# Math behind the Genetic Relationship Matrix

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The genetic relationship matrix (GRM) can estimate the genetic relationship between two individuals (\$j\$ and \$k\$) over \$m\$ SNPs and \$i\$ representing a specific SNP. What I don't understand from their equation is why we divide our summation by \$m\$ (the number of SNPs). \$x_{ij}\$ is the number of copies of the minor allele for the \$j\$-th individual in SNP \$i\$. \$p_i\$ is the frequency of the minor allele for SNP \$i\$.

This expression is a mean

$$frac{1}{m}sum_{i=1}^m…$$

($$m$$ is the number of SNPs) of the ratio

$$frac{numerator}{denominator}$$

where the numerator is a covariance

$$(a_i-C)(b_i-C)$$

and the denominator is the expected heterozygosity (it is also the variance of binomial distribution with`n=2`)

$$2p_i(1-p_i)$$

Therefore, it represents how much do two individuals covary $$(x_{ij})(x_{ik}-p_i)$$ respectively to what is expected on average $$2p_i(1-p_i)$$ averaged over all SNPs $$frac{1}{m}sum_{i=1}^m…$$, where $$m$$ is the number of SNPs.

It is a relative measure (relative to the expected heterozygosity) of covariance between each individual (averaged over all SNPs).

Does it help?

\$2p_i\$ is the expectation of \$SNP_i\$:

\$\$E(SNP_i) = 0 imes (1-p_i)^2 + 1 imes 2p_i(1 - p_i) + 2 imes p_i^2 = 2 p_i\$\$

\$(x_{ij} - 2p_i)(x_{ik} - 2p_i)\$ measures how the two SNPs covary. I have no idea why they divide it by \$2p_i(1 - p_i)\$, but if you leave that one out, you have the plain definition of covariance.

https://en.wikipedia.org/wiki/Expected_value

https://en.wikipedia.org/wiki/Covariance

Though the question was posted a long ago, I feel that a clarification could be beneficial.

Heterozygosity / gene diversity
The factor in the denominator is twice the variance of the number of major alleles at site $$i$$, also known as heterozygosity or gene diversity: $$H_i = 2p_i(1-p_i).$$ What might appear confusing here is the factor $$1-p_i$$ that is explicitly written instead of traditional $$q_i=1-p_i$$: $$H=2pq$$ (since the probabilities of minor and major alleles sum to $$1$$).

Covariance
In statistical terms this factor can be interpreted as twice the variance of the average number of major/minor alleles on the site, which is the motivation for including it in the denominator of the covariance function, as, e.g., when converting covariance to a correlation coefficient. It is necessary however to note that this inclusion can be done in different ways, depending on what statistical aspects one wants to emphasize.

• In particular, the equation given in the question, taken from this paper (referenced in the comments), is the covariance of a standartized genotype matrix, defined as $$W_{ij} = frac{x_{ij}-2p_i}{sqrt{2p_i(1-p_i)}}.$$ In other words the standartization is done prior to calculating the covariance $$cov_{jk} = frac{1}{m}sum_i W_{ij}W_{ik}.$$

This is however different from a more traditional definition of Genetic relationship matrix, as in VanRaden's work, where such division is done after calculating the covariance: $$G_{jk} = frac{frac{1}{m}sum_i(x_{ij}-2p_i)(x_{ik}-2p_i)}{frac{1}{m}sum_i 2p_i(1-p_i)} = frac{sum_i(x_{ij}-2p_i)(x_{ik}-2p_i)}{sum_i 2p_i(1-p_i)}.$$

Finally, as a word of caution, let me mention that $$x_{ij}, x_{ik}$$ in these expressions can take only values $$0$$ or $$1$$.

The matrix gives you an estimate of the average linear relationship between any two individuals genomes, it's essentially taking the average of the betas (like linear regression betas) across each locus. One of the formulas for 'beta' is covariance divided by the sample variance, which is exactly what is happening. Each locus beta predicts the state of a person's genome at that locus from another person's genome at the same locus. Taking the average of these betas across the entire genome gives you a coefficient that can be thought of as a measure of how well we can predict one person's genome from another.

## Math behind the Genetic Relationship Matrix - Biology

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## Explained: Matrices

Among the most common tools in electrical engineering and computer science are rectangular grids of numbers known as matrices. The numbers in a matrix can represent data, and they can also represent mathematical equations. In many time-sensitive engineering applications, multiplying matrices can give quick but good approximations of much more complicated calculations.

Matrices arose originally as a way to describe systems of linear equations, a type of problem familiar to anyone who took grade-school algebra. “Linear” just means that the variables in the equations don’t have any exponents, so their graphs will always be straight lines.

