1. On higher-order thinking ability

With the development of knowledge age, the requirements for talents' quality focus on the following nine abilities: innovation ability, decision-making ability, critical thinking ability, information literacy, teamwork ability, compatibility ability, tacit knowledge acquisition ability, self-management ability and sustainable development ability. These nine abilities are called higher-order abilities. The so-called higher-order ability is based on higher-order thinking. The so-called higher-order thinking refers to mental activities or higher-level cognitive abilities that occur at a higher cognitive level. For example, the ability to show a higher cognitive level in the classification of teaching objectives, such as analysis, synthesis and evaluation. These abilities are necessary to meet the needs of the future information society. People with these skills will become leaders in the information age. Therefore, a lasting and long-term goal of modern education is to help students transcend the current low thinking ability and gain a higher level of thinking ability.

D. Perkins (1992+0992), a professor of psychology at Harvard University, believes that everyday thinking is just like our ordinary walking ability, and everyone is born with it. But good thinking ability, like the 100-meter race, is the result of technology and technical training. Runners need training to master the 100 meter sprint skills. Similarly, good thinking ability needs corresponding teaching support, including a series of targeted exercises. Therefore, as long as the methods are proper, students' higher-order thinking ability can be cultivated and trained. The key to the problem is how to cultivate and train students' higher-order thinking and what tools to use to cultivate it. Therefore, exploring the instructional design hypothesis that promotes the development of learners' higher-order thinking is one of the important topics in contemporary instructional design research.

2. Higher-order thinking ability of mathematics

Combined with the characteristics of mathematics, the so-called higher-order thinking in mathematics refers to the mental activity or cognitive ability at a higher cognitive level in mathematical thinking activities, which is characterized by analysis, synthesis, evaluation and creation in the classification of teaching objectives. It is rigorous, profound, quantitative, critical, original and flexible.

(1) profound. Have a thorough understanding of mathematical concepts and a good grasp of mathematical theorems; You can freely translate other languages into mathematical languages equivalently; Be able to use thinking operations such as analysis, comparison and generalization to discover the internal relations between mathematical objects with different forms but the same essence; Even if the problem-solving conditions are not clearly given, we can't be bothered by superficial phenomena, dig out implicit conditions from the appearance and find suitable problem-solving conditions;

(2) flexibility. The starting point of thinking is flexible, and you can consider problems from various angles and directions related to the topic; It is easier to have a psychological turning point, from positive thinking to reverse thinking, and the alternate use of analytical method and comprehensive method in solving problems is free; Thinking changes quickly, and it is not affected by the previous problem-solving methods. It can overcome the negative effects of thinking set and its own psychological limitations, so as to solve problems with a clear aim. Being good at transformation in the process of thinking, you can easily transform into maturity, turn parts into whole, and turn parts into parts.

(3) originality. Ability to think and analyze mathematical objects independently; Can observe problems from different "new" angles, can find unusual places in seemingly ordinary information, thus discovering hidden special connections and producing different problem-solving methods and results from others; Free from conventional restrictions and constraints, rich associations, active contact with different branches of mathematics, other disciplines, real life and even thinking jumps when solving problems, often produce creative ideas.

(4) critical. Usually study with suspicion, agree with other people's views without thinking, can stick to their own reasonable views but are willing to correct and accept lessons; Be able to compare the similarities and differences between different objects and distinguish some confusing concepts and forms; It can evaluate the reliability of information resources and judge the adequacy of deriving another conclusion from one conclusion, so it can find mistakes in others' problem-solving process or conclusion;

(5) agility. Can quickly and correctly complete the text understanding of the topic; Able to consciously use simple operation methods to perform faster operations on numbers; Can quickly identify the pattern of the topic; Can have a clear memory of the topics I have done recently; Be able to quickly judge and make a decision whether to give up solving this problem under the tight time.

These five aspects of advanced mathematics thinking are not completely separated and independent from each other, they are interrelated and infiltrated. Among them, profundity is the basis of advanced thinking in mathematics; Flexibility and originality are developed on a profound basis; The key is also based on profundity; Criticality directly restricts originality; Agility is based on four other factors.

Second, the teaching characteristics of college mathematics and the development of higher-order thinking ability

Rhomberg (1990) thinks that the purpose of mathematics teaching is not to master mathematics knowledge, but to cultivate students' advanced thinking ability by learning mathematics knowledge. The most effective way to develop learners' higher-order thinking ability is to combine it with course content and teaching methods, so that learners can engage in learning activities that require the use of higher-order thinking ability, which is generally called higher-order learning. In the process of college mathematics teaching, it is an effective way to cultivate students' higher-order thinking ability by designing teaching from two aspects: teaching and learning, making full use of modern information education means and carrying out a series of thinking teaching activities suitable for curriculum characteristics. Combining the characteristics of higher-order thinking in mathematics with college mathematics teaching, we can cultivate students' higher-order thinking ability from the following aspects:

1. Innovative teaching content provides a platform for cultivating higher-order thinking.

First, realize modernization in content. Change the past tendency of attaching importance to classics and neglecting modernity, and introduce necessary modern mathematics knowledge. First, the content of mutual penetration and organic combination. Combining algebra with geometry, the spatial analytic geometry in the original higher mathematics is inserted into linear algebra to form a whole; Linear algebra is arranged between univariate function calculus and multivariate function calculus, which is convenient to use.