Category: Skills and Learning

Have you ever struggled with teaching statistics? Do you and your students share a sense of apprehension when data lessons appear in the scheme of work? You’re not alone. Anecdotally, many teachers tell me that statistics is one of the topics they like teaching the least, and I am no exception to this myself. In my mathematics degree, I took the minimum number of statistics-related courses allowed, following a very poor diet of data at school, and carried this negative association into my teaching. Looking back on my career in the classroom, I did not do a good job of teaching statistics, but having had the luxury of spending many years at Cambridge Mathematics immersed in research from excellent statistics teachers and education academics I now understand why!

So now, of course, the question has been posed. Why is statistics hard to teach well? In part, I believe that it stems from viewing statistics through a mathematical lens – understandably, given that we are delivering it alongside quadratic equations, Pythagoras’ theorem, fractions, decimals and percentages. But while statistical analysis would not exist without the mathematical concepts and techniques underpinning it, we have a tendency within curricula to make the mathematical techniques the whole point, and reduce the statistical analysis part to an afterthought or an added extra. Students find the more subjective analysis hard, so it is tempting to make sure everyone can manage the techniques and then focus on the interpretation as something only the most able have time to spend on (although, there is always the additional temptation to move on to other, more properly ‘maths-y’ topics as soon as possible).

This approach is at odds with how education researchers suggest students should encounter statistical ideas. In the early 1990s, George Cobbⁱ and other researchers recommended that statistics should

• emphasise statistical thinking,
• include more real data,
• encourage the exploration of genuine statistical problems, and
• reduce emphasis on calculations and techniques.

Since then, much subsequent research has refined these recommendations to account for new technology tools and new ideas, but the core principles have remained the same. In much of my reading of education research, three ways of seeing or interacting with data keep appearing:

·        Data modelling – the idea that data can be used to create models of the world in order to pose and answer questions

·        Informal inference – the idea that data can be used to make predictions about something outside of the data itself with some attempt made to describe how likely the prediction is to be true

·        Exploratory data analysis – the idea that data can be explored, manipulated and represented to identify and make visible patterns and associations that can be interpreted

In the abstract, these ways of seeing, while distinct, have a degree of overlap, and all students may benefit from multiple experiences of all three approaches to data work from their very earliest encounters with data through to advanced-level study.

Imagine the following classroom activity that could be given to very young students (e.g., in primary school). A class of students is given a list of snacks and treats and the students are asked to rank them on a scale of one to five based on how much they like each item. How could this data be worked with through each of the three approaches?

Firstly, we will consider data modelling. Students could be asked to plan a class party with a limited budget. They can buy some but not all of the items listed and must decide what they should buy so that the maximum number of students get to have things they like. In this activity, students must create a model from the data that identifies those things they should buy more of, and those things they should buy least of, along with how many of each thing they should get – perhaps considering these quantities proportionally. This activity uses the data as a model but inevitably requires some assumptions and the creation of some principles. Is the goal to ensure everyone gets the thing they like most? Or is it to minimise the inclusion of the things students like least? What if everyone gets their favourite thing except one student who gets nothing they like?

Secondly, we will think about this as an activity in informal inference. Imagine a new student is joining the class and the class wants to make a welcome pack of a few treats for this student, but they don’t know which treats the student likes. Can they use the data to decide which five items an unknown student is most likely to choose? What if they know some small details about the student; would that additional information allow them to decide based on ‘similar’ students in the class? While the second part of this activity must be handled with a degree of sensitivity, it is an excellent primer for how purchasing algorithms, which are common in online shops, work.

Finally, we turn to exploratory data analysis. In this approach students are encouraged to look for patterns in the data, perhaps by creating representations. This approach may come from asking questions – e.g., do students who like one type of chocolate snacks rate the other chocolate snacks highly too? Is a certain brand of snack popular with everyone in the class? What is the least popular snack? Alternatively, the analysis may generate questions from patterns that are spotted – e.g. why do students seem to rate a certain snack highly? What are the common characteristics of the three most popular snacks?

Each of these approaches could be engaged in as separate and isolated activities, but there is also the scope to combine them and use the results of one approach to inform another. For example, exploratory data analysis may usefully contribute both to model building and inference making and support students’ justifications for their decisions in those activities. Similarly, data modelling activities can be extended into inferential tasks very easily, simply by shifting the use of the model from the population of the data (e.g., the students in the class it was collected from) to some secondary population (e.g., another class in the school, or as in the example, a new student joining the class).

Looking back on my time in the classroom, I wish that my understanding of these approaches and their importance for developing statistical reasoning skills in my students had been better. While not made explicit as important in many curricula, there are ample opportunities to embed these approaches and make them a fundamental part of the statistics teacher’s pedagogy.

Do you currently use any of these approaches in your lessons? Can you see where you might use them in the future? And how might you adapt activities to allow your students opportunities to engage in data modelling, informal inference and exploratory data analysis?

