Month: March 2020

Can three cubed numbers be added up to give 42? Until now, we didn’t know

It might not tell us the meaning of life, the universe, and everything, but mathematicians have cracked an elusive problem involving the number 42.

Since the 1950s, mathematicians have been puzzling over whether any integer – or whole number – can be represented as the sum of three cubed numbers.

Put another way: are there integers k, x, y and z such that k = x³ + y³ + z³ for each possible value of k?

Andrew Booker at Bristol University, UK, and Andrew Sutherland at the Massachusetts Institute of Technology, US, have solved the problem for the number 42, the only number less than 100 for which a solution had not been found.

Some numbers have simple solutions. The number 3, for example, can be expressed as 1³ + 1³ + 1³ and 4³ + 4³ + (-5) ³ . But solving the problem for other numbers requires vast strings of digits and, correspondingly, computing power.

The solution for 42, which Booker and Sutherland found using an algorithm, is:

42 = (-80538738812075974)³ + 80435758145817515³ + 12602123297335631³

They worked with software firm Charity Engine to run the program across more than 400,000 volunteers’ idle computers, using computing power that would otherwise be wasted. The amount of processing time is equivalent to a single computer processor running continuously for more than 50 years, says Sutherland.

Earlier this year, Booker found a sum of cubes for the number 33, which was previously the lowest unsolved example.

We know for certain that some numbers, such as 4, 5 and 13, can’t be expressed as the sum of three cubes.

The problem is still unsolved for 10 numbers less than 1000, the smallest of which is 114.

The team will next search for another solution to the number 3.

“It’s possible we’ll find it in the next few months; it’s possible it won’t be for another hundred years,” says Booker.

People interested in aiding the search can can volunteer computing power through Charity Engine, says Sutherland.

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

*Credit for article given to Donna Lu*

Planning is a key ingredient for effective teaching. But why is it so important? What is ‘good’ planning? How do we make our planning purposeful and focused? Let’s delve further into planning, and what it encompasses.

Curriculum is central to planning. It guides what we teach. The other crucial factor is to know the needs of your students. Effective planning begins with finding where these two elements meet.

The Purposeful planning podcast has been developed to complement the Explicit Teaching in Maths professional learning modules. In this podcast, Dr Emily Ross from The University of Queensland uses the analogy of planning being like a road trip. Dr Ross explains that the curriculum (the knowledge and skills taught) is the destination of this road trip. It’s where you want your students to get to by the end of the lesson or topic.

The road trip itself includes the places you see and the stops on the way to your destination, and this is likened to the teaching and learning. It’s the steps in the lesson or unit plan that enable your students to reach their destination.

Using this road trip analogy, we can ask ourselves two big questions in terms of planning: where are we going? and how will we get there?

Let’s breakdown this analogy further.

• Some people like to be very well planned and outline the detailed steps required to reach the destination.
• Some people like to make a more general plan and they outline the main signposts required to travel past to get to the destination.
• Sometimes you may need to take some detours along the way depending on the needs of your learners. Listen to your learners: Are you moving too slowly? Are you going too fast? Or are you on the wrong road?There are different ways of getting to your end destination, and understanding your students and their needs will determine the path you take.

The process of planning

Planning involves interpreting the curriculum and working out how we can support our students to learn knowledge and develop skills. So, what might ‘good’ planning look like?

Start with the learner

What do your students know about the topic? What prior knowledge do they have? This isn’t always straightforward as students in your class will bring a range of skills and knowledge to each topic, and your planning needs to reflect this.

The curriculum

Use your curriculum knowledge and understandings and know exactly what you want your students to achieve.

The steps

Break the learning into small steps. Think about the chunks of knowledge and skills the students need to learn and build upon this throughout the lesson and topic.

Learning sequences

Build authentic teaching and learning sequences to support students to learn knowledge, develop skills, and understand and apply concepts.

Learning intentions and success criteria

Planning is enhanced by including purposeful learning intentions and success criteria.

A learning intention states the goal of the lesson. What will you learn?  The success criteria outlines how the students will know they have succeeded. How do you know you have learnt it? How do you know you can now do it?