The equation x - 2y = 0, for instance, has an infinite number of solutions for both x and y, which can be depicted as a straight line that passes through the points (0,0), (2,1), (4,2), and so on. But if you combine it with the equation x - y = 1, then there’s only one solution: x = 2 and y = 1. The point (2,1) is also where the graphs of the two equations intersect.

The matrix that depicts those two equations would be a two-by-two grid of numbers: The top row would be [1 -2], and the bottom row would be [1 -1], to correspond to the coefficients of the variables in the two equations.

In a range of applications from image processing to genetic analysis, computers are often called upon to solve systems of linear equations — usually with many more than two variables. Even more frequently, they’re called upon to multiply matrices.

Matrix multiplication can be thought of as solving linear equations for particular variables. Suppose, for instance, that the expressions t + 2p + 3h 4t + 5p + 6h and 7t + 8p + 9h describe three different mathematical operations involving temperature, pressure, and humidity measurements. They could be represented as a matrix with three rows: [1 2 3], [4 5 6], and [7 8 9].

Now suppose that, at two different times, you take temperature, pressure, and humidity readings outside your home. Those readings could be represented as a matrix as well, with the first set of readings in one column and the second in the other. Multiplying these matrices together means matching up rows from the first matrix — the one describing the equations — and columns from the second — the one representing the measurements — multiplying the corresponding terms, adding them all up, and entering the results in a new matrix. The numbers in the final matrix might, for instance, predict the trajectory of a low-pressure system.

Of course, reducing the complex dynamics of weather-system models to a system of linear equations is itself a difficult task. But that points to one of the reasons that matrices are so common in computer science: They allow computers to, in effect, do a lot of the computational heavy lifting in advance. Creating a matrix that yields useful computational results may be difficult, but performing matrix multiplication generally isn’t.

One of the areas of computer science in which matrix multiplication is particularly useful is graphics, since a digital image is basically a matrix to begin with: The rows and columns of the matrix correspond to rows and columns of pixels, and the numerical entries correspond to the pixels’ color values. Decoding digital video, for instance, requires matrix multiplication earlier this year, MIT researchers were able to build one of the first chips to implement the new high-efficiency video-coding standard for ultrahigh-definition TVs, in part because of patterns they discerned in the matrices it employs.

In the same way that matrix multiplication can help process digital video, it can help process digital sound. A digital audio signal is basically a sequence of numbers, representing the variation over time of the air pressure of an acoustic audio signal. Many techniques for filtering or compressing digital audio signals, such as the Fourier transform, rely on matrix multiplication.

Another reason that matrices are so useful in computer science is that graphs are. In this context, a graph is a mathematical construct consisting of nodes, usually depicted as circles, and edges, usually depicted as lines between them. Network diagrams and family trees are familiar examples of graphs, but in computer science they’re used to represent everything from operations performed during the execution of a computer program to the relationships characteristic of logistics problems.

Every graph can be represented as a matrix, however, where each column and each row represents a node, and the value at their intersection represents the strength of the connection between them (which might frequently be zero). Often, the most efficient way to analyze graphs is to convert them to matrices first, and the solutions to problems involving graphs are frequently solutions to systems of linear equations.

## Math behind the Genetic Relationship Matrix - Biology

Students will distinguish between natural and artificial selection and use a student-centered learning activity to see how science and genetics have been used to artificially select apples for specific traits like color, texture, taste, and crispness.

##### Estimated Time

Two 50 min class periods. 10-15 min per station

##### Materials Needed
• Apples and the Science of Selection handout, 1 copy per student
• Station Cards, 1 copy per class printed front to back
• Paper plates, utensils, and napkins as needed for taste testing
• Supplies for each station:
• Station 1:
• Station 1 card (Johnny Appleseed)
• Crabapples, apple cider, or crabapple jelly to taste.
• Station 2 card video
• Device to view video and use for research.
• Station 3 card
• Device to browse A Guide to the Most Popular Apple Varieties webpage
• ingredients to make homemade applesauce in the microwave, slow cooker, or pressure cooker. Choose the recipe and method that meets your classroom needs.
• Station 4 card
• Device to view Have We Engineered the Perfect Apple? video
• Samples of Honeycrisp and Red Delicious apples to taste.
• Tip: Expand the taste test to include varieties of apples grown in your region. Check local farmers markets or grocery stores to discover varieties unique to your area.
• Station 5 card
• Device to view videos:
##### Vocabulary Words

artificial selection: the intentional breeding of plants and animals to produce specific, desirable traits

asexual reproduction: a type of reproduction by which offspring arise from a single organism and inherit the genes of that parent only

clone: an organism or cell produced asexually from one ancestor or stock to which they are genetically identical

crossbreeding: to produce a hybrid by breeding two breeds or varieties

evolution: the process by which different kinds of living organisms are thought to have developed and diversified from earlier forms during the history of the earth

natural selection: the process whereby organisms better adapted to their environment tend to survive and produce more offspring a theory first introduced by Charles Darwin

propagation: the breeding of specimens of a plant or animal by natural processes from the parent stock

sexual reproduction: production of new living organisms by combining genetic information from two individuals of different types resulting in a genetically similar, but different offspring

