjects, and by their

environment at home. Where reading and writing

are not reinforced at home, the progress of

learning is much slower.

Lynn Authur Steen, St. Olaf College

- (Mathematics) is rarely reinforced, neither in

school nor at home. Parents, coaches, and

teachers of other subjects seldom make the effort

to engage children in activities that would use

mathematical or statistical methods--perhaps

because the adults themselves tend to avoid such

methods.

U.S. Ranked 24th out of 29 OECD Countries in

Mathematics

Organization for Economic Cooperation and

Development (OECD), PISA 2003

How can we fix it?

"Somebody has to do something, and it's

just incredibly pathetic that it has to be us."

- Jerry Garcia of the Grateful Dead

As any wise old farmer can tell you, you dont

fatten your lambs simply by weighing them

Common Misconceptions about Learners Sue E.

Berryman and Thomas Bailey, The Double Helix of

Education and the Economy (New York Institute on

Education and The Economy, Columbia University,

1992), 45-68

- People predictably transfer learning from one

situation to another. - Learners are passive receivers of wisdom-empty

vessels into which knowledge is poured. - Learning is the strengthening of bonds between

stimuli and correct responses. - What matters is getting the right answer.
- Skills and knowledge, to be transferable to new

situations, should be acquired independent of the

contexts of uses.

Cognitive Science Questions about the Teaching

and Learning Process

- How do the human mind and body work in their

learning capacity? - How can an understanding of the mind/bodys way

of learning be used in educational settings?

A

How Do New Pieces of Information Fit?

Learning often occurs only when students process

new information or knowledge in such a way that

it makes sense in their frame of reference

Students can increase high level mathematics

understanding through experiences.

And by Collaborating with Their Peers.

Learning in Context Involves

- Linking new information to students familiar

frame of reference - Hands on activities combined with teacher support

to allow students to discover new understandings - Application of new knowledge to real world

situations - Working in collaborative groups to solve problems
- Transfer understanding to new situations and

problems

Why is relevancy important?

- Example Learning a Code

A

Now

- Spell the word FACE in Code.
- Lets see how you did!

a

Algebra students feel the same way about what

they learn in algebra. The out of context codes

for letters make just as much sense to them as

using x and y as variables.

Traditional mathematics education presents the

concepts first and applications second

This logical approach is successful for a

limited segment of abstract thinkers in our

student population. Many students have difficulty

assimilating abstract theories. These students

learn from educational programs that emphasize

hands on learning. They need to experience it.

Learning Mathematics should be based on

understanding concepts and on doing mathematics,

not on memorizing rote procedures

Why Should CTE Teachers Care About Mathematics?

- NCLB / AYP
- Students indicate they do not like mathematics

because they do not see the use for it. CTE

courses fill that need. - CTE Teachers can provide the relevance for

motivation and the frame of reference so that CTE

students value mathematics. - The CTE classroom also provides the environment

where students can develop high level math skills - Mathematics is one of the new basic skills for

industry. - Mathematical literacy is required of anyone

entering a workplace or seeking advancement in a

career.

CTE Classrooms Provide the Perfect Learning

Environment

- The purpose of Perkins IV- The purpose of this

Act is to - develop more fully the academic and career and

technical skills of secondary - education students and postsecondary education

students who elect to enroll in - career and technical education programs, By
- 1st of 7 building on the efforts of States and

localities to develop challenging academic and

technical standards and to assist students in

meeting such standards, including preparation for

high skill, high wage, or high demand occupations

in current or emerging professions - 2nd of 7 promoting the development of services

and activities that integrate rigorous and

challenging academic and career and technical

instruction, and that link secondary education

and postsecondary education for participating

career and technical education students

Did You Know?

- Daggett, 2005
- Studies have shown that students understand and

retain knowledge best when they have applied it

in a practical, relevant setting. - According to Daggett, at the high school level,

CTE programs provide the most effective learning

opportunities. - Not only are students applying skills and

knowledge to real-world situations in their CTE

programs, but also they are drawing on knowledge

learned in their core subjects.

Often, mathematical and CTE skills are chunked,

making transfer to the mathematics classroom and

standardized tests difficult

Pythagorean Theorem?

3-4-5 Angle?

Often the mathematics in CTE is bypassed or

minimized by teaching students shortcuts.

