# data[].attributes.body
1. What is meant by the term 'proof' in mathematics, and how is this similar to, or different from what is meant by this term in other areas of knowledge?

This video shows that "proof" in mathmatics can be different than in other areas of knowledge as you need to both reasoning and practical use of the math for it to be confirmed true.

# data[].attributes.attachment_link.title
The Scientific Way to Cut a Cake - Numberphile

# data[].attributes.attachment_link.description
SUPPORT US ON PATREON: http://www.patreon.com/numberphile Alex Bellos on cutting a cake using scientific principles.

# data[].attributes.body
1. What is meant by the term 'proof' in mathematics, and how is this similar to, or different from what is meant by this term in other areas of knowledge?

# data[].attributes.attachment_link.readable_text
Featuring Tom Crawford.

Check out opportunities with Jane Street at https://www.janestreet.com/join-jane-street/ (episode sponsor)

Tom discusses a series that, by one definition, has an unexpected property.

Tom Crawford's website, with links to his work and other outreach: https://tomrocksmaths.com

More Tom videos on Numberphile: http://bit.ly/Crawford_Videos

Tom on the Numberphile Podcast: https://youtu.be/_HwKGncsGo0

# data[].attributes.attachment_link.title
The Distance Between Numbers - Numberphile

# data[].attributes.attachment_link.description
Featuring Tom Crawford. Check out opportunities with Jane Street at https://www.janestreet.com/join-jane-street/ (episode sponsor) Tom discusses a series that, by one definition, has an unexpected property. Tom Crawford's website, with links to his work and other outreach: https://tomrocksmaths.com More Tom videos on Numberphile: http://bit.ly/Crawford_Videos Tom on the Numberphile Podcast: https://youtu.be/_HwKGncsGo0

# data[].attributes.body
"What is meant by the term 'proof' in mathematics, and how is this similar to, or different from what is meant by this term in other areas of knowledge?"
This article relates to using "proof" as they have used their previous knowledge in order to come to a conclusion, and be able to reuse their previous data in order to predict water flows in forest, in a manner that can be repeated and proven. Although math can be properly used here, there are still many manners that hydrologists cannot measure yet, so they must keep on using mathematics in order to come to a conclusion.

# data[].attributes.attachment_link.readable_text
Abstract

Water flows through forests in many ways, so it is difficult to understand and predict where and when it will flow. Understanding how water flows through forests is important, as it affects many of the services that forests offer to people, like lumber for houses and cleaner air. Water scientists (called hydrologists) have a way to reduce the complex water flow of a forest into something we can explain with simple math: they turn forests into buckets! In this article, we describe a forest "water park," where leafy and bark buckets fill and empty into each other or onto the forest floor. These buckets help us describe and predict water flows in forests using simple arithmetic and statistics. We focus on the flow of rainwater as it drains through the canopy to the ground, contributing water needed for tree growth and filling the groundwater that we drink.

Water Scientists and Their Love of Buckets

Water is an important part of nature, our societies, and our individual lives. It is cycling all the time-from land to the clouds; returning as precipitation; filling rivers, lakes, and seas; recharging groundwater aquifers; forming surface runoff; wetting fields and forests; and watering plants and trees. This movement of water is called the water cycle and it is studied by scientists called hydrologists. Hydrologists observe the various water processes in nature, trying to understand and predict them. Some water flows are hard to follow-for example, it is hard to watch liquid water become an invisible gas through evaporation. This makes the water cycle difficult to understand and even more difficult to explain or predict. To think about these complex processes, hydrologists imagine that all these processes act as one big water park full of interconnected buckets. Some buckets are being filled with rainfall, others with snow, and some are already full of water, which starts to overflow.

