Estimate: Given 497 and 624, estimate the products to the nearest thousand. Sample Response

1. 
   2x + 3x = 47 – x
   
2. 
   6x² – 96
   
3. 
   2 x² + 4x + 25
   
4. 
   3x – 23 = 5x + x²
   
5. 
   5x³ – 2x² + 3x – 5 = 0
   

25, 3/8, .314, –17, π, 3.14, –64/4, .375, ¼%

–17, –64/4, ¼%, .314, 3/8, .375, 3.14, π, 25

In making this prediction, I calculated that 5² = 25, 5³ = 125, 5⁴ = 625, and noticed that when 5 is raised to each of these powers, the last digit was always 5. Then I realized that when 5 is raised to any power, the last digit must always be 5. So, 5⁵⁰⁰, when expanded, must have a units digit of 5.

In this activity you will analyze how higher order thinking is promoted in your classroom for the purpose of applying new knowledge to your current lesson planning.

1. 
   Answer the following questions in the space provided, focusing on how ideas could be implemented in your current lesson. Be prepared to add this information to your Learning Log.
   
   1. 
      In what ways do you currently ask your students to play the role of "receiver," for the goal of learning factual information?
      

2. 
   What percentage of your lessons/lesson planning would you estimate is consumed by activities that focus on factual/information recall?
   

3. 
   In what ways are students required to manipulate information and ideas in the selected lesson?
   

4. 
   With what opportunities are students provided in order to combine facts and ideas for the purpose of analysis, evaluation, or creation?
   

5. 
   How would you describe the "level of uncertainty" that is evident in your lesson, where instructional outcomes may be unpredictable?
   

6. 
   In the chart below, identify the ways in which you utilize the higher order technology tools available to you for the purpose of engaging your students in higher order thinking skills. Remember, this is not simply a web search, but prescribed use of technology for the purpose of analyzing, evaluating, or creating something. As you are searching, think of the benefits and possibilities of Web 2.0 technologies in light of the information provided.
   

7. 
   Of the following (please see glossary or search the Internet if terms are unfamiliar to you), which could you see adding value to your lessons in terms of promoting higher order thinking? Please provide an explanation.
    
   

2. 
   How would you specifically apply new knowledge about higher order thinking skills to your current lesson plans?
   

3. 
   How would you take advantage of the interactive nature of Web 2.0 to facilitate conversations, collaborations, and sharing of ideas and media, in your current lesson plans?
   

4. 
   Summarize your responses in the space provided, and be prepared to enter your responses in your learning log.
   

5. 
   Enter your summary in your Learning Log by clicking on "Resources" and then "Learning Log." (Label your entry "Developing Higher Order Thinking Skills.")
   
6. 
   Close the Learning Log window to return to the course.
   

Teachers will, without a doubt, tell you that they never have enough time to cover the curriculum and that they feel like they are knowledge slaves, fitting in as much depth as possible before the bell rings. They know that the next day will often bring a new topic to cram into the next 42-minute period. To free the educator from this shallowness, we must step back and organize our "big rocks," gravel, sand, and water.

Borrowing a page from First Things First (Covey, Merrill, and Merrill, 1994), the facilitator began carefully stacking the smooth river rocks into the jar until the last rock was placed near the top. She asked the group if the jar was now full. "Yes, it is full," the group answered, "but you still have the other materials. I thought we were supposed to use all of the materials." The facilitator carefully poured in the brightly colored pea-gravel, tipping the jar from side to side, her audience gazing in amazement as every last pebble fell into place through the cracks between the bigger rocks. "Now is it full?" she queried. There appeared to be no more space, so when the fine-grained sand was carefully poured in, filtering down and settling throughout, there were audible gasps. At last, the water trickled over the rocks, gurgling between the gravel and wetting the sand. The pickle jar was finally filled to the brim with the very last drop of water.

The participant who originally claimed that he did not see the point now exclaimed, "Okay! Now I get it!" He started to share his revelation, but the facilitator asked him to hold onto his thoughts and instructed each group to come up with their own meanings for the analogy. Though comments such as, "You have to be able to effectively plan and prioritize," and, "As teachers, we need to try harder to fill in the gaps," were included by most groups, every single one placed, "You have to put the big rocks in first!" at the top of their list!