- CTE students often learn shortcuts to bypass or

memorize mathematical processes. CTE courses can

be used to create teachable moments for

understanding mathematics.

Reading, Mathematics, and Science HSTW Scores for

Students

CTE Programs Provide Focus Foundation

Context

Why Is It Important That CTE And Academics Work

Together?

- Example Picture Recall

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We bring different pieces of information to the

team!

Team Member Roles

- Math Teachers
- Collaborate to find and emphasize embedded math

in CTE. - CTE Teachers
- Collaborate to provide relevancy in Academics.
- CTE and Math Teachers
- Consult with each other to implement successful

instructional strategies.

Important to remember

- CTE teachers should introduce the CTE concepts

first and then emphasize the math concepts - Mathematics teachers should introduce the math

concepts first and then emphasize the CTE

connections to provide relevancy

The teaming of math teachers and CTE teachers

merges context and content.

- Historically, this has not happened naturally.

There must be a concentrated effort.

What Can Mathematics and CTE teachers do?

- Implement authentic anchor projects that become

places where students use mathematics - Require students to solve authentic adult-like

problems - Have students use tools of the trade to complete

tasks

Good reasons for integration

- Integration is how people work in the real world.
- Academic and CTE teachers expand their repertoire

of teaching strategies. - Enhances student motivation.
- Enhances student career planning.
- Promotes professionalism among teachers.
- Improves academic achievement.

What is integrated embedded CTE learning?

- NOT inserting mathematics in the CTE curriculum
- Is unwrapping embedded mathematics in the CTE

Authentic Anchor Projects students complete - NOT turning CTE teachers into mathematics

teachers - Is looking at how the mathematics can be

emphasized, made more rigorous, and academic

language included in the projects and problems

CTE teachers teach.

What is Authentic Academic Learning?

- Not setting aside the math curriculum
- Is providing real life scenarios and projects for

math concepts - Not turning math teachers into CTE teachers
- Is bridging the language of math to the language

of CTE and providing opportunities to practice

authentic problem solving and application

Important to remember

- CTE teachers should introduce the CTE concepts

first and then emphasize the math concepts and

necessary components - Mathematics teachers should introduce the math

concepts first and then emphasize the CTE

connections to provide relevancy

SREBs Criteria for Authentic Anchor Project

Units

Overview Eight Steps to Develop Authentic

Integrated Projects

Seven Elements for teaching mathematics through

authentic integrated project units

- CTE teacher introduces CTE lesson.
- CTE teacher assesses students math awareness.
- CTE teacher works through embedded example.
- Math teachers work through traditional examples.
- CTE and math teachers work through related,

contextual examples. - Students demonstrate understanding in CTE and

math classes. - CTE and math teachers formally assess students.

Step One Identify a major project with embedded

mathematics content.

- In your teams, CTE teachers share a major project

they will complete with students. The project

selected should be rich in embedded mathematics. - Break down into the activities that make up the

project and consider making adjustments that add

math rigor. - Identify the CTE concepts in each activity.
- Identify the CTE vocabulary.

Is the Math in CTE Courses the Same as that in

Academic Math Courses?

Usually the math is the same but the terminology

is often different.

Pitch or slope?

Bridging the Gap Between Academic and CTE

Vocabulary

- We often use
- Different words for the same concept (pitch,

slope, grade, incline, steepness, elevation) - Different meanings for the same word
- Directions
- Before going any further, you will need a piece

of paper and something to write with. - You will see four words. Write each word on your

paper as you see it and then write the first

definition of that word that comes to mind.

Define

- AXIS
- SOLUTION
- RANGE
- YARD

AXIS

- Science the second vertebra in the neck, which

acts as the pivot on which the head and first

vertebra turn - Agriculture central part of plant the main part

of a plant, usually the stem and the root, from

which all subsidiary parts develop - Geometry one of two or more lines on which

coordinates are measured. Often on a graph two

axes form its left and lower margins.

RANGE

- Transportation distance traveled without

refueling the farthest distance that a vehicle

or aircraft can travel without refueling. - Agriculture open land for grazing farm

animals a large area of open land on which farm

animals can graze. - Construction north-south strip of townships a

north-south strip of townships six miles square

and numbered east and west from a meridian in a

U.S. public land survey.

