Teaching Math, Science, and Technology in Schools Today (Adams & Hamm, 2014): Chapter 2: Creative and Inventive Thinking

Teaching Math, Science, and Technology in Schools Today (Dennis Adams and Mary Hamm, 2014)

NOTES & QUOTES FROM CHAPTER 2: Creative and Inventive Thinking: Collaborative Inquiry, Open-Ended Problem Solving, and Innovation

Innovation is a place where thinking meets up with problems to do new things and do old things in new ways...Imaginative problem solving, technology-related applications, and collaborative inquiry have important roles to play in creating tomorrow's innovators (31).

See Destination Imagination Extra-Curricular Curriculum :) here

Thoughtful Problem Solving and Inquiry
Problem-solving is at the heart of mathematics: inquiry is equally important to science education. Both are compatible with developing highly refined reasoning skills...Creative and innovative thinking are important skills needed to meet the challenges of living and succeeding in the twenty-first century (Robinson, 2011). These skills should be near the center of education; they amplify enthusiasm for learning and generate new solutions for the most pressing problems of today and tomorrow (32).

Creative and innovative thinking is now such an important part of analytical problem-solving in math and collaborative inquiry in science that it cannot be ignored. As the content standards and Common Core suggest, the worth of the ideas created has a lot to do with how the problem being solved is defined (33).

Teachers can promote creativity by making collaboration a normal part of the daily routine and encouraging students to express themselves using multiple media (34).

The Societal Impact on Creative and Critical Thinking 
Developing a deep understanding of a subject or process naturally leads to an application. Along the way, it is important that neither science nor its mathematical and technological tools be separated from humanism... there is often a tendency to put too much weight on information you like and too little on data that contradicts your assumptions (35).

As students develop critical thinking skills, it becomes more natural to approach a task in a realistic way, while leaving room for unconventional, spontaneous, flexible, and original ideas. Sometimes this is done within a preexisting paradigm, and at other times, it means breaking out of conventional boundaries (35).

Being good at thinking means being able to form alternative explanations and demonstrate intellectual curiosity in a manner that is flexible, elaborate, and novel to the thinker (36)...Reasoning, criticism, logical analysis, searching for supporting evidence, and evaluating outcomes might all be considered part of critical thinking. Activities that support this involve clarifying problems, considering the alternatives, strategic planning, problem solving, and analyzing the results. Creative thinking may be viewed as fluency, flexibility, originality, and elaboration (36)

The skills developed in this area would result in the creation of unique expressions, original conceptions, novel approaches, and demonstration of the ability to see things in imaginative and unusual ways (36).

Thinking Across Subjects and in Everyday Life
One way to look at modes of thought across disciplines is through symbolic, imagic, and affective thinking. Symbolic includes words, numbers and other symbol systems. Imagic is visual, spatial, tonal, and kinesthetic. It involves the kind of imagery used by mathematicians and architects, sound relationships explored by musicians, and the movement found in sports and dance. Affective thinking works with emotions and feeling to direct inquiry. All three modes of thinking build on reasoning and intuiting to connect the analytic to the intuitive (37).

Bill Gates-- "You need to understand things in order to invent things beyond them."

Effective instruction in mathematics, science, and technology provokes students to create their own questions and think of innovative applications in the world outside of school. As students become interested in such intellectual invention, it is important that the teachers hold off on their judgements and let the evidence itself be the judge (37).

Mathematicians, scientists, and technology workers use the tools of science and mathematics to collect, examine and think about data. Conclusions are formulated and outcomes explained. Like scientists, students can reason, analyze, criticize, and advocate--while avoiding dangerous materials and problems that are developmentally inappropriate (37).

Stanford's D.School teaches people to be innovative collaborators

Multiple Thinking Points Prior to Knowledge
There is a tendency to think of the scientific method and mathematics problem solving as clear and clean; you formulate hypothesis, organize experiments, collect and analyze data, and interpret the findings...the reality is far less clear-cut and tidy (38).

