preface
Mathematics is an abstract art of thinking. The abstraction of mathematics is to refine and simplify natural phenomena and life experiences, remove the shell of phenomena and extract the skeleton of principles. This feature enables mathematics to cross the boundary of human cognitive range and pursue the mysterious truth of the universe. Junior high school mathematics plays a connecting role in the process of cultivating students' abstract thinking. On the one hand, it is abstract, and the introduction of every knowledge point is based on real examples in life. For example, quadratic function, starting from the surface area of a cube A=6x2, from the surface of a cube that can be divided into six squares to 6x2, is a highly abstract process. The other direction is to abstract and show the original abstract theory in an intuitive way, one of which is visualization.
The concept of visualization originally comes from information graphics, including but not limited to the presentation and application of visualization means in science or knowledge dissemination, which makes information/knowledge easier to understand, spread and control. For example, statistical chart is an extensive and important position for data visualization. Another well-known application is the realistic rendering of phase, shape, nature, surface, volume, light source and other aspects in complex systems of science and engineering, such as meteorology, architecture or biology, which are static or contain dynamic time components.
The application of visualization in mathematics can be traced back to the source of mathematics. It is said that Archimedes was drawing geometric figures in the sand when he was killed. In the final analysis, geometry is a visual presentation. A point in the mathematical sense has no size or dimension, and a point in a drawing is the result of drawing for the convenience of observation. In more than two thousand years of geometry communication and teaching, this intuitive presentation has proved to be fruitful.
The new curriculum standard of junior high school mathematics puts forward that learning is a lively process, and students should have enough time to experience the process of observation and experiment. Drawing statistical chart is the main content of "statistics and probability" and one of the visualization means. For geometric intuition, the new curriculum standard emphasizes that "geometric intuition mainly refers to the use of graphics to describe and analyze problems, and with the help of geometric intuition, complex mathematical problems are simplified and visualized". In algebra, under the guidance of the combination of numbers and shapes, it is another application of visualization to understand functions with images.
Combined with junior high school mathematics textbooks published by People's Education Press, visualization can be used as an effective teaching tool to help students understand mathematics intuitively and play an important role in mathematics learning. The author believes that the application of classroom visualization teaching can be divided into the following categories:
Turn abstraction into intuition
The eighth grade textbook of People's Education Edition introduces the inverse theorem of Pythagorean theorem and the story of Egyptians constructing right angles. After building pyramids and measuring land after the Nile flooded, Egyptians used Pythagorean theorem to construct right triangle. Proposition and proposition belong to abstract logical reasoning concepts, which are unfamiliar to students who are in contact for the first time. Visualization can make abstract concepts intuitive.
Experiment: Prepare a long enough thin cotton thread, scale and cardboard. Please ask two students to go on stage and mark the lengths of 1 5cm, 20cm and 25cm on the cotton thread1to form a closed rope. Line segments with lengths of 24cm, 10cm and 26cm are marked on cotton thread 2 to form a closed rope. Please ask the third student to come to the stage, straighten the two ropes and fix them on the thick paper board with a pin. It is not difficult for students to find that the shapes of the two triangles are unique and fixed, and they both form a right triangle, as shown in figure 1(a).
Commentary: 32+42=52 and 122+52= 132 are special cases of integer pythagorean numbers. But is the true proposition always true? Please look at the following example.
Demonstration: Mark four lengths of rope 10cm on cotton thread 3 to form a closed rope. Straighten and fix with a pin to obtain the shape shown in Figure 1(b). We already know a true proposition 2: If a quadrilateral a=b=c=d is a square, then its four sides are a=b=c=d. Its inverse proposition is, ask questions and students will answer: If four sides of a quadrilateral meet the requirements of A = B = C = D, then the quadrilateral is a square. Does this inverse proposition hold?
(a) (b) (c)
Figure 1 Visualizing the Inverse Theorem of Pythagorean Theorem with Rope
We move the pin position to A', B', C', D', and a=b=c=d remains unchanged. Obviously, the quadrangle is no longer a square, but a diamond. The inverse proposition of proposition 2 does not hold.
Using simple and accessible equipment to design classroom mathematics experiments, and using visualization technology to improve students' participation and reduce knowledge abstraction.
Third, construct the concept of space.
The new curriculum standard defines the concept of space as "the concept of space mainly refers to abstracting geometric figures according to the characteristics of objects, and imagining the actual objects described according to the geometric figures;" Imagine the orientation of objects and the positional relationship between them. "
Using multimedia tools and three-dimensional modeling software, three views and projections of geometric shapes and complex models can be dynamically displayed, so that students can intuitively establish the concept of three-dimensional space. For example, Google SketchUp, a free 3D software, comes with a rich model library, which can be imported into airplane models. You can easily switch between top view, front view and left view by using shortcut keys, as shown in Figure 2. With the light source, the projection of each geometry on the plane is also clear, as shown in Figure 3. It is also convenient to realize transformation operations such as spatial translation, rotation and axial symmetry.
Fig. 2 Three views of the airplane model
Three views and projection are students' initial contact with three-dimensional space. Using the powerful visualization function of 3D software, students can successfully complete the transition from two-dimensional space view to three-dimensional space view.
Four. Visualized data
The collection, arrangement and description of data are introduced in the preliminary statistics of grade seven. Bar charts, line charts, fan charts and histograms are all ways to describe data. Taking histogram as an example, this paper introduces the fusion application of statistical graph in classroom.
Exercises in the textbook of People's Education Press: Use the age data of Fields Prize winners up to 2002 (data omitted). Please draw the histogram of frequency distribution according to different grouping methods, group spacing 2, group spacing 5 and group spacing 10, as shown in Figure 4.
Fig. 4 Histogram of age distribution of two groups of Faldts Prize winners.
In the lesson example, students have been given square paper, and students are familiar with frequency statistics by drawing histograms by hand. Here, the frequency statistics of different groups are generated by combining the elective content "Computer Drawing Statistical Chart".
Demonstration: Open Excel and other spreadsheet software, enter the age in column A, 28, 30, …, 40 groups with a group spacing of 2 in column B, 25, 30, 35, 40 groups with a group spacing of 5 in column C, and 20, 30, 40 groups with a group spacing of 10 in column D. Select Data-Data Analysis-Histogram. Take column A as the input area and columns B, C and D as the receiving areas respectively, and generate histogram and frequency statistics, as shown in Figure 4.
Using spreadsheet software to generate histogram to change the interval between groups is simple, which saves time and effort than drawing with graphic paper. Through statistical charts, at first glance, the data without clues shows the law. For example, the age of the winner of the Fields Medal reaches its peak around the age of 38, which is of course related to the stipulation that the Fields Medal is awarded to young mathematicians, and only the mathematics prize of no more than 40 years old is awarded.
Through visualization technology, intuitive charts are organically combined with the interpretation of data.
Five summaries
Based on the above three examples, this paper summarizes the application of visualization technology in junior high school mathematics classroom. Visualization can communicate abstract mathematical thinking with intuitive cognitive process, simplify the complex and simplify the complex, improve students' interest in learning, acquire knowledge in smooth experience, and receive excellent teaching results. Making full use of the materials and equipment around us, whether it is teaching AIDS or software, ancient rulers or cutting-edge computer graphics, are all applicable teaching resources.