![\"\"](\"https://emilymath.com/media/posts/10/image.png\")
In the competitive exam-oriented educational system, it is common for Chinese students to dedicate their summer and winter breaks to advance their understanding of upcoming semester’s subject matter, aiming to maintain their advantageous position. Students who have already delved into algebra may enjoy certain benefits over their counterparts who have not done so. For instance, there are certain steps and strategies to solve equations, which guarantees that one can master the skills with diligent practice. In contrast, geometric proof, reliant on robust deductive reasoning rather than the adherence to routine formulas, present a greater degree of flexibility, challenge, and even occasional mystique. Does a head start in geometric proofs also yield advantages? More precisely, what role does such a head start play in cultivating the deductive reasoning abilities required for the cognitively demanding Chinese mathematics curriculum?
\n\n\n\nStudents’ Deductive Reasoning Abilities
Low-Cognitive
Having been a math teacher for a decade, I would say that a head start in geometry does provide benefits, but confined to concepts and problems of low-cognitive complexity. Learning all the contents in a short time period is possible, but only at an entry level, far from the cognitive demands imposed by the Chinese mathematics curriculum. Annually, approximately half of the students in my honor math class preemptively engage with geometric proofs, including Jerry. Surely, the head start enables students such as Jerry to outperform their classmates who learn the contents for the first time in the upcoming semester. Those who have already mastered the three methodologies for prove congruent triangles can remember the theorems and directly apply them to establish congruence in varied scenarios. Quesiton 1 in Table 1 is a typical low-cognitive question in the Chinese mathematics curriculum, which requires only the knowledge of congruent triangles and equilateral triangles; moreover, the given conditions are adequate to straightforwardly deduce the congruence of triangle AOE and COE. Witnessing their newfound mathematical abilities and progress in contrast to their past performance, I used to be surprised and glad with them when I was a new teacher. However, the advantages of a head start dissipate quickly within the first month of the semester. This transition invariably transpires as the content escalates in complexity, exemplified by Question 2 in Table 1.
After a month of study, it becomes evident that all students, irrespective of whether they commenced with a head start or not, converge at the same juncture upon mastery of the essential theorems, poised to embark upon the subsequent phase of geometric proof. Question 2 serves as an introduction to the application of “auxiliary lines” in solving intricate geometric problems. In the beginning, it is challenging for almost all the students in my class, even though these students are in honor level. The learning process predominantly unfolds through two paths: rote memorization and deductive reasoning. Jerry is a representative of the students who embrace geometric proof mainly by rote memorization rather than deductive reasoning. I’m really touched by his effort that he notes down every detailed step written on the whiteboard and each seemingly pivotal strategy imparted during lectures. After class, he also works on extra exercises in addition to homework in order to reinforce his memorization and comprehension. If he sees “midpoint” in a similar question, without further contemplation, he will extend the line segment bisecting the midpoint to construct congruent triangles, just because that’s what the class example said. If he gets stuck with a new problem such as Question 3 in Table 1, he knows that he may add the auxiliary line(s) in the wrong way or he just can’t move further with the correct auxiliary lines, but he feels frustrated to find another way to solve the problem. In essence, he also uses deductive reasoning upon mere memorization, but his deductive reasoning ability is limited to low-cognitive problems. While rote memorization yields some efficacy, it does not help much with solving novel and challenging problems.
\n\n\n\nHigh-Cognitive
Students prefer deductive reasoning also memorize the key hints: midpoint implies extending the line segment to creative congruent triangles. Other than memorization, they also try backward and forward logical reasoning — bridge the gap between the given premises and the statement through trial-and-error. In Question 2,
1. What does the equality AM=CM+CD imply?
2. How can the algebraic expression be translated into geometric relations?
3. How important is the midpoint N and square in relating the given information and the unknown?
4. Angle bisector implies two congruent angles; The measurements of the angles are not given, so to prove two angles are equal, students can use either congruent triangles or isosceles triangle. Which is more feasible?
To answer the first two questions, students notice the requisite alignment of line segments CM and CD. The alignment can then be used to compare the combined length of line segments CM and CD with that of AM, which provide clues for the introduction of auxiliary lines. By extending MC and AN, two congruent triangles ADN and ECN are constructed, which subsequently paves the road to reach the final statement.
Question 3 is given to the 8th graders after right triangle and the properties of median lines are introduced. While akin to Question 2 in its utilization of squares and midpoints, Question 3 covers more theorems and needs more steps to unlock.
1. If DP is equal to AB, then DP is equal to any side of the square. Which side can simplify the problem?
2. DP and AP are in the same triangle ADP, but it is hard to prove that angle DAP is equal to angle DPA (It’s an alternative way but students haven’t learned trigonometry yet).
3. There are two midpoints E and F. If auxiliary lines are based on midpoints like Question 2, which midpoint is better?
4. Extending BF and AD creates two congruent triangles BCF and GDF and corresponding sides BC and DG. Prove DP=DG?
5. It’s hard to prove DP=DG. Does it mean the auxiliary lines are not useful?
6. Are all the given information used? What does midpoint E indicate?
7. Is point P special? It is the intersection point of AE and BF. What does this tell?
8. AE is perpendicular to BF and point P is the foot, therefore, triangle APG is a right triangle.
The breakthrough lies in the median of the hypotenuse of the right triangle APG. In my annual honor-level math class, no more than three students are able to independently solve Question 3 successfully each year. Among all these students, some of them not only possess a head start but are consistent top performers; there also exist students who, despite not initiating with a head start, promptly assimilate novel concepts and showcase a profound grasp of underlying principles. Other students, including those learn in ahead, either construct the auxiliary lines in the wrong way or encounter obstacles when proving DP=DG. Upon knowing all the relevant knowledge, students also need to make efficient interrelations between the premises and the theorems, which is the essence of deductive reasoning, the most challenging part of geometric proof. Thus far, the advantages conferred by a “head start” have been expended. What then distinguishes students from one another is their ability in deductive reasoning.
![\"\"](\"https://emilymath.com/media/posts/10/image-1.png\")
A Head Start in Deductive Reasoning?
Then why not provide students with a head start in deductive reasoning? Indeed, students engage in early learning during the summer or the winter breaks also learn reasoning skills, but this reasoning is relatively at a low-cognitive level. However, in the selective exam-oriented educational system in China (take Shanghai as an example), geometry, especially geometric proof, serves as a tool to discern students’ academic potential. Geometry accounts for over 40% in the Shanghai High School Entrance Examination each year (Shanghai Zhongkao Mathematics). The last question, the one that distinguishes students, remains to be the most challenging comprehensive geometric proof inquiry. Take the 2019 Shanghai Zhongkao Mathematics as an example, Question 25 (Figure 1) is composed of three sub-questions. According to the cognitive demand framework outlined in the 2019 Shanghai Middle School Mathematics Curriculum Final Guide, the first sub-question is at level II understanding, while the other two are both at level III exploration. The last sub-question covers eight topics spanning from algebra to geometry and necessitates the employment of 11 theorems throughout the course of its proof (as detailed Table 2). Evidently, this question is more cognitively demanding than Question 3 in Table 1. Honestly, the journey from questions in Table 1 to the last question in the high school entrance examination is an arduous endeavor spanning about three years filled with repetitive practice. However, even with the same amount of practice, fewer than 10% of the students (out of about 50,000 students took the exam in 2019) successfully solved the last question. So, does a head start in deductive reasoning offer a significant advantage? Regrettably, a head start doesn’t make any difference in solving the high-cognitive challenging problems.
![\"\"](\"https://emilymath.com/media/posts/10/image.jpeg\")
Ironically, it’s noteworthy that not every secondary school mathematics teacher can tackle the final question, despite their wealth of accumulated experience and prior knowledge. This situation is akin to my own inability to solve the geometry questions presented in the International Mathematics Olympiad. Does it mean that deductive reasoning is exclusively an innate ability, and we should give up if we were not born into a geometry person?