Creative and critical thinkers tend to be reflective as they think problems through, flexible when they consider original solutions, and curious as they pose new questions. The research evidence suggests that giving students multiple perspectives and entry points into subject matter increases thinking and learning (Costa and Kallick, 2009).

Almost any important concept can be approached from multiple directions (to be continued....)

Teaching Math, Science, and Technology in Schools Today (Adams & Hamm, 2014): Chapter 1: Helping All Students Learn about Math and Science

Teaching Math, Science, and Technology in Schools Today: Guidelines for Engaging Both Eager and Reluctant Learners (2nd Edition)

Notes from the Preface

Having a negative attitude toward certain content is bound to have a similar negative impact on students (v).

Learning math, science, and technology is compared by Montreal teacher Eugen Pascu to learning to drive a car: "First you've got to memorize the rules of the road. Then you've got to apply them to get behind the wheel and actually move the car....You need to learn formulas but you also have to understand how they work. Learn the rules, then see how the rules work (Pertiz, 2013). (v)

Questions, engagement, and curiosity are viewed as natural partners for mathematical problem solving, scientific inquiry, and technology-rich learning. With teacher assistance, even students who are having learning problems can move from believing they "can't do" or "don't like" these subjects to having a sense of genuine achievement and confidence (vii).

In the workd outside of school, creative and collaborative engagement is central to mathematical problem solving, scientific inquiry, and technological innovation. So why not use a similar approach for all students in K-8 classrooms? (Vii)

Chapter 1: Helping All Students Learn About Math and Science

The Common Core State Standards recognize the need for changing societal attitudes toward numeracy and literacy. They also suggest that in a wired world, students of all ages need to learn about previously understated dimensions of math and science (CCSS, 2013). (1)

In spite of their obvious importance, math and science are just about the only topics for which more than a few well-educated adults will freely admit ignorance. Teachers reflect the general population. So it is little wonder than when it comes to teaching these subjects, many of the characteristics of effective instructions fall by the wayside (2).

When teaching math and science, asking disaffected learners to reason, solve problems, and maintain a positive disposition may be a tall order. So we should not be surprised when teachers sometimes pay more attention to procedural knowledge than they do to reflection and understanding (2).

In spite of the difficulties, the primary goal of instruction should be getting every student in the classroom to develop and use the higher-level thinking skills associated with problems solving and inquiry (2).

Math and Science Instruction for All Students
Identifying some reasons behind the reluctance to learn math and science is essential if we are going to engage student interest and help them succeed. The key is finding something that will spark every student's interest. The next step is turning that spark into a flame (3).

Everyone Needs to Understand Math and Science
The need to understand math and science in everyday life has never been greater...Data enhanced decisions are ore than ever part of human decision making. Statisticians and data analysts pay more attention to correlations than human predispositions (Mayer-Schonberger and Culier, 2013).  (5)

Effective use of visuals, manipulatives, and learning aids often help overcome various problems. Working in pairs or small groups is a good motivators. Peer involvement expands and strengthens language skills and increases students' confidence.

When students fail to see the connections among concepts, math and science become a rote exercise, and understanding is limited. As experienced teachers will tell you, simply memorizing terms without knowing what they mean is not useful. Comprehension is the goal (8).

Many struggling students do not understand that being successful in math and science involved employing problem-solving strategies. Teachers have to teach them how to be metacognitive learners and help them recognize the thinking strategies they are using. Metacognition strategies can amplify self-reliance and creativity for struggling learners. Teachers who model thoughtfulness and encourage students to go a long way toward fully engaging struggling students (8-9).