This mental "model" of nature may sound crazy, but it is actually quite useful in helping us understand and manage water. How? Well, let us begin by admitting something obvious: nature is very complex, so it can be hard to understand and predict. Even when we focus on one important piece of the water cycle-like, "how does rain recharge the soil in a forest?"-a lot of complexity remains []. For example, to reach the soil, rainwater must pass through the forest's canopy, which has leaves and branches that stick out in all directions and angles ( Figure 1A). At any point during a storm, rainwater could be stuck or moving or splashing anywhere throughout that forest canopy! How can we keep track of all the water-at all times during the storm and at all places in the canopy-to understand how much rain reaches the soil ( Figure 1B)? We cannot, at least not without enormous expense and difficulty. But hydrologists using the "water park" approach investigate this complex process using a scientific technique called reductionism, which is a way to describe something complex in a simple way. All those branches and leaves can be thought of as buckets ( Figure 1C)! No matter where the rainwater is sitting in the forest's canopy, it is now simply sitting in a bucket. When a canopy bucket fills up, the extra rainwater empties out onto the soil below. Another benefit of this reduction (turning a canopy into a bucket) is that we can now use math to describe the evaporation (E) process, called rainfall interception, with some simple arithmetic:

Figure 1 - (A) A complex forest canopy.

(B) In real life, the flow of rainwater through the canopy to the ground is a messy process. Blue arrows show rainfall and red arrows are evaporation. (C) Using reductionism, hydrologists can create a simple bucket model to make water flow in the forest easier to understand. This allows us to use simple math to describe the process of rainfall interception as the filling and emptying of the bucket.

In this equation, evaporation from rainfall interception (E) is really hard to measure (in fact, hydrologists still have not figured out how to measure this consistently []), but each of the letters on the right side of the equation represent something we can measure directly. If we measure the rain (R) falling in an open area next to trees, the rain that drips to the ground beneath the trees as throughfall (R T), and the rain that drains down the tree trunks as (R S), then we can estimate E.

Poking Holes in Hydrologists' Buckets

Most of a hydrologists' buckets have a big hole at the top, so that storms can fill the buckets and evaporation can empty them. This is the case for a canopy bucket-the rain fills it up, while some water is constantly evaporating back into the atmosphere. But, to make the canopy bucket reduction more useful, we must poke two holes in it: one for throughfall and another for stemflow. These holes need to change based on how much water enters the canopy bucket over time. And all canopy buckets are not the same because there are many different kinds of trees.

Let us say that the bucket model is being used by hydrologists to understand and predict stemflow. The amount and timing of stemflow mostly depend on how much water the tree bucket can hold, how well the tree can drain rainwater to the trunk, and how the storm delivers the rainfall. All three of these categories are described by what are called , describing characteristics or values that can be measured or counted. For example, some trees have flat leaves, others have needles, and the leaves of some trees fall off during the winter. Trees can have a lot of trunks and branches or only one main trunk, and some trees have smooth bark while the others have thick, rough bark. All of these are different variables/tree properties that determine the size of the bucket and how well the canopy will drain rainwater to that bucket ( Figure 2). That is a lot of properties, so let us reduce them all to a single number that will adjust the rainfall amount by the portion of rainfall that is captured by the canopy and drained to the trunk. Hydrologists call this the trunk drainage coefficient (p t), and it can range from 0 to 100%! Thus, the bucket that overflows to make stemflow only receives the rainfall (R) times this coefficient, and minus the evaporation. Since the amount of water in the bucket (C) can sometimes be less than the total capacity of the bucket (S), we can adjust the evaporation rate []. Using math, we can write it like this:

Figure 2 - Different types of trees have different canopy properties that affect the flow of rainwater to the ground.

(A) This tree has a smaller water storage bucket that drains greater amounts of rainwater to the trunk, compared to the tree in (B), which has a larger water storage bucket that poorly drains rainwater to the trunk.

To estimate the evaporation rate, we need also information about the weather. Water will run down the trunk differently during a short, intense summer rainstorm, a long winter rainstorm, or a gentle spring drizzle. It is as if different holes are poked in the bucket depending on how much rain falls during the storm, how long it falls for, how much wind is blowing, whether it is warm or cold, and whether raindrops are large, small, numerous, or scattered. So, what is a useful way to deal with all these interacting conditions?

Statistics: Crafting Buckets for all Weather and Trees

If we are not careful, all of these interacting forest and weather variables can turn our simple bucket water park back into a complex mess. Then, the equations we described above will seem too simple to be useful. But this does not happen because hydrologists pair these reductions with other mathematical methods, called statistics. This field of math helps us to make sense of large amounts of data, including its collection, organization, and analysis.

