| Memorization | Procedures without Connections | Procedures with Connections | Doing Mathematics | What's the Level of Demand? |
|---|---|---|---|---|
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False.
Tasks at the memorization level involve using previously learned facts, rules, formulae, or definitions.
True or False
Memorization tasks require the use of algorithms or procedures to solve them. |
True.
Tasks at this level require no explanations or simple explanations that focus on describing the procedure that was used.
True or False
Tasks at the procedures without connections level do not require in depth explanations. |
True.
Tasks at this level can be represented using visual diagrams, manipulatives, symbols, or problem situations. Connecting these different representations helps develop meaning.
True or False
Procedures with connections tasks can often be represented in multiple ways. |
False.
Doing mathematics tasks require complex, non-algorithmic thinking. There are not explicit pathways or predictable approaches suggested in the problem.
True or False
Tasks that are at the doing mathematics level can be solved using algorithms or explicit approaches. |
Memorization
- Reproducing previously learned facts - Does not require procedures - No connection to concepts of meaning - No ambiguous
Complete the following problems in a minute or less.
2 x 3= _____ 4 x 9= _____ 5 x 6= _____ 10 x 3= _____ |
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No.
This would be a procedures without connections problem because the solutions are too large to be easily memorized. An algorithm must be used to determine the answers.
Is the following a memorization task?
Solve the following problems. 24 x 19= _______ 13 x 12= ________ 18 x 21= ________ |
Yes.
There is an algorithmic way to solve the problem and it would be implied by previous instruction. There is little ambiguity and no connection to underlying meaning. The explanation is solely focused on the procedure.
Is the following a procedures without connections task?
A dress originally costs $65. For the weekend sale, it was marked 40% off the original price. What was the price of the sweater during the sale? Explain the procedure used to find the sale price. |
No.
This would be a procedures without connections task. There is an algorithmic step-by-step procedure shown and a calculator can be used, which lowers the cognitive demand. There is also no connection to the underlying concepts.
Is the following a procedures with connections task?
(materials: calculator) 2 x 2 = 4 2 x 2 x 2 = 8 2 x 2 x 2 x 2 = 16 2 x 2 x 2 x 2 x 2 = 32 Could 275 be in this pattern? Why or why not? |
No.
This would be a procedures with connections task because a general procedure must be followed but the student must come up with this procedure on their own, creating some cognitive effort. This problem can also be shown using multiple representations.
Is the following a doing mathematics task?
Megan is putting new carpet in her bedroom, which is 15 feet long and 10 feet wide. How many square feet of carpet will Megan need? |
Procedures without connections
- Algorithmic - Focused on correct answers - No explanation required - No connection to underlying concepts - Limited cognitive demand
Jamie and Oliver made 120 cookies for the school bake sale. They packaged these cookies into 24 bags. How many cookies are in each bag?
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- Involves recalling already learned facts
- Cannot be solved using an algorithm - Very clear, not ambiguous - Has no connection to underlying concepts or meaning.
What makes the following problem a memorization task?
State the highest place value of the given number. 3748 7 658 34 |
- Limited cognitive demand
- Uses a set, rehearsed procedure. - Focused on correct answers - No explanation is required
What makes the following problem a procedures without connections task?
Using base-ten blocks, show that .497 is less than .5. |
- There is a procedural way to find the answer, but focuses on gaining a deeper understanding of the problem.
- There is an implicitly suggested path to solving the problem. - Can be represented in multiples ways. - Requires some cognitive effort.
What makes the following problem a procedures with connections task?
Monica is 4 years older than James. James is twice as old as Frank. If Monica is 10 years old, how old is Frank? |
- Non-algorithmic thinking
- Demands self-regulation of one's own abilities - Access relevant knowledge about spatial awareness - Examining task constraints - May cause some anxiety
What makes the following problem a doing mathematics task?
Draw a rectangle and shade approximately 4/5 of the rectangle. What other fractions are near 4/5 in size? |
Procedures with connections
- Uses procedures to gain a deeper understanding - Suggests pathways - Can make a connection using multiple representations. - Requires some cognitive effort because procedures isn't explicitly stated.
8 is 10% of a number
The square root of 64. Create an expression comparing these two numbers. |
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Answers may vary
1) Involves reproducing already learned facts, rules, formulae, or definitions or committing these to memory. 2) Cannot be solved using procedures. 3) Not ambiguous.
Name three features of a memorization task.
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Answers may vary
1) Algorithmic procedure is specifically called for or evident. 2) Limited cognitive demand for completion. 3) No connection to concepts that underlie the procedure.
Name three features of a procedures without connections task.
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Answers may vary
1) Uses procedures to gain a deeper level of understanding. 2) Suggests pathways to follow that have connections to underlying conceptual ideas. 3) Represented in multiple ways and makes connections between representations.
Name three features of a procedures with connections task.
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Answers may vary
1) Complex non-algorithmic thinking. 2) Explore and understand the nature of mathematical concepts, processes, and relationships. 3) Self-monitoring or self-regulation of one's own cognitive processes.
Name three features of a doing mathematics task.
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Procedures without connections
- Focuses on using procedures to understand concepts. - Implicitly suggests pathways - Can be represented in multiple ways - Requires some cognitive effort
A class has 30 students. A teacher wants to set up the desks in rows. How many ways can the desks be arranged to that all the rows are equal?
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Answers may vary
Give an example of a memorization task.
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Answers may vary
Give an example of a procedures without task.
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Answers may vary
Give an example of a procedures with connections task.
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Answers may vary
Give an example of a doing mathematics task.
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Doing mathematics
- Requires complex, non-algorithmic thinking - Explore and understand math concepts - Access relevant knowledge - Examine task constraints - Can cause some anxiety
A pattern of dots is shown on the board.
Amy wants to figure out the number of dots in the 20th pattern but doesn't want to draw them all. Explain how she can figure this out and tell how many dots will be in the 20th pattern. |
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