21st Century Mathematics Grade 7 Answers
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| Skills for the 21st Century: teaching higher-order thinkingThis article originally appeared in the April 2014 edition of ISQ Briefings, a publication of Independent Schools Queensland. Republished with permission. It is hard to imagine a teacher or school leader who is not aware of the importance of teaching higher-order thinking skills to prepare young men and women to live in the 21st Century. However, the extent to which higher-order thinking skills are taught and assessed continues to be an area of debate, with many teachers and employers expressing concern that young people 'cannot think'. What are we talking about when we talk about 'higher-order thinking'? Brookhart (2010) identifies definitions of higher-order thinking as falling into three categories: (1) those that define higher-order thinking in terms of transfer, (2) those that define it in terms of critical thinking, and (3) those that define it in terms of problem solving. In the category of transfer, Anderson, Krathwohl et al (2001) define transfer in how it differs from retention: Two of the most important educational goals are to promote retention and to promote transfer (which, when it occurs, indicates meaningful learning) … retention requires that students remember what they have learned, whereas transfer requires students not only to remember but also to make sense of and be able to use what they have learned. While learning for recall requires thinking, the higher-order thinking is in 'transfer'. That is, students not only acquire the knowledge and skills, but also can apply them to new situations. It is this kind of thinking, according to Brookhart (2010) that applies to life outside of school where thinking is characterised by 'a series of transfer opportunities (rather) than as a series of recall assignments to be done'. The critical thinking category includes definitions that refer to 'reasonable, reflective thinking that is focused on deciding what to believe or do' (Norris & Ennis, 1989) and 'artful thinking', which includes reasoning, questioning and investigating, observing and describing, comparing and connecting, finding complexity, and exploring viewpoints (Barahal, 2008). In critical thinking, being able 'to think' means students can apply wise judgment or produce a reasoned critique. The goal of teaching is then to equip students to be wise by guiding them towards how to make sound decisions and exercise reasoned judgment. The skills students need to be taught to do this include: the ability to judge the credibility of a source; identify assumptions, generalisation and bias; identify connotation in language use; understand the purpose of a written or spoken text; identify the audience; and to make critical judgments about the relative effectiveness of various strategies used to meet the purpose of the text. In the problem-solving category Brookhart provides the following definition: A student incurs a problem when the student wants to reach a specific outcome or goal but does not automatically recognize the proper path or solution to use to reach it. The problem to solve is how to reach the desired goal. Because a student cannot automatically recognize the proper way to reach the desired goal, she must use one or more higher-order thinking processes. These thinking processes are called problem solving (Nitko & Brookhart, 2007). They may include remembering information, learning with understanding, critically evaluating ideas, formulating creative alternatives, and communicating effectively. The broad definition of problem solving is that it is the skill that enables a person to find a solution for a problem that cannot be solved simply by memorising (ibid). While there are many closed problems–in maths for example–that require students to use memory to repeatedly practice a particular algorithm, many problems are open-ended and cannot be solved from memory alone. Or they may have more than one solution. Or they may be genuine problems where an answer is not yet known. Problems may also be open-ended in that solutions change as circumstances change. For example, living within a budget is an open-ended problem. Bransford and Stein (1984) point out that problem solving is the general mechanism behind all thinking, including recall, critical thinking, creative thinking, and effective communication. They assert that to recall something, students have to identify it as a problem ("I need to memorise the planets, a poem, a list of capital cities. How can I do that?") and devise a solution that works for them. Similarly, critical thinking is a matter of problem solving–how well does Shakespeare develop this character?–and communication also involves problem solving–who is my audience? How do I need to best communicate with them? What words might I use to persuade? Project Zero developed at Harvard University, provides an example of how teachers might help students to think by viewing works of art, using an 'Artful Thinking Palette' (Barahal, 2008). Students were asked to use six thinking dispositions to view art: exploring viewpoints, reasoning, questioning and investigating, observing and describing, comparing and connecting, and finding complexity. Teaching students these thinking skills is not only useful in art but in other disciplines, and in preparation for standardised tests such as the Queensland Core Skills Test. Brookhart (2010) argues that if teachers think of higher-order thinking as problem solving they can set lesson goals to teach students how to identify and solve problems at school and in life. This, she says, involves not just solving problems set by the teacher but solving new problems that 'they define themselves, creating something new as the solution'. How do we teach higher-order thinking? While Bloom's Taxonomy is not the only framework for teaching thinking, it is the most widely used, and subsequent frameworks tend to be closely linked to Bloom's work. A committee under the leadership of Dr Benjamin Bloom created the Taxonomy in 1956. Bloom's aim was to promote higher forms of thinking in education, such as analysing and evaluating, rather than just teaching students to remember facts (rote learning). Learning was divided into three domains of educational activity:
While all three domains are important for a 'rounded' person, it is the first domain (Cognitive) that is the subject of this paper. The cognitive domain involves 'knowledge and the development of intellectual skills' (Bloom, 1956). The abilities and skills within the domain are listed in six major categories starting from the simplest thinking behaviour to the most complex (see Table 1). It is generally accepted that each behaviour needs to be mastered before the next one can take place. This is useful knowledge in assisting teachers in their lesson planning. 1. Specifically teaching the language and concepts of higher-order thinking Teachers should not only teach the language and concepts but also tell students what they are doing and why higher-order thinking skills are necessary for them to problem-solve at school and in life. For example, by using a common language, students can recognise the skill they are exercising and the level of complexity of a question. When they see words like 'define', 'recognise', 'recall', 'identify', 'label', 'understand', 'examine', or 'collect', they know they are being asked to recall facts and demonstrate their knowledge of content. When they see words like, 'apply', 'solve', 'experiment', 'show', or 'predict', they understand they are being asked to demonstrate application. And when a question begins with 'appraise', 'judge', 'criticise', or 'decide', they understand the higher-order thinking skill they are practising is 'evaluation'. Teachers also have an instant checklist of whether or not the level of work they require from students is of sufficient complexity. Students should begin to practice higher-order thinking skills from primary school, but by the time they reach year 11 and 12 the bulk of class and assessment questions and discussion should be in the higher levels of the taxonomy. The important word here is 'plan'. Teachers, on the whole, are very good at 'thinking on their feet'; however, without meticulous planning they are likely to ask recall questions rather than questions that require higher-order thinking. Similarly, discussions can be de-railed if they are not planned with a higher-order thinking learning objective in mind. While this does not mean every question or discussion has to be pitched at higher-order thinking, a good proportion should be. By carefully planning lessons and discussions, teachers can ensure the proportion is right. It is useful to ask a colleague to observe a class with a view to recording the percentage of higher-order thinking skills practiced in a lesson; or even to ask students to use the knowledge they have gained in learning the language of thinking to record the teacher's use of higher-order terms; or to observe and assess their classmates in planned activities. Teachers should also encourage students to reflect on their learning so they understand their thinking strengths and weaknesses. The research is overwhelmingly in favour of explicit, direct instruction (Hattie, 2005; Marzano, 2011). This is particularly so in the teaching of concepts. Students need to understand the critical features that define what higher-order thinking skills they are practising. Once again, Bloom's Taxonomy (or the Core Skills of the Queensland curriculum) is a useful place to start. In any subject area, students should be aware of the key concepts they must learn. They must be able to identify them and they must practice them. Teachers can help by alerting students when a key concept is being introduced, and identifying the explicit characteristics of the concept. Students need to understand whether the concept is concrete, abstract, verbal, nonverbal, or process. For example, often students who perform poorly in mathematics have difficulty with nonverbal concepts. Simply working problems again and again with no verbal explanation will do little to help these students to understand maths concepts. Teachers have to spend time helping students to make strong connections between the manipulation of the symbols, the associated language and some form of concrete materials and images. By working through problems with students and verbalising the appropriate language, students begin to understand mathematical procedures. Conversely, students who have difficulty with verbal concept formation need multiple examples with relatively less language, which may confuse them. That is, some students need to be shown how to solve a problem, some students told, and some need both. In countries where attainment in maths is particularly high, it seems teachers ensure students have mastered basic concepts before proceeding to more sophisticated ones. Where students do not master basic concepts they are likely to attempt to memorise rather than to understand. While this works for them in the early years, it leads to misunderstanding and the inability to apply knowledge in the later years of schooling. It is also possibly the reason why many students 'turn off' maths. Thomas and Thorne (2009) suggest a multi-step process for teaching and learning concepts, which includes: Scaffolding involves giving students support at the beginning of a lesson and then gradually turning over responsibility to the students to operate on their own (Slavin, 1995). Without this limited temporary support students are unlikely to develop higher-order thinking skills; however too much scaffolding can be as detrimental as not enough. Kauchan and Eggen (1998) suggest teachers should provide 'only enough support so that learners make progress on their own'. Too much or too little support can interfere with the development of higher-order thinking skills. Too little support, and students are left floundering; provide support even though students don't ask for it, and they get the message they cannot do the task on their own. Kauchan and Eggins (1998), propose the following guidelines: 1. Use scaffolding:
2. Provide structured representations and discussions of thinking tasks:
3. Provide opportunities for practice in solving problems
5. Consciously teach to encourage higher order thinking In order to foster deep conceptual understanding, consider using the following strategies:
Using assignments and assessments that require intellectual work and critical thinking is associated with increased student achievement. These increases have been shown on a variety of achievement outcomes, including standardised test scores, classroom grades, and research instruments. The increases have been demonstrated in reading, mathematics, science, and social studies. And they have been documented particularly for low-achieving students. Evidence from both the National Assessment of Educational Progress (NAEP) and the Trends in International Mathematics and Science Study (TIMSS) report clear evidence that in mathematics and science instruction emphasising reasoning is associated with higher scores in all grade levels tested, while in reading, teaching for meaning (including thinking about main ideas, author's purpose, theme, and using real texts) is associated with higher NAEP performance, Wenglinsky (2004). Higgins et al (2005), for example, did a meta-analysis of studies of thinking-skills interventions on student cognition, achievement, and attitudes. He and his colleagues found 29 studies, mostly from the United States and the United Kingdom that reported enough data to calculate effect sizes. They found very strong effects. The average effect of thinking-skills instruction was:
Assessing higher-order thinking skills has also been shown to assist disadvantaged students. The 'Higher Order Thinking Skills' (HOTS) program designed by Pogrow (2005) specifically for educationally disadvantaged students, is based on four kinds of thinking skills: (1) metacognition, or the ability to think about thinking; (2) making inferences; (3) transfer, or generalising ideas across contexts; and (4) synthesising information. The project is a pure thinking skills approach to assist disadvantaged students in grades 4–8 in the United States. It combines the use of Socratic dialogue, drama, and technology, and has been used in approximately 2,600 schools in 48 states. It produced student gains in standardised tests, on measures of metacognition, in writing, in problem solving, and in grade point average. Furthermore, Pogrow found that in studies contrasting the efficacy of teaching higher-order thinking skills with teaching enhanced content instruction, the former was much better at setting up students to be flexible, allowing them to 'understand understanding' and to handle a variety of content. Finally, research has shown that student motivation increases when teachers hold them accountable for higher-order thinking. This seems to be so, because teaching students higher-order thinking tasks forces them to engage in thinking about particular things, and undertaking assessment that requires intellectual work and critical thinking. Memorising, while it is useful in some cases, does not increase students' autonomy and, to a large extent, does not contribute to mastery, although it might be argued that knowing basic facts is essential in providing building blocks for understanding. Also, it should be noted that 'knowing things' for immediate recall is a relatively unimportant skill. In most things we do, it is not the facts that are important but how we apply knowledge. For example, knowing the times table is useful to save time, to help in estimating and because rote learning builds useful pathways in the brain, but it is only when we use our knowledge of tables to manage our finances, plan a budget, or make decisions about whether one item is more expensive than another, that we exercise problem solving and higher-order thinking. Therefore, in order to assess so that students can demonstrate mastery, teachers need to plan assessment items that allow students to use all the skills of the Taxonomy: analysis, evaluation, and creation (the "top end" of Bloom's Taxonomy); logical reasoning; judgment and critical thinking; problem solving; and creativity and creative thinking. There are countless resources online and on paper to assist in the teaching of higher-order thinking, and while these are useful, an effective teacher needs to make few changes to programs already in place in order to ensure that students are encouraged to think. The research suggests that constant awareness of the language teachers use, and reflection on how the skills might be incorporated in every lesson, are pivotal in making the difference. Bibliography Anderson, L., Krathwohl, D., Airasian, P. et al (2001), A Taxonomy for Learning, Teaching, and Assessing: A revision of Bloom's Taxonomy of Educational Objectives, New York: Pearson, Allyn & Bacon Further Reading Brookhart begins by laying out principles for assessment in general and for assessment of higher-order thinking in particular. She then defines and describes aspects of higher-order thinking according to the categories established in leading taxonomies, giving specific guidance on how to assess students in the following areas:
Included on this site are specific conversations with students to test their thinking; examples of how to increase higher-order thinking; comments about evaluation; and recommended resources. KLA Subject HeadingsThought and thinking |
21st Century Mathematics Grade 7 Answers
Source: http://www.curriculum.edu.au/leader/teaching_higher_order_thinking,37431.html?issueID=12910
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