For example, hydrologists will often look to see if stemflow from a certain tree has a relationship with one weather variable, like the amount of rainfall. This relationship, called a correlation, can help us calculate the size of the bucket, how well the canopy drains rainwater to the trunk, and even how much evaporation occurs on the bark! Figure 3A shows an example tree that required 10 mm of rainfall to fill the canopy bucket and to get the bark wet enough to start stemflow flowing. Once stemflow began, the tree canopy could drain 1 mm of every additional 10 mm of rain to the trunk as stemflow. So, we now have the bucket size (10 mm) and the stem drainage coefficient (1/10 = 10%) for this tree!

Figure 3 - Statistical analysis of stemflow data for an example tree.

(A) A simple correlation plot between rainfall and stemflow that shows how a bucket can be sized (the lowest point on the horizontal axis) and the drainage coefficient estimated (blue and red arrows). (B) A simplification of a more complex statistical method, showing consideration of more variables in determining the most appropriate type of bucket [4]. Using more observations of stemflow across more types of storms, we can discover what types of buckets would be better for windy or calm conditions or for storms of different durations (longer or shorter).

This example uses only two variables, but we can look for correlations between stemflow and all the tree and weather variables at the same time, using more complex statistical methods. For example, hydrologists can use statistics to represent a tree canopy as different types of buckets according to different weather conditions ( Figure 3B). In this way, hydrologists can still use the bucket concept even if the tree canopy responds differently in big windy storms compared to small windless storms. Using these methods, hydrologists can not only poke the right kinds of holes in the bucket, but they can also understand why some holes should be larger or why they should be positioned differently. These methods transform the messy reality of nature into the neat and orderly bucket water park.

The Importance of Hydrologists' Buckets

Water is needed for all life on Earth, so it is no wonder that there is water all over our planet-including inside the Earth and inside us! It is therefore difficult to know where water is, how it moves around, and when it is where. Still, it is important to follow, understand, and predict the water cycle and how human actions will influence it. To reduce this complex water cycle into something we humans can follow, understand, and predict, the reductionist bucket approach is very useful. In this article, we focused on one type of bucket (the forest canopy) and one type of water flow that occurs during storms in the forest (stemflow). The forest bucket and its water flows are extremely important to understand because forests provide many services to humanity-from producing lumber used to build houses to cooling the air. Many of these services depend on the way water flows through the forest "water park." By reducing the complexity of the forest water cycle, hydrologists can describe those complex processes using math, which you now know is simple enough to be learned by kids all over the world!

Glossary

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

The paper was prepared as part of the activities at the UNESCO Chair of Water-related Disaster Risk Reduction, University of Ljubljana, Slovenia.

References

[1] ↑ Allen, S. T., Aubrey, D. P., Bader, M. Y., Coenders-Gerrits, M., Friesen, J., Gutmann, E. D., et al. 2020. "Key questions on the evaporation and transport of intercepted precipitation," in Precipitation Partitioning by Vegetation (Cham: Springer). p. 269-80.

[2] ↑ Coenders-Gerrits, M., Schilperoort, B., and Jiménez-Rodríguez, C. 2020. "Evaporative processes on vegetation: an inside look," in Precipitation Partitioning by Vegetation (Cham: Springer). p. 35-48.

[3] ↑ Rutter, A. J., Kershaw, K. A., Robins, P. C., and Morton, A. J. 1971. A predictive model of rainfall interception in forests, 1. Derivation of the model from observations in a plantation of Corsican pine. Agric. Meteorol. 9:367-84. doi: 10.1016/0002-1571(71)90034-3

[4] ↑ Bezak, N., Zabret, K., and Šraj, M. 2018. Application of copula functions for rainfall interception modelling. Water. 10:995. doi: 10.3390/w10080995

# data[].attributes.attachment_link.title
How Math Helps us Predict Water Flows in Forests

# data[].attributes.attachment_link.description
Water flows through forests in many ways, so it is difficult to understand and predict where and when it will flow. Understanding how water flows through forests is important, as it affects many of the services that forests offer to people, like lumber for houses and cleaner air.