 For more insights like this, visit our website at www.international-maths-challenge.com.

Credit for the article given to Darren Macey

The importance of mathematics in daily and professional life has been increasing with the contribution of developing technology. The level of mathematical knowledge and skills directly influence the quality standards of our individual and social life. However, mathematics the importance of which we feel in every aspect of our life is unfortunately not learned enough by many individuals for many reasons. The leading reasons regarding this issue are as follows: the abstract and hierarchical structure of mathematics, methods and strategies in learning mathematics, and the learning difficulties in mathematics. Developmental Dyscalculia (DD)/Mathematics Learning Difficulty (MLD) is a brain-based condition that negatively affects mathematics acquisition.

The mathematical performance of a student with MLD is much lower than expected for age, intelligence, and education, although there are no conditions such as intellectual disability, emotional disturbances, cultural deprivation, or lack of education. Difficulties in mathematics result from a number of cognitive and emotional factors. Math anxiety is one of the emotional factors that may severely disrupt a significant number of children and adults in learning and achievement in math.

Math anxiety is defined as “the feelings of tension and anxiety that interfere with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations”. Sherard describes math anxiety as the fear of math or an intense and negative emotional response to mathematics. There are many reasons for the cause of the math anxiety. These include lack of the appropriate mathematical background of the students, study habits of memorizing formulas, problems and applications that are not related to real life, challenging and time-limited exams, lack of concrete materials, the difficulty of some subjects in mathematics, type of personality, negative approach on mathematics, lack of confidence, the approaches, feelings, and thoughts of teachers and parents on mathematics.

The negative relationship between math anxiety and math performance is an international issue. The PISA (Programme for International Student Assessment) statistics measuring a wide variety of countries and cultures depict that the high level of negative correlation between math anxiety and mathematical performance is remarkable. Some studies showed that highly math-anxious individuals are worse than those with low mathematics anxiety in terms of solving mathematical problems. These differences are not typically observed in simple arithmetic operations such as 7 + 9 and 6 × 8, but it is more evident when more difficult arithmetic problems are tested.

Math anxiety is associated with cognitive information processing resources during arithmetic task performance in a developing brain. It is generally accepted that math anxiety negatively affects mathematical performance by distorting sources of working memory. The working memory is conceptualized as a limited source of cognitive systems responsible for the temporary storage and processing of information in momentary awareness.

The learning difficulties in mathematics relate to deficiencies in the central executive component of the working memory. Many studies suggest that individuals with learning difficulties in mathematics have a lack of working memory. It is stated that students with learning difficulties in mathematics use more inferior strategies than their peers for solving basic (4 + 3) and complex (16 + 8) addition and fall two years behind their peers while they fall a year behind in their peers’ working memory capacities.

Highly math-anxious individuals showed smaller working memory spans, especially when evaluated with a computationally based task. This reduced working memory capacity, when implemented simultaneously with a memory load task, resulting in a significant increase in the reaction time and errors. A number of studies showed that working memory capacity is a robust predictor of arithmetic problem-solving and solution strategies.

Although it is not clear to what extent math anxiety affects mathematical difficulties and how much of the experience of mathematical difficulties causes mathematical anxiety, there is considerable evidence that math anxiety affects mathematical performance that requires working memory. Figure below depicts these reciprocal relationships among math anxiety, poor math performance, and lack of working memory. The findings of the studies mentioned above, make it possible to draw this figure.

Basic numerical and mathematical skills have been crucial predictors of an individual’s vital success. When anxiety is controlled, it is seen that the mathematical performance of the students increases significantly. Hence, early identification and treatment of math anxiety is of importance. Otherwise, early anxieties can have a snowball effect and eventually lead students to avoid mathematics courses and career options for math majors. Although many studies confirm that math anxiety is present at high levels in primary school children, it is seen that the studies conducted at this level are relatively less when the literature on math anxiety is examined. In this context, this study aims to determine the dimensions of the relationship between math anxiety and mathematics achievement of third graders by their mathematics achievement levels.

Methods

The study was conducted by descriptive method. The purpose of the descriptive method is to reveal an existing situation as it is. This study aims to examine the relationship between math anxiety and mathematics achievement of third graders in primary school in terms of student achievement levels.

Participants

Researchers of mathematics learning difficulties (MLD) commonly use cutoff scores to determine which participants have MLD. These cutoff scores vary between -2 ss and -0.68 ss. Some researchers apply more restrictive cutoffs than others (e.g., performance below the 10th percentile or below the 35th percentile). The present study adopted the math achievement test to determine children with MLD based below the 10th percentile. The unit of analysis was third graders of an elementary school located in a low socioeconomic area. The study reached 288 students using math anxiety scale and math achievement test tools. The students were classified into four groups by their mathematics achievement test scores: math learning difficulties (0-10%), low achievers (11-25%), normal achievers (26-95%), and high achievers (96-100%).