\n\n\n\nTeaching Strategies and Heuristics to Improve Reasoning Abilities on Individual Basis
Drawing from my experience teaching secondary school students and my personal learning experience, I contend that while innate ability may indeed play a role, the transformative impact of appropriate instruction and effective practice should not be underestimated. Those who possess stronger reasoning abilities than their peer groups also improve through practice. They seem to have intuitions when making decisions during problem solving, but this intuition does not come naturally, rather, it comes from experience and prior knowledge (Schoenfeld, 1982). Therefore, our emphasis should center on enhancing individual deductive reasoning skills rather than engaging in comparative evaluations with others. In sum, deductive reasoning skills are teachable, although the extent to which they can be effectively imparted on an individual basis may vary. Educators should actively employ strategies and heuristics aimed at nurturing and cultivating students’ capacities for sound reasoning.
References
\n\n\n\nSchoenfeld, A. H. (1982). Some thoughts on problem-solving research and mathematics education. In F. K. Lester & J. Garofalo (Eds.), Mathematical problem solving: Issues in research (pp. 27–37). Philadelphia: Franklin Institute Press.
\n\n\n\n2019 Shanghai Middle School Mathematics Curriculum Final Guide.
\n", "author": { "name": "Emily" }, "tags": [ ], "date_published": "2023-08-27T03:53:20-04:00", "date_modified": "2023-08-27T03:53:20-04:00" }, { "id": "https://emilymath.com/intuition-can-be-cultivated-but-is-there-a-limit-insights-from-chinese-school-geometry-education/", "url": "https://emilymath.com/intuition-can-be-cultivated-but-is-there-a-limit-insights-from-chinese-school-geometry-education/", "title": "Intuition Can Be Cultivated, But Is There A Limit? — Insights from Chinese School Geometry Education", "summary": "Drawing from my decade-long experience as a math teacher, I’ve observed that the introduction of geometric proof often marks a fresh beginning for middle school…", "content_html": "\nDrawing from my decade-long experience as a math teacher, I’ve observed that the introduction of geometric proof often marks a fresh beginning for middle school students. Interestingly, students who are good at algebra may not necessarily excel in geometry, while those under performers in elementry school may surprisingly develop a fondness for geometry. In contrast to school algebra, geometry presents distinct cognitive challenges mainly involving spatial reasoning and deductive reasoning. The process of solving high-cognitive geometric questions often mirrors the thinking patterns mathematicians experience. During the process of problem solving, intuition is considered of great importance to mathematicians (Burton, 1999). Similarly, intuition also plays a pivotal role in solving challenging geometric problems, especially when the application of auxiliary lines is essential. However, intuition is not an innate trait; rather, intuition is built upon a foundation of prior knowledge, attempts and thought (Schoenfeld, 1982). Therefore, intuition is a skill that can be nurtured and developed over time. Through exposure to various mathematical concepts, problem-solving scenarios, and real-world applications, individuals gradually cultivate the ability to perceive patterns, make informed guesses, and formulate conjectures. The teaching and learning of geometry in Chinese middle schools appear to offer a platform for unveiling the enigmatic nature of intuition.
\n\n\n\nGeometry curriculum of Shanghai Middle schools
\n\n\n\nGeometry education holds a pivotal role within the mathematics curriculum of Chinese middle schools. Its significance is underscored by the fact that geometry constitutes 40% of the High School Entrance Exam. To excel in the selective exam, students undergo rigorous and intensive long-term training. From 7th to 9th grade in China (i.e., Shanghai), students delve into the realm of geometric proofs, exploring concepts such as parallel lines, congruent triangles, parallelograms, rectangles, squares, rhombi, similar triangles, and circles. Table 1 provides an overview of the geometry-focused curriculum in Shanghai’s middle schools, with sections highlighted in blue denoting topics involving geometric proof. In a typical public middle school, math classes are given daily and there are about seven 40-min math classes each week. In addition to taking math classes, students also do homework to reinforce their grasp of the daily lessons everyday. In fact, many schools and math teachers spend more time on instruction to achieve the teaching tasks.
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/11/image-2.png\")
Aligned with the van Hiele Model of Geometric Thinking, Shanghai middle school students are expected to attain Level 3 Deduction, which is a prerequisite for success in the High School Entrance Examination. For instance, the majority of students are expected to successfully solve the first two sub-questions presented in Figure 1, however, only a small part of students can handle the last sub-question. The challenging nature is not confined to this selective middle school graduation examination, but is reflected in the difficulty of practice questions encountered throughout the three-year period. A notable strategy employed to tackle these cognitively demanding problems involves the strategic utilization of auxiliary lines.
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/11/image-3.png\")
Teaching and learning of auxiliary lines in school
\n\n\n\nThe strategic placement of auxiliary lines, often guided by deductive reasoning, forms a crucial aspect of geometric problem-solving. Chinese educators distill these auxiliary line techniques into practical rules that students apply extensively through focused practice to attain mastery. For example, when a midpoint is given, a congruent triangle can be constructed by extending the line bisecting the midpoint. Figure 2 demonstrates two methods of adding auxiliary lines involving midpoints to solve the last sub-question of question 25. The congruent triangle acts as a link to unveil hidden information within the problem. While various methods for incorporating auxiliary lines exist, for instance, extending line BG to intersect line AC exemplifies an alternative strategy, the utilization of lines related to midpoints often presents a promising approach to uncover additional conditions conducive to proving a given statement.
\n\n\n\nVarious other techniques are employed as well, including extending lines to create intersections, establishing parallel lines, introducing triangle heights, and utilizing transformations such as reflection, rotation, and dilation. These rules serve as a foundation, accompanied by comprehensive practice enabling students to adeptly employ them in resolving analogous problems. While these rules are advantageous for handling similar problems, they offer limited assistance in navigating intricate, comprehensive, and high-cognitive ones. To solve the challenging novice problems, students must often engage in authentic mathematical activities akin to mathematicians (Stylianides, 2022), involving a movement between conjecturing, refutation, and constructing coherent arguments.
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/11/image-4.png\")
Students’ growth through learning geometric proof
\n\n\n\nMost students experience a hard transition from Algebra to Geometry in the second semester of Grade 7. Unlike equations or area calculations, geometric proofs require students to demonstrate the validity of a given proposition, a task that initially brings about discomfort. In Shanghai middle schools, geometric proofs starts with identifying parallel lines, which serves as a relatively straightforward introduction. However, despite its apparent simplicity, students spend morethan an hour on homework involving two or three moderately complex geometric proof questions. The composition of proofs and the application of logical reasoning present novel challenges that requires considerable time and effort to become adapted. Congruent triangles are taught right after parallel lines. With some experience, students begin to comprehend the formal structure of geometric proofs. Nevertheless, as they confront more cognitively demanding problems, they struggle a lot more, not in the format of writing but in figuring out the hidden information necessary for solving the problem. On average, when faced with challenging questions, students invest 1.5 to 2 hours in their homework, sometimes even without arriving at a resolution. However, the positive part of this practice struggling is that it fosters substantial growth in deductive reasoning skill and perseverance. In the eighth grade, concepts such as right triangles, angle bisectors, and quadrilaterals come into focus. Students become less anxious about challenging questions, striving to employ strategies imparted in class to solve intricate problems. With considerable and consistent effort on problems necessitating numerous reasoning steps and the consideration of diverse scenarios, students advance to the rigorous nature of geometric proofs. This process of proving can be described as “Lakastos-style proving” (Komatsu & Jones, 2019), which demonstrates repetitive trial-and-error in exploration of conjecture and arguments. Grade 9, the last year of middle school, covers the two most demanding topics: similar triangles and circles. Students commit themselves diligently to memorizing strategies, practicing consistently, and reflecting on their progress. The culmination of intensive and protracted training equips most students to master theorems comprehensively, accumulating a wealth of experiential knowledge in preparation for the examination.
\n\n\n\nDiscussion 1: Intuition can be cultivated
\n\n\n\nIn my article “Is A Head Start Helpful in Geometric Proof and Deductive Reasoning?”, I describe two learning paths of students in my honor math class: rote memorization and deductive reasoning. Students prefer rote memorization rely heavily in worked out examples initially, remembering that midpoints imply extending the line segment bisecting the midpoint; students apply deductive reasoning advance through metacognitive practice, bridging the gap between conjectures and evidence. Mevarech 2003 indicates that the effect of metacognitive training outperforms that of worked-out examples on students’ mathematical reasoning. My conclusion that a head start is not help in deductive reasoning aligns with his observation but through an alternative perspective. However, when considering students’ overall progress and, more specifically, their development of intuitive abilities to tackle challenging questions, both groups of students exhibited noticeable advancements in deductive reasoning over the course of the three-year study. This underscores that intuition is indeed cultivable. While the paths taken by different students may vary, the common thread lies in educators’ ability to foster an environment conducive to cultivating intuition. Surely, the exam-oriented Chinese educational system has its own drawbacks. Educators should look for more efficient ways to improve students deductive reasoning abilities, such as encouraging students to engage in exploratory thinking and profound metacognitive training.