##### Did You Know? (Ag Facts)
• More than 100 varieties of apples are grown commercially in the United States, but a total of 15 popular varieties account for almost 90% of the production of apples. 1
• The crabapple is native to North America. 2
• If planted from a seed, an apple tree would take four to five years to produce its first fruit. 2
##### Background Agricultural Connections

Species of plants and animals change through time. This evolutionary process can occur as a result of mutation, migration, or genetic drift. However, natural selection is the most probable and accepted cause of evolution. Natural selection can also be known as "survival of the fittest." In other words, the fittest organism in every situation (weather, temperature, protection from predators or disease, etc.) is most likely to survive and pass its genetic traits to the next generation. Artificial selection (or selective breeding), results in changes over time as well, but rather than nature choosing the desirable characteristics, humans do.

The only apples that are native to the United States are crab apples. Apples originated in the Old World and were brought to the Americas as part of the Columbian Exchange. Thousands of varieties of apple exist, though only a handful are likely to be familiar to most consumers. For a detailed history of apples, read A Curious Tale: The Apple in North America.

Washington, New York, and Michigan are the top apple producing states in America. Although apples are grown in almost every state, not every state produces apples on a commercial level. Apple trees require an annual cold period in order to set fruit and produce a viable crop. Therefore, states with a year-round warm climate are not suited for apple production. Apple trees are propagated by cuttings, a method of asexual reproduction. This allows apple producers to make every Granny Smith apple look and taste the same, since each new tree is produced from a cutting, or clone of the previous. New and improved apple varieties may be developed by propagating apple trees from seed in order to produce an offspring that is genetically different than the parent plants (sexual reproduction). Once a desirable variety is developed through crossbreeding, the tree is propagated by cuttings to produce a uniform crop of apples.

Apples offer an authentic learning connection into scientific concepts, such as investigating states or phases of matter (applesauce), exploring methods of plant propagation (grafting), discovering applications of plant breeding techniques, understanding genetics and heredity, and biotechnology. In this lesson, students will rotate between five stations that introduce them to these scientific connections.

 This lesson could be used to introduce a phenomena storyline to investigate several questions (episodes) in relation to apples and science phenomena. For example: Why do apples brown? Why do apple farmers use grafting to propagate new trees? How does every [Granny Smith] apple look and taste the same? etc. For more information about phenomena storylines visit nextgenstorylines.org.

##### Interest Approach - Engagement

This lesson has been adapted for online instruction and can be found on the 9-12th grade eLearning site.

1. Ask students to raise their hand if they have recently eaten a Red Delicious apple. Ask a student to describe what it looked and tasted like.
2. Ask students to raise their hand if they have recently eaten a Honeycrisp apple. Ask a student to describe what it looked and tasted like.
3. Continue a class discussion comparing varieties of apples. What qualities make a good apple? What qualities make a poor apple? How many varieties of apples are there? How are different varieties of apples developed? What will apples be like in the future? Can science explain why different varieties of apples taste different?
4. Conclude your discussion with the final question, "Are apples different today because of something humans have done or because of something that occurred naturally (without human intervention)?" Leave the question open-ended and inform students that you will return to it after the activity.
##### Procedures

Preparation: Prior to class, set up five stations around the classroom. Each station should have the supplies listed above in the Materials section of the lesson plan.

1. Give each student one copy of the Apples and the Science of Selection handout.
2. Divide the class into 5 equal groups and assign each group a specific station for their first rotation.
3. Give a brief introduction to students by explaining that they will be rotating through 5 stations. They will have approximately 10-15 minutes at each station to read the station card and complete the three tasks listed on the back. Students will need their handout and a writing utensil to begin.
4. Set a timer. Consider projecting it in the classroom to allow students to gauge their time at each station. After the time is up, groups should move in a sequential direction. Reset the timer and continue until all groups visit all five stations.
5. Once students have come back together, re-ask the question, "Are apples different today because of something humans have done or because of something that occurred naturally (without human intervention)?" (They are a result of what humans have done, also known as artificial selection.)
6. Show the video, Natural Selection vs Artificial Selection.

Natural Selection: Natural selection occurs only if there is both (1) variation in the genetic information between organisms in a population and (2) variation in the expression of that genetic information - that is trait variation that leads to differences in performance among individuals.