Table 1. Distribution of participants by gender and groups

Data Collection Tools

Two copyrighted survey scales, consisting of 29 items were used to construct a survey questionnaire. The first scale is the Math Anxiety Scale developed by Mutlu & Söylemez for 3rd and 4th graders with a 3-factor structure of 13 items. The Cronbach’s Alpha coefficient is adopted by the study to evaluate the extent to which a measurement produces reliable results at different times. The Cronbach Alpha coefficient of the scale is .75 which confirms the reliability of and internal consistency of the study. The response set was designed in accordance with the three- point Likert scale with agree, neutral, and disagree. Of the 13 items in the scale, 5 were positive and 8 were negative. Positive items were rated as 3-2-1, while negative items were rated as 1-2-3. The highest score on the scale was 39 and the lowest on the scale was 13.

The second data collection tool adopted by this study is the math achievement test for third graders developed by Fidan (2013). It has 16 items designed in accordance with the national math curriculum. Correct responses were scored one point while wrong responses were scored zero point.

Data Analysis

The study mainly utilized five statistical analyses which are descriptive analysis, independent samples t-test, Pearson product-moment correlation analysis, linear regression and ANOVA. First, an independent samples t-test was performed to determine whether there was a significant difference between the levels of math anxiety by gender. Then, a Pearson product-moment correlation analysis was performed to determine the relationship between the math anxiety and mathematics achievement of the students. After that, a linear regression analysis was performed to predict the mathematics achievement of the participants based on their math anxiety. Finally, an ANOVA was performed to determine if there was a significant difference between the math anxiety of the groups determined in terms of mathematics achievement.

Results

The findings of the math anxiety scores by gender of the study found no significant difference between the averages [t(286)= 1.790, p< .05]. This result shows that the math anxiety levels of girls and boys are close to each other. Since there is no difference between math anxiety scores by gender, the data in the study were combined.

Table 2. Comparison of anxiety scores by gender

There was a strong and negative correlation between math anxiety and mathematics achievement with the values of r= -0.597, n= 288, and p= .00. This result indicates that the highly math-anxious students and decreases in math anxiety were correlated with increases in rating of math achievement.

A simple linear regression was calculated to predict math achievement level based on the math anxiety. A significant regression equation was found (F(1,286)= 158.691, p< .000) with an R2 of .357. Participants’ predicted math achievement is equal to 20.153 + -6.611 when math anxiety is measured in unit. Math achievement decreased -6.611 for each unit of the math anxiety.

Figure below shows the relationship between the math anxiety of the children and their mathematics achievement on a group basis. Figure 1 provides us that there is a negative correlation between mathematical performance and math anxiety. The results depict that the HA group has the lowest math anxiety score, while the MLD group has the highest math anxiety.

Table 3. Comparison of the mathematical anxiety scores of the groups

The table indicates that there is a statistically significant difference between groups as determined at the p<.05 level by one-way ANOVA (F(3,284)= 36.584, p= .000). Post hoc comparisons using the Tukey test indicated that the mean score for MLD group (M= 1.96, sd= 0.30) was significantly different than the NA group (M= 1.41, sd= 0.84) and HA group (M= 1.24, sd= 0.28). However, the MLD group (M= 1.96, sd= 0.30) did not significantly differ from the LA group (M= 1.76, sd= 0.27).

Discussion and Conclusion

Math anxiety is a problem that can adversely affect the academic success and employment prospects of children. Although the literature on math anxiety is largely focused on adults, recent studies have reported that some children begin to encounter math anxiety at the elementary school level. The findings of the study depict that the correlation level of math anxiety and math achievement is -.597 among students. In a meta-analysis study of Hembre and Ma, found that the level of relationship between mathematical success and math anxiety is -.34 and -.27, respectively. In a similar meta-analysis study performed in Turkey, the correlation coefficient was found to be -.44. The different occurrence of the coefficients is probably dependent on the scales used and the sample variety.

The participants of the study were classified into four groups: math learning difficulties (0-10%), low success (11-25%), normal (26-95%), and high success (96-100%) by the mathematics achievement test scores. The study compared the math anxiety scores of the groups and found no significant difference between the mean scores of the math anxiety of the lower two groups (mean of MLD math anxiety, .196; mean of LA math anxiety .177) as it was between the upper two groups (mean of NA math anxiety, .142; mean of HA math anxiety .125). This indicates that the math anxiety level of the students with learning difficulties in math does not differ from the low math students. However, a significant difference was found between the mean scores of math anxiety of the low successful and the normal group.

It may be better for some students to maintain moderate levels of math anxiety to make their learning and testing materials moderately challenging, but it can be clearly said that high math anxiety has detrimental effects on the mathematical performance of the individuals. Especially for students with learning difficulties in math, the high level of math anxiety will lead to destructive effects in many dimensions, primarily a lack of working memory.