\n\n\n\nDiscussion 2: Is there a limit?
\n\n\n\nI can often hear G6er’s and G7er’s parents say, “My child doesn’t work hard enough. If he also works as hard as A, he will be as good as A.” The answer usually emerges vividly in Grade 8 when most students realize the importance of the High School Entrance Examination and work hard to prepare for it. Many of those parents realize that learning outcomes can vary significantly even with similar amounts of practice. Innate abilities including intelligence and creativity do play a role (Meier, 2022), but the Chinese geometry education proves that dedication, perseverance, and targeted training can push those limits further. Yet, reflecting on my own experience with the 2023 International Olympiad Mathematics, it seems that I don’t possess the intuition to solve these exceedingly challenging questions. Can I transcend them? Could professional training and the accumulation of substantial experience enable me to overcome these boundaries? If I can indeed surpass these constraints, where is my next limit? Are there certain limits that remain impervious to improvement, irrespective of the strategies applied to individual students? How can teachers stimulate students’ potentials to the greatest extent?
\n\n\n\nReference
\n\n\n\nBurton, L. (1999). Why Is Intuition so Important to Mathematicians but Missing from Mathematics Education? For the Learning of Mathematics, 19(3), 27–32. http://www.jstor.org/stable/40248307
\n\n\n\nKomatsu, K., & Jones, K. (2018). Task design principles for heuristic refutation in Dynamic Geometry Environments. International Journal of Science and Mathematics Education, 17(4), 801–824. https://doi.org/10.1007/s10763-018-9892-0
\n\n\n\nMeier, M. A., & Grabner, R. H. (2022). The roles of Intelligence and Creativity for Learning Mathematics. Handbook of Cognitive Mathematics, 648–679. https://doi.org/10.1007/978-3-030-44982-7_6-1
\n\n\n\nMevarech, Z. R., & Kramarski, B. (2003). The effects of metacognitive training versus worked-out examples on students’ mathematical reasoning. British Journal of Educational Psychology, 73(4), 449–471. https://doi.org/10.1348/000709903322591181
\n\n\n\nSchoenfeld, A. H. (1982). Some thoughts on problem-solving research and mathematics education. In F. K. Lester & J. Garofalo (Eds.), Mathematical problem solving: Issues in research (pp. 27–37). Philadelphia: Franklin Institute Press.
\n\n\n\nStylianides, A. J., Komatsu, K., Weber, K., & Stylianides, G. J. (2022). Teaching and learning authentic mathematics: The case of proving. Handbook of Cognitive Mathematics, 727–761. https://doi.org/10.1007/978-3-031-03945-4_9
\n", "author": { "name": "Emily" }, "tags": [ ], "date_published": "2023-08-26T03:57:00-04:00", "date_modified": "2023-08-27T03:58:39-04:00" }, { "id": "https://emilymath.com/a-journey-of-self-growth-in-learning-mathematics/", "url": "https://emilymath.com/a-journey-of-self-growth-in-learning-mathematics/", "title": "A Journey of Self-growth in Learning Mathematics", "summary": "After receiving the mid-term exam paper, J.L., a student in my math class, cried to me, “I’ve tried my best to learn geometric proofs, but…", "content_html": "\nAfter receiving the mid-term exam paper, J.L., a student in my math class, cried to me, “I’ve tried my best to learn geometric proofs, but I just don’t know when and where to add auxiliary lines. I’m not as smart as D.G. (D.G. is the best performer in the class). Mathematics frustrates me”. J.L. is one of the many students who have similar struggling experiences when learning mathematics, especially in the transition from primary school to secondary school. Some of the students overcome the challenges and move on, but others hold the stereotype that math only belongs to those smart minds and stop putting effort into the subject. Those who believe that math is based on one’s innate ability and that efforts don’t make much difference are said to have a fixed mindset, which negatively affects their study and life (Dweck). On the other hand, a growing mindset enables students to take challenges and work hard consistently to acquire specific abilities. Therefore, it is vitally important to prepare students with a growing mindset, which is almost common sense nowadays. Nevertheless, the theory does little to help ease the “math hurts”, particularly in high-demanding and competitive test-oriented learning environments like East Asia. To cultivate a growing mindset, students first need to prepare for the challenges with positive mental attitudes towards practice, mistakes, and self-growth.
\n\n\n\nLearning math, just like mastering any life skill, requires a certain amount of practice to achieve expertise. A baby isn’t born with the ability to walk but needs lots of practice to balance the body. To get a driver’s license, a beginner needs months of driving practice to pass the road test, even though he or she may be crystal clear with the rules. Likewise, on top of understanding that solving a quadratic equation is to find the intersection points of the parabola and the x-axis, it still takes intense practice to be adept with all three methods: the quadratic formula, cross multiplication, and completing the squares. In the book Outliers, Gladwell reveals that it takes 10,000 hours to achieve expertise, and it also applies to those widely known geniuses. After Fermat’s Last Theorem was published in 1670, it remained unproved for more than 300 years, until Professor Andrew Wiles finally published his proof in 1995 (Wiles). He is definitely a genius in mathematics, yet he spent decades of consistent effort on proving the theorem. In addition to gaining expertise, practice enhances understanding. Rote memorization and repetition are often blamed; however, nearly half (44.3%) of Chinese students who took the 2018 PISA Mathematics were at level 5 or level 6, the two highest levels in PISA mathematics, compared to the OECD average of 10.9% (105). The two highest levels show that Chinese students possess strong mathematical abilities, such as deep understanding and logical reasoning, rather than just memorizing those subject-related concepts. Whether genius or not, consistent effort is critical to acquiring skills and gaining expertise.
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/12/image-5.png\")
In the competitive test-oriented education system, students tend to resent the mistakes they make in the pursuit of high grades, and only a few students recognize mistakes as opportunities to strengthen their understanding. In fact, trial and error is a fundamental strategy of problem solving in secondary school mathematics. Taking geometric proof as an example, an experienced geometry teacher may also need to take several tries to determine where to add one or more auxiliary lines before arriving at the correct solution. Surely, students need to avoid those careless mistakes. However, most of the mistakes in secondary school or college mathematics are caused by misunderstanding of concepts and lack of logical reasoning abilities. The question “how many two-digit numbers have at least one digit that is a 4?” requires students to think completely. It’s not easy to include all the cases without repetition. Usually, 44 is the number that many students miss and 04 is not a two-digit number which is easily over counted. Nevertheless, through such mistakes, students learn the logic to identify all cases. Sometimes it is hard to come up with an idea to solve a problem, let alone make a mistake. Students get frustrated if that happens. In terms of success, “no idea” surely is unsatisfactory, but it is not bad at all. Research shows that “unsuccessful retrieval attempts enhance subsequence learning (Kornell et al.). Set the test scores aside, and focus only on learning itself. Understanding that true learning is journey with ups and downs.
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/12/image-6.png\")
The problem is, despite all the advice, students still struggle with mathematics. Peer pressure is one of the main reasons. J.Z. was not the top performer but quite well-performed in mathematics, yet he was still sad about himself for not being the top student. Actually, he was better than the top math student in writing, public speaking, and socializing, but he chose to zoom in on his weakness. Comparing one’s weakness with the strengths of others only makes students more distressed. Human beings are born with equal rights but distinct gifts. One of my undergraduate classmates has been a professor in a top research institute, I am just about to start my research career in mathematics education. My classmate could easily prove the well-known primitive tests used in cryptography, but I can’t even understand the Riemann Hypothesis. Aiming at proving Riemann Hypothesis is unrealistic for me, but this does not prevent me from learning number theory. Regardless of my limitations compared to my classmate, I’m glad to see my progress in mathematical proofs, from Diophantine Equation to Pell Equation and more. Students who suffer from peer pressure are inherently eager to improve. Parents and teachers should encourage teenagers to feel satisfied with the progress in their learning process, no matter how slow it is.