Concept Elaboration and Evaluation:

After completing these activities, review and summarize the following key concepts:

• Apples have a long history of importance and selective breeding in the United States.
• Most apple trees today are grown through grafting, a method of asexual propagation to reduce genetic variability in apple varieties. This allows every apple of a specific variety to taste and look the same.
• In apples, characteristics such as color, texture, sweetness/tartness, juiciness, and crunchiness are determined by the genetic make-up of the apple.
• Scientists use a knowledge of genetics and heredity to crossbreed apples (using seed, or sexual propagation) to produce new varieties of apples.
• Genetic engineering is a tool used in plant and sometimes animal breeding. One variety of apple, the Arctic® apple, was genetically modified so that it does not brown when it is cut. All other apple varieties were created through crossbreeding and artificial selection.
##### Enriching Activities

Have students interview staff and administration about their knowledge and opinions on genetically engineered products.

Have students visit the US Apple Association's Popular Varieties webpage to explore apple recipes and watch short video clips about popular apple varieties in the United States.

Listen to the NPR Planet Money podcast The Miracle Apple to hear the story of the development of new varieties of apples.

Research the crossbreeding behind apple varieties. Suggested varieties include: Honeycrisp, Zestar, SweeTango, SnowSweet, Frostbite, and Minnehaha.

Tour a local apple orchard to view apple production first hand.

Practice grafting with apple trees, bring in a community expert for additional help.

Pair students and assign each pair a breeding technology from the Crop Modification Techniques Infographic. Have them research and present their findings in a gallery walk.

## Graphic Uses of Matrix Mathematics

Graphic software uses matrix mathematics to process linear transformations to render images. A square matrix, one with exactly as many rows as columns, can represent a linear transformation of a geometric object. For example, in the Cartesian X-Y plane, the matrix reflects an object in the vertical Y axis. In a video game, this would render the upside-down mirror image of a castle reflected in a lake.

If the video game has curved reflecting surfaces, such as a shiny silver goblet, the linear transformation matrix would be more complicated, to stretch or shrink the reflection.

## Is Innate Talent a Myth?

Elite-level performance can leave us awestruck. This summer, in Rio, Simone Biles appeared to defy gravity in her gymnastics routines, and Michelle Carter seemed to harness super-human strength to win gold in the shot put. Michael Phelps, meanwhile, collected 5 gold medals, bringing his career total to 23.

In everyday conversation, we say that elite performers like Biles, Carter, and Phelps must be &ldquonaturals&rdquo who possess a &ldquogift&rdquo that &ldquocan&rsquot be taught.&rdquo What does science say? Is innate talent a myth? This question is the focus of the new book Peak: Secrets from the New Science of Expertise by Florida State University psychologist Anders Ericsson and science writer Robert Pool. Ericsson and Pool argue that, with the exception of height and body size, the idea that we are limited by genetic factors&mdashinnate talent&mdashis a pernicious myth. &ldquoThe belief that one&rsquos abilities are limited by one&rsquos genetically prescribed characteristics. manifests itself in all sorts of &lsquoI can&rsquot&rsquo or &lsquoI&rsquom not&rsquo statements,&rdquo Ericsson and Pool write. The key to extraordinary performance, they argue, is &ldquothousands and thousands of hours of hard, focused work.&rdquo

To make their case, Ericsson and Pool review evidence from a wide range of studies demonstrating the effects of training on performance. In one study, Ericsson and his late colleague William Chase found that, through over 230 hours of practice, a college student was able to increase his digit span&mdashthe number of random digits he could recall&mdashfrom a normal 7 to nearly 80. In another study, the Japanese psychologist Ayako Sakakibara enrolled 24 children from a private Tokyo music school in a training program designed to train &ldquoperfect pitch&rdquo&mdashthe ability to name the pitch of a tone without hearing another tone for reference. With a trainer playing a piano, the children learned to identify chords using colored flags&mdashfor example, a red flag for CEG and a green flag for DGH. Then, the children were tested on their ability to identify the pitches of individual notes until they reached a criterion level of proficiency. By the end of the study, the children had seemed to acquire perfect pitch. Based on these findings, Ericsson and Pool conclude that the &ldquoclear implication is that perfect pitch, far from being a gift bestowed upon only a lucky few, is an ability that pretty much anyone can develop with the right exposure and training.&rdquo

This sort of evidence makes a compelling case for the importance of training in becoming an expert. No one becomes an expert overnight, and the effects of extended training on performance can be larger than might seem possible. This is something that psychologists have long recognized. In 1912, Edward Thorndike, the founder of educational psychology, wrote that &ldquowe stay far below our own possibilities in almost everything that we do&hellip.not because proper practice would not improve us further, but because we do not take the training or because we take it with too little zeal.&rdquo And, in Peak, Ericsson and Pool write that in &ldquopretty much any area of human endeavor, people have a tremendous capacity to improve their performance, as long as they train in the right way.&rdquo