Many of the techniques employed to reduce or eliminate the link between math anxiety and poor math performance involve addressing the anxiety rather than training math itself. Some methods for reducing math anxiety can be used in teaching mathematics. For instance, effective instruction for struggling mathematics learners includes instructional explicitness, a strong conceptual basis, cumulative review and practice, and motivators to help maintain student interest and engagement.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit for the article given to Yılmaz Mutlu

Most middle and high school math curricula follow a well-defined path:

Pre-algebra → Algebra 1 → Geometry → Algebra 2 → Trig / Precalculus → Calculus

Other middle and high schools prefer an “integrated” curriculum, wherein elements of algebra,

geometry and trigonometry are mixed together over a 3-year or 4-year sequence. However, both of these approaches generally lack a great deal of emphasis on discrete math: topics such as combinatorics, probability, number theory, set theory, logic, algorithms, and graph theory. Because discrete math does not figure prominently in most states’ middle or high school “high-stakes” progress exams, and because it also does not figure prominently on college admissions exams such as the SAT, it is often overlooked.

However, discrete math has become increasingly important in recent years, for a number of reasons:

Discrete math is essential to college-level mathematics and beyond.

Discrete math—together with calculus and abstract algebra—is one of the core components of mathematics at the undergraduate level. Students who learn a significant quantity of discrete math before entering college will be at a significant advantage when taking undergraduate-level math courses.

Discrete math is the mathematics of computing.

The mathematics of modern computer science is built almost entirely on discrete math, in particular combinatorics and graph theory. This means that in order to learn the fundamental algorithms used by computer programmers, students will need a solid background in these subjects. Indeed, at most universities, an undergraduate-level course in discrete mathematics is a required part of pursuing a computer science degree.

Discrete math is very much “real-world” mathematics.

Many students’ complaints about traditional high school math—algebra, geometry, trigonometry, and the like—are, “What good is this for?” The somewhat abstract nature of these subjects often turns off students. By contrast, discrete math, in particular counting and probability, allows students—even at the middle school level—to very quickly explore non-trivial “real world” problems that are challenging and interesting.

Discrete math shows up in most middle and high school math contests.

Prominent math competitions such as the International Maths Olympiad feature discrete math questions as a significant portion of their contests. On harder high school contests, such as the AIME, the quantity of discrete math is even larger. Students who do not have a discrete math background will be at a significant disadvantage in these contests. In fact, one prominent Math coach tells us that he spends nearly 50% of his preparation time with his students covering counting and probability topics, because of their importance in global contests.

Discrete math teaches mathematical reasoning and proof techniques.

Algebra is often taught as a series of formulas and algorithms for students to memorize (for example, the quadratic formula, solving systems of linear equations by substitution, etc.), and geometry is often taught as a series of “definition-theorem-proof” exercises that are often done by rote (for example, the infamous “two-column proof”). While undoubtedly the subject matter being taught is important, the material (at least at the introductory level) does not lend itself to a great deal of creative mathematical thinking. By contrast, with discrete mathematics, students will be thinking flexibly and creatively right out of the box. There are relatively few formulas to memorize; rather, there are a number of fundamental concepts to be mastered and applied in many different ways.

Discrete math is fun.

Many students, especially bright and motivated students, find algebra, geometry, and even calculus dull and uninspiring. Rarely is this the case with most discrete math topics. When we ask students what their favourite topic is, most respond either “combinatorics” or “number theory.” (When we ask them what their least favourite topic is, the overwhelming response is “geometry.”) Simply put, most students find discrete math more fun than algebra or geometry.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit for the article given to David Patrick

Sudoku, the classic number puzzle, has captured the minds of puzzle enthusiasts around the globe. Originating from Japan, the name Sudoku translates to “single number,” reflecting the puzzle’s core principle: each number should appear only once in each row, column, and grid. This article will introduce you to the basics of Sudoku, walk you through some examples, and offer tips to enhance your solving skills.

What is Sudoku?

Sudoku is a logic-based puzzle game typically played on a 9×9 grid divided into nine 3×3 subgrids. The objective is to fill the grid with numbers from 1 to 9, ensuring that each row, each column, and each 3×3 subgrid contains all the numbers from 1 to 9 without repetition.

Basic Rules of Sudoku:

1. Each row must contain the numbers 1 to 9, without repetition.
2. Each column must contain the numbers 1 to 9, without repetition.
3. Each 3×3 subgrid must contain the numbers 1 to 9, without repetition.

How to Solve a Sudoku Puzzle

Step-by-Step Example

Let’s walk through a simple Sudoku puzzle to understand the solving process.