Collaborative Inquiry in Math and Science
All students can flourish when good teaching is combined with collaborative inquiry and an engaging curriculum (Tomlinson and Imbeau, 2010). Collaborative inquiry is a form of reasoning and peer cooperation that begins with a problem and ends with a solution. It generally involves asking questions, observing, examining information, investigating, arriving at answers, and communicating the results. (9)

A collaborative inquiry approach to the teaching of math and science has been found to work well with struggling learners. Among other things, it helps these students experience the excitement of mathematics and science activities in learning groups (9).

Knowledge of math and science has always been constructed in association with others. At all levels mathematical and scientific inquiry is much more than an individual endeavor. So it is best if elementary and middle school students employ procedures similar to the collaborative procedures that mathematicians and scientists actually use when they work.

The collaborative inquiry approach is a student-centered process of cooperative discovery. The teacher often gives the students directions and materials--but does not tell the small group exactly how to go about doing their work. The teacher encourages conversation and provides activities that help students understand how math and science are applies in the world outside of the school. The teacher might also give a brief-whole class presentation and then move from small group to small group, encouraging questions and guiding student observations (10).

Collaboration, Inquiry, and Reluctant Learners
Inquiry is sometimes thought of as the way people study the world and propose explanations based on the evidence they have accumulated. It involves actively seeking information, truth, and knowledge. When collaboration is added to the process, it helps build the positive relationships that are at the heart of the learning community.

Collaborative inquiry may be thought of as a range of concepts and techniques for enhancing interactive questioning, investigation, and learning. When questions that connect to student student experiences are raised collectively, ideas and strengths are shared in a manner that supports the struggling students' search for understanding (Snow, 2005). (11)

Science classes will do Collaborative seed example on pages 12-13

Making Instructional Decisions with Differentiated Learning
Differentiated learning is an organized approach through which teachers and students work together in planning, setting goals, and monitoring progress. In such classrooms, the teacher draws on the cultural knowledge of student by using culturally and personally relevant examples. They show great respect for learners by valuing their similarities and differences, not by treating everybody the same (13).

The most useful teaching approach for the struggling learner is often well-organized differentiated instruction (Tomlinson and Cunningham Eidson, 2003). A teacher who is organized examines the conditions surrounding the child such as curriculum content, the classroom environment, and the student's academic and social behaviors. The ways students react to information and respond to feedback are also important....But teachers know that no approach is effective in every situation, so it is important to be flexible.

Discovering Ways to Differentiate Instruction
What is clear is that struggling students seem to have have the hardest time with the traditional classroom setting (straight desks, teacher lectures, textbooks, worksheets, lots of listening, waiting, following directions, reading, and writing.) In other environments, students who struggle have much less difficulty, for example in an art classroom, a wood shop, a dance floor, or the outdoors. In these differentiated classroom settings where students have opportunities to engage in movement, hands-on learning, and other new learning approaches, their interest and desire to learn have been shown to be at or above average (Gardner, 1993).(15)

Meeting the Principles and Standards
Equity: High-quality math and science require raising expectations for students' learning. All students must have opportunities to study and learn mathematics and science. This does not mean that every student should receive identical instruction: indeed, it demands that appropriate accommodations be made for all students. Resources and classroom supplies are also a large part of equity (17).

Curriculum: coherent, focused on math and science, and articulated across grade levels....building deeper understandings provides a map for guiding teachers through the different levels of learning.

Technology: Tools such as calculators and computers provide visual images of math and science ideas. They facilitate learning by organizing and analyzing data, and they compute accurately.

Assessment: Assessment should support the learning of math and science and provide useful information to students and teachers.

Teaching: Effective teachers understand mathematics and science, comprehend what underachieving students know and need to learn, and challenge and support them through learning experiences. Teachers need to know multiple kinds of knowledge: knowledge of the subject, pedagogical knowledge, and an understanding of how children learn. Different techniques and instructional materials also affect how well their students learn mathematics and science. Struggling learners are often inundated with only practice materials to help them master the "basic skills."They quite often lack the conceptual procedures of real understanding. Students frequently forget procedures and are referred back to the same uninteresting skill-based drill work. The learner is not the focus rather the basic skill drill is the center of attention (17).

