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the assumption that there is a single established body of mathematical skills that everyone needs to know to be prepared for 21st-century careers. This assumption is wrong. The truth is that different sets of math skills are useful for different careers, and our math education should be changed to reflect this fact.

Today, American high schools offer a sequence of algebra, geometry, more algebra, pre-calculus and calculus (or a “reform” version in which these topics are interwoven). This has been codified by the Common Core State Standards, recently adopted by more than 40 states. This highly abstract curriculum is simply not the best way to prepare a vast majority of high school students for life.

A math curriculum that focused on real-life problems would still expose students to the abstract tools of mathematics, especially the manipulation of unknown quantities. But there is a world of difference between teaching “pure” math, with no context, and teaching relevant problems that will lead students to appreciate how a mathematical formula models and clarifies real-world situations.

Are in-depth group discussions provided during class time? Are students coached to question their thinking processes and those of their classmates? Are students afforded opportunities to evaluate their progress regularly? Are students encouraged to pose questions regularly in class? Are students provided with guides to help them reflect on their thinking (such as Bloom’s Taxonomy)? Do class projects engage students in analysis, synthesis, and evaluation? Are students given opportunities to consider various opinions and to justify their own beliefs?

Table 1 Bloom's Cognitive Taxonomy Competence Description Question/Prompt Knowledge Dates, events, formulas, other facts When did the United States become an independent country from England? What is the formula for area? Comprehension Recognize meaning, sequence, events, interpret information, compare ideas, make inferences, predict ideas What is the author's purpose? How are these numbers related? Is water of sunshine more critical to plant life? Human life? Application Use of information and concepts to solve problems Using your knowledge of calories and your physical makeup, calculate how much energy you must exert to lose three pounds per month. Demonstrate your understanding of how to create a Web site. Analysis Recognize patterns, parts, components Considering the stock market, examine which investments were the most lucrative this quarter. Organize these games by level of difficulty. Synthesis Use of information to create a new system, generalize, draw conclusions When did the United States become an independent country from England? What is the formula for area? Evaluation Assess concepts, weigh opinions for subjectivity, select items, judge Which type of dog would be best suited for your family, given your lifestyle and housing? Which local newspaper is written the most objectively?

In the 1950's Benjamin Bloom developed his taxonomy of cognitive objectives, Bloom's Taxonomy. This categorized and ordered thinking skills and objectives. His taxonomy follows the thinking process. You can not understand a concept if you do not first remember it, similarly you can not apply knowledge and concepts if you do not understand them. It is a continuum from Lower Order Thinking Skills (LOTS) to Higher Order Thinking Skills (HOTS). Bloom labels each category with a gerund.

In the 1990's, a former student of Bloom, Lorin Anderson, revised Bloom's Taxonomy and published this- Bloom's Revised Taxonomy in 2001.Key to this is the use of verbs rather than nouns for each of the categories and a rearrangement of the sequence within the taxonomy. They are arranged below in increasing order, from low to high.

Each of the categories or taxonomic elements has a number of key verbs associated with it Lower Order Thinking Skills (LOTS) Remembering - Recognising, listing, describing, identifying, retrieving, naming, locating, finding Understanding - Interpreting, Summarising, inferring, paraphrasing, classifying, comparing, explaining, exemplifying Applying - Implementing, carrying out, using, executing Analysing - Comparing, organising, deconstructing, Attributing, outlining, finding, structuring, integrating Evaluating - Checking, hypothesising, critiquing, Experimenting, judging, testing, Detecting, Monitoring Creating - designing, constructing, planning, producing, inventing, devising, making

at least 45 percent of undergraduates demonstrated "no improvement in critical thinking, complex reasoning, and writing skills in the first two years of college, and 36 percent showed no progress in four years."

What good does it do to increase the number of students in college if the ones who are already there are not learning much? Would it not make more sense to improve the quality of education before we increase the quantity of students?

students in math, science, humanities, and social sciences—rather than those in more directly career-oriented fields—tend to show the most growth in the areas measured by the Collegiate Learning Assessment, the primary tool used in their study. Also, students learn more from professors with high expectations who interact with them outside of the classroom. If you do more reading, writing, and thinking, you tend to get better at those things, particularly if you have a lot of support from your teachers.