Initial Puzzle:

5 3 _ | _ 7 _ | _ _ _

6 _ _ | 1 9 5 | _ _ _

_ 9 8 | _ _ _ | _ 6 _

——+——-+——

8 _ _ | _ 6 _ | _ _ 3

4 _ _ | 8 _ 3 | _ _ 1

7 _ _ | _ 2 _ | _ _ 6

——+——-+——

_ 6 _ | _ _ _ | 2 8 _

_ _ _ | 4 1 9 | _ _ 5

_ _ _ | _ 8 _ | _ 7 9

Step 1: Start with the Easy Ones

Begin by looking for rows, columns, or subgrids where only one number is missing. For instance, in the first row, the missing numbers are 1, 2, 4, 6, 8, and 9. However, given the other numbers in the row and subgrid, you can often narrow down the possibilities.

Step 2: Use the Process of Elimination

In the first 3×3 subgrid (top left), we are missing the numbers 1, 2, 4, 6, 7. By checking the rows and columns intersecting the empty cells, we can often deduce which numbers go where.

Example Fill-In:

• For the empty cell in the first row, third column, the possible numbers are 1, 2, 4, 6, 8, 9. However, since 6 and 9 are in the same column and 8 is in the same row, the number for this cell must be 2.

Step 3: Repeat the Process

Continue using the process of elimination and logical deduction for the remaining cells. Let’s fill in a few more:

• For the second row, second column, the missing numbers are 2, 3, 4, 7, 8. Since 3 and 8 are already in the same subgrid, we need to see which other numbers fit based on the column and row.

Intermediate Puzzle:

5 3 2 | _ 7 _ | _ _ _

6 _ _ | 1 9 5 | _ _ _

_ 9 8 | _ _ _ | _ 6 _

——+——-+——

8 _ _ | _ 6 _ | _ _ 3

4 _ _ | 8 _ 3 | _ _ 1

7 _ _ | _ 2 _ | _ _ 6

——+——-+——

_ 6 _ | _ _ _ | 2 8 _

_ _ _ | 4 1 9 | _ _ 5

_ _ _ | _ 8 _ | _ 7 9

Step 4: Solve the Puzzle

By continuing to apply these methods, you will gradually fill in the entire grid. Here’s the completed puzzle for reference:

5 3 4 | 6 7 8 | 9 1 2

6 7 2 | 1 9 5 | 3 4 8

1 9 8 | 3 4 2 | 5 6 7

——+——-+——

8 5 9 | 7 6 1 | 4 2 3

4 2 6 | 8 5 3 | 7 9 1

7 1 3 | 9 2 4 | 8 5 6

——+——-+——

9 6 1 | 5 3 7 | 2 8 4

2 8 7 | 4 1 9 | 6 3 5

3 4 5 | 2 8 6 | 1 7 9

Tips for Solving Sudoku

1. Start with the obvious: Fill in the easy cells first to gain momentum.
2. Use pencil marks: Write possible numbers in cells to keep track of your thoughts.
3. Look for patterns: Familiarize yourself with common patterns and techniques, such as naked pairs, hidden pairs, and X-Wing.
4. Stay organized: Work methodically through rows, columns, and sub grids.
5. Practice regularly: The more you practice, the better you’ll get at spotting solutions quickly.

Conclusion

Sudoku is a fantastic way to challenge your brain, improve your logical thinking, and enjoy a bit of quiet time. Whether you’re a beginner or an experienced solver, the satisfaction of completing a Sudoku puzzle is a reward in itself. So, grab a puzzle, follow these steps, and immerse yourself in the fascinating world of Sudoku!

Three factors—the needs of the subject, the child, and the society—have influenced what mathematics is to be taught in schools. Many people think that “math is math” and never changes. In this three-part series, we briefly discuss these three factors and paint a different picture: mathematics is a subject that is ever-changing.

In this third and final part, we discuss the-

Needs of the Society

The usefulness of mathematics in everyday life and in many vocations has also affected what is taught and when it is taught. In early America, mathematics was considered necessary primarily for clerks and bookkeepers. The curriculum was limited to counting, the simpler procedures for addition, subtraction, and multiplication, and some facts about measures and fractions. By the late nineteenth century, business and commerce had advanced to the point. that mathematics was considered important for everyone. The arithmetic curriculum expanded to include such topics as percentages, ratios and proportions, powers, roots, and series.

This emphasis on social utility, on teaching what was needed for use in occupations, continued into the twentieth century. One of the most vocal advocates of social utility was Guy Wilson. He and his students conducted numerous surveys to determine what arithmetic was actually used by carpenters, shopkeepers, and other workers. He believed that the dominating aim of the school mathematics program should be to teach those skills and only those skills.

In the 1950s, the outburst of public concern over the “space race” resulted in a wave of research and development in mathematics curricula. Much of this effort was focused on teaching the mathematically talented student. By the mid-1960s, however, concern was also being expressed for the disadvantaged student as U.S. society renewed its commitment to equality of opportunity. With each of these changes, more and better mathematical achievement was promised.