RANGE Continued

- Mathematics set of values the set of values

that can be taken by a function or a variable. - Statistics extent of frequency distribution the

difference between the smallest and the largest

value in a frequency distribution.

SOLUTION

- Science fluid with substance dissolved in it a

substance consisting of two or more substances

mixed together and uniformly dispersed, most

commonly the result of dissolving a solid, fluid,

or gas in a liquid. It is also, however, possible

to form a solution by dissolving a gas or solid

in a solid or one gas in another gas. - Mathematics value satisfying an equation a

value for a variable that satisfies an equation.

YARD

- Business Slang for one billion dollars. Used

particularly in currency trading, e.g. for

Japanese yen since on billion yen only equals

approximately US10 million. It is clearer to

say, " I'm a buyer of a yard of yen," than to

say, "I'm a buyer of a billion yen," which could

be misheard as, "I'm a buyer of a million yen. - Agriculture livestock enclosure an enclosed

area of land for livestock.

YARD Continued

- Agriculture land around a house the area of

land immediately surrounding a house, often

covered with grass or landscaping. - Agriculture winter grazing area an area of land

where deer, moose, or other animals graze in

winter. - Mathematics imperial unit of length a unit of

length equal to 0.9144 m (3 ft).

Authentic Anchor Project Unit Development

Template

- Step One Describe a CTE project rich with

embedded mathematics that you will be teaching

soon. - Include the
- essential questions for the unit.

Chart It

Guiding Questions/Essential Questions

- Open-ended
- Drive all instruction
- Higher order and stand the test of time. Even

when curriculum changes, the essential questions

are still there. - At least one connects to students real life
- Examples
- What is the price of eggs in China?
- How would the price of eggs in China effect the

cost of things in the US?

AUTHENTIC INTEGRATED PROJECT UNIT CONCEPTS Step

One

Identify, in order, the activities that make up the project CTE Concepts to be Covered Mathematics Concepts to be Covered Tools Needed Habits of Success/Literacy Vocabulary

Chart It

SEE SAMPLE

Step Two Identify the embedded mathematics to

be taught through the authentic integrated

project unit.

- CTE teachers explain project objectives so that

mathematics teachers can help discover the

embedded mathematics. - Identify the math tools and vocabulary
- Tools to help with this process
- School/district pacing guides
- Competency Checklist

What do the embedded math standards look like?

AUTHENTIC INTEGRATED PROJECT UNIT CONCEPTS Step

Two

Identify, in order, the activities that make up the project CTE Concepts to be Covered Mathematics Concepts to be Covered Tools Needed Habits of Success/Literacy Vocabulary

Chart It

Step Three Choose literacy and habits of

success strategies you will use as you teach the

project unit.

Listening

Speaking

The Big Six Literacy Strategies Any Teacher Can

and Should Use

- Summarizing
- Paraphrasing
- Categorizing
- Inferring
- Predicting
- Recognizing Academic/Technical vocabulary

Habits of Success .

- 1. Create Relationships
- 2. Study, Manage Time, Organize
- 3. Improve Reading/Writing Skills
- 4. Improve Mathematics Skills
- 5. Set Goals/Plan
- 6. Access Resources

Exploring Habits of Success- INSERT Strategy

- Read the description of the Habits of Success

assigned to your team. - Place a by each bulleted item that you

have a strategy for. - Place a ? by each bulleted item that you do not

have strategy for.

Exploring Habits of Success- INSERT Strategy

- 3. Look for people in the room who have a where

you have ? and interview them. - 4. The goal is to have at least one new strategy

under each Habit of Success category.

AUTHENTIC INTEGRATED PROJECT UNIT CONCEPTS Step

Three

Identify, in order, the activities that make up the project CTE Concepts to be Covered Mathematics Concepts to be Covered Tools Needed Habits of Success/Literacy

Chart It

Step Four Develop a summative unit exam to

assess students understanding of mathematics and

CTE concepts used in the project.

- Dictates what students should know and be

able to do, includes a traditional paper and

pencil assessment, and a rubric for the

Culminating Task. - Should also Include
- Performance assessment (if appropriate)
- Traditional problems found on college placement

and state level exams - Authentic Problems from the pathway

Step Five Pre-assess students mathematics and

CTE knowledge and skills that are embedded in the

authentic integrated project.