But does the fact that training leads to improvements&mdasheven massive improvements&mdashin skill level mean that innate talent is a myth? This is a much harder scientific argument to make, and is where Peak runs into trouble. Ericsson and Pool gloss over or omit critical details of research they review that undermine the anti-talent argument. As one example, although they claim that the results of Sakakibara&rsquos training study imply that &ldquopretty much anyone&rdquo can acquire perfect pitch, the sample in that study did not include pretty much anyone. It included children who had been enrolled in a private music school from a very young age (the average age at which training began was 4). It does not seem likely that this non-random sample was representative of the general population in music aptitude or interest&mdashfactors that are known to be influenced by genetic factors. It&rsquos also not clear whether the children had acquired true perfect pitch, because there was no comparison of the children after training to people who possess this rare ability&mdashfor example, in terms of speed of identifying notes or neural correlates of performance.

As another example, describing the results of a study of ballet dancers by Ericsson and colleagues, Ericsson and Pool claim that &ldquothe only significant factor determining an individual ballet dancer&rsquos ultimate skill level was the total number of hours devoted to practice&rdquo and that there was &ldquono sign of anyone born with the sort of talent that would make it possible to reach the upper levels of ballet without working as hard or harder than anyone else.&rdquo Not mentioned is the exact magnitude of the correlation&mdasha value of .42, where 1.0 is perfect. The fact that the correlation was modest in magnitude means that factors not measured in the study&mdashincluding heritable aptitudes&mdashcould have actually accounted for more of the differences in ballet skill than deliberate practice did. As it always is in scientific debates, the devil is in the details in the debate over the origins of expertise.

Ericsson and Pool also leave out a good deal of evidence that runs counter to the anti-talent argument. For example, they claim that professional baseball players have &ldquono better eyesight than an average person,&rdquo but there is evidence to suggest otherwise. In a study published in the American Journal of Ophthalmology, Daniel Laby and colleagues assessed the vision of major and minor league baseball players in the Los Angeles Dodgers organization over the course of four spring training seasons. As David Epstein recounts in his book The Sports Gene, in the first year of the study the researchers used a standard test of visual acuity, and it turned out to be too easy. Over 80% of the players got a perfect score of 20/15, meaning that they could see at 20 feet what an average person can see at 15 feet. In the following seasons, using a custom test, Laby and colleagues found that 77% of the 600 eyes tested had visual acuity of 20/15 or better, with a median of about 20/13. Even for young adults, this is excellent vision. Overall, Laby and colleagues concluded that &ldquo[p]rofessional baseball players have excellent visual skills. Mean visual acuity, distance stereoacuity, and contrast sensitivity are significantly better than those of the general population.&rdquo

Another notable omission from Peak is a study of 18 prodigies by Joanne Ruthsatz and colleagues&mdashto date, the largest study of the intellectual abilities of prodigies. (Given the rarity of prodigies, a sample size of 18 is very large in this area of research.) The researchers gave the prodigies a standardized IQ test, and found that all scored very high on working memory (most were above the 99th percentile, and the average score for the sample was in the top 1%). A major factor underlying the ability to acquire complex skills, working memory is substantially heritable. There is also no discussion of the landmark Study of Mathematically Precocious Youth, started in the 1970s by the Johns Hopkins psychologist Julian Stanley and now co-directed by Camilla Benbow and David Lubinski at Vanderbilt. Now in its forty-fifth year, this longitudinal study has revealed that, even in the top 1%, cognitive ability in childhood is a significant predictor of objective occupational achievements in adulthood, such as earning advanced degrees, publishing scientific articles, and patent awards.

Based on our own evaluation of the evidence, we argue in a recent Psychological Bulletin article that training is necessary to become an expert, but that genetic factors may play an important role at all levels of expertise, from beginner to elite. There is both indirect and direct evidence to support this &ldquomultifactorial&rdquo view of expertise. (We call the model the Multifactorial Gene-Environment Interaction Model, or MGIM.) The indirect evidence comes in the form of large individual differences in the effects of training on performance. In other words, some people take much more training than other people to acquire a given level of skill. As it happens, Sakakibara&rsquos pitch training study provides some of the most compelling evidence of this type. There was a large amount of variability in how long it took the children to pass the test for perfect pitch&mdashfrom around 2 years to 8 years. As Sakakibara notes in her article, this evidence implies that factors other than training may be involved in acquiring perfect pitch, including genetic factors. This finding is consistent with the results of recent reviews of the relationship between deliberate practice and skill, which include numerous studies Ericsson and colleagues have used to argue for the importance of deliberate practice. Regardless of domain, deliberate practice leaves a large amount of individual differences in skill unexplained, indicating that other factors contribute to expertise.