In the 1970s, when it became apparent that the promise of greater achievement had not fully materialised, another swing in curriculum development occurred. Emphasis was again placed on the skills needed for success in the real world. The minimal competency movement stressed the basics. As embodied in sets of objectives and in tests, the basics were considered to be primarily addition, subtraction, multiplication, and division with whole numbers and fractions. Thus, the skills needed in colonial times were again being considered by many to be the sole necessities, even though children were now living in a world with calculators, computers, and other features of a much more technological society.

By the 1980s, it was acknowledged that no one knew exactly what skills were needed for the future but that everyone needed to be able to solve problems. The emphasis on problem-solving matured through the last 20 years of the century to the point where problem-solving was not seen as a separate topic but as a way to learn and use mathematics.

Today, one need of our society is for a workforce that is competitive in the world. There is a call for school mathematics to ensure that students are ready for workforce training programs or college.

Conclusion

International Mathematics Olympiad play a vital role in shaping the intellectual and analytical landscape of society. They not only foster critical thinking, problem-solving skills, and creativity among students but also prepare them to tackle complex real-world issues. By encouraging young minds to engage with challenging mathematical concepts, Olympiads help cultivate a future generation of scientists, engineers, economists, and leaders who can drive innovation and progress. Moreover, the collaborative and competitive nature of these competitions promotes a culture of academic excellence and perseverance.

As we face increasingly complex global challenges, the importance of nurturing a strong foundation in mathematics through Olympiads cannot be overstated. They are not just competitions; they are essential platforms for equipping society with the tools and mindset needed to build a better, more informed, and innovative world.

 

 

Three factors—the needs of the subject, the child, and the society—have influenced what mathematics is to be taught in schools. Many people think that “math is math” and never changes. This three-part series briefly discusses these three factors and paints a different picture: mathematics is an ever-changing subject.

We have already discussed the Needs of the Subject in the previous blog. In this second part, we discuss-

Needs of the Child

The mathematics curriculum has been influenced by beliefs and knowledge about how children learn and, ultimately, about how they should be taught. Before the early years of the twentieth century, mathematics was taught to train “mental faculties” or provide “mental discipline.” Struggling with mathematical procedures was thought to exercise the mind (like muscles are exercised), helping children’s brains work more effectively. Around the turn of the twentieth century, “mental discipline” was replaced by connectionism, the belief that learning established bonds, or connections, between a stimulus and responses. This led teachers to the endless use of drills aimed at establishing important mathematical connections.

In the 1920s, the Progressive movement advocated incidental learning, reflecting the belief that children would learn as much arithmetic as they needed and would learn it better if it was not systematically taught. The teacher’s role was to take advantage of situations when they occurred naturally as well as to create situations in which arithmetic would arise.

During the late 1920s, the Committee of Seven, a committee of school superintendents and principals from midwestern cities, surveyed pupils to find out when they mastered various topics. Based on that survey, the committee recommended teaching mathematics topics according to students’ mental age. For example, subtraction facts under 10 were to be taught to children with a mental age of 6 years 7 months and facts over 10 at 7 years 8 months; subtraction with borrowing or carrying was to be taught at 8 years 9 months. The recommendations of the Committee of Seven had a strong impact on the sequencing of the curriculum for years afterward.

Another change in thinking occurred in the mid-1930s, under the influence of field theory, or Gestalt theory. A 1954 article by William A. Brownell (2006), a prominent mathematics education researcher, showed the benefits of encouraging insight and the understanding of relationships, structures, patterns, interpretations, and principles. His research contributed to an increased focus on learning as a process that led to meaning and understanding. The value of drill was acknowledged, but it was given less importance than understanding; drill was no longer the major means of providing instruction.

The relative importance of drill and understanding is still debated today. In this debate, people often treat understanding and learning skills as if they are opposites, but this is not the case. The drill is necessary to build speed and accuracy and to make skills automatic. But equally clearly, you need to know why as well as how. Both skills and understanding must be developed, and they can be developed together with the help of International Maths Challenge sample questions.

Changes in the field of psychology have continued to affect education. During the second half of the twentieth century, educators came to understand that the developmental level of the child is a major factor in determining the sequence of the curriculum. Topics cannot be taught until children are developmentally ready to learn them. Or, from another point of view, topics must be taught in such a way that children at a given developmental level are ready to learn them.

Research has provided increasing evidence that children construct their own knowledge. In so doing, they make sense of the mathematics and feel that they can tackle new problems. Thus, helping children learn mathematics means being aware of how children have constructed mathematics from their experiences both in and out of school.

End Note

As we have explored, a child’s journey through mathematics is deeply intertwined with their cognitive development, critical thinking skills, and overall academic success. By addressing their individual needs, providing appropriate support, and fostering a positive learning environment, we lay the foundation for a lifelong appreciation and understanding of mathematics. But what about the broader context? How does mathematics serve society at large, and what influences has it made in history? In our next blog, we will delve into these questions, examining the societal needs in mathematics and its profound impact on the course of human history.

