Month: April 2023

When President Ronald Reagan proclaimed the first National Math Awareness Week in April 1986, one of the problems he cited was that too few students were devoted to the study of math.

“Despite the increasing importance of mathematics to the progress of our economy and society, enrollment in mathematics programs has been declining at all levels of the American educational system,” Reagan wrote in his proclamation.

Nearly 40 years later, the problem that Reagan lamented during the first National Math Awareness Week—which has since evolved to become “Mathematics and Statistics Awareness Month”—not only remains but has gotten worse.

Whereas 1.63%, or about 16,000, of the nearly 1 million bachelor’s degrees awarded in the U.S. in the 1985–1986 school year went to math majors, in 2020, just 1.4%, or about 27,000, of the 1.9 million bachelor’s degrees were awarded in the field of math—a small but significant decrease in the proportion.

Post-pandemic data suggests the number of students majoring in math in the U.S. is likely to decrease in the future.

A key factor is the dramatic decline in math learning that took place during the lockdown. For instance, whereas 34% of eighth graders were proficient in math in 2019, test data shows the percentage dropped to 26% after the pandemic.

These declines will undoubtedly affect how much math U.S. students can do at the college level. For instance, in 2022, only 31% of graduating high school seniors were ready for college-level math—down from 39% in 2019.

These declines will also affect how many U.S. students are able to take advantage of the growing number of high-paying math occupations, such as data scientists and actuaries. Employment in math occupations is projected to increase by 29% in the period from 2021 to 2031.

About 30,600 math jobs are expected to open up per year from growth and replacement needs. That exceeds the 27,000 or so math graduates being produced each year—and not all math degree holders go into math fields. Shortages will also arise in several other areas, since math is a gateway to many STEM fields.

For all of these reasons and more, as a mathematician who thinks deeply about the importance of math and what it means to our world—and even to our existence as human beings—I believe this year, and probably for the foreseeable future, educators, policymakers and employers need to take Mathematics and Statistics Awareness Month more seriously than ever before.

Struggles with mastery

Subpar math achievement has been endemic in the U.S. for a long time.

Data from the National Assessment of Educational Progress shows that no more than 26% of 12th graders have been rated proficient in math since 2005.

The pandemic disproportionately affected racially and economically disadvantaged groups. During the lockdown, these groups had less access to the internet and quiet studying spaces than their peers. So securing Wi-Fi and places to study are key parts of the battle to improve math learning.

Some people believe math teaching techniques need to be revamped, as they were through the Common Core, a new set of educational standards that stressed alternative ways to solve math problems. Others want a return to more traditional methods. Advocates also argue there is a need for colleges to produce better-prepared teachers.

Other observers believe the problem lies with the “fixed mindset” many students have—where failure leads to the conviction that they can’t do math—and say the solution is to foster a “growth” mindset—by which failure spurs students to try harder.

Although all these factors are relevant, none address what in my opinion is a root cause of math underachievement: our nation’s ambivalent relationship with mathematics.

Low visibility

Many observers worry about how U.S. children fare in international rankings, even though math anxiety makes many adults in the U.S. steer clear of the subject themselves.

Mathematics is not like art or music, which people regularly enjoy all over the country by visiting museums or attending concerts. It’s true that there is a National Museum of Mathematics in New York, and some science centers in the U.S. devote exhibit space to mathematics, but these can be geographically inaccessible for many.

A 2020 study on media portrayals of math found an overall “invisibility of mathematics” in popular culture. Other findings were that math is presented as being irrelevant to the real world and of little interest to most people, while mathematicians are stereotyped to be singular geniuses or socially inept nerds, and white and male.

Math is tough and typically takes much discipline and perseverance to succeed in. It also calls for a cumulative learning approach—you need to master lessons at each level because you’re going to need them later.

While research in neuroscience shows almost everyone’s brain is equipped to take up the challenge, many students balk at putting in the effort when they don’t score well on tests. The myth that math is just about procedures and memorization can make it easier for students to give up. So can negative opinions about math ability conveyed by peers and parents, such as declarations of not being “a math person.”

A positive experience

Here’s the good news. A 2017 Pew poll found that despite the bad rap the subject gets, 58% of U.S. adults enjoyed their school math classes. It’s members of this legion who would make excellent recruits to help promote April’s math awareness. The initial charge is simple: Think of something you liked about math—a topic, a puzzle, a fun fact—and go over it with someone. It could be a child, a student, or just one of the many adults who have left school with a negative view of math.

Can something that sounds so simplistic make a difference? Based on my years of experience as a mathematician, I believe it can—if nothing else, for the person you talk to. The goal is to stimulate curiosity and convey that mathematics is much more about exhilarating ideas that inform our universe than it is about the school homework-type calculations so many dread.

Raising math awareness is a first step toward making sure people possess the basic math skills required not only for employment, but also to understand math-related issues—such as gerrymandering or climate change—well enough to be an informed and participating citizen. However, it’s not something that can be done in one month.

Given the decline in both math scores and the percentage of students studying math, it may take many years before America realizes the stronger relationship with math that President Reagan’s proclamation called for during the first National Math Awareness Week in 1986.

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

Credit of the article given to Manil Suri, The Conversation

Australian students’ performance and engagement in mathematics is an ongoing issue.

International studies show Australian students’ mean performance in maths has steadily declined since 2003. The latest Program for International Student Assessment (PISA) in 2018 showed only 10% of Australian teenagers scored in the top two levels, compared to 44% in China and 37% in Singapore.

Despite attempts to reform how we teach maths, it is unlikely students’ performance will improve if they are not engaging with their lessons.