\n\n\n\nIf math ever hurts you or you are resentful of not having a math brain, it may be just because you haven’t developed a positive mental attitude towards learning mathematics. Considering learning a journey of self-growth consisting of intense practice and correction from mistakes can prepare you for the challenges. This mental preparation is not only applied to learning mathematics but also applied to learning in any disciplines. More importantly, in the process of learning, students can develop perseverance and resilience that goes beyond just academic achievements.
\n\n\n\nWorks Cited
\n\n\n\nDweck, Carol S. Mindset: The New Psychology of Success, Ballantine Books, 2007.
\n\n\n\nGladwell, Malcolm. Outliers: The Story of Success. Back Bay Books, Little, Brown and Company, 2019.
\n\n\n\nKornell, Nate, et al. “Unsuccessful Retrieval Attempts Enhance Subsequent Learning.” Journal of Experimental Psychology: Learning, Memory, and Cognition, vol. 35, no. 4, 2009, pp. 989–998, https://doi.org/10.1037/a0015729.
\n\n\n\nPISA 2018 Results (Volume I): What Students Know and Can Do, PISA, OECD Publishing, Paris, 2019, p.105 https://doi.org/10.1787/5f07c754-en.
\n\n\n\nWiles, Andrew. “Modular Elliptic Curves and Fermat’s Last Theorem.” The Annals of Mathematics, vol. 141, no. 3, 1995, p. 443, https://doi.org/10.2307/2118559.
\n", "author": { "name": "Emily" }, "tags": [ ], "date_published": "2023-08-24T03:59:00-04:00", "date_modified": "2023-08-27T04:01:41-04:00" }, { "id": "https://emilymath.com/area-model-from-distributive-property-to-calculus/", "url": "https://emilymath.com/area-model-from-distributive-property-to-calculus/", "title": "Area Model: from Distributive Property to Calculus", "summary": "“Mommy, do you know how many bars of chocolate are there?”, Shawn asked when he opened the pack of chocolate. “Hmm, there are 24 bars”,…", "content_html": "\n“Mommy, do you know how many bars of chocolate are there?”, Shawn asked when he opened the pack of chocolate.
“Hmm, there are 24 bars”, I replied in about 2 seconds.
Shawn, “how can you count the total numbers so quickly? I didn’t count but tried to add 6 bars four times, yet I’m still slow, much slower than you”.
Me, “Honey, I used 6*4=24. This is multiplication, which is more efficient than adding the same number repeatedly.”
![\"\"](\"https://emilymath.com/media/posts/9/1fafSl7cSsMv1mtMMzpNaEA.png\")
Since then, Shawn understands the meaning of multiplication 6+6+6+6=6*4=24
When it comes to multiplication, I naturally think of multiplication tables and rely on the method of ‘long multiplication’. However, he does multiplication by drawing and adding all the time.
If Shawn is asked what is the product of 7 and 13, he would draw 13 sticks in a line and repeat 7 times (Figure 1). Later on, he noticed some kind of shortcuts to do the repeated addition quickly. Instead of adding 13 seven times, he first added 13 to 13 and get 26, then 26+26+26=78, so far, he added 13 six times, and finally added the last 13 to 52 to obtain 91. Although still slow, he actually used “distributive property”, this cognition through geometric illustration surprised me and ignited my thinking of teaching kids long multiplication through the distributive property. If I’m asked what is 17 times 3, I would just follow the multiplication formula, mentally. And now I realized how abrupt to introduce the formula to a novice and how important teachers understand and respect kids’ cognition process.
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/9/1l6ZydpniOHZwWsO3ZM95nA.png\")
It was such a coincidence that I was taking the course named “Mathematics for Machine Learning: Multivariate Calculus” offered by Imperial College London on Coursera. The professor visualized the product rule using the area model and it is intrinsically the same as distributive property in elementary school. Ironically, it is the first time that I get to understand ‘Calculus’ after finishing my undergrad and graduate study for over 10 years. Calculus to me used to be barely formulas and hundreds of algebraic manipulations upon the formulas, e.g., chain rules (product rule). Given f(x)=xsinx, I can find the derivative using the formula easily, even without thinking:
f’(x)=(xsinx)’=x’sinx+xsin(x)’=sinx+xcosx.
As a student, I didn’t ever think about the logic behind all the formulas, in another way, formulas are enough to solve calculus problems, so why is logic important? One can solve difficult differential equations with strong algebraic skills but can’t apply them well without knowing the underlying logic. Identically, the basic logic under machine learning is the foundation for further exploration and advancement. Teaching strategies in advanced math can also affect undergraduate learning. Geometric illustration of an algebraic problem is essential to enforce students’ understanding. Technology also provides more possibilities for educators to tailor the appropriate demonstration to students.
From elementary to advanced mathematics, the same geometric illustration, as well as the same thinking method, is used. Looking back, the connection between knowledge and the human cognition process may provide new insights for K-12 mathematics education.
\n\n\n\nAs Figure 2 shows, it’s natural to make 13 columns into 10 and 3 columns not only because any natural number multiplied by 10 is easy to get but also because we use the base 10 system. The grey part holds 10 columns and 7 rows of squares, 7*10=70; the blue part is composed of 3 columns and 7 rows of squares, 7*3=21. The total number of squares is the sum of 70 and 21, which gives 91. The distributive property involved here can be written:
7*13
=7*(10+3)
=7*10+7*3
=70+21
=91
![\"\"](\"https://emilymath.com/media/posts/9/1_nz8IZAjH-EAN75db4gN6Q.png\")
This perfectly explains the long multiplication method below. Instead of using the traditional way of expression called regrouping tens, it makes more sense to write it following the distributive property to a new learner, especially when larger numbers are introduced.
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/9/10dgwKi8o_5wwlozhUrVeTw.png\")
Take 13*12 as an example. 13=10+3 and 12=10+2. The geometric illustration (Figure 3) shows that the 13*12 squares can be distributed into four parts: the grey part with 10 columns and 10 rows, the yellow part with 10 columns and 2 rows, the blue part with 3 columns and 10 rows, and the pink part with 3 columns and 2 rows. The whole calculation process can be written:
13*12
=(10+3)*(10+2)
=10*10+10*2+3*10+3*2
=100+20+30+6
=156
![\"\"](\"https://emilymath.com/media/posts/9/1x4TnYslgNSi9AcdDAM7GUw.png\")
It is exactly how long multiplication works.
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/9/104WSmH2PN2SLLuEV_Y-Tug.png\")
Many students have struggled with the transition from numbers to variables. This probably is the first transition from elementary to secondary school math worldwide. Many students feel it is hard to step over. Researches reveal that the leap can be smooth with a continuous and connected curriculum (H.Wu, 2001). Chinese students have a strong arithmetic background and start to learn to solve simple linear equations in Grade 5/6 but still suffered more or less. Curriculum design is one aspect, but knowledge presentation is somewhat more realistic in teaching. Geometric illustration undoubtedly helps ease the leap from concrete to abstract. For example, the multiplication of polynomials and hence factoring polynomials can be well demonstrated by the combination of rectangles. As an adult, we intuitively choose the FOIL method to multiply polynomials, e.g.,(x+1)(x+3)=x²+3x+x+3=x²+4x+3
which is a conditional reflex after repeated practice. FOIL is introduced in some classes directly and surely students can master the steps whatever, however, many students just automatically remember the steps without understanding, only some may discover that it is actually the same as the way distributive property is used to calculate 13*12. Adding the areas of the four parts x², 3x, x, and 3, the total area, the value of (x+1)(x+3) is obtained (Figure 4).
![\"\"](\"https://emilymath.com/media/posts/9/1VDwLOaQfk3IlU465KMh58Q.png\")
What’s the difference then? For multiplication, there’s no big difference whether you understand it or not, but for factoring and solving quadratic equations, it matters a lot. Factoring a quadratic trinomial is to write it in the form of a product of two linear expressions. x²+4x+3 is a quadratic trinomial factorable in integers, meaning that algebraically we can easily use cross multiplication to factor it. What is the basis for cross multiplication? Generally,
x²+bx+c=(x+x1)(x+x2) where x1+x2=b and x1*x2=c.
If a quadratic trinomial is the area of a rectangle, then Factoring is expressed in width times length. Geometrically, the blocks or tiles have to be arranged appropriately to fit the area. Figure 4 is a perfect fit. In contrast, with a corner missing, Figure 5 is not even a rectangle, though the total area is also x²+4x+3. Now, it is clear how cross multiplication works, we first need to find two numbers whose product is c, in the meantime, the sum of the two numbers must be b, in order to form a rectangle given the area.