Three factors—the needs of the subject, the child, and the society—have influenced what mathematics is to be taught in schools. Many people think that “math is math” and never changes. This three-part series briefly discusses these three factors and paints a different picture: mathematics is an ever-changing subject.

In the first part, we discuss-

Needs of the Subject

The nature of mathematics helps determine what is taught and when it is taught in elementary grades. For example, number work begins with whole numbers, then fractions and decimals. Length is studied before area. Such seemingly natural sequences are the result of long years of curricular evolution. This process has involved much analysis of what constitutes a progression from easy to difficult, based in part on what is deemed necessary at one level to develop ideas at later levels. Once a curriculum is in place for a long time, however, people tend to consider it the only proper sequence. Thus, omitting a topic or changing the sequence of issues often involves a struggle for acceptance. However, research shows that all students do not always learn in the sequence that has been ingrained in our curriculum.

Sometimes, the process of change is the result of an event, such as when the Soviet Union sent the first Sputnik into orbit. The shock of this evidence of another country’s technological superiority sped curriculum change in the United States. The “new math” of the 1950s and 1960s was the result, and millions of dollars were channeled into mathematics and science education to strengthen school programs. Mathematicians became integrally involved. Because of their interests and the perceived weaknesses of previous curricula, they developed curricula based on the needs of the subject. The emphasis shifted from social usefulness to such unifying themes as the structure of mathematics, operations and their inverses, systems of notation, properties of numbers, and set language. New content was added at the elementary school level, and other topics were introduced at earlier grade levels.

Mathematics continues to change; new mathematics is created, and new uses of mathematics are discovered. As part of this change, technology has made some mathematics obsolete and has opened the door for other mathematics to be accessible to students. Think about all the mathematics you learned in elementary school. How much of this can be done on a simple calculator? What mathematics is now important because of the technology available today?

As mathematical research progresses and new theories and applications emerge, the curriculum must adapt to incorporate these advancements. For example, the development of computer science has introduced concepts such as algorithms and computational thinking into mathematics education. These topics were not traditionally part of the elementary curriculum but have become essential due to their relevance in today’s technology-driven world. Additionally, as interdisciplinary fields like data science and quantitative biology grow, there is a pressing need to equip students with skills in statistics, probability, and data analysis from an early age, and here, the International Maths Challenge is playing a crucial role. This integration ensures that students are prepared for future academic and career opportunities that increasingly rely on mathematical literacy. Furthermore, globalization and the interconnected nature of modern societies require students to understand complex systems and patterns, necessitating the introduction of topics such as systems theory and network analysis. Consequently, the curriculum evolves not only to preserve the integrity and progression of mathematical concepts but also to reflect the dynamic and ever-expanding landscape of mathematical applications in the real world.

End Note

In the next blog, we will move towards the second factor: understanding the need for mathematics in a child’s education can set a foundation for problem-solving, logical thinking, and even everyday decision-making. In the following blog, we will delve into why mathematics is not just a subject but a vital tool for a child’s overall development and future success. Stay tuned for further updates!

A pair of mathematicians studied the UK National Lottery and figured out a combination of 27 tickets that guarantees you will always win, but they tell New Scientist they don’t bother to play.

David Cushing and David Stewart calculate a winning solution

Earlier this year, two mathematicians revealed that it is possible to guarantee a win on the UK national lottery by buying just 27 tickets, despite there being 45,057,474 possible draw combinations. The pair were shocked to see their findings make headlines around the world and inspire numerous people to play these 27 tickets – with mixed results – and say they don’t bother to play themselves.

David Cushing and David Stewart at the University of Manchester, UK, used a mathematical field called finite geometry to prove that particular sets of 27 tickets would guarantee a win.

They placed each of the lottery numbers from 1 to 59 in pairs or triplets on a point within one of five geometrical shapes, then used these to generate lottery tickets based on the lines within the shapes. The five shapes offer 27 such lines, meaning that 27 tickets will cover every possible winning combination of two numbers, the minimum needed to win a prize. Each ticket costs £2.

It was an elegant and intuitive solution to a tricky problem, but also an irresistible headline that attracted newspapers, radio stations and television channels from around the world. And it also led many people to chance their luck – despite the researchers always pointing out that it was, statistically speaking, a very good way to lose money, as the winnings were in no way guaranteed to even cover the cost of the tickets.

Cushing says he has received numerous emails since the paper was released from people who cheerily announce that they have won tiny amounts, like two free lucky dips – essentially another free go on the lottery. “They were very happy to tell me how much they’d lost basically,” he says.

The pair did calculate that their method would have won them £1810 if they had played on one night during the writing of their research paper – 21 June. Both Cushing and Stewart had decided not to play the numbers themselves that night, but they have since found that a member of their research group “went rogue” and bought the right tickets – putting himself £1756 in profit.