What teachers, parents, and policymakers may not be aware of is research shows students are using “non-thinking behaviours” to avoid engaging with maths.

That is, when your child says they didn’t do anything in maths today, our research shows they’re probably right.

What are non-thinking behaviours?

There are four main non-thinking behaviours. These are:

• slacking: where there is no attempt at a task. The student may talk or do nothing
• stalling: where there is no real attempt at a task. This may involve legitimate off-task behaviours, such as sharpening a pencil
• faking: where a student pretends to do a task, but achieves nothing. This may involve legitimate on-task behaviours such as drawing pictures or writing numbers
• mimicking: this includes attempts to complete a task and can often involve completing it. It involves referring to others or previous examples.

Peter Liljedahl studied Canadian maths lessons in all years of school, over 15 years. This research found up to 80% of students exhibit non-thinking behaviours for 100% of the time in a typical hour-long lesson.

The most common behaviour was mimicking (53%), reflecting a trend of the teacher doing all the thinking, rather than the students.

It also found when students were given “now you try one” tasks (a teacher demonstrates something, then asks students to try it), the majority of students engaged in non-thinking behaviours.

Australian students are ‘non-thinking’ too

Tracey Muir conducted a smaller-scale study in 2021 with a Year ¾ class.

Some 63% of students were observed engaged in non-thinking behaviours, with slacking and stalling (54%) being the most common. These behaviours included rubbing out, sharpening pencils, and playing with counters, and were especially prevalent in unsupervised small groups.

One explanation for students slacking and stalling is teachers are doing most of the talking and directing, and not providing enough opportunities for students to think.

How can we build “thinking” maths classrooms and reduce the prevalence of non-thinking behaviours?

Here are two research-based ideas.

Form random groups

Often students are placed in groups to work through new skills or lessons. Sometimes these are arranged by the teacher or by the students themselves.

Students know why they have been placed in groups with certain individuals (even if this is not explicitly stated). Here they tend to “live down” to expectations.

If they are with their friends they also tend to distract each other.

Our studies found random groupings improved students’ willingness to collaborate, reduced social stress often caused by self-selecting groups, and increased enthusiasm for mathematics learning.

As one student told us:

“I’m starting to like maths now, and working with random people is better for me so I don’t get off track.”

Get kids to stand up

Classroom learning is often done at desks or sitting on the floor. This encourages passive behaviour and we know from physiology that standing is better than sitting. But we found groups of about three students standing together and working on a whiteboard can promote thinking behaviours. Just the physical act of standing can eliminate slacking, stalling, and faking behaviours. As one student said, “Standing helps me concentrate more because if I’m sitting down I’m just fiddling with stuff, but if I’m standing up, the only thing you can do is write and do maths. ”

The additional strategy of only allowing the student with the pen to record others’ thinking and not their own, has shown to be especially beneficial. As one teacher told us: “The people that don’t have the pen have to do the thinking […] so it’s a real group effort and they don’t have the ability to slack off as much.”

Simple changes can work

While our studies were conducted in maths classrooms, our strategies would be transferable to other discipline areas.

So, while parents and educators may feel concerned about Australia’s declining mathsresults, by introducing simple changes to the classroom, we can ensure students are not only learning and thinking deeply about mathematics, but hopefully, enjoying it, too.

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

Credit of the article given to Tracey Muir and Peter Liljedahl, The Conversation

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

Luis Caffarelli has been awarded the most prestigious prize in mathematics for his work on nonlinear partial differential equations, which have many applications in the real world.

Luis Caffarelli has won the 2023 Abel prize, unofficially called the Nobel prize for mathematics, for his work on a class of equations that describe many real-world physical systems, from melting ice to jet engines.

Caffarelli was having breakfast with his wife when he found out the news. “The breakfast was better all of a sudden,” he says. “My wife was happy, I was happy — it was an emotional moment.”

Based at the University of Texas at Austin, Caffarelli started work on partial differential equations (PDEs) in the late 1970s and has contributed to hundreds of papers since. He is known for making connections between seemingly distant mathematical concepts, such as how a theory describing the smallest possible areas that surfaces can occupy can be used to describe PDEs in extreme cases.

PDEs have been studied for hundreds of years and describe almost every sort of physical process, ranging from fluids to combustion engines to financial models. Caffarelli’s most important work concerned nonlinear PDEs, which describe complex relationships between several variables. These equations are more difficult to solve than other PDEs, and often produce solutions that don’t make sense in the physical world.

Caffarelli helped tackle these problems with regularity theory, which sets out how to deal with problematic solutions by borrowing ideas from geometry. His approach carefully elucidated the troublesome parts of the equations, solving a wide range of problems over his more than four-decade career.

“Forty years after these papers appeared, we have digested them and we know how to do some of these things more efficiently,” says Francesco Maggi at the University of Texas at Austin. “But when they appeared back in the day, in the 80s, these were alien mathematics.”

Many of the nonlinear PDEs that Caffarelli helped describe were so-called free boundary problems, which describe physical scenarios where two objects in contact share a changing surface, like ice melting into water or water seeping through a filter.

“He has used insights that combined ingenuity, and sometimes methods that are not ultra-complicated, but which are used in a manner that others could not see — and he has done that time and time again,” says Thomas Chen at the University of Texas at Austin.

These insights have also helped other researchers translate equations so that they can be solved on supercomputers. “He has been one of the most prominent people in bringing this theory to a point where it’s really useful for applications,” says Maggi.

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

*Credit for article given to Alex Wilkins*