![\"\"](\"https://emilymath.com/media/posts/9/1d3eadpUX4Z3BX0BQCZQQzQ.png\")
Graphically, factoring means that given the area of a rectangle, find the dimension. To factor x²+4x+3, first focus on 3, which can only be written in the product of 1 and 3, second, the sum of 1 and 3 is 4. Algebraically, we also solved
x²+4x+3=x²+x+3x+1*3=x(x+1)+3(x+1)=(x+1)(x+3)
Turn to Figure 5 again, although the tiles can’t add up to x²+4x+3, it reveals that a perfect square (x+2)² is formed by adding 1 to x²+4x+3. This explains the method of completing the square.
\n\n\n\nIn addition to factoring quadratic trinomials, the area model also helps understand solving quadratic equations geometrically. Given x²+4x+3, the area of the rectangle in Figure 4 is 0, either width or length is or both of them are 0. When the width is 0, x+1=0, x=-1; when the length is 0, x+3=0, x=-3. Therefore, the solution to the quadratic equation is
x=-1 or x=-3
As I mentioned in the beginning, the graphical demonstration of the product rule enlightens and refreshes my understanding of formulas. Given h(x)=f(x)g(x), you may know the derivative of h(x) is
h’(x)=(f(x)g(x))’=f’(x)g(x)+f(x)g’(x)
Do you also know why?
![\"\"](\"https://emilymath.com/media/posts/9/1q5sQcdJw8ZlaLse8KuPXnA.png\")
![\"\"](\"https://emilymath.com/media/posts/9/1yhOXVO45ETtWAkbr_2TKRg.png\")
![\"\"](\"https://emilymath.com/media/posts/9/1ij4BDs5Bqizv_76I4e2l-A.png\")
When delta x (dx) is near 0, f(x+dx) is nearly the same as f(x) and g(x+dx) is also extremely close to 0. Therefore, [f(x+dx)-f(x)][g(x+dx)-g(x)] is near 0, meaning the area of the pink rectangle can be neglected compared with the blue and yellow area. The expression can then be written as follows:
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/9/1kP-29FbaiLw2tXRPPdYiMA.png\")
Now, it’s clear why h’(x)=(f(x)g(x))’=f’(x)g(x)+f(x)g’(x).
\n\n\n\nReference
H. Wu, How to prepare students for algebra, American Educator, Summer 2001, Vol. 25, №2, pp. 10–17. http://www.aft.org/pubs- reports/american educator/summer2001/index.html
Question
Three females and three males are applying for three identical positions with a company. The hiring committee wants to hire at least one woman. How many different ways can the four positions be filled?
Student answer
Since at least one woman will be hired, choose 1 woman out of 3 firstly to secure a woman C(3,1). Secondly, choose the other two candidates from the other five persons, including 2 females and 3 males, C(5,2). Multiplying C(3,1) and C(5,2) leads to an answer of 30 ways.
I. What’s the logic mistake?
\n\n\n\nAt a first glance, the logic offered by the student seems to be reasonable. Selecting one woman at the first step surely can gaurantee that at least one woman will be hired. Unfortunately, the answer is wrong, not even close to the correct one. Can you identify the logic mistake?
\n\n\n\nThe most straightforward way to identify the problem is through a tree diagram. As shown in the table, F1, F2, and F3 represent three females, there are three ways to choose 1 female out of 3 females. In step 2, there are three distinct cases to choose 2 persons out of the other 5 persons: 1 female and 1 male, 2 males, and 2 females. Different colors are used to identify the repeated combinations in the table. Exactly, there are 9 colors representing 9 repeated combinations when choosing 1 female and 1 male in step 2. Moreover, look at the column of choosing 2 females in step 2, all of them are the same. Why? There are 3 females and there’s only one way to choose 3 females out of 3 females: choose all of them.
Why are there repeated ones?
When steps are involved, you naturally introduce ‘ORDER’ to the problem. For example, F1 F2 M1 is the same combination as F2 F1 M1 even though they are in different orders. This is the mistake! In mathematics, it is called ‘OVERCOUNTING’. Only combination matters here. Permutation will lead to overcounting.
![\"\"](\"https://emilymath.com/media/posts/8/1_12k1VUX3PfBiWAUykSffa3A.png\")
II. The correct logic
\n\n\n\nAt least one woman indicates the following 3 cases:
> 1 female + 2 males: C(3,1)*C(3,2)=9
> 2 females + 1 male: C(3,2)*C(3,1)=9
> 3 females + 0 male: C(3,3)*C(3,0)=1
So, the problem can be solved by adding the combinations in three different cases:
9+9+1 = 19.
To clearly show the combinations, a table is used again below for your reference.
![\"\"](\"https://emilymath.com/media/posts/8/1Y1JE8tYaF9DkFKJQCoj5xQ.png\")
III. Alternative Strategy?
\n\n\n\nIn general, ‘At least’ problems are better solved by the complement of this event. Event and the complement of the event form a complete whole.
*** Event + Complement of this event = Whole ***
Therefore, subtracting the complement of the event from the whole leaves the event, the desired answer.
![\"\"](\"https://emilymath.com/media/posts/8/1nGiI1mLWO2z8Y7ZbqFvsig.png\")
What is the “WHOLE” of this problem?
choosing 3 candidates out of 6 people is the “WHOLE”: C(6,3)=20
The “WHOLE” is actually composed of 4 distinct cases:
> 0 female + 3 males: C(3,0)*C(3,3)=1
> 1 female + 2 males: C(3,1)*C(3,2)=9
> 2 females + 1 male: C(3,2)*C(3,1)=9
> 3 femlaes + 0 male: C(3,3)*C(3,0)=1
Again, the WHOLE is 1+9+9+1=20 ways.
The event in the problem is “At least one woman”. So, the complement of this event is “no woman”:
0 female + 3 males: C(3,0)*C(3,3)=1
Answer : WHOLE — No woman = 20–1=19
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/8/1oVIY7_7fABCia7izqqTPLg.png\")
Counting Techniques are all about logical thinking.
\n", "author": { "name": "Emily" }, "tags": [ ], "date_published": "2022-07-31T02:10:32-04:00", "date_modified": "2022-07-31T02:12:56-04:00" }, { "id": "https://emilymath.com/square-numbers-and-odd-numbers/", "url": "https://emilymath.com/square-numbers-and-odd-numbers/", "title": "Square Numbers and Odd Numbers", "summary": "When we were having dinner, Shawn’s dad casually tested his calculation. Dad: 1 plus 3 is? Shawn: 4 Dad: How about 1 plus 3 plus…", "content_html": "\nWhen we were having dinner, Shawn’s dad casually tested his calculation.
Dad: 1 plus 3 is?
Shawn: 4
Dad: How about 1 plus 3 plus 5?
Shawn: 9
Dad: Add the next odd number 7?
Shawn: Well, it is 9 plus 7, and it equals 16.
Shawn’s dad suddenly realized that square numbers could be formed by adding consecutive odd numbers together. Why is it so magical? We know that odd numbers form an arithmetic sequence, the sum can be easily obtained through the formula, and for this case, it is n squared. Shawn was curious about but confused with our conversation. Exactly, it is such an interesting mathematical relationship that can naturally trigger a young kid to explore further. However, to a kid in Grade One, explaining by algebraic expressions is like playing the piano to a cow (Chinese old saying). So, I used SQUAREs to visualize SQUARE Numbers.
\n\n\n\nI. Activate Prior Knowledge
\n\n\n\nShawn learned multiplication through drawing columns and rows. He’s really sick of remembering the multiplication table. How? He’d rather draw tables and repeatedly add the same number a certain amount of times. To some extent, thanks to his laziness, he has developed a sense of geometrical illustration to represent numbers and operations, which my generation didn’t have at his age. Given the multiplication table below, the highlighted numbers are the product of the same integers, such kinds of numbers are square numbers. Shawn quickly related square numbers to graphics: the same number of columns and rows, this is how a square is formed. Great, it is time to move forward.
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/7/1kd6t4FEHMC2E395-yqN-Qw.png\")
II. Geometric Illustration
\n\n\n\nLego, you are popular for a reason!