“He said what convinced him to definitely put them on was that it was summer solstice. He said he had this feeling,” says Cushing, shaking his head as he speaks. “He’s a professional statistician. He is incredibly lucky with it; he claims he once found a lottery ticket in the street and it won £10.”

Cushing and Stewart say that while their winning colleague – who would prefer to remain nameless – has not even bought them lunch as a thank you for their efforts, he has continued to play the 27 lottery tickets. However, he now randomly permutes the tickets to alternative 27-ticket, guaranteed-win sets in case others have also been inspired by the set that was made public. Avoiding that set could avert a situation where a future jackpot win would be shared with dozens or even hundreds of mathematically-inclined players.

Stewart says there is no way to know how many people are doing the same because Camelot, which runs the lottery, doesn’t release that information. “If the jackpot comes up and it happens to match exactly one of the [set of] tickets and it gets split a thousand ways, that will be some indication,” he says.

Nonetheless, Cushing says that he no longer has any interest in playing the 27 tickets. “I came to the conclusion that whenever we were involved, they didn’t make any money, and then they made money when we decided not to put them on. That’s not very mathematical, but it seemed to be what was happening,” he says.

And Stewart is keen to stress that mathematics, no matter how neat a proof, can never make the UK lottery a wise investment. “If every single man, woman and child in the UK bought a separate ticket, we’d only have a quarter chance of someone winning the jackpot,” he says.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Matthew Sparkes*

The International Mathematical Olympiad is an annual mathematics competition for primary and high school students. The first IMO was held in 1959 in Romania, and since then, it has become the most prestigious international mathematics competition for high school students. The competition involves solving a series of challenging mathematical problems over two days. Each participating country sends a team of up to six students, who compete individually and as a team.

The problems in the IMO require students to demonstrate their problem-solving skills and mathematical creativity, often involving advanced topics in algebra, geometry, number theory, and combinatorics. The IMO aims to encourage and inspire young students to develop their mathematical skills and pursue careers in mathematics and related fields.

Preparing for the International Mathematical Olympiad (IMO) is a significant undertaking and requires a lot of hard work and dedication. Here are some essential tips to help you prepare for the Maths Olympiad:

Master the Basics

You need to have a strong foundation in mathematics to excel in the IMO. Make sure you have a good grasp of the fundamentals, including algebra, geometry, number theory, and combinatorics.

Practice, Practice, Practice

The key to success in the IMO is practice. Work through as many problems as you can and try to solve them using different methods. You can find plenty of practice problems in math books, online resources, and previous IMO papers.

Join a Study Group

Joining a study group is an excellent way to exchange ideas and learn from others. It can also help you stay motivated and focused. You can find study groups online or through your school or local math club.

Attend a Math Camp

Math camps are intensive programs that offer specialized training for math competitions like the IMO. They can provide you with the opportunity to work with experienced coaches and other talented students.

Stay Up-to-Date

Keep yourself updated with the latest news and information about the International Maths Olympiad. Check out the official website and other math resources for updates, past papers, and other relevant information.

Learn from Your Mistakes

Analyze your mistakes and learn from them. Understanding where you went wrong can help you avoid making the same mistake in the future.

Stay Calm and Confident

The IMO is a challenging competition, but it’s essential to stay calm and confident. Believe in your abilities and trust your preparation.

Remember that preparing for the International Maths Olympiad requires patience, perseverance, and hard work. Be consistent in your preparation, and with the right mindset and dedication, you can achieve great success.

Mathematics is an essential part of our daily lives, and it is crucial for children to develop strong math skills from a young age. Here are some reasons why math is important in kids’ daily lives:

Problem-solving skills: Mathematics teaches children how to solve problems, both in math-related situations and in real-life situations. The logical and analytical skills they develop through math help them find solutions to problems and make informed decisions.

Money management: Math skills are essential for managing finances. Children need to learn how to add, subtract, multiply, and divide money to manage their allowances and understand the value of different amounts.

Time management: Math skills also play a critical role in time management. Children need to be able to tell time, calculate elapsed time, and understand the concept of time zones to manage their schedules and keep appointments.

Measurements: Measurements are everywhere, from cooking to construction. Math skills are necessary for children to understand the different units of measurement and use them in everyday situations.

Technology: Math is essential for understanding and using technology. Programming, robotics, and computer science are all based on math concepts. Register for the International Maths Olympiad Challenge to improve your kid’s skill and thinking level.

Academic and career success: Strong math skills are essential for success in academic and career fields such as engineering, science, finance, and technology. Building a strong foundation in math from a young age can set children up for future success.

In summary, math is an essential subject that plays a crucial role in students’ academic and personal development. It helps students develop problem-solving and critical thinking skills, enhances their quantitative abilities, improves decision-making abilities, advances career opportunities, and improves overall academic performance.