# data[].attributes.body
This video investigates math as a language. If words are combined together to form a language, can't symbols of math be combined together to form their own language. By watching this video you will be able to understand how math is a language and what exactly the meaning of mathematical symbols are. It breaks down the different components of mathematics as a language and how the cohesively come together. In addition to explaining how math is a language, they also provide valid examples as to how it applies to the real world. This is important when making an argument or case. Once watching the video, one can then compare and contrast the meaning and use of mathematical symbols to the meaning and use of words. Since the video argues that both are a language in their own, it is their components that we must investigate to see how they compare.

# data[].attributes.attachment_link.title
THE LANGUAGE OF MATHEMATICS AND ITS SYMBOLS

# data[].attributes.body
There are many different mathematical proofs and concepts that can be used to prove the existence of black holes. Black holes are impossible to see and observe. Thus, mathematics is the only possible way to measure and understand it. Mathematics turns the black hole which would be something invisible to us into something that is measurable. The application of physics allows scientists to gain a grasp of the behavior of those objects by understanding its effects on space and time.

# data[].attributes.attachment_link.readable_text
Abstract

Gravity is a universally attractive force, which tries to cause matter to clump. This means that in diverse physical systems, ranging from stars at the end of their lives burning nuclear fuel, to gas clouds which collapse under the weight of their own gravity, a natural endpoint can be the development of a black hole. This is an object where matter has been packed so densely that even light cannot escape gravity's pull at its surface, or event horizon. Black holes are both commonplace-our own Milky Way galaxy hosts a giant black hole in its center!-and mysterious. In this article we discuss current thoughts about one of the mysteries of black holes-how one can explain the number of distinct (but similar) physical states they hide behind their event horizon. Physicists call this the black hole entropy.

1. What is Entropy?

The in a physical system is a measure of the amount of messiness or arbitrariness characterizing the system. Systems in very special states-almost any hunk of normal matter at very low temperatures, or a very clean bedroom, for instance-have low entropy. In both cases this is because the constituents of the system have no choice about where to be. At low temperature, in a chunk of material, every atom sits at a place that minimizes the energy of the system. In a very clean room, every piece of clothing is folded and stacked in a drawer. In contrast, matter at high temperature, or a messy room, has high entropy. The atoms can be flying about every which way in hot matter. In a messy bedroom, shirts and shorts can be flung about the room willy nilly. The entropy of a system is defined as the logarithm of the number of possible configurations of its constituents (atoms for matter; shirts and shorts for the messy room) that would look roughly the same, to a casual observer.

For conventional objects like tables, rooms, or slabs of lead, it is no surprise that one can associate an entropy to the system. We know what these systems are made of, and how to count configurations of the underlying parts. To the great surprise of theoretical physicists, it was discovered in the early 1970s that to this list of commonplace system with an entropy, you can add black holes.

2. Black Holes and Bekenstein's Insight

What is a black hole? Imagine taking a lump of matter at some fixed mass M, and crushing it. Einstein's theory of gravity says that when you squeeze it into a small enough region of space-with a size known as the Schwarzschild radius, ( M)-the object becomes a qualitatively new kind of beast, a black hole. A black hole is a clump of matter so dense that even light cannot escape its surface, or . Therefore, its interior workings remain a mystery to the outside world, including us! If you want to know in more detail what a "small enough region" means, we can say

where G is the gravitational constant and c is the speed of light. It is okay if you do not follow the details of the equation; the basic point is that if you put an amount of mass M or larger in a sphere of radius smaller than ( M), then voila-it will become a black hole.

We are talking about really dense matter. For instance, to compress the Earth to the size required to make a black hole, you would have to reduce its radius from today's 4,000 miles to ~9 mm! Happily, Nature has provided us with real life examples of black holes, so we do not need to do the work of making one. In fact, now you can even see a black hole-or more precisely, its surrounding neighborhood. A spectacular image of the black hole sitting in the galaxy M87-a huge black hole weighing almost 7 billion times as much as our sun-was recently produced by the Event Horizon Telescope collaboration []!

As a graduate student at Princeton in the 1970s, Jacob Bekenstein pondered the following puzzling question. An observer outside of a black hole can only see the event horizon-the interior is hidden. The physical properties of the event horizon are totally characterized by just a few numbers: in the simplest case, the mass M of the black hole.

But, thought Bekenstein, could not I have formed the black hole of mass M in many different ways? I could have formed it by throwing in many elephants, or alternatively, many wombats. What happened to the information about how the black hole was formed? The black hole must have an entropy!