I used a square lego block to draw a yellow square first.
Me: Can you make a bigger square if I give you one more square?
Shawn: No, I need 4 squares to make a bigger square.
I then asked Shawn to draw the second square and color the extra three squares blue. This square is expressed as 2 times 2.
Me: How many square blocks are needed to form a third square?
Shawn tried to place more square blocks together and he used 9 blocks. He noticed that there are three columns and rows so that 3*3=9. The same as square 2*2, I asked Shawn to color the third one (Figure 2).
Me: How many more square blocks are needed compared to the second one?
Shawn counted and replied, ‘5 more’.
Before moving on to the fourth square, I asked, ‘how many more square blocks are needed to form the fourth square?’ By asking this question, I was leading him to find the pattern of adding consecutive odd numbers. I’m too fast or aggressive here. For a pattern to be obvious to anyone, a minimum of three or four numbers are essential. Shawn repeated the steps in the squares 2*2 and 3*3 and found that 7 blocks were added to form a 4*4 square. He seems to be more interested in both drawing and calculating. Before I asked him to create the fifth square, he drew the 5*5 square by himself and grasped the pattern: 9 blocks should be added to the fourth square as a new layer to form the fifth one.
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/7/1JYjoJgXf3YsB3AKHA6JPxg.jpg\")
III. Discover the Patterns
\n\n\n\nIn order for the pattern to be clearer, I marked 1, 3, 5, 7, and 9 on the correspondent layers shown by distinct colors. Take the 2*2 square as an example, 1 block surrounded by another 3 blocks is expressed as 1+3=4, and 4=2²
Me: Could you write down the expressions for the 3rd, 4th, and 5th squares following my example?
Shawn wrote 1+3+5=9=3³.
He doesn’t know square numbers and of course, he doesn’t know that 2*2 is abbreviated by 2². He wrote 3³ as he thought the 3 in the superscript means the second 3. I briefly explained that 3+3=3*2 showing the same number was added twice can be abbreviated by the number multiplied by 2. Similar to this, 3*3 shows that the same number is multiplied twice, which can be expressed by the number squared and this square is a 2 in the superscript. I don’t expect a 1st grader to truly understand exponents but planting a seed at an appropriate time is an art.
He then correctly wrote 1+3+5+7=16=4² and 1+3+5+7+9=25=5²
![\"\"](\"https://emilymath.com/media/posts/7/1W4_TslHexc7aVvWqryfk8w.jpg\")
To sum up, he decided to draw a big square on a whole piece of paper. He did it all by himself, from drawing the squares, coloring each layer, adding the numbers, to writing the square number.
\n\n\n\nInstant Review
Making an immediate summary of what we’ve just learned is a good way to consolidate our understanding of the new discoveries. Shawn thought that he knew everything now and that I should record a video for him to share his new knowledge on youtube. We all know that knowing and teaching are basically two stories. I asked him what he learned from all of these illustrations and expressions. He replied, ‘squares and square numbers’. Should I be happy? The idea came from the addition of consecutive odd integers and it seemed that he doesn’t care about it that much. Well, at least he learned that square numbers were perfectly demonstrated by squares. To remind him of the key knowledge, I made a chart (Figure 3) to show:
1. the number of unit squares used to form the nth square is n*n=n².
2. a square number is the sum of the consecutive odd integers. 3² is obtained by adding the first three positive odd integers; 4² is the sum of the first four positive integers;
Arithmetic Sequence
How about the nth square?
Why are there such magical relationships between square numbers and odd numbers?
It is where the formula of arithmetic series comes from. The arithmetic sequence is 1 + 3 + 5 +…+ 2n-1
Write it in the reversed order 2n-1+2n-3+2n-5+…+1
Add the two expressions 2n+2n+2n+…+2n
There are n ‘2n: n*2n=2n²
Since we add the same sequence twice, divide the above expression by 2: 2n²/2=n².
Proved! Amazing, isn’t it?
Variables and arithmetic sequences are way beyond Shawn’s age. Again, planting a seed at an appropriate time.
\n\n\n\nIV. Turn 2D to 3D
\n\n\n\nMe: Shawn, you built a Lego pyramid last week, right?
Shawn: Yeah. Sister broke it but I can build a new one.
Me: How about we build all the 6 squares together?
Shawn: Is it a pyramid?
Me: You can find the answer by adding the 6 squares and I can help you.
![\"\"](\"https://emilymath.com/media/posts/7/gallery/WechatIMG512-2-1024x908.jpeg\")
![\"\"](\"https://emilymath.com/media/posts/7/gallery/WechatIMG513-1-1024x731.jpeg\")
Figure 4
\n\n\n\nWe spend about 20 minutes building the 3d figure together (Figure 4). Is it a pyramid? It is not the traditional Egyptian pyramid but it is a special pyramid.
On one hand, the process of building the pyramid is a favored tool to consolidate kids’ understanding of abstract numbers.
On the other hand, how many unit squares in total are there in the pyramid? In other words, 1²+2²+3²+…+n²=?
Lego can help to reveal the mystery. This allows me to design the next topic for Shawn.
Given that P is a point in parallelogram ABCD, line GF passes through point P and is parallel to AB, line EH passes through point P and is parallel to AD. If the area of BFPE =3，the area of PHDG = 5, find the area of ▵APC.
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/6/p1.png\")
Method 1: Use Triangle ABC as a whole
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/6/gallery/screen-shot-2020-03-30-at-9.11.00-pm-1.png\")
![\"\"](\"https://emilymath.com/media/posts/6/gallery/screen-shot-2020-03-31-at-12.04.39-pm-1.png\")
Method 2: use Quadrilateral APCD as a whole
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/6/gallery/screen-shot-2020-03-30-at-9.39.52-pm.png\")
![\"\"](\"https://emilymath.com/media/posts/6/gallery/screen-shot-2020-03-30-at-9.22.06-pm.png\")
![\"\"](\"https://emilymath.com/media/posts/6/gallery/picture4.png\")
![\"\"](\"https://emilymath.com/media/posts/6/gallery/p2.png\")
![\"\"](\"https://emilymath.com/media/posts/6/gallery/picture3.png\")
Geometry is art!
\n\n\n\nA visual representation will open a new door for you.
\n", "author": { "name": "Emily" }, "tags": [ ], "date_published": "2021-12-28T19:26:09-05:00", "date_modified": "2021-12-28T19:26:09-05:00" }, { "id": "https://emilymath.com/explore-sides-and-interior-angles-of-a-polygon/", "url": "https://emilymath.com/explore-sides-and-interior-angles-of-a-polygon/", "title": "Explore: sides and interior angles of a polygon", "summary": "Question: Suppose a convex polygon has exactly 4 obtuse interior angles. What is the maximum number of the sides of such a polygon? Let’s use regular polygons to explore as all the regular…", "content_html": "\nQuestion: Suppose a convex polygon has exactly 4 obtuse interior angles. What is the maximum number of the sides of such a polygon?
\n\n\n\nLet’s use regular polygons to explore as all the regular polygons are convex polygons.
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/5/screen-shot-2020-03-29-at-9.00.15-pm.png\")
Observation 1:
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/5/screen-shot-2020-03-29-at-9.13.27-pm.png\")
Observation 2:
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/5/screen-shot-2020-03-29-at-9.13.50-pm.png\")
Observation 3:
\n\n\n\nConclusion about sides and interior angles:
\n\n\n\nSo far, we can conclude that to get the maximum number of sides, the interior angles should be as great as possible.
\n\n\n\n4 obtuse angles + 3 right angles = 7 angels –> 7 sides
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/5/screen-shot-2021-12-28-at-12.09.56-pm.png\")
The sum of interior angles = (n-2)∙180°
\n\n\n\nThis sum should be less than the sum under the maximum situation. But what is the maximum situation?
\n\n\n\nexactly 4 obtuse interior angles implies:
\n\n\n\nInequality: Sum < the maximum situation
\n\n\n\nThis polygon has 7 sides at most. It can be a heptagon.
\n\n\n\nA convex polygon with 4n+2 sides can be expressed as A1A2A3—A(4n+2). If each interior angle is a multiple of 30 degrees and∠𝐴1=∠𝐴2=∠𝐴3=90°, find the value of n.
\n\n\n\nCan you use the above-mentioned methods to solve this question?
