Scaffolding Math Benchmarks with Blooms Taxonomy

I'm in the middle of a PBL unit and am designing a series of scaffolded formative assessments to help students achieve a mastery of understanding without the traditional summative test at the end. The walk in assessments have given some good insight to student abilities and I'm using small group peer evaluation and teacher one-on-one sessions while students work on their project to ensure that students are making progress.

I just saw "Waiting for Superman" and one of the points in the documentary was that there was an inconsistency between states, districts, and even schools of what "mastery" means. They used an example wherein a student could fail an assessment but then cross into a neighboring state and pass the same one. People are hoping that common core will eliminate the disparity but I think it will take 2-3 more years of implementation and resource acquisition before we see a more level playing field. Basically, people had different ideas of 'mastery'. 

What does "Mastery" Mean?
Why is there such disagreement about what "Mastery" means? I like to think that mastery means an "in-depth understanding of a mathematical concept that can be explained with critical thinking and reasoning" In short, something that cannot be easily measured through standardization. As a math teacher of 10 years, I've learned that there are many different answers and supporting reasons why something could be correct. Take this problem from properties of exponents wherein students learn some of the simplification processes of working with exponents with the same base:
Lower level rungs on Bloom's Taxonomy involve remembering principles, and summarizing or demonstrating that students can do them. Good for entry level understanding.

If they can simplify this to the correct answer which is 7 to the 7th power, they are correct. But have they "mastered" this understanding? I think not. All they have done is recalled the process of adding exponents with the same base and there is no explanation why this is supported. In fact, our textbook makes no mention of this as well.

Enter Blooms Taxonomy
Blooms taxonomy is a great model for designing assessments that work their way up towards the higher order skills of "evaluation" which we see in the pinnacle below. It can serve as a model to make the curriculum and learning of it "attainable" to entry level students in the beginning, but provide continual spiraling and scaffolding through subsequent more difficult tasks to develop cognitive abilities.

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In Practice with Formative Assessments
As my walk in "entry interview" I had a couple questions from the previous lesson. They all started at a very attainable level such as "Knowledge" and "Comprehension" but got progressively harder. In the case of properties of exponents, I tried to scaffold problems in a way that would lead to the higher order skills that I thought resembled "mastery".

Day 1-A low level question to check for basic understanding before moving on to higher applications

Day 2-An application and or analysis with many answers. Open ended questions can be differentiated by asking for supporting reasoning if needed.

Day 3-Evaluation and Synthesis to show reasoning and mathematical principles.

By day 5 within the unit, problems from our first lesson had been phased out, and students had been given one-on-one help with earlier concepts to tailor to a very individualized learning experience.

Final Stage of the Unit
By the time the end of the unit had rolled around, students had had enough practice with earlier concepts through spiraling to eliminate the need for a final summative test. The cyclical nature of using progressively harder formative assessments did the following:
  • Gave multiple opportunities to show learning though mistakes and peer and teacher conferencing
  • Allowed the opportunity to apply previous learning foundations to new situations
  • Eliminated the mandate for a final test
  • Provided inflection points to raise the bar with higher order thinking questions
  • Gave indicators over what benchmarks students were beginning to understand, developing their understanding or mastering their understanding
  •  Got rid of the "Remember this, because you'll be tested on it in two weeks" mentality.
Since the above criteria were met, I could focus on an in-depth understanding of the Pythagorean Theorum for the final project which contained elements of roots and exponents and our own PBL slant on the unit with some integrated artistic principles and technology expression as well. I didn't feel chained to the 15 benchmarks of the unit and had essentially broken down the test in a series of warm ups.

More Ideas on Scaffolding
There are other ways that others are exploring the use of scaffolding for understanding. Here are some ideas that I've learned from educators around the world.
  1. Provide "Challenge by Choice" materials. CBC is offering the curriculum through different tracts. entry level, standard level and advanced level. It's not modifying the curriculum in any way, but rather making it more accessible or challenging to any heterogeneous population of students. 
  2. Experiment with math stations. Such activity centers give opportunities to apply learning to a real-world authentic problem which can be shared to the class or online.
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Formative Assessment in the Math Classroom
Turning Student Failure into Information
Challenge by Choice

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