Learning: Math and science must be learned with understanding. Students actively build new knowledge from prior experience. Students should have the ability to use knowledge in a flexible manner, applying what is learned, and melding factual knowledge with conceptual understanding


Struggling Learners and the Math and Science Standards

If students can't learn the way we teach, we must teach them the way they learn.
                                                                                                    -- Carol Ann Tomlinson

We want all students, particularly struggling learners, to be involved in high-quality engaging mathematics and science instruction. High expectations should be set for all, with accommodations for those who need them (18). The National Council of Teachers of Mathematics and the National Science Foundation have developed standards that serve as guides for focused and enduring efforts to improve students' school mathematics and science education. These content standards provide a comprehensive set of standards for teaching mathematics and science from kindergarten through grade 12 (18). (See overview in NTCM (2000), and NGSS (2013).

Going Beyond Skill Mastery
Students who complete their math and science lessons with little understanding quickly forget of confuse the procedures. Understanding important ideas and accurately completing problems are some of the first steps in becoming mathematically and scientifically skillful. Mathematics and science contains five strands of thought:
1. Understanding ideas and being able to comprehend important content.
2. Being flexible and using accurate procedures.
3. Poising and solving problems.
4. Reflecting and evaluating knowledge.
5. Reasoning and making sense and value out of what is learner. (20)

Organizing Successful Lessons
Stage 1: Review: access prior knowledge--make connections between familiar and new information
Stage 2: Demonstrate skill or knowledge- increase student engagement and promote independent student activities
Stage 3: Guided practice- reinforce language skills, partner, and share. Do a variety of problems.
Stage 4: Check for Understanding and provide feedback--summarize strategies and evaluate

Teacher provides continuous feedback at each stage so that errors can be found and corrected (21-22).

Summary, Conclusion and Looking Ahead:
At every age level, it is important to think big and risk failure to make new ideas and positive change possible. It does not have to happen at once---teachers can think big and start small  as they weave new ideas into the educational fabric (26).

When it comes to student learning, there is no substitute for a good teacher who develops lessons that  combine thinking and feeling in a way that reaches both the head and the heart (28).

Assessment is Essential, Chapter 1, part II: Action Research & Inquiry

ACTION RESEARCH & THE INQUIRY STANCE

Notes from Assessment is Essential, Chapter 1

--this section provides a framework for problem solving in your classroom that uses assessment as it's foundation (19)

Inquiry Stance
In contrast to a "caring stance" or a "best practices stance," an inquiry stance prepares teacher candidates to identify problems, collect relevant data, make judgements, and modify practices to bring about improvement in teaching and learning (19).

Inquiry Stance:An approach to dealing with challenges in the classroom that involves identifying problems, collecting relevant data, making judgements, and then modifying practices to improve teaching and learning.

In 1910, John Dewey, one of the founders of modern educational practices, described a similar process for dealing with problematic situations. He suggested designing a solution, observing and experimenting to test the solution, and then accepting or rejecting it.

If you want to read exactly what John Dewey said in How We Think (2010), I am including a link for the free e-book  here.

One key use of the inquiry stance and problem-solving process is analyzing and then improving student learning outcomes in each classroom...Assessment gives you information every step of the way on what your students know and are able to do and whether your instructional strategies are working (19).

Inquiry Stance and Action Research
The process of examining and improving our teaching practices and our outcomes using an inquiry stance is often termed action research.

Action Research: The process of examining and improving teaching practices and student outcomes using an inquiry stance.

Assessment is Essential from Chapter 1 (Pages 1-19) Kinds of Assessment

Why is Assessment Essential?
by Susan K. Green and Robert L. Johnson (2010)

Please note: text in black comes verbatim from our textbook, text in red are glossary words from the reading; text in orange are my notes/thoughts/instructions.

