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The art of teaching is the art of assisting discovery.

-Mark Van Doren

Instructional planning involves the preparation necessary to meet the individual needs of the classroom members. A Mathematics Lesson Evaluation is included to show the ability to interpret the goals, strengths, and weaknesses of a lesson plan. Constant changes are required to build the best possible lessons from those that were initially successful.

This section also contains notes on Problem Solving, Transfer, and Information Processing. These a excellent reminders of the methods that are utilized by students to reach their goals.


Curriculum Evaluation

             The Authors of the lesson, Counting Cubes Suggest that the teacher begin by leading a discussion with the students that explores the advantages of using cubes over other materials such as popcorn  to measure volume. The teacher is then asked to model for the students how to build shapes with the cubes and produce a two-dimensional symbol that represents the three-dimensional shape. Following the discussion, the students should be allowed to move around the room and investigate the volumes of various classroom objects with counting cubes. Once the learners are armed with this background knowledge, the class is broken into pairs and asked to construct an original shape using the counting cubes. Students are then asked to draw a two-dimensional representation of their creation on grid paper, exchange shapes with their partner, draw a two-dimensional depiction of their partner’s creation, and compare the similarities and differences between the two shapes. After constructing then representing shapes, students are asked to reverse the process and create a building from a pictorial representation they are given. Each pair of students is given two drawings that appear to be different, but they represent the same three-dimensional shape. After the learners discover that three-dimensional objects can be illustrated in more than one way in two-dimensional space, they are requested to find the volume of the shape they made. In the last step in this part of the lesson, students build another prism and attempt to find two different representations for it that can be produced on grid paper. The second portion of the Counting Cubes lesson has students create irregular shapes and investigate how rotating the objects alters the top-view representation that would be drawn when recording the characteristics of the shape. During this activity, teachers are expected to circulate and guide students having trouble interpreting the diagrams. Students are then presented with a series of practice problems to reinforce what they just discovered. After everyone has finished working out their problems, the class reconvenes and discusses the various strategies that were used when solving the problems. The students are then led through a series of problems on worksheets that continually build on the prior information in order to help learners develop strong connections and encode the conceptual understanding to long-term memory.

            The table of standards and expectations of appropriate content in the NCTM standards suggests that students should “identify, compare, and analyze attributes of two and three-dimensional shapes and develop vocabulary to describe the attributes.” The lesson that was reviewed complies with this standard in its exercises that guide students to discover the similarities and differences that exist among objects that are depicted in a concrete three-dimensional models and two-dimensional maps. Similarly, the lesson connects vocabulary such as cubic unit, volume, two-dimensional, and three-dimensional to the concepts they represent by using hands-on learning activities as reinforcement. Secondly, the NCTM content standard that suggests students should “investigate, describe, and reason about the results of subdividing, combining and transforming shapes,” is apparent in the lesson in the exercise that asks learners to transform the cubes by removing cubic units and determine the resulting change. Skill worksheet 22, which also covers content included in this NCTM standard, explores breaking shapes into parts by shading the target area and determining the characteristics of that portion of the figure. The third content standard in the 3rd-5th grade band that describes content included in this lesson states that students should “make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.” This speculation is evident in the beginning of the lesson with discussions that explore the best ways to calculate the volume of something, when students make their original conjectures and justify their reasoning, then again later when the class discusses some of the successful strategies that the students used. The next NCTM content standard that is touched on by participation in this lesson says, “Predict and describe the results of sliding, flipping, and turning two-dimensional shapes.” Part of this exploration in geometry involves building a three-dimensional model from a two-dimensional map that is provided, flipping the three-dimensional model that is created, and drawing a two-dimensional illustration of the flipped figure. When doing this part, young explorers can see for themselves how a two-dimensional representation of a three-dimensional object changes after a flip is made to the object. The fifth content standard that Counting Cubes covers suggests students should “build and draw geometric objects.” The entire beginning portion of the geometry lesson involves building and drawing geometric objects in order to build strong conceptual understanding of the mathematical ideas. Likewise, the standard: “identify and build a three-dimensional object from two-dimensional representations of that object” is represented in almost exactly that context several times in the lesson. The inverse of that standard, “Identify and build a two-dimensional representation of a three-dimensional object,” is similarly repeatedly addressed; with every example, the lesson’s creators have the learners represent the objects with both a two and three-dimensional representation so the participants gain proficiency at moving between the different types of representations. The eighth content standard from the 3rd-5th grade band from which ideas are taught by the lesson suggests students should “use geometric models to solve problems in other areas of mathematics, such as number and measurement.” This standard is basically echoed by the activities contained in the lesson. Students are guided to explore measurement of number of cubic units that make up volume through three-dimensional geometric shapes that are meant as concrete representations of what the containers could hold. Finally, the NCTM content standard that states students should “recognize geometric ideas and relationships and apply them to other disciplines and to problems that rise in the classroom or in every day life, is applied at the end of the lesson when the worksheet activities culminate with a tie-in to discovering the volumes of cartons that might be found in the students everyday life.