The more direct evidence for the multifactorial view of expertise comes from &ldquogenetically informative&rdquo research on skill&mdashstudies that estimate the contribution of genetic factors to variation across people in factors that may influence expert performance. In a study of over 10,000 twins, two of us found that music aptitude was substantially heritable, with genes accounting for around half of the differences across people on a test of music aptitude. As another example, in a pioneering series of studies, the Australian geneticist Kathryn North and her colleagues found a significant association between a variant of a gene (called ACTN3) expressed in fast-twitch muscle fibers and elite performance in sprinting events such as the 100 meter dash. There is no denying the importance of training for becoming an elite athlete, but this evidence (which is not discussed in Peak) provides compelling evidence that genetic factors matter, too.

Based on this sort of evidence, we have argued that the experts are &ldquoborn versus made&rdquo debate is over&mdashor at least that it should be. There is no doubt that training is required to become an expert. Notwithstanding a report by North Korea&rsquos state-run news agency that Kim Jong-il made five holes-in-one his first time playing golf and rolled a perfect 300 his first time bowling, no one is literally born an expert. Expertise is acquired gradually, often over many years. However, as science is making increasingly clear, there is more to becoming an expert than training. Moving ahead, the goal for scientific research on expertise is to identify all of the remaining factors that matter.

## Math + culture = gender gap?

Researchers have all but debunked the idea that girls are innately worse at math than boys. But psychologists have identified other factors that might set girls back.

July/August 2010, Vol 41, No. 7

We’ve come a long way since the days when 19th century mathematician Sophie Germain’s parents confiscated her candles to keep her from studying mathematics because it was considered “unsuitable” for a woman. But the long-standing debate over gender differences in mathematics is alive and well, and continues to be a lively topic within psychology.

Most experts agree that if gender differences do exist, they are small and likely to affect specific areas of math skill at the highest end of the spectrum — and there’s no indication that women cannot succeed in mathematically demanding fields. Still, women continue to be underrepresented in math, science and engineering-related careers, and there’s evidence that girls can lose ground in math under certain circumstances.

One factor inhibiting girls is self-confidence, says University of Wisconsin psychologist Janet Hyde, PhD. “Even when girls are getting better grades, boys are more confident in math. It’s important to understand what might be sapping girls’ confidence.”

And that lack of self-assurance likely stems from culture, research suggests. After reviewing decades of research on gender differences, Cornell University psychologists Steven Ceci, PhD, and Wendy Williams, PhD, conclude that while there’s probably some genetic basis for small differences between the sexes in math and spatial ability, culture plays by far the bigger role in men and boys’ higher interest and achievement.

“If you look at the students scoring in the top one in 10,000 in mathematics in 1983, there were 13 boys for every girl,” says Ceci. “Since then, until 2007, that gap has shrunk to somewhere between 2.8 and four boys for every girl.

So if the difference were just in the genome, there would not be that improvement. Rather, shifts like that are due in large part to increases in the number of girls who take higher level math courses in high school, where girls traditionally began falling behind boys. They appear to be taking more math courses because changing cultural norms make it more acceptable.

Research by Hyde supports that idea. In a January article in Psychological Bulletin (Vol. 136, No. 1), she and her colleagues found that the more gender equity a country had — measured by school enrollment, women’s share of research jobs and women’s parliamentary representation — the smaller its math gender gap.

“When girls see opportunities for themselves in science, technology, engineering and math, they’re more likely to take higher math in high school and more likely to pursue those careers,” says Hyde.

In fact, women in the United States now earn 48 percent of bachelor’s degrees in mathematics and 30 percent of the doctorates, says Hyde. “If they can’t do math, how are they doing this? They can do math just fine.”

That doesn’t mean, however, that just because girls and women can do the math, they want to. When Vanderbilt University psychologist David Lubinski, PhD, and his colleagues interviewed a group of more than 5,000 intellectually precocious girls and boys they’d followed from childhood into their mid-30s, they noticed that while men and women earned equal proportions of advanced degrees, there were gender differences in the areas people decided to study.

He found that just as many women as men started college planning to go into physical sciences and math. However, women more than men later switched to humanities and social science majors. Every one of these study participants had the ability to succeed in math-related careers, but many of them were more likely to choose law school or medicine, Lubinski says.

“The sexes are making different choices,” he says. “But when we look at how satisfied these people are with their career choices, they’re equally satisfied and equally successful.”

Ceci and Williams posit that girls are more attracted to a variety of careers because they tend to have both strong math and verbal skills. “Boys who are really good at math say, ‘This is who I am, I’m a mathematician,’” says Ceci. “Girls who are really good at math are more likely to be really good at verbal skills, too, and they ask themselves, ‘I wonder what I want to do?’”