Work of Bekenstein-and later more precise work of Bardeen, Carter, and Hawking-succeeded in proving beyond reasonable doubt that black holes do indeed have an entropy. And they were able to produce a formula for the entropy of a black hole as a function of its mass S( M) []. But the precise nature of the physical states that could account for S( M) remained mysterious.

3. Counting With Hardy and Ramanujan

Let us briefly change the subject, to another pair of heroes. One of the most dramatic stories in the mathematics of the 20th century is that of Hardy and Ramanujan []. Ramanujan, self-educated in higher mathematics and working a day job as a clerk in the Port of Madras while doing mathematical research at night, sent letters describing his results to a few well-known mathematicians. His letters were finally answered by the prominent Cambridge University number theorist G.H. Hardy, who said of the formulae Ramanujan included in the letters:

A single look at them is enough to show that they could only be written by a mathematician of the highest class. They must be true because, if they were not true, no one would have the imagination to invent them.

On the strength of these formulae, Hardy brought Ramanujan to Cambridge, where they engaged in one of the most famous collaborations in the history of mathematics.

One of their central results is easily explained: suppose you have 5 oranges, and you wish to divide them up among various friends. How many ways could you imagine dividing them up? Well:

So we see that there are 7 ways of dividing up 5 oranges: into one group of 5, two groups of 4 and 1, and so forth. We say that the number of partitions of 5 is 7. If we write a function that associates to each whole number the number of partitions p( n), then we can re-write this as

For small numbers ( n), like 3, 4, or 5, computing p( n) is easy. But how about p(100)? If you start trying to figure this out, you will see that the partition numbers quickly grow out of hand! How are we to determine them?

Hardy and Ramanujan succeeded in doing this by developing an ingenious trick called "the circle method." They came up with a formula that allowed them to roughly determine p( n) and, using this formula, a graph like that in Figure 1 could be created. On the x-axis we have n, while on the y-axis we have the accuracy of their estimate for p( n). (Partitions only make sense for whole numbers n, but their formula spits out an answer-and can be plotted-for all values of n.)

Figure 1 - A plot of the accuracy of the Hardy-Ramanujan estimate for p(n) vs. n, for values of n up to 100.

The label on the y-axis shows the fractional error of the estimate (so the y-value of 0.05 close to x = 100 means that the formula gets p(100) to within about 5% accuracy). Image credit to John D. Cook.

4. What About Black Holes?

We started with black holes, and then detoured into the partitions of positive integers. How are these two problems related?

To put these two subjects together, we first have to talk about a third subject, called string theory. String theory is one of the theories scientists use to describe the smallest things-even smaller than atoms-that make up the universe. In string theory, these fundamental objects are (tiny) loops of string. The lowest energy state of a tiny loop of string would be perfect stillness. You can also add energy to the string to create waves that can travel with various around the loop. For instance, from Figure 2 (where we use a guitar string instead of a closed loop of string for ease of illustration), we see that the lowest frequency wave you can put on the string has one peak, while higher frequencies (often called higher harmonics in music!) have two, three, or more peaks. This allows multiple possible different waves on the string, with frequencies given by positive integers.

Figure 2 - A guitar string has a lowest frequency of vibration, as well as higher harmonics which are integer multiples of this lowest frequency.

Image credit to cronodon.com.

Importantly, we can also excite the string with two waves of the lowest frequency, or three, or more. Here, we are using the fact that energies present in the waves of a given frequency on the string are subject to quantum mechanics. A classical guitar string allows one to pluck the lowest frequency with any (and hence put any amount of energy into the string at that frequency). But quantum mechanics quantizes the energy present at a given frequency into discrete lumps-so one has a whole number (like 1.2, or 1,729, but not 2.718) times the energy associated to a wave at a given frequency.

Now, we are just a few steps away from figuring out how to compute black hole entropy in string theory! We rely on the following observations:

1. We could imagine waves with many different frequencies being simultaneously present on the string. Remember that all frequencies are positive integers, and we want to keep track of the number of "quantum lumps" present at each frequency.

2. The energy cost of these quantum lumps increases with k. Higher frequencies use more energy, with the kth frequency using k times as much energy as the lowest frequency.