The lesson counting cubes does a good job of meeting the five NCTM process standards. The first process standard, problem solving, involves creating new mathematical knowledge through problem solving; which is, as Van De Walle explains, “the vehicle through which children develop mathematical ideas.” This problem solving is the basis for the discovery learning that is supplied by the lesson. The lesson begins by having students reflect on the ideas they already have about the amount of space inside an object and ways that it could be measured. It then asks them to explore means of demonstrating three-dimensional space and construct concrete models to proceed, one step at a time, through the exercises discovering connections to new concepts and understanding. The second process standard, reasoning and proof, is characterized by logically thinking about presented problems, deciding on a logical method for finding a solution, and determine if the answers are logical and make sense. The safe pairs that this lesson provides for students to reason through the small jumps in conceptual understanding that the worksheets asks, allow for a comfortable atmosphere and predetermined points at which to periodically compare results and verify that the reasoning being used is a valid strategy that is leading the learner to reasonable answers. The third NCTM process standard, communication, highlights, as Van De Walle informs, “the importance of being able to talk about, write about, describe, and explain mathematical ideas.” This process standard also stresses communication as another means of active participation in the search for answers. Communication is included in this lesson in various forms; verbal communication is used in the introductory lecture when concepts, methods and strategies are being discussed. Verbal communication is also used when the pairs are exploring the counting cubes to find the answers for the worksheets provided. Written communication is required in this lesson when the learners are required to fill out the worksheets and create the two-dimensional representations of the shapes that are created and demonstrate that they posses an understanding of the ideas presented. The fourth process standard, connections, is concerned with the interrelations among mathematical ideas, how these mathematical ideas build on one another, and the connections to the world outside the classroom. Counting Cubes’ worksheets do a nice job of creating a steady, progressive growth toward building a true conceptual understanding of the reasons why length times width times height will explain the volume of a rectangular prism and a good conceptual base to which other connections can be connected in order to determine the volume of other three-dimensional objects. Another strong point of this lesson that connects to this standard is the tie to the volume of a puzzle box and a cereal box, thus providing a connection outside the classroom to the mathematical ideas presented in the lesson. The final NCTM process standard, representation, involves the ability to illustrate mathematical ideas in order to demonstrate understanding and communicate ideas. This geometry lesson moves excellently from one representation to another to help strengthen students’ understanding that there are multiple ways to represent the same object and some representations can be better models for certain problems. On one of the extra worksheets, the lesson even points out some of the limitations of reproducing a three-dimensional figure in two-dimensional space; it has students explore the possible representations, and has the students discover some of the drawbacks of each type of representation.

The lesson plan provides several opportunities for assessment of the students’ understanding of the concepts that are presented and the effectiveness of the lesson’s teaching. The first exercise requires learners to visualize missing parts and determine what was removed, determine how many cubic units make up a larger cube, and determine the volumes of various shapes. This exercise will show if the student understands the construction of the objects and the relationship between the cubes and a cubic unit. Skill worksheet 21 presents learners with pictures of  three-dimensional shapes and corresponding two-dimensional plans of the object and asks them to find the volumes. This exercise further reinforces the learning to the student, measures their understanding, and measures the effectiveness of the instruction. Skill worksheet 22 similarly assesses conceptual understanding by asking learners to find the volume of the shaded portion of objects, thus illustrating rather or not students understand the goal of the questions. The next ancillary assessment worksheet provided by the lesson has students predict the number of cubes that would be needed to construct objects that are pictured; by not showing the students the entire view of the objects or a two-dimensional plan, the focus starts to move the students away from simply counting the blocks and toward demonstrating they are gaining understanding. The next alternate worksheet removes the individual cubic units and replaces them with totals for length, width, and height in order to determine that students still understand that a volume is still a measure of the total cubic units it can hold even though they can not see each individual unit. The final worksheet leads learners through interpretations of different physical arrangements of the same number of cubes and moves on to representations of real-world examples. This worksheet assesses that students really understand that the concept of volume is independent of shape. Even though these assessments do a good job of covering important mathematics, enhancing learning, promoting equity, providing an open process, promoting questioning, and providing coherency to the lesson, I would probably supplement these assessments with a small quiz on volume at a later time to help ensure encoding to long-term memory. 