What are the big rocks in math? For math teachers, what does it mean to put the big rocks in first? How does putting in the big rocks first promote depth of knowledge and understanding for math students? How might all teachers use the "big rocks" analogy to reach their personal and professional productivity goals that focus their practice and define them as exemplary educators?

As the facilitator passed out permanent markers and pointed out the resources on the reference table, the participants formed content groups, and each group drew a card from a small bowl. The cards were marked with either "MATH PROCESS SKILLS" or "MATH CONTENT KNOWLEDGE." Each group was asked to think about the words on the card they had drawn and then to engage in scholarly discourse to identify what they collectively thought to be the "big rocks" for their content. After reaching agreement in their own group, participants would join other groups with the same topic, to defend their choices for what was key and essential. The goal was to reach an overall consensus, identifying those concepts that are most important and then to mark a set of river rocks accordingly. The big rocks would ultimately be placed in their appropriate jar to represent those non-negotiable components of each content area that are essential to provide a solid foundation and to serve as a filter for subordinate concepts.

Though initially at a loss, in light of the daunting task, the teachers were quick to recover and took a look at the resources available to them. There were multiple copies of The Principles and Standards for School Mathematics (NCTM1996), and Quantitative Literacy: Why Numeracy Matters for Schools and Colleges (MAA 2003) would certainly be useful. For the math teacher who strives to identify the "big rocks" in math, readily available resources abound and many of these can be found online, printed in their entirety!

The Principles and Standards for School Mathematics, published by the National Council of Teachers of Mathematics (NCTM), specifies what students should know and be able to do at various grade levels if they are to achieve mathematical literacy. They provide a guiding beacon for all math teachers and are clearly expressed as six principles:

• 
  Equity
  
• 
  Curriculum
  
• 
  Teaching
  
• 
  Learning
  
• 
  Assessment
  
• 
  Technology
  

• 
  Number and operations
  
• 
  Algebra
  
• 
  Geometry
  
• 
  Measurement
  
• 
  Data analysis and probability
  
• 
  Problem solving
  
• 
  Reasoning and proof
  
• 
  Communication
  
• 
  Connections
  
• 
  Representation
  

The NCTM has published a series entitled, "Navigating through ... in Grades ...-...", to address the Math Standards for each grade level. The standards cover all math content areas (i.e., algebra, geometry, numbers & operations, etc.), and serve as practical guidebooks to teaching standards-based math. Acknowledging the unique, divergent, and distinctive interests and talents of math teachers, these resources do not attempt to address how the content knowledge and process skills are to be taught. Instead, they help focus educators on the essential learning related to content standards, which must be thoroughly understood by students so that they can achieve the goal of math literacy.

• 
  The recall of information such as a fact, definition, term, simple procedure, performance of a simple algorithm, or application of a formula.
  
• 
  Key words that suggest Level 1 include "identify," "recall," "recognize," "use," and "measure."
  
• 
  Verbs such as "describe" and "explain" could be classified at different levels depending on what is to be described and explained.
  

• 
  Engagement of some mental processing.
  
• 
  Requires students to make a few decisions as to how to approach the problem.
  
• 
  Keywords that generally distinguish a Level 2 item include "classify," "organize," "estimate," "make observations," "collect and display data," and "compare data."
  
• 
  Student actions imply more than one step.
  
• 
  Action verbs, such as "explain," "describe," or "interpret" could be classified at different levels, depending on the object of the action.
  
• 
  Students indicate a relationship between two concepts.
  
• 
  Interpreting information from a simple graph.
  
• 
  Interpreting information from a complex graph that requires some decisions on what features of the graph need to be considered.
  
• 
  Carrying out experimental procedures; making observations and collecting data; classifying, organizing, and comparing data.
  

• 
  Reasoning, planning, and using evidence.
  
• 
  Requires students to explain their thinking.
  
• 
  Students making conjectures.
  
• 
  Complex and abstract cognitive demands.
  
• 
  Tasks require more demanding reasoning.
  
• 
  Drawing conclusions from observations; citing evidence and developing a logical argument for concepts; explaining phenomena in terms of concepts; and using concepts to solve problems.
  

• 
  Requires complex reasoning, planning, developing, and thinking—most likely over an extended period of time.
  