It doesn’t help that the corporate culture of many math-centered careers speaks more to boys’ well-documented tendency to be interested in “things” than girls’ tendency to be interested in working with people, says Hyde. “Engineering portrays itself as being about things,” she says. “Maybe if engineering professors made better connections to how engineering helps people, women would be more enticed.”

Classroom influences

To explore why girls are less confident than boys in their math abilities, University of Georgia psychologist Martha Carr, PhD, studies first-graders, and has found that girls use different strategies and have different motivations to do math.

Boys, Carr says, tend to use memory to retrieve sums and are motivated by a sense of competition to get the answer fast, even if they sacrifice accuracy. Girls care less about speed than accuracy and more often rely on “manipulatives” — counting on their fingers or a counting board.

“Girls will use manipulatives even when they might be able to retrieve [the answer],” says Carr. “They need an added push that boys don’t need to start using cognitive strategies.”

That’s important because while using manipulatives is an excellent strategy when students first learn math, it slows them down as problems get more difficult. In fact, in a study that followed students from second grade through fourth grade, Carr found that becoming fluent, and therefore faster, at basic math is directly linked to math performance. The study also found that girls were less fluent than boys.

“If we make sure all children are fluent [in math facts], we will eliminate most gender differences,” she says.

But what if girls’ confidence and their interest in becoming “fluent” are influenced by math anxiety among their predominantly female elementary school teachers? A 2010 study (PNAS, Vol. 107, No. 5) by University of Chicago psychologist Sian Beilock, PhD, suggests that this may well be the case for some girls. She and her colleagues started with these facts: More than 90 percent of elementary school teachers are women, and studies show that elementary education majors have higher levels of math anxiety than any other major. The researchers then assessed math anxiety in 17 female first- and second-grade teachers, as well as math achievement and gender stereotypes among 52 boys and 65 girls from their classes. At the start of the school year, the researchers found no link between teacher anxiety and student math achievement. But by school year’s end, the more anxious teachers were about math, the more likely girls, but not boys, agreed with the statement, “Boys are good at math and girls are good at reading.” In addition, girls who accepted this stereotype performed significantly worse on math achievement measures than girls who did not and boys overall.

Interestingly, on average, girls and boys performed the same, says Beilock. Only the girls who endorsed the stereotype showed a drop in math performance. That finding supports work Beilock and others have done on “stereotype threat,” which shows that people perform poorly when a negative stereotype is in play.

It’s also not surprising that girls picked up on their teachers’ anxiety and not boys because research shows that young children are more likely to emulate adults of the same gender.

In the end, though, it’s not just girls who need math help, emphasizes University of Missouri psychologist David Geary, PhD, an expert on mathematical development and author of “Male, Female: The Evolution of Human Sex Differences, Second Edition” (APA, 2009). He believes all the focus on gender distracts from the more serious problem that U.S. math achievement is abysmal compared with that of other countries.

Hyde agrees. “We need to look toward better math instruction for the United States, not specifically for boys or girls.”

Beth Azar is a writer in Portland, Ore.

Ceci, S. & Williams, W. (2010) “The Mathematics of Sex: How Biology and Society Conspire to Limit Talented Women and Girls.” Oxford University Press.

Else-Quest, N., Hyde, J.S., Linn, M. (2010) Cross-National Patterns of Gender Differences in Mathematics: A Meta-Analysis. Psychological Bulletin, 136(1) 103.

Ceci, S., Williams, W., & Barnett, S. (2009) Women’s Underrepresentation in Science: Sociocultural and Biological Considerations. Psychological Bulletin, 135(2) 218.

Halpern, et al. (2007) The Science of Sex Differences in Science and Mathematics. Psychological Science in the Public Interest, 8(1) 1.

## What's the Universe Made Of? Math, Says Scientist

BROOKLYN, N.Y. — Scientists have long used mathematics to describe the physical properties of the universe. But what if the universe itself is math? That's what cosmologist Max Tegmark believes.

In Tegmark's view, everything in the universe — humans included — is part of a mathematical structure. All matter is made up of particles, which have properties such as charge and spin, but these properties are purely mathematical, he says. And space itself has properties such as dimensions, but is still ultimately a mathematical structure.

"If you accept the idea that both space itself, and all the stuff in space, have no properties at all except mathematical properties," then the idea that everything is mathematical "starts to sound a little bit less insane," Tegmark said in a talk given Jan. 15 here at The Bell House. The talk was based on his book "Our Mathematical Universe: My Quest for the Ultimate Nature of Reality" (Knopf, 2014).

"If my idea is wrong, physics is ultimately doomed," Tegmark said. But if the universe really is mathematics, he added, "There's nothing we can't, in principle, understand." [7 Surprising Things About the Universe]

Nature is full of math

The idea follows the observation that nature is full of patterns, such as the Fibonacci sequence, a series of numbers in which each number is the sum of the previous two numbers. The flowering of an artichoke follows this sequence, for example, with the distance between each petal and the next matching the ratio of the numbers in the sequence.