I believe that Counting Cubes, in its current form, would be beneficial to learners at a wide range of ability levels. I do not believe that the lesson itself would need to be altered when teaching special needs children; however, instructors will need to show diligence and flexibility in the amount of additional support they provide. Significant scaffolding will be needed to guide learners that are more challenged to discover the knowledge and understand the reasons behind the knowledge and it may be possible to place students in groups to explore the more difficult worksheets. If using the lesson plan with gifted students, instructors would attempt to challenge enough to hold interest by using a hands-off approach, only using open-ended questioning when absolutely necessary. In addition to the easy adaptability, I believe that using the same lesson for the entire class has the added benefits of creating a sense of community in the classroom and giving traditionally low-achievers an improved self-efficacy in mathematics as they are successful on the same lessons given to fellow students that are known to be gifted. 



Problem Solving

Well-defined problem: only one solution and a certain method of finding it.

Ill-defined problem: more than one acceptable solution and no agreed on strategy.

 Strategies for improving problem solving abilities:

bulletDiscuss a general model that can be applied in a variety of cases.
bulletDescribe characteristics of expert problem solvers to use as model.
bulletGive specific suggestions for problem solving improvement.
bulletUse technology to help students become better problem solvers.

 General problem solving model:

           Identify Problem

         Represent Problem

               Select Strategy

        Implement Strategy

           Evaluate Strategy

 Obstacles to identifying the problem:

bulletLack of experience in defining problems
bullettendency to rush to solution before problem has been clearly defined
bullettendency to think convergently rather than explore strategies divergently
bulletLack of domain-specific knowledge (background knowledge)

 Representing the Problem: using a visual or written representation decreases load on working memory.

 Selecting a strategy:

Algorithm: specified set of steps used in solving a problem (for well-defined problems)

Heuristics: general, widely applicable strategies.

bulletTrial and error: inefficient but valuable in gaining experience.
bulletMeans-end analysis: breaking problems into smaller parts
bulletDrawing analogies: solve by comparing to similar problems that learner is familiar with.

 Implementing strategy: being able to perform the chosen strategy.

 Evaluating strategy: assessing the effectiveness of the chosen strategy

 * Help students find value in evaluating their results by having them estimate the results before starting the problem.

 Critical thinking: ability and tendency to make and assess conclusions based on evidence.

Elements of critical thinking:

        Basic processes


 Domain-specific knowledge


      Motivational factors






Basic processes: the fundamental components of thinking.

bulletFinding patterns and generalizing
bulletcomparing and contrasting
bulletidentifying ir/relevant information
bulletForming conclusions based on patterns
bulletAssessing conclusions based on observation
bulletchecking consistency
bulletidentifying bias, stereotypes, clichés and propaganda
bulletidentifying unstated assumptions
bulletRecognizing under/overgeneralizations
bulletconfirming conclusions with facts

Metacognition: knowing when to use the basic processes, how they relate to prior knowledge, and why they are used.

Motivational factors: the attitude and disposition the learner brings to the learning experience.

bulletThe inclination to rely on evidence in making conclusions.
bulletThe willingness to respect different opinions.
bulletSense of curiosity, inquisitiveness, and a desire to be informed.
bulletTendency to reflect before acting.
bulletThe willingness to replace ideas, beliefs, and assumptions.



Transfer: When prior experience and knowledge effect learning or problem solving in a new situation.


Positive transfer: When learning in one situation facilitates performance in another.


Negative transfer: When one situation hinders performance in another.


General transfer: Ability to take knowledge or skills learned in one situation and apply them to a wide range of different situations.


Specific transfer: Ability to use information in a setting similar to the one in which the information is originally learned.