• 
  The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher order thinking.
  
• 
  The cognitive demands of the task should be high, and the work should be very complex.
  
• 
  Students should be required to make several connections—relate ideas within the content area or among content areas—and have to select one approach among many alternatives on how the situation should be solved.
  
• 
  Activities include designing and conducting experiments; making connections between a finding and related concepts and phenomena; combining and synthesizing ideas into new concepts; and critiquing experimental designs.
  

Higher order thinking occurs frequently when the teacher lets the students take over the project or task, with structure, allowing the students to understand how and on what they will be assessed (Tricia, 2007). An example would be when a group of students teach a lesson to the class.

This project is designed for the latter half of the year. At this point, students have become very comfortable with what is expected from classroom lessons, with pair and group work, and with each other. This classroom organization has been reinforced consistently by the teacher throughout the year. Each group's lesson would be structured by several criteria. During a chapter, one group would prepare and present their lesson for a section of a chapter. They would also create review and assessment items for the chapter review and test. The presenting group would be responsible to evaluate the rest of the class' comprehension of their assigned section's objective(s).

The rubric associated with this project would be created in their "buddy pair" first, and then they would meet with their respective "helper pair," creating a team. Once the team has comprised a rubric, a classroom discussion of the created rubrics would follow. The discussion would allow individual teams to add or subtract from their individual team rubric. The team would be graded on their rubric, after approval from the class and the teacher. Rubrics would then be posted around the classroom for a social grading activity of the team's actual lesson presentation.

• 
  Associating the state standard with the objectives they are assigned. They will connect how the standard(s) relates to their assigned objective(s), and will incorporate the meaning into the lesson.
  
• 
  Stating of the team's assigned objective(s).
  
• 
  Relaying an understanding of the assigned content and connecting it to the "real" world. This could be done by using the history of the concept, or by its use in everyday life. This is an important part of the project. The team must be able to creatively connect their assigned objective(s) to the world around us.
  
• 
  Utilizing scaffolding examples in order for the class to gain the skills necessary to comprehend the concept.
  
• 
  Demonstration of the students' comprehension of the concept(s), rather than just the procedure(s).
  
• 
  Having a summative activity to allow the class to reflect on their understandings of the objective(s), as well as relaying the students' perception of the connection of the topic to the "real" world.
  
• 
  Using the presenting team's rubric to be assessed by the rest of the class.
  

While carefully listening, as the groups debated the merits of their self-defined "big rocks," it became apparent to the facilitator that some redirection was warranted. For these teachers to fully comprehend the point of the session, she needed to make sure she had driven home the importance of the big rocks. She feared that, perhaps, the point of the analogy was being lost in the activity's deliberations. "As you reach consensus on what all your students must know, with the help of these resources, make sure you have clearly identified the big ideas that are most worth spending time on. These and only these are the big rocks. The purpose of this activity is to identify the essential math content knowledge and process skills to be covered in-depth. Once determined, these essentials become the big rocks to serve as the foundation for solid math learning. If an activity being considered does not serve to provide your students with a greater depth of understanding of the particular big rock, it is not worthy of the time and resources it will take to implement. Stick with the big rocks! Go deep rather than wasting time on things not related to the big rocks. If you follow this plan, you will find that your students are capable of both learning and retaining the essential content and you will not find yourself behind the curve as you certainly will if you spread the content too thin!"

For each of the content groups, the "big rocks" were becoming more apparent. The teachers began to agree to let go of some of their favorite units that just did not fall into the realm of an aligned, standards-based curriculum within the scope and sequence, in order to provide continuity and depth of understanding. One teacher pointed out, "The need-to-knows have got to be the big rocks. The nice-to-knows or fun-to-dos might have a place somewhere in the model, represented by the pebbles, sand, or water, but only if they are in support of—and not at the expense of—the big rocks!"