\n", "author": { "name": "Emily" }, "tags": [ ], "date_published": "2021-12-28T18:51:50-05:00", "date_modified": "2021-12-28T18:51:50-05:00" }, { "id": "https://emilymath.com/build-a-math-brain-through-deliberate-training/", "url": "https://emilymath.com/build-a-math-brain-through-deliberate-training/", "title": "Build a Math Brain Through Deliberate Training", "summary": "My son could say numbers from 1 to 10 in both Chinese and English at age 3. However, numbers to him were nearly the same…", "content_html": "\nMy son could say numbers from 1 to 10 in both Chinese and English at age 3. However, numbers to him were nearly the same as Peppa Pig to him, which were merely words with distinct pronunciation. He couldn’t count the number of toy cars he owned due to a lack of sense on numbers or the numerical orders. This raised a question: if kids are not taught numbers, can they obtain the numerical skills by themselves one day?
\n\n\n\nI didn’t realize how unrealistic I was until I read the book The Story of Mathematics. The number system we take for granted today is built upon thousands of years of accumulation of human wisdom. The role of Mathematics, in particular, is always the catalyst of human civilization.
\n\n\n\nTake counting numbers as an example, there is a story. Once upon a time, there was a small kingdom in the foothills of the Youshan Mountains in China, and the prince of the kingdom was a young man surnamed Chen. One day, the prince caught a tortoise by the sea, and when he returned to the village, he said to the soldiers: “I caught a lot of tortoises today.” At that time, he proudly lifted the lid, but there were only 3 turtles in the cage. The soldiers still cheered: “Wow! There are so many turtles!” As people in this kingdom had no idea about numbers such as 3 and 4, they only divided all numbers into 3 categories. Except for 1 and 2, the others were “many”.
\n\n\n\nEven with the decimal system commonly used in the modern world, Babylon once described the numbers like this:
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/4/unnamed-file.numbers.gif\")
It is inconceivable that by how long a series of patterns must be drawn in this way to exactly describe the housing price of tens of millions of dollars nowadays in Shanghai. Even though the Arabic numerals are dominant in the world, the west and the east use quite different place values:
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/4/e4b8ade59bbde695b0e4bd8d.jpg\")
![\"\"](\"https://emilymath.com/media/posts/4/place-value.png\")
Obviously, even the most basic part of mathematics implies the powerful logical ability. Different from visual stimulation, mathematical abilities refresh human cognition through deeper thinking and reasoning. And this kind of ability, begins to show up and permeate in middle school in Shanghai, is precisly the essential difference between elementary school mathematics and middle school mathematics.
\n\n\n\nFrom Arithmetic to Mathematical thinking
\n\n\n\nSome students struggle with the transition from calculation to deep thinking in middle school mathematics. There are three main reasons:
\n\n\n\nIn elementary school, few mathematical theorems are involved in raising the level of human cognition, instead, “numbers” and “calculation” are the main abilities which can be improved through repeated mechanical training. However, in the context of Chinese exam-oriented education, if a student fails to build reflexive computing skills in elementary school, he or she will be unlikely to follow up with more complicated calculation in middle school. As a result, such students are still making up for calculation when the talented students have upgraded to analysis, reasoning, and speculation.
\n\n\n\n2. Taking notes without thinking
\n\n\n\nThese students have developed a “good” study habit in elementary school. They immerse themselves in taking notes and memorizing the steps of each question word by word in class. Their notes can always attract many other students to borrow, especially when a final exam is to come. In sharp contrast to their detailed notes, their grades were just mediocre. In middle school, this “good” study habit needs to extend from “mechanical repetition” to “analytical thinking”. Yes, it is the ability to think deeply, critically and thoroughly that distinguishes students as the grades go up. Truly, students have to step out of their comfort zone to go across the leap in advanced thinking.
\n\n\n\n3. NOT strive for perfection
\n\n\n\nUnderstanding all the concepts and theorems may be enough to achieve an outstanding score on SAT but far from adequate to succeed in the national exams in China. Some students possess strong comprehension ability and enable to put forward new ideas sometimes but their test scores are just at the average level. It is not the so called IQ that separates them from top students, rather, it is the attitude to detail every single step that widens the gap between average students and top ones. Why is that? Chinese math tests, unlike most western countries where multiple choice questions are prevalent, emphasize on detailed steps (format), geometry proof in particular, prioritizing students’ thinking process over final results. Without deliberate practicing through daily homework, it is impossible to meet the requirements of the tests. To put it another way, the quality of perseverance and the extent of details eventually form ever-increasing disparities among people.
\n\n\n\nMathematical Thinking plays a key role in Chinese math education
\n\n\n\nMalcolm Gladwell, the author of Outlier, put forward the 10,000-hour theory, revealing the accumulation of knowledge behind Bill Gates’ success. The inspirational film The King’s Speech also reveals that consistent deliberate practice is the key to success. In fact, the 10,000-hour theory is a perfect demonstration of the traditional Chinese testing. However, even though students spent the same amount of time and solved the same number of questions, their way of learning associated with efficiency eventually lead to radically different test results and overall ability. Some parents think it wrongly that the reason of children scoring higher is because they have practiced a lot more. Actually, it is the way of learning efficiently and effectively that leads to the gap among students in terms of knowledge and wisdom.
\n\n\n\nLet’s first look at the skills and abilities required by the Shanghai High School Entrance Examination (2018 Zhongkao):
Basic knowledge and basic skills 49.3%
Logical reasoning 14.7%
Calculation 10.0%
Space concept 11.3%
Problem solving ability 14.7%
Most students were in the 136-140 score segment. That is to say, in the high school entrance examination with nearly 50% basic knowledge and basic skills, it is not difficult for students to get 136 points in the test as long as they study hard. As a selective exam, what distinguishes students are these abilities: logical reasoning, spatial concepts, and problem-solving skills. Students got 140-145 have a good mastery of the above but may fail in one or two extremely hard questions whereas those above 145 to 150 not only have a “math brain” but also “extraordinary qualities”.
\n\n\n\n4 Stages of learning math
\n\n\n\nThe duration of math education for a typical university graduate majored in science or engineering in China is 16 years, beginning with 6 years in elementary school, 3 years in middle school followed by another 3 years in high school and finally 4 years in university. To get a Bachelor’s degree in science or enginering, one has to firstly outperform peer groups in the high school entrance exam and then the national college entrance exam 3 years later. On top of that, mathematics is generally considered the subject that tells students apart from different tiers of universities.
Based on my own learning experience as a reference, I would like to divide the learning process into four stages:
Linear algebra is a compulsory course in university, the most difficult one for me. On the one hand, the transformation of matrix follows no formula. On the other hand, I don’t understand the meaning of this transformation. Many years after graduation I saw the picture shown on Matrix67’s blog – the “linear pull” of the plane is transformed through matrix multiplication, I instantly hit the essence of linear algebra. Moreover, I suddenly understand what a Fourier transform means.
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In fact, many students often question why they have to learn mathematics? With students’ limited knowledge and experience, it is difficult to understand the knowledge framework, knowledge progression and knowledge extension. Questioning is valuable. When we question, we might as well explore the history of mathematics. Why did the ancestors do this and what is the story behind it? Is there a better way to deal with it? Perhaps, these amazing discoveries will arouse our interest and broaden our knowledge.
\n\n\n\n2. Training: quantity changes lead to quality changes
\n\n\n\nSome researchers doubt the importance of homework. Homework, classwork, quizzes, or any other forms of training, are absolutely necessary in math education. Is understanding of the concepts enough? Knowing the theorems to prove congruent triangles only helps students solve straightforward one step geometry proof questions as these questions don’t involve any deep logical thinking. However, a typical congruent triangle question in China is like this:
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/4/screen-shot-2021-12-26-at-8.16.11-pm.png\")
Is this question asking to prove congruent triangles? Why congruent triangles are needed to relate the three line segments? Exactly, all the conditions are given in a hidden way. Without training, most students even don’t know what the question is asking for? Honestly, this question is merely medium level in Chinese math. Only through repeated training can a student quickly get the key clue and therefore finish a test in limited amount of time.
\n\n\n\n3. Summarize: from memorization to understanding
\n\n\n\nAfter a certain amount of practices, we can calculate the GCF greatest common factor and LCM least common multiple by “short division” quickly and accurately. Let’s take a look at the two word problems.