- Identify prerequisite and new skills students

need to be successful in the unit. - Include
- Reading problems
- Procedural mathematics problems
- Assess various vocabulary, skills and

understanding of mathematics content - Varying levels of mathematics problems

What does varying levels mean?

- Getting to mastery
- at the proficient level is the key!
- Basic
- Proficient
- Advanced
- Stem Questions from Blooms

Where can we find good examples of assessment

items?

- Released NAEP items
- http//nces.ed.gov/nationsreportcard/nde
- http//nces.ed.gov/nationsreportcard/itmrls/start

search.asp - State accountability tests-released items
- SkillsUSA test items
- http//skillsusa.org/compete/math.shtml
- Textbooks (enrichment sections)
- www.micron.com

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www.micron.com

- MACHINIST
- A worker in a lawnmower factory is asked to make

the part shown below for a new lawnmower design.

To program a machine to cut the part out, the

worker must find the slope and length of edge D.

Standardized Tests

- The project assessment should include items that

might appear on - state assessments
- ACT
- SAT

ACT- The edges of a cube are each 3 inches long.

What is the surface area, in square inches, of

this cube? F. 9 G. 18 H. 27 J. 36 K. 54

- In Context- The youth center has installed a

swimming pool on level ground. The pool is a

right circular cylinder with a diameter of 24

feet and a height of 6 feet. A diagram of the

pool and its entry ladder is shown below. To the

nearest cubic foot, what is the volume of water

that will be in the pool when it is filled with

water to a depth of 5 feet? - A. 942
- B. 1,885
- C. 2,262
- D. 9,047
- E. 11,310

Guidelines for Developing Authentic Problems

- Apply desired math content.
- Use a non-contrived scenario.
- Include real-world numbers with appropriate units

of measure. - Remain faithful to the selected occupational

area. - Include some extraneous data.
- Avoid hand-holding or step-by-step guidance.
- Examples

Process Form for Creating Mathematics/CTE

Authentic Problems

- Examples
- Work as a team
- to write an authentic
- problem

How Do We Help Students Learn Problem Solving?

- RAP Sheet
- Think Aloud
- Socratic
- Questioning

Steps 6 7 Designing Engaging Instructional

Strategies

Step Six Identify instructional strategies CTE

teachers will use to engage students with

mathematics embedded in the Authentic Anchor

Project Unit. Step Seven Identify

instructional strategies math teachers will use

to engage students by providing relevance for the

math content.

The Corandic

Engaging Instructional Strategies that aim toward

college readiness.

- Cooperative learning
- Project-based learning
- Socratic method
- Anticipation guides
- Videos
- Readings
- Demonstrations
- Multi-intelligences approach

- Technology integration
- Blogs
- YouTube
- Graphing calculators
- CBLs and CBRs
- Excel
- Literacy Strategies
- Use of manipulatives
- Others?

Notes

- Bring both CTE and mathematics knowledge base

into discussions. - Mathematics teachers help CTE teachers develop

ways to teach mathematics. - CTE teachers help mathematics teachers understand

the CTE well enough to develop contextual

problems for use in the mathematics classroomto

give mathematics students a reason to understand

the mathematics. - Both CTE and mathematics need to share in

preparation of students for high-stakes exams.

Remember the seven elements

- CTE teacher introduces CTE lesson.
- CTE teacher assesses students math awareness.
- CTE teacher works through embedded example.
- Math teacher works through traditional math

examples. - CTE and math teachers work through related,

contextual examples. - Students demonstrate understanding in CTE and

math classes. - CTE and math teachers formally assess students.

Seven Elements A sample from the building trades

- CTE teacher points out that carpenters (students)

must ensure 90-degree (square) corners on their

building project. - Assess students math awareness as it relates to

CTE. - CTE teachers walks through the original problem

of ensuring a wall frames rectangular shape,

gradually introducing the math formula and

terminology of the Pythagorean theorem. CTE

teachers uses both math and CTE vocabulary. - Mathematics teachers teaches Pythagorean Theorem

in the traditional manner. - CTE and mathematics teachers assign construction

problems to which the Pythagorean theorem can be

applied. - Students demonstrate their understanding of

Pythagorean theorem through project completion. - The students complete their building project, as

well as a worksheet practicing the Pythagorean

theoremboth in CTE and traditional math

problemsas a part of a formal assessment. In

the mathematics class, students apply the

Pythagorean theorem in traditional and contextual

problems as a part of a formal assessment.