While the teachers finished building and defining their models, the topics of rigor and relevance began popping up, and a discussion soon broke out among the groups. The facilitator suggested that perhaps each group could decide how they might include not only rigor and relevance, but also the third of the "Big R's" of an effective program—relationships. Both rigor and relevance must be incorporated into standards-based instruction and assessment if students are to be successful learners who are prepared to accept the challenges they face at each juncture of their academic career. It is important for teachers to understand that rigor is not only for the gifted student, nor is it merely more busy work. Rigor means challenging all students. Rigor involves developing higher order thinking skills and engaging students in rich problem solving experiences that provide an abundance of opportunities to analyze, evaluate, and create. Relevance applies to those meaningful connections that exemplary teachers make between students and curriculum. The teacher ensures that topics are interesting and culturally aligned. Students are given choices through differentiated learning and assessment strategies. They are encouraged to express their opinions, and they are invited to take certain risks without fear of ridicule from teachers or peers. They are provided with opportunities to be responsible and are guided through the process of taking initiative. Relevance involves identifying problems and engaging in projects that are valued by their community in a real-world sense. The groups acknowledged the importance of every classroom teacher taking the initiative to establish supportive relationships between students and their peers and between students and adults. It is essential that teachers incorporate rigor, relevance, and relationships into their standards-based programs.

Finally, the groups came together to share their pickle jar models. The facilitator asked the teachers, "Can you find the significance in using the rather narrow and deep pickle jars instead of a shorter, wider aquarium?" By now, the teachers were beginning to fully understand the significance of the workshop. A geometry teacher, who had earlier voiced his frustration because his students just didn't "get" geometry, had the answer. "We are trying to cram too much into each class," he lamented. "I teach the standards, the book, and I do a bunch of whiz-bang demonstrations to keep the kids interested. We do interactive constructions and activities, but the students are not really that interested in geometry... only in the entertainment! What I have been doing is a lot of stuff that lacks continuity. I spend a little time here and a little time there, just to get it all in. The students never really see the connections that are vital to understanding their world, much less how geometry concepts can be utilized in the real world and in future studies. And, I tend to go back to teaching the same things over and over, year after year, without really considering what my students already know or what they really need to know or are capable of learning. No wonder my students mostly seem to hate geometry!"

The facilitator asked each teacher to reflect on his or her own practice. Were they teaching content that was a mile wide and only an inch deep, as the geometry instructor suggested? "How might you move towards changing that? How about teaching content that is an inch wide and a mile deep? Wouldn't that encourage the development of the depth of knowledge to which you aspire for your students?" she asked. "It just might require your own leap of faith," she affirmed, "A leap that will likely take you right out of your comfort zone, at least in the beginning. But, if you persevere, you may finally realize or remember why you became a teacher in the first place!"

As the participants presented their models, each group pointed out the importance of identifying the highest priorities and placing them first. They all agreed that lower priorities needed to be positioned to fit in and around the non-negotiable, high priority items, and then only when they were clearly in support of the "big rocks!" The teachers elaborated on the importance of collaborating with their peers to identify and reach consensus on what constitutes the "big rocks." They agreed that teachers should be able to defend the value of the high priority items in terms of the content standards and the scope and sequence, rather than merely teach according to personal preference, expedience in covering the material, or the fun factor.

Participants soon found themselves considering ways to incorporate the big rocks across grade levels through the development and the diligent maintenance of a vertically aligned scope and sequence. The facilitator contributed to the discussion by pointing out, "Collaborative efforts are vital in promoting the depth of knowledge necessary to move students' understanding along from one grade to the next. One benefit of successful vertical planning is that students are better prepared to succeed at each level of instruction because each year builds upon the previous year. Another benefit is that less time is wasted on needless repetition. While students' depth of understanding should be the primary goal for each individual teacher, this can only be accomplished if connections to the curriculum lead seamlessly from one grade level to the next. With sufficient attention to vertical alignment, the curriculum flows through developmentally appropriate activities at each level, building upon the depth of knowledge attained at the previous level and ideally leaving no gaps in the students' acquisition of knowledge."

As the session closed, the participants began making plans to contact their individual school districts to make sure a vertically aligned scope and sequence could be developed and that all teachers, as stakeholders, were aware of the importance of teaching the content standards for which they were responsible. Participants agreed that in cases where these essential pieces were lacking, colleagues must encourage each other to take a proactive role to initiate reform and model best practices. The teachers carried their pickle jar models that clearly identified the "big rocks" they would feature at the heart of their curriculum in the coming school year. As the facilitator turned out the lights and closed the door, she let out a sigh of satisfaction, knowing this group of teachers was well on their way to success, and their students would greatly benefit from their efforts.