\n\n\n\nAlice works for a florist shop. She has 96 lilies and 72 white roses. She wants to put all the flowers into bouquets with each bouquet having the same number of flowers in it. Without leftovers, what is the greatest number of flower bouquets? GCF(96, 72)=24
\n\n\n\nBrian has swimming lessons every fifth day and dancing lessons every third day. If he had a swimming lesson and a dancing lesson on July 5, when will be the next date on which he has both swimming and dancing lessons? LCM (5, 3) =15
\n\n\n\nIf students are trained to solve the two word problems by those so called exam skills – find GCF when the word greatest is in the problem, similarly, find LCM if the word least is there, students would not explore the essence themselves.
\n\n\n\nOnly when we truly understand mathematical knowledge and make our own generalizations can we turn knowledge into wisdom.
\n\n\n\n4. Comprehensive Application: Extracting Information-Integrating Information
\n\n\n\n![\"\"](\"https://emilymath.com/media/posts/4/screen-shot-2021-12-26-at-6.54.50-pm.png\")
This is the last question of the 2019 middle school math proficiency test in Shanghai (Zhongkao). The three questions cover many contents, such as angle bisector, special trigonometric ratio, similar triangle and triangle incenter. Question 3) is challengeable in two ways. Firstly, there should be two cases according to the given information. Secondly, none of the side length is given but is needed to get the area ratio of the two similar triangles. The key to find the side ratio lies in the special right triangle
What information is given?
What is unknown?
Can the unknown information be derived from the given ones?
If not, are auxiliary conditions available?
If not, can the problem be transformed into an accessible one?
Though hard, it is just a question on the test. The real world is more complicated than a test. Therefore, the skills of extracting information and integrating the known to infer the unknown will be pivotal in adulthood. Furthermore, interdisciplinary knowledge is inextricably tied together to tackle the real cases in the evolving world. This is a process that involves modern heuristics, which not only tests the mathematical thinking ability, but also requires a person’s perseverance.
There is no shortcut! Deliberate training is essential to transform knowledge into “wisdom.”
At a young age, it’s hard for us to understand the need to learn those abstract knowledge. But when we forget what we have learned, The rest is the “wisdom” that we have obtained along the way.
\n", "author": { "name": "Emily" }, "tags": [ ], "date_published": "2021-12-27T14:29:24-05:00", "date_modified": "2021-12-27T14:29:24-05:00" }, { "id": "https://emilymath.com/new-volunteer-opportunities/", "url": "https://emilymath.com/new-volunteer-opportunities/", "title": "Shanghai Middle School Math Proficiency Test", "summary": "To figure out why Chinese students top in PISA and other math competitions, let us turn to ‘Zhongkao’, which appears to be completely different from…", "content_html": "\nTo figure out why Chinese students top in PISA and other math competitions, let us turn to ‘Zhongkao’, which appears to be completely different from PISA but reveals how the math abilities required in PISA are obtained through math education. PISA’s targeted students are those 15-year-olds and meanwhile enrolled at grade 7 or higher. In the B-S-J-Z population, students at 9th grade and 10th grade count for 96.9%[1]. In the Chinese education system, 9th grade is the last year of middle school and 10th grade is the first year of high school. From middle school to high school, students need to climb the first mountain in their lifetime – Middle School Academic Proficiency Tests, locally known as ‘Zhongkao’, given once a year in the mid-June for all the registered students in Grade 9 (around 15 years old). In Shanghai, subjects such as Chinese, Mathematics, English, Physics and Chemistry, and Sports, are tested and counted toward the overall score. More subjects are added including History, Civics and Law, and Interdisciplinary Case Analysis, thus more abilities and skills are required since 2021. On one hand, this is a national graduation exam after 9-year compulsory education. On the other hand, it is also the High School Entrance Examination due to its selective essence. In reality, students are ranked according to the overall test scores of all the subjects and matched with high schools from top 1 to the last. Among all the subjects, Mathematics is widely recognized as the most challengeable one. Shanghai has attended PISA each year ever since 2009, a close look at Shanghai Middle School Mathematics Proficiency Test will walk us into math education in Shanghai, China.
\n\n\n\nThe exam structure is exactly the same each year, composed of three parts: multiple choice, fill in the blanks and student-produced response questions. Students are given 100min to finish 25 questions, with some of them containing two or three sub-questions. Refer to Table 1 for detailed composition of the question types and the associated scores. The contents covered throughout middle school math includes five aspects (Figure 1) with a focus on geometry, equations and functions. Five types of mathematical abilities are tested (Figure 2): understanding basic math concepts, doing some complex calculation without a calculator, having the sense of space, able to think and reason logically, and applying the knowledge to solve real world problems. Middle school graduates with 9 years’ compulsory education in Shanghai are supposed to have acquired these five abilities. Among the three question types, student-produced response questions account for the highest proportion, mainly composed of mixed calculation, solving equations, the application of functions in word problems, quadratic function and analysis, triangle or quadrilateral proof, and geometric questions combined with circles and triangles, covering the main contents taught in middle school. Therefore, this section will thoroughly demonstrate student-produced response questions selected from the former proficiency tests in order that readers get to know the details and requirements of math teaching in Shanghai.
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[Comment]
\n\n\n\nThe first question is straightforward with a difficulty factor of 0.85. Students can substitute the given points to the function to get the analytical expression. The second question (difficulty factor 0.70) requires students to have a sense of space, imagine a moving point and relate graphics to the quadratic function. The last question (difficulty factor 0.40) is about translation of the parabola. Students need to find out the corresponding points to sketch out the shape of the quadrilateral. Finally, students are supposed to be rigorous enough to include both cases.
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[Comment]
\n\n\n\nThe three questions cover many contents, such as angle bisector, special trigonometric ratio, similar triangle and triangle incenter. Question 3) is challengeable in two ways. Firstly, there should be two cases according to the given information. Secondly, none of the side length is given but is needed to get the area ratio of the two similar triangles. The key to find the side ratio lies in the special right triangle 30-60-90 degrees and 45-45-90 degrees. Besides, the property of the incenter is the adhesive of the sides in two similar triangles
\n\n\n\nAccording to the Municipal Education Commission, the difficulty of the test paper is originally designed into 3 levels, 80% easy, 10% medium and 10% hard. As the full mark of the test is 150 points, 80% easy means that most students can secure a score of 120. Table 2 shows the statistics of students’ performance in Math Proficiency Test in 2018 and 2019. If students get 130 (roughly the median of each year) on the Shanghai Middle School Math Proficiency Test, they should have mastered the basic concepts and skills such as solving rational equations, simplifying algebraic expressions with exponential, radical and/or rational expressions, able to solve easy to medium level functions and geometry and so on. Please note that the level of difficulty is defined by the Chinese education system. Only by finishing the test paper in 100min can one explicitly understand the distribution of difficulty.
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Here, we also need to look into the high school acceptance rate to understand the competency of the test. In the past ten years, the average high school acceptance rate is around 55% to 65% with some slight fluctuations year by year. However, only about 25% of the students can be admitted into municipal key high schools which will almost guarantee a ticket to a university three years later. To be the top 25% on the whole, students are most likely to be top 25% in math as well. The third quartile reveals that students should reach 140 or higher, losing 10 points top. To get 140 or more on the test, mastery of the basic concepts and skills are far from enough. Usually, quadratic functions and geometry proof are the main parts setting students apart from median to top. To succeed in the 10% hard questions of functions and geometry, students need to have strong
\n\n\n\nTruly, these abilities are similar to, or to some extent more challenging than, the corresponding content of IB DP higher level mathematics, or AMC 8/10, but are imposed on the 15-year-olds in Shanghai. Indeed, extraordinary abilities are necessary to stand out of such a huge population.
\n\n\n\nBibliography
\n\n\n\n[1] PISA 2018 Results (Volume I): What Students Know and Can Do. P48
\n\n\n\n[2] http://www.shmeea.edu.cn/page/03500/index.html
\n", "author": { "name": "Emily" }, "tags": [ ], "date_published": "2021-12-02T14:32:47-05:00", "date_modified": "2021-12-02T14:32:47-05:00" } ] }![](\"https://mathpowerme.wordpress.com/e16ddd2a-dcb4-409c-8be3-9faf7ea3ef3c\")
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