3. Einstein's most famous equation tells us that

That is, the energy of the vibrating string shows up as mass.

We conclude that we can compute the number of massive string states at an energy N times that of the lowest energy mode, in the following way, using the method of Hardy-Ramanujan described earlier. We can reach energy level N by having any number of excitations of the lowest frequency n(1), any number n(2) in the second lowest frequency, any number n(3) with the third frequency, and so forth. However, to attain total energy N we must have

But each choice of the set of numbers [ n(1), n(2), etc.] appearing on the right-hand side of this formula is precisely exhibiting a partition of N! The number of massive string states at string mass level N, is given by the number of partitions of N into positive integers, p(N).

This is related to black holes because a highly excited, wiggling string in string theory, can theoretically be made to lie within its Schwarzschild radius and form a black hole. Since we know the mass of the string states as a function of N, and we know the number of such string states from p( N), we can produce a formula for the entropy S( M). This gives a (correct) prediction for the entropy of suitable black holes in terms of partition numbers []!

Glossary

Entropy: ↑ The amount of messiness or arbitrariness that goes into specifying a given physical system (or roughly, the number of ways it could be made).

Event Horizon: ↑ The surface of a black hole. If you are at the event horizon-or closer to the center-you cannot possibly escape being sucked in, even if you are a light ray, the fastest moving object known to modern physics!

Frequency: ↑ For a wave (e.g., of water, or displacement of a string), this is the number of times the wave reaches its maximum height per unit time.

Amplitude: ↑ Amplitude is a measure of the height of peaks of, e.g., waves of water or waves of displacement of strings.

Conflict of Interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

This work was supported in part by the National Science Foundation under grant PHY-1720397.

References

[1] ↑ Event Horizon Telescope. Available online at: https://eventhorizontelescope.org (accessed April 24, 2020).

[2] ↑ Hawking, S. 1998. A Brief History of Time. New York, NY: Bantam.

[3] ↑ Kanigel, R. 2016. The Man Who Knew Infinity. New York, NY: Washington Square Press.

[4] ↑ Susskind, L. 2009. The Black Hole War. New York, NY: Back Bay Books.

# data[].attributes.attachment_link.title
How Many Ways Are There to Make a Black Hole?

# data[].attributes.attachment_link.description
Gravity is a universally attractive force, which tries to cause matter to clump. This means that in diverse physical systems, ranging from stars at the end of their lives burning nuclear fuel, to gas clouds which collapse under the weight of their own gravity, a natural endpoint can be the development of a black hole.

# data[].attributes.body
"How do mathematicians reconcile the fact that some conclusions seem to conflict with our intuitions?"

This video helps with this question because intuitively, there should be no way to contain so many mathematical concepts in one sequence. However, this videos proves it possible.

# data[].attributes.attachment_link.title
A Number Sequence with Everything - Numberphile

# data[].attributes.body
1. What is meant by the term 'proof' in mathematics, and how is this similar to, or different from what is meant by this term in other areas of knowledge?

It helps with the question 1 because it connects what is proof in math to the medicine field

# data[].attributes.attachment_link.authors[].name
Authors William J. Parnell, William D. Rowley, Naomi R. M. Curati

# data[].attributes.attachment_link.description
Maths is a crucial part of medicine. All the graphs, equations, statistics, and general maths we learn at school help us to understand important aspects of human and veterinary medicine, biology, and science in general. People always think that biology and chemistry are important for doctors, nurses…

# data[].attributes.body
The following link introduces a conversation that outlines how children have preferred ways of completing mathematics that they find easier, or more instinctual to do. Many math methods are difficult to new learners, and may not work for every student. This relates to question #2 directly because it shows how some mathematical conclusions and methods conflict with our instincts and intuitions as humans. Not only may this be true for the results of the certain math problem, but also for the method of obtaining that information or equation. This article shows how children, the newest learners of each generation, are now being allowed to solve math problems in a variety of ways that better suit their intuitions. This could make the math itself, and the results, more understandable for the learner:  https://theconversation.com/why-teachers-are-letting-students-solve-math-problems-in-lots-of-different-ways-197601

# data[].attributes.attachment_link.readable_text
Families might be wondering why their child's math classroom looks so different from what they remember in school.

