optimizing instruction and learning for vocational technical students – part two building capacity...

Optimizing instruction and learning for vocational technical students – Part two

Building capacity in technical labs and and related classrooms for strong Tier I instruction, systematic differentiation, and building the framework for accommodations and interventions.

K. Rocke for the Pioneer Valley DSAC Tech / SPED leadership institute presentation for 2/7/13 1

Plan for today

• Describe one possible approach to eliminating learning barriers for students in technical programs (integration of academic and technical learning)

• Identify ‘leading edge’ projects and methodologies that might be useful to technical schools

• Analyze one kind of math ‘unit’ from that perspective


Unique set of circumstances – new frameworks in academics and CTE

Common Core

• Mass. Math Frameworks

• PARCC Model Curriculum Units

Technical Frameworks

• CVTE Technical Curriculum Frameworks

• Technical Curriculum model units


What would the ideal look like? (modeling mathematical learning)


What would the ideal look like?

• Technical teachers understand the frameworks, mathematical concepts, techniques and language of math

• Math teachers know specific and authentic applications of key math concepts and practices in each technical area

• Special education teachers are familiar with both, and are able to coach both teachers and students to ‘make the connections’


What would the ideal look like?

• Students engage in integrated, inter-disciplinary projects that bring together math concepts, math skills, and technical applications.

• Assessments are based on these projects, and curriculum and instruction is UbD-designed to build the skills and knowledge necessary.

• Technical, academic and special education teachers collaborate to accomplish this, and then explicitly model and teach their shared methodology.


What strategies can we employ to get there?

• Multiple means of engagement (theoretical, practical) – technical area motivation

• Multiple means of representation (theory to practice) – curriculum design

• Higher order thinking skills will require higher-order collaboration strategies…

• Finding the time to meet is just the first step – what will they do when they get together…?



Skills and Knowledge

What carpenters need to know about geometry (practical level)

Rectangles (area)

Rectangular solids (volume)

triangles

circles


Lines (length and distance)

…and the formulas

• Length • Perimeter• Area• Volume• Diameter• Radius• Hypotenuse • Circumference (sometimes)


Pythagorean Theorem:A2 + B2 = C2

B

AC


Pythagorean Theorem (area representation)


What do the standards say about this?

• Math standards• Carpentry standards


Math frameworks (excerpt): what 8th grade students are expected to know and be able to do:

Geometry• Understand congruence and similarity using physical

models, transparencies, or geometry software.

• Understand and apply the Pythagorean Theorem.

• Solve real-world and mathematical problems involving volume of cylinders, cones and spheres.


And from the frameworks glossary:

• Pythagorean theorem. For any right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse.


What the new draft CVTE standards say about the Pythagorean Theorem:


Group Discussion: Comparing the Standards • What are the similarities between the 8th

grade math standards and the draft CVTE standards about the Pythagorean Theorem?

• What are the differences?


3-4-5 storyHarry and I were told to build a ‘dust house’ on the factory floor. Piping would be connected to it to collect the cotton dust extracted from machines during the production process. Harry was an experienced and able carpenter.

The floor was concrete, and uneven. The sketch for the structure showed a wooden building 8’ by 10’, with diagonal board decking.

The first step was to frame the deck, using KD lumber – two 2x6x8’ and two 2x6x10’

pieces of lumber.

We cut the stock to length, nailed the corners using 16d spikes, and were ready to square the frame so that we could begin laying the decking.

Most commonly you square a frame by measuring the diagonals, and adjusting the frame until it is reasonably square (maybe +/- 1/8” over a frame this size).

However, in this case, there was a round pipe in the center of our deck, protruding through the plane of the deck surface-to-be, so we could not use this method.

Harry said that he would measure down 3’ feet on one side, and I should measure 4’ down the other (he was the foreman on the job), and we would check the diagonal to square the deck.

I suggested we use 6’, 8’ and 10’ instead, in case the lumber wasn’t perfectly straight, and to get a truer corner to work from.

Harry’s response was that that would not work, and we would stick with the 3-4-5 method.

Questions:

• What did Harry understand?

• What didn’t Harry understand?

• What kind of education (and assessment) would raise the likelihood that a graduate of a carpentry program in vocational technical schools would not make this mistake?

• When squaring a frame by measuring the diagonals, what geometric principles, ideas or theorems are we employing?

• Why are parallelograms bad when you’re a builder?


Discussion: Think-Pair-Share

• What did Harry understand? • What didn’t he understand?• What kind of education (and assessment)

would raise the likelihood that a graduate of a carpentry program in vocational technical schools would not make this mistake?


What he knew…

4

35


3-4-5 works

What he thought he knew…

8

610


6-8-10 does not work

Did he know?

40

3050


Did he know that:Similar triangles yield similar results?

Similar triangles yield similar ratios

• 3-4-5• 6-8-10• 9-12-15• 12-16-20• 30-40-50

Do they really work?

• 9 + 16 = 25• 36 + 64 = 100• 81 + 144 = 225• 144 + 256 = 400• 900 + 1600 = 2500


Did he know this?


Or how to apply it to other right triangles?


Did not appear to understand similar triangles…


Who knew?