(clipart uploaded July 6, 2014 from http://assessment.uconn.edu/images/Assessmen_06.jpg)

CHAPTER 1 NOTES

"But we have come to see assessment as an amazingly flexible and comprehensive tool that has measurably improved our own teaching. Even more important, we believe learning to design good assessments also helps students for participation as citizens in a democracy" (page 3).

Achievement Gap: The disparity in performance between student groups (e.g. ethnicity, gender, and or SES) on achievement measures such as large-scale tests or graduation rates. 

The current achievement gaps have begun to erode our belief that anyone who conscientiously tries hard, regardless of race, color, creed or wealth can succeed in our schools and in our country (4). Achievement gap exists before children even enter kindergarten.

Good assessment practices provide the opportunity for teachers, working in the realm where they have primary impact--their own classroom-- to maximize learning for their students. Equal access does not mean that every child receives the same instruction, rather it suggests that some may need extra accommodations or differentiated resources and instructional opportunities to be able to reach mastery on the learning goals for the class (5).

Self-Governing Skills for Participation in a Democracy
Performance Goals: Academic Goals held by students that focus on performing well in front of others and that assume ability is fixed.

Students with strong performance goals are motivated to perform well in the eyes of other people (parents, teachers, peers). Those schools focused on performance goals have students who decide it's better to be bad than stupid, if they don't perform at the top of the class.

Shift is toward mastery goals, that every child can improve, so no one falls in the cracks (7). Students want to learn to improve, to better themselves, take pleasure in learning. They persist when the work gets harder.

Mastery Goals: Academic goals help by students that focus on a desire to understand the task and that assume ability can increase.

Assessment Tasks that Enhance Mastery Goals (page 9)
1. Varied, meaningful, challenging tasks
Strategy: Many different types of assessments (papers, projects, brief oral and written-check ups)
Challenging, novel, relevant assessments (eg using data with students collected on their friends for math, etc.)

2. Students participate in decision making.
Strategies: Provide choices for ways for students to show what they have learned.
Use student participation in designing scoring guides for assessments.
Have students engage in self-assessment and peer assessment.

3. Focus on personal goals, own improvement
Strategies: Keep assessment goals private rather than public
Focus on individual improvement across time using benchmarks
Allow students to develop goals and keep track of their progress using a chart/graph
Develop assignments where mistakes are an expected par of learning and improvement is expected across time (multiple drafts)
Recognize student effort.


Effort Optimism: The idea that effort brings rewards. For example, if you work hard in school, you will learn more. The idea that effort has a payoff. Students enter kindergarten this way, and then they experience failure, discrimination, etc and those the optimism they once had. (10)

The the classroom, teachers must make the connection explicit and strong between student effort and academic success.

Promoting Mastery Goals through Assessment: Examples
The ability to assess oneself is one of the goals of education.

Self-assessment-
--the ability to assess oneself is one of the primary goals of education (11). It teaches objectivity--being able to get beyond your own point of view and look at yourself in relation to a standard...allows you to become open to feedback from a variety of sources...helps you decide which sources of information are valuable and which are not..become active in connecting their work to criteria used for evaluating it. (11).

Metacognition: The process of analyzing and thinking about one's own thinking enabling such skills monitoring progress, staying on task, and self-correcting errors...the ability to step back from merely listening to a lecture or doing an assigned task, to thinking about what is happening in a more critical way. Metacognition must be taught by teachers (11-12).

Focus on Individual Progress Across Time
Importance of goal setting and conferencing about student progress (12).

IMPORTANCE OF CRITICAL THINKING
Analyzing arguments, seeing both sides of an issue before choosing a position, and discerning what is left unsaid are key critical-thinking skills related to democratic participation that must be taught. In our view, providing opportunities to learn these skills is one of the essential functions of schools (14),

Designing assessment questions that address only basic concepts and definitions is a common problem in the teacher-made assessments that we see. Questions about facts and definitions are the easiest kinds of questions to design, and they are also easier to grade. But if you want your students to learn more than the facts, if you want them to learn critical-thinking skills and master the requirements of the standards you are teaching, you need to develop assessments that do this too (14).