Why aren't teachers putting students on the spot and getting them to prove that they know the math? Why are teachers letting students solve problems in lots of different ways instead of just telling them how to do it?

Teachers are moving beyond traditional ways of teaching math and shifting towards more evidence-based equitable teaching and assessment practices. This means every student is provided with what they need as opposed to being provided with the exact same resources and assessment practices.

For example, instead of giving all students in the classroom the exact same test, a student with test anxiety will be given an alternative format to showcase their learning.

Math is not 'neutral'

Mathematics is not a "neutral" subject - biases in math exist that are grounded in colonial ideals.

For example, what is traditionally considered a standard measurement in a Canadian math curricula is only standard in western cultures.

Traditional mathematics teaching focuses on the teacher's way of solving a problem as the right way. Students who might think about the problem in a different way are then at a disadvantage. In some cases, their cultural way of doing mathematics is then viewed as incorrect.

Many strategies can be used

One study examined the multiplication strategies Filipino immigrant children living in Japan used. In Japanese culture, children learn a formal multiplication method (kuku) that involves learning to recite times tables, whereas in the Philippines, it is common to use finger multiplication.

Read more: Indigenous basket-weaving makes an excellent digital math lesson

The study found that immigrant Filipino families only taught their children finger multiplication if they found that their child was struggling with multiplication. The children would hide this method from their teachers.

When only one strategy for math is taught as universally correct, this marginalizes students and compounds inequities.

Draw on existing knowledge

What are some signs teachers are practising equitable teaching in math?

Teachers draw on students' existing knowledge. In order to serve each student and their unique learning styles, teachers ideally get to know their students and how they learn best.

This means teachers view their students as having knowledge that can be shared within the learning community. This is in contrast to traditional teaching practices that work on the assumption that students are empty vessels to be filled with knowledge.

Peer collaboration

When you were in school, what did participating in classroom activities look like? Did you raise your hand? Did you turn to your elbow partner? Were you sitting in rows and expected to be quiet until you were called upon?

Equitable classrooms promote student voice and agency, peer collaboration and values learning from each other.

This creates an equitable learning environment because it provides all students with the platform to have their voices heard and validates students' cultural ways of knowing math that may have otherwise been hidden.

An added bonus is that if all students have multiple ways to solve a problem, they have more strategies to rely on.

What an equitable math classroom looks like

Teachers get to know students through student interest surveys where students can explain what they like about math and what they need from their teachers to be successful in math.

Students can write narratives about their experiences with math to help identify students' attitudes towards math (how they feel about themselves as a math learner, what they think math is about, whether they like math or not).

Teachers focus on how students solve problems and their solution strategies instead of just getting the right answer. Teachers allow students to show their thinking in a variety of ways (like in conversations and group work).

Instructional time (when the teacher is speaking about the ideas being taught) happens after students engage with their learning. This communicates to the students that they are capable of solving the problem, and that their ideas are both important and valuable.

There are a variety of strategies and solutions generated by the students.

Students have lots of choice about where they are working (standing at whiteboards, working collaboratively with peers or in groups, building hands-on models).

Instead of students only writing tests, the teacher is able to use diverse and different types of data to assess student learning (conversations, group work, models).

What an equitable math classroom sounds like

Teachers ask questions about what students are thinking, rather than giving them the answers.

Teachers wait for students to ask for help before assistance is offered to communicate that they are able to solve the problem (though adults can remind students they are there to help when students need it).

Students talk more than the teacher; these classrooms are loud with student talk.

Teachers ask questions about student thinking ("How do you know that's the correct answer?"; "Can you tell me more about your thinking?").

The teacher uses the students' ideas to consolidate the learning after the students have worked with the problem.

What an equitable math classroom feels like

Equitable math classrooms can be best realized when families, teachers and students support the importance and benefits of equitable classrooms.

# data[].attributes.attachment_link.title
Why teachers are letting students solve math problems in lots of different ways

# data[].attributes.attachment_link.description
Families might be wondering why their child's math classroom looks so different from what they remember in school. Why aren't teachers putting students on the spot and getting them to prove that they know the math? Why are teachers letting students solve problems in lots of different ways instead of just telling them how to do it?