*Research has shown transfer is specific.


Factors affecting the transfer of learning:

bulletSimilarity between the two learning situations.
bulletDepth of learners’ original understanding.
bulletQuality of learning experiences.
bulletContext for learners’ experiences.
bulletVariety of learning experiences.
bulletEmphasis on metacognition.


Information processing

Information processing: cognitive theory examining the way information is stored and used.

bulletInformation stores: storage places for information
  1. Sensory memory: location for briefly holding stimuli received by our senses until it can be processed further.
    bulletFades fast, .1-1 second visual, 2-4 sec. hearing.
  2. Working memory: (short-term memory) location for storing information as it is being worked with.
    bulletOnly holds 7 items or 2 or 3 when being processed simultaneously.
    bulletCognitive load theory: recognizes limitations of memory and stresses teaching that can enhance capacity.
    bulletChunking: combining items into more meaningful groups.
    bulletAutomaticity: mental operations done with little conscious effort.
    bulletDual processing: describes working memory as visual and auditory
  3. Long-term memory: permanent memory store.
    bulletDeclarative knowledge: knowledge of facts, definitions, procedures, and rules.

* can be determined from a person’s comments.

bulletProcedural knowledge: knowledge of how to perform tasks.
bulletDeclarative stage: can not perform task but knows rules.
bulletAssociative stage: can perform but must think about task.
bulletAutomatic stage: can perform with little thought.

* can be determined by person’s performance.

* Efficiency depends of organization and development of schemas.

bulletSchemas: complex networks of connected information. (help to reduce load on working memory)
bulletScripts: plans of action for certain situations.


Cognitive processes: way information moves from one memory store to another.

  1. Attention: process of focusing on stimulus.
    bulletAttracting attention in the classroom.
    bulletDisplays, Pictures, Maps, and Graphs
    bulletThought-provoking questions
    bulletMove around the classroom
    bulletChange rate and pitch and intensity of speech
    bulletUse gestures and energetic motions
    bulletUse students’ names
  2. Perception: process of attaching meaning to experience.

* can be determined by asking open-ended questions.

1.      Rehearsal: repeating information without altering its form.

* Not an efficient method but one of the first developed.

2.      Encoding: placing information into long-term memory.

3.      Meaningfulness: description of the number of connections to information stored in long-term memory.

§         Enhancing meaningfulness

·        Organization: grouping related items into categories or patterns

o       Charts and matrices

o       Hierarchies

o       Models

o       Outlines

* For organization to be effective the relationship must be understood.

·        Elaboration: process of making information meaningful by forming additional links to existing knowledge.

o       Examples: specific cases that illustrate an idea.

o       Analogies: comparisons made between otherwise dissimilar ideas.

o       Mnemonic devices:

§         Method of loci: ties items to familiar items

§         Peg-word method: ties to “peg words” from something in learners memory.

§         Link method: visually linking items.

§         Key-word method: using imagery and rhyming words to recall unfamiliar words.

§         First-letter method: remembering the first letter

·        Activity: being actively involved in the processing.

o       Encouraging active students.

§         Put content in problems to be solved instead of information to memorize.

§         Ask questions that require analyzing rather than recall.

§         Make students provide evidence for conclusions

§         Develop lessons around examples and applications instead of definitions.

§         Use tests and homework that require application instead of rote memory.

4.      Levels of processing: view that the more deeply information is processed the more meaningful it becomes.

5.      Forgetting: loss or inability to retrieve information from memory.

§         Information retained in working memory only through rehearsal

§         Long-term must be encoded

§         Interference: loss of information because other knowledge detracts from learning.

·        Minimizing interference.

o       When new subject is introduced, compare with closely related info. Already studied pointing out easily confused similarities.

o       Teach related ideas together.

o       Highlight relationships, emphasize differences, and identify areas easily confused.

6.      Metacognition: knowing and controlling our cognitive process.

§         Strategies: plans for accomplishing learning goals.

·        Start lesson by asking what is already known about subject.

·        Provide background experiences with rich examples of content.

·        Use open-ended questions to assess perceptions of examples.

·        Use student’s experiences to augment those lacking experience.

7.      Criticisms of information processing theory.

§         Learners have a variety of experience, emotions, beliefs, expectations, and goals that influence them. LPT fails to factor these.

§         Fails to consider social contexts that learning takes place.