Assessment FOR learning promotes democratic values by guiding the development of tools to help teachers address achievement gaps by promoting skills such as independence and responsibility, and by designing assessment tools essential in your classroom (14).

AN OVERVIEW OF ASSESSMENT
Assessment: The variety of methods used to determine what students know and are able to do before, during, and after instruction (14).

See fifty second video clip of educator Carol Ann Tomlinson: Tomlinson

Purposes of Assessment (page 15)
from table on page 15

Diagnostic Assessment: 
Why: Getting a sense of strengths and needs for planning instruction.
When: Before Instruction
How: School records, teacher observations, teacher-made questionnaires and pre-tests

Let's look at higher education a minute: SNHU administers the CIRP Freshmen Survey every five years. See the instrument here. What questions do you want to know about your peers that could be answered with this survey??

Formative Assessment: 
Why: Monitoring Growth as you teach (assessment FOR learning)
When: During Instruction
How: Teacher Observations, quizzes, skill checklists, homework, student self-assessments, systematic teacher questioning

Summative Assessment
Why: Determining what students have learned after instruction or for accountability purposes (assessment OF learning)
When: After instruction
How: Teacher Observations, quizzes, skill checklists, homework, student self-assessments, systematic teacher questioning

Educator Rick Wormeli distinguishes between summative and diagnostic assessment in the video above.

Purpose 1: Diagnostic Assessment

Diagnostic Assessment: Assessment at the early stages of a school year or unit that provides the teacher with information about what students already know and are able to do (16)

Accurately understanding your what your students know and are able to do from the beginning of the year is crucial in designing instruction for them that challenges but does not overwhelm them (16).

Differentiation: Using students' current understanding, readiness levels, skills, and interests to tailor instruction and meet individual needs (16).

Purpose 2: Formative Assessment
--Links goal-oriented instruction and assessment 
•  Promotes learning as well as informs 
instruction 
•  Used continuously- before and throughout 
instruction 
•  Encourages students to become more aware of their 
own learning (metacognition) and the ideas of 
others 
•  Transforms the learning environment 

from Page Keeley (2012) accessed on July 6, 2014 at http://esd113.org/cms/lib3/WA01001093/Centricity/Domain/46/Page%20Keeley%20Presentation%20May%2022%202012.pdf

A key aspect of formative assessment is giving students feedback on their growth toward the learning goals they need to master...It helps students understand where they are now compared to where they should be going and it gives them suggestions for getting there (17). A video of how this is done in Sweden can be seen here. Despite its cheesy guitar rock, it's quite good.

Many [college] students have a difficult time wit the distinction between formative assessment and assessment for grades (summative assessment). After explaining formative assessment and its use to enhance learning, we often have students write a paragraph for homework explaining how they saw a concept we discussed in class demonstrated in the classroom in which they are currently observing (17).

For our purposes, and since this class isn't tied to a field experience classroom, this week when you go to your other classes, write down an example of a formative assessment a professor gave. If you are certain there were no formative assessments, explain how a professor used a summative assessment.

Purpose 3: Summative Assessment
Summative Assessment: A summing up if what students know and are able to do after instruction is completed.

It is assessment OF learning...includes exams, projects, and term papers.  Grading is based on information from summative assessment at the classroom level.  It can also be large-scale state or national tests administered infrequently and used by local, state, and national decision makers (17)...Large scale assessment requires covering a wide range of content. These kinds of assessments are administered differently (see table on pg. 18 in text). (We would not use the SAT as a graduation requirement!)  Large-scale tests must be standardized.

Standardized: administered, scored, and interpreted exactly the same for all test takers.