Practical triples (isosceles right triangles)


5

5

7-1/16

12

12

17

Technical and academic language and concepts are not the same…

Academic– Pythagorean

Theorem– Similar triangles– Pythagorean triples– Determining the

hypotenuse of a right triangle

Technical (carpentry)– 3-4-5– 6-8-10– Squaring a frame– Laying-out a

common rafter


Group Discussion:

We often say that students laying-out rafters are learning the Pythagorean Theorem

• Is it true?• How do we know?• How could we know?


Laying out a rafter without using the Pythagorean Theorem


assessment


Are our assessment practices adequate to the task?

• We have…

– Academic assessments (MCAS, math tests)

– Assessments of technical competency (competency profiles)

– Instruments and rubrics for observation of academic classrooms (walkthroughs, evaluation rubrics)


But do we have a way of assessing…

For students

• Student ability to apply math theory to practical applications?

• Student ability to understand the theory behind practical technical applications?

For teachers and schools

• Teachers’ ability to teach to both theory and practice, and to assess same?

• Do we know who is ‘doing the math’ in technical labs and related classrooms?


Discuss: What would these assessments look like?

Task predicts performance.

What evidence would demonstrate that students are being asked to (and have) accomplished this?

For students

• That is has been learned:

Comprehensive, rigorous assessments, incorporating technical expertise and explicit assessment of embedded mathematics.

For teachers

• That it is being taught and practiced:

Observations of the ‘instructional core’ in technical labs and related classrooms


Discuss: What would these tasks look like?

What would a comprehensive assessment of this ‘unit’ of knowledge and skills (Pythagorean Theorem) look like?

• Theoretical knowledge and understanding

• Practical applications and ability to perform

• Ability to apply theory and practical applications to a new situation


A quick first attempt at modeling a comprehensive assessment

1. State the formula for the Pythagorean Theorem (algebraically). 2. State the Pythagorean Theorem (in words).3. Can a right triangle also be an isosceles triangle?4. What are Pythagorean Triples?5. Given two sides of a right triangle, calculate the length of the third side:

• Given both legs;• Given one leg and the hypotenuse.

6. Given the squares of two sides of a triangle, calculate the length of all three sides:

• Given the square of one leg and the square of the hypotenuse;• Given the squares of both legs.


Which are equivalent concepts, vocabulary?• Trade vocabulary• Common rafter• Framing square• Rise• Run• Total Rise• Total Run• Plumb cut• Seat cut• Bird’s mouth• Tongue• Blade• Line length• 3-4-5• 10ths scale• 12ths scale• 16ths scale

• Math vocabulary• Triangle• Right triangle• Isosceles right triangle• Leg (of triangle)• Hypotenuse• Pythagorean Theorem• Proof• Geometric proof• Algebraic proof• Square• Square root• Similar (triangles)• Ratio (of sides of triangles)• Parallel• Perpendicular• Pythagorean Triple• Mixed fraction• Decimal• Square root


More of the same…

1. Why are some Pythagorean Triples more useful than others in the field of carpentry?

2. Given a height on a wall and a length from that wall on a deck, calculate the length of a 45 degree brace.

3. Given the total run and rise of a rafter, calculate the length of stock needed if there is a 1’ horizontal overhang.

4. Determine the line length of a common rafter using the Pythagorean Theorem. (from the standards)

5. When laying-out a hip rafter with a framing square, why is the number 17 used for the run?


Calculate a set of triangles similar to a 3-4-5 triangle, containing only whole numbers for sides:


Observation


What might we look for in observation?

Who’s doing the work?

• Who’s doing the thinking?• Who’s doing the math?• Who’s doing the research?• Who’s doing the planning?• Who’s managing?• How is self-regulation

occurring?

Levels of engagement

• Do students understand what they’re working on?

• Are they motivated?• Are they engaged?


Observation can take many forms…

methods

• Instructional Rounds

• Learning Walkthroughs

• Peer Observation

• Educator Evaluation Observations for team goals

instruments

• Characteristics of a Standards-based classroom:– Student thinking– Applying new knowledge


Student thinking


Applying new knowledge


It’s the teachable moment, but…

• Do technical teachers have the skills to explicitly teach math, language, etc.?

• Are resources available to support technical teachers in the acquisition of these skills?

• Do we have pre- and post- assessments that will measure students’ acquisition of embedded math, science, literacy, planning skills?


What approach to curriculum design might get us there?


Understanding by Design (UbD)

• Desired results• Evidence of those results (assessment,

observation)• Learning plan• Lesson plan• Scaffolding plan


What kinds of collaboration strategies could teachers practice to integrate their teaching?


Structures to promote this adult learning and collaboration – under development by the DSACs

• Teacher-centric Learning Walkthroughs and peer observations (team goals) in labs and related classrooms

• PLCs, with hybrid teams of technical, academic and special education teachers

• Math and assessment courses for technical teachers

• Regional networks of technical educators (like this one!)


Theoretical understanding

(academic)

Practical applications (technical)

Goal: transferrable knowledge

and skills

School and teacher practices promote integration of learning for all students

Essential for career

success

Essential for further education


Our students and teachers work at all levels, as needed…


optimizing instruction and learning for vocational technical students – part two building capacity...

Documents

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vocational technical

technical labs

technical schoolsanalyze

learning barriers

kind of learning path

teachers there1april

powerpoint presentation