Project Activity Worksheet

Day Career/Technical Course Instructional Activities Math Course Instructional Activities Field Trips Guest Speakers

Chart It

Example

Project Activity Worksheet

Day Career/Technical Course Instructional Activities Math Course Instructional Activities Field Trips Guest Speakers

Chart It

Team Planning Time for Steps 6 and 7

- Consider the strategies that CTE teachers might

use to teach the mathematics - Consider the strategies that academic teachers

might use to relate the mathematics embedded in

the project to traditional mathematics concepts.

Chart It

- Make sure your work is charted and hung before

you leave!

Step Eight Describe how students will

demonstrate their understanding of CTE and

mathematics knowledge and skills as they complete

the project.

Example?

- This involves
- Coordinating the instructional activities and

assignments from previous steps into a coherent

plan which students complete.

Step Eight Describe how students will

demonstrate their understanding of CTE and

mathematics knowledge and skills by completing

the project.

- This also involves
- Creating rubrics for mathematics and CTE

components of the project. - Determining when co-teaching can and will take

place - Developing community resources
- Providing performance guidelines
- Writing a scenario or problem for students to

solve - Developing extra help strategies for students who

do not meet mastery at the proficient level

Step Eight Describe how students will

demonstrate their understanding of mathematics

knowledge and skills by completing the project.

- Most important to remember
- Require students to demonstrate understanding at

varying levels of difficulty (basic, proficient,

advanced) - Assess learning through performance-based

assessments AND using test items found in college

placement exams and state level accountability

tests

Team Planning Time

- With your community, complete a Project Outline

which thoroughly articulates how students will

demonstrate understanding of both technical

competencies and mathematics competencies.

Reflections on Work Using SREB Criteria

- As a team, analyze your project using the
- Checklist for Quality Authentic Integrated

Project Units - Note any adjustments or improvement that need to

be made.

What is the Cycle of Learning?

- Get started
- Engage
- Explore
- Explain
- Practice together
- Practice in pairs
- Practice alone
- Evaluate
- Close

Cycle of Learning

- For all classes
- Creates a pattern of learning
- Moves students into deeper understanding
- Allows for formative assessment

First Getting Started

- Quick
- All students can be successful.
- Not new learning.
- Establishes routine.

Second Engage

- Introduces focus for the day
- Provides relevance for the lesson
- Provides motivation for learning

Third Explore

- Creates personal learning
- Sets the stage for content
- Allows for interaction
- Creates investment in content

Fourth Explain

- Multiple types
- Short bursts of new content
- Within context of exploration
- Teacher talk

Fifth Stages of Practice

- Practice together
- Practice in small groups
- Practice alone

Sixth Evaluation

- Formative assessment usually
- Indicates mastery or re-teaching
- Usually brief
- Often informal

Seventh Closing activities

- Bring closure to each class
- Enhance retention
- Improve parent relations

The Cycle can take more than one day!

- Sample two day cycle
- Day One Day Two
- Getting Started Getting Started
- Engage Engage
- Explore Practice in teams
- Explain Practice alone
- Practice together Evaluate
- Closure Closure

Project Planning

- This is where you put everything together into

a clearly articulated series of daily that ensure

instructional time is used wisely. - Example

Using the Cycle of Learning

- Lets build a daily lesson plan together for one

of the authentic integrated units developed

today.

Write a Daily Lesson with your school team!

- Using your authentic integrated unit, develop a

daily lesson plan for the CTE teacher and a daily

lesson plan for the mathematics teacher. - One Day for Math and One Day for CTE

Complete Daily Lesson Plans!

- As team members, continue working on the unit
- Email me if you have questions

Instructionally..

- Does the project present learning activities to

students following the seven elements?

Action Planning Using the Team Action Plan

- Identify next steps to completion of the

skeleton unit, including implementation of the

unit. - How will you finish planning together?
- When will you teach the project?
- How will you share with others what you are

learning? - What will you bring to the follow up meeting?
- What data will you share with your community in

follow up meetings?

Remember the goal

- Supersize Opportunities for Students to Embrace

Mathematics

Ticket out the door!

- Last page in your handout!
- Thank You for Being Here!
- See ya next time.