This is a topic that I used to teach in a full day lecture and have presented a profession conference. I'm a big fan of cognitive theory and the things we can gain from it. While this isn't one of the "student development" theories it is a very real one when it comes to college. It also explains why algebra is required for college degrees. You might have seen it before in regards to note taking, asking questions, but I think that a basic understanding of this concept teaches students what their professors expect from them.
This pyramid was developed by Benjamin Bloom back in the 1950s and is referred to as Bloom's Taxonomy of Learning. The basic idea of the taxonomy is that not all learning is equal. As we learn new things we progress through the different levels of leaning. We cannot move into higher levels of learning without first achieving the levels below them. The levels are broken down as follows:
Knowledge (basic learning) - being able to repeat information as it was given to you. Defining terms, listing components, repeating concepts you have heard.
Understanding (simple comprehension) - the ability to explain in your own words. Taking the information out of context and replaying it to someone else.
Application (initiation of action) - using the information to accomplish a task or other such action. seeing how the information works in the context of the material.
Analysis (Deep understanding) - breaking the information down into its respective components and seeing how the parts interact. Also, seeing how the information fits into the greater context and outside of the initial context. Identifying relationships between the information and itself and other topics and facts.
Synthesis - (deep application, creation) - using the information to create new ideas, conclusions and applications. Using the component pieces to produce new ends and means.
Evaluation (defending your conclusions and actions) - having the ability to defend your conclusions and actions to others using clear and developed arguments and evidences.
The best way to break down the levels is with the use of action verbs - verbs that require action or demonstrate clear objectives. These verbs are often found in the assignments, tests, and research proposals that professors give. Being able to associate the verb with the level helps students understand what kind of depth or quality the professor is looking for. There are many lists of action verbs, but I like this one from Clemson University. It's got a nice layout and a good list of verbs.
Now, the expectation that I explain to college students. In high school the basic standard of teaching and learning is that students are expected to demonstrate that they know and understand the material as it was presented to them. Teachers and students do not progress much beyond the first to layers of the pyramid. However, when student arrive at college, the professors will often times help them achieve those same levels of a topic, but then will expect work on the higher levels. They try to give the tools to help students move from one to the next, but they are not content if students just linger in the levels of knowledge and understanding. This is often why new freshman will complain that their tests are not fair. "The test was about stuff that we didn't cover in class." or "I don't remember seeing this in class." These responses are often the case because the student leaned the material on a minimal level but the professor is testing their higher level learning.
What does this have to do with algebra? If the goal of college is to get people to think on a higher level then algebra is the natural gateway. Example:
Knowledge level - Define addition: calculating the total of two or more amounts (citation)
Understanding level - Explain addition: what you get when you combine two numbers
Application level - 2+2=? answer: 4
Analysis level - 2+?=4 answer: 2
Basic algebra, even in such a simple calculation, elevates a persons thinking - requiring them to analyze, break down, and find the relationships in the equation. When my old math teach said, "This is to make you think" the desire was not just a cognitive process. Bloom shows us that not all thinking is the same. This is to get you to think deeper then you might normally. This is to stretch your brain so that it doesn't return to its original, limited state, but to enlarge your capabilities. Like I said last week "This is to make you think" is the most accurate response to the question "why do I need to study algebra." i also think that it is the most valid and the best. This is why colleges set a standard on the minimum level of mathematics students must complete. So they can be sure that students are going to have an opportunity to think on those levels that can actually empower students to be proactive and not just reactive.
This pyramid was developed by Benjamin Bloom back in the 1950s and is referred to as Bloom's Taxonomy of Learning. The basic idea of the taxonomy is that not all learning is equal. As we learn new things we progress through the different levels of leaning. We cannot move into higher levels of learning without first achieving the levels below them. The levels are broken down as follows:
Knowledge (basic learning) - being able to repeat information as it was given to you. Defining terms, listing components, repeating concepts you have heard.
Understanding (simple comprehension) - the ability to explain in your own words. Taking the information out of context and replaying it to someone else.
Application (initiation of action) - using the information to accomplish a task or other such action. seeing how the information works in the context of the material.
Analysis (Deep understanding) - breaking the information down into its respective components and seeing how the parts interact. Also, seeing how the information fits into the greater context and outside of the initial context. Identifying relationships between the information and itself and other topics and facts.
Synthesis - (deep application, creation) - using the information to create new ideas, conclusions and applications. Using the component pieces to produce new ends and means.
Evaluation (defending your conclusions and actions) - having the ability to defend your conclusions and actions to others using clear and developed arguments and evidences.
The best way to break down the levels is with the use of action verbs - verbs that require action or demonstrate clear objectives. These verbs are often found in the assignments, tests, and research proposals that professors give. Being able to associate the verb with the level helps students understand what kind of depth or quality the professor is looking for. There are many lists of action verbs, but I like this one from Clemson University. It's got a nice layout and a good list of verbs.
Now, the expectation that I explain to college students. In high school the basic standard of teaching and learning is that students are expected to demonstrate that they know and understand the material as it was presented to them. Teachers and students do not progress much beyond the first to layers of the pyramid. However, when student arrive at college, the professors will often times help them achieve those same levels of a topic, but then will expect work on the higher levels. They try to give the tools to help students move from one to the next, but they are not content if students just linger in the levels of knowledge and understanding. This is often why new freshman will complain that their tests are not fair. "The test was about stuff that we didn't cover in class." or "I don't remember seeing this in class." These responses are often the case because the student leaned the material on a minimal level but the professor is testing their higher level learning.
What does this have to do with algebra? If the goal of college is to get people to think on a higher level then algebra is the natural gateway. Example:
Knowledge level - Define addition: calculating the total of two or more amounts (citation)
Understanding level - Explain addition: what you get when you combine two numbers
Application level - 2+2=? answer: 4
Analysis level - 2+?=4 answer: 2
Basic algebra, even in such a simple calculation, elevates a persons thinking - requiring them to analyze, break down, and find the relationships in the equation. When my old math teach said, "This is to make you think" the desire was not just a cognitive process. Bloom shows us that not all thinking is the same. This is to get you to think deeper then you might normally. This is to stretch your brain so that it doesn't return to its original, limited state, but to enlarge your capabilities. Like I said last week "This is to make you think" is the most accurate response to the question "why do I need to study algebra." i also think that it is the most valid and the best. This is why colleges set a standard on the minimum level of mathematics students must complete. So they can be sure that students are going to have an opportunity to think on those levels that can actually empower students to be proactive and not just reactive.
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