Learning intention (LI) and success criteria (SC) checklist

• Sharethe LI and SC with your students.
• Make the LI and SC explicitso that students know exactly what is expected.
• Make the LI challenging,but not too difficult. Students need to be learning new knowledge and skills, and experience success in doing so.
• Make the SC measurableso students can easily see if they have been successful or not.
• Provide feedbackto students throughout the lesson, so they know what to do next to achieve the learning intention.

Good planning is essential for quality maths teaching and learning. If you’d like to know more about planning:

• sign-up to The Maths in schools: Explicit teaching in Maths learning modules. This self-paced, professional leaning course offers five modules that are designed around the seven components of explicit teaching. The modules are aligned to the Australian Institute and School Leadership (ASITSL) professional standards, and they include lessons and activities you can use to teach maths concepts from the Australian Curriculum.
• listen to the Maths hub podcast, Episode 1: Purposeful planning. This engaging and informative podcast is hosted by Allan Dougan, the CEO of the Australian Association of Mathematics Teachers (AAMT). Allan chats to expert Dr Emily Ross about planning. The podcast provides practical ideas you can readily use as you plan your maths lessons. A highlight is the discussion on how to incorporate learning intentions and success criteria in creative and flexible ways.
• watch the Explicit teaching in mathematics: purposeful planning webinar. Associate Professor Helen Chick, University of Tasmania, discusses purposeful planning when thinking through and constructing maths lessons using the explicit teaching model and how careful planning of lessons is just as important as teaching the lesson and can enable explicit teaching to be successful.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to The Mathematics Hub

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

Mathematicians don’t need to worry about AI taking over their jobs just yet

You don’t need a human brain to do maths — even artificial intelligence can write airtight proofs of mathematical theorems.

An AI created by a team at Google has proven more than 1200 mathematical theorems. Mathematicians already knew proofs for these particular theorems, but eventually the AI could start working on more difficult problems.

One of the core pillars of maths is the concept of proof. It is an argument based on known statements, assumptions, or rules, that a certain mathematical statement, such as a theorem, is true.

To train their AI, the Google team started with a database of more than 10,000 human-written mathematical proofs, along with the reasoning behind each step known as a tactic. Tactics could include using a known property about numbers, such as the fact that multiplying x by y is the same as multiplying y by x, or applying the chain rule.

Then, they tested the AI on 3225 theorems it hadn’t seen before and it successfully proved 1253 of them. Those that it couldn’t prove were because it had only 41 tactics at its disposal.

To prove each theorem, the AI split them into smaller and smaller components using the list of tactics. Eventually each of the smaller components could be proven with a single tactic, thus proving the larger theorem.

“Most of the proofs we used are relatively short, so they don’t require a lot of long complicated reasoning, but this is a start,” says Christian Szegedy at Google. “Where we want to get to is a system that can prove all the theorems that humans can prove, and maybe even more.”

Tackling harder problems

While this particular algorithm is focused on linear algebra and complex calculus, changing its training set could allow it to do any sort of mathematics, says Szgedy. For now, the AI’s main application is filling in the details of long and arduous proofs with extreme precision.

Mathematicians often make intellectual jumps in their proofs without spelling out the exact tactics used to get from one step to the next, and provers like this could walk through the intermediate work automatically, without requiring a human mathematician to fill in each exact tactic used.

“You get the maximum of precision and correctness all really spelled out, but you don’t have to do the work of filling in the details,” says Jeremy Avigad at Carnegie Mellon University in Pennsylvania. “Maybe offloading some things that we used to do by hand frees us up for looking for new concepts and asking new questions.”

AIs like this could one day even solve maths problems we don’t know how to solve or that are too long and complicated. But that will take a much larger training set, more tactics, and a simpler way to plug the theorems into the computer. “That’s far away, but I think it could happen in our lifetime,” says Szgedy.

“Pretty much anything that you can state and try to prove mathematically, you can put into this system,” says Avigad. “You can distill just about all of mathematics down to very basic rules and assumptions, and these systems implement those rules and assumptions.”

All of this happens in a matter of seconds per proof and the only source of error is the translation of the theorem into formal language the computer can understand. Szegedy says that the team is now working on the problem of automatic translation so that it’s easier for mathematicians to interact with the system.

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

*Credit for article given to Leah Crane*