The nonliving worldalso behaves in a mathematical way. If you throw a baseball in the air, it follows a roughly parabolic trajectory. Planets and other astrophysical bodies follow elliptical orbits.

"There's an elegant simplicity and beauty in nature revealed by mathematical patterns and shapes, which our minds have been able to figure out," said Tegmark, who loves math so much he has framed pictures of famous equations in his living room.

One consequence of the mathematical nature of the universe is that scientists could in theory predict every observation or measurement in physics. Tegmark pointed out that mathematics predicted the existence of the planet Neptune, radio waves and the Higgs boson particle thought to explain how other particles get their mass.

Some people argue that math is just a tool invented by scientists to explain the natural world. But Tegmark contends the mathematical structure found in the natural world shows that math exists in reality, not just in the human mind.

And speaking of the human mind, could we use math to explain the brain?

Mathematics of consciousness

Some have described the human brain as the most complex structure in the universe. Indeed, the human mind has made possible all of the great leaps in understanding our world.

Someday, Tegmark said, scientists will probably be able to describe even consciousness using math. (Carl Sagan is quoted as having said, "the brain is a very big place, in a very small space.")

"Consciousness is probably the way information feels when it's being processed in certain, very complicated ways," Tegmark said. He pointed out that many great breakthroughs in physics have involved unifying two things once thought to be separate: energy and matter, space and time, electricity and magnetism. He said he suspects the mind, which is the feeling of a conscious self, will ultimately be unified with the body, which is a collection of moving particles.

But if the brain is just math, does that mean free will doesn't exist, because the movements of particles could be calculated using equations? Not necessarily, he said.

One way to think of it is, if a computer tried to simulate what a person will do, the computation would take at least the same amount of time as performing the action. So some people have suggested defining free will as an inability to predict what one is going to do before the event occurs.

But that doesn't mean humans are powerless. Tegmark concluded his talk with a call to action: "Humans have the power not only to understand our world, but to shape and improve it."

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## Math behind the Genetic Relationship Matrix - Biology

allele: a variant of a gene

dominant allele: an allele whose trait always shows up in the organism when the allele is present (written as uppercase letter)

gene: a section of DNA that codes for a certain trait

genotype: an organism's genetic makeup or allele combinations

heredity: the passing of traits from parents to offspring

heterozygous: having 2 different alleles for a trait.

homozygous: having two identical alleles for a trait.

phenotype: an organism's physical appearance or visible trait

probability: a number that describes how likely it is that an event will occur

Punnet Square: a diagram used to predict an outcome of a particular cross or breeding experiment

recessive allele: an allele that is masked when a dominant allele is present (written as lower case letter)

trait: a characteristic that an organism can pass on to its offspring through its genes

##### Did You Know? (Ag Facts)
• Apples are a member of the rose family. 1
• More than 2,500 varieties of apples are grown in the United States, but only the crabapple is native to North America. 1
• The average person eats 65 apples per year. 1
• Apples are 25% air, which is why they float in water. 1
##### Background Agricultural Connections

This lesson can be nested into a storyline as an episode exploring the phenomena of taste and other characteristics that can be observed in apples. In this episode, students investigate the question, "What makes apple characteristics different?" Phenomena-based lessons include storylines which emerge based upon student questions. Other lesson plans in the National Agricultural Literacy Curriculum Matrix may be used as episodes to investigate student questions needing science-based explanations. For more information about phenomena storylines visit nextgenstorylines.org.

Prior to this lesson, students should have a basic understanding of inherited traits and know that all cells of an organism have DNA. DNA is the blueprint providing the organism with coded instructions for proper function and development. Students should also know that genes are sections of DNA that are responsible for passing specific traits from parent to offspring. Students will need to be familiar with vocabulary such as phenotype, genotype, homozygous, and heterozygous to successfully complete the lesson and student worksheet and determine probabilities associated with possible offspring using a Punnett Square. Students will be introduced to several varieties of apples and discover how new varieties can be created through crossbreeding.

Key STEM Ideas

Genetics is the study of heredity, while heredity is the passing of traits from parents to offspring. This lesson will help solidify key genetics vocabulary words.

The main idea of this lesson is to show the application of genetic crossing for the benefit of agriculture by producing apples with a variety of traits.

Gregor Mendel was a priest who worked with the genetic crossing of pea plants. He would cross purebred short pea plants with purebred tall pea plants. Through his experiments he determined that some traits were visible in the plant (dominant traits) while others were not, but were still able to be passed on to future generations (recessive traits). Understanding what we see and what the genetic makeup of an organism is can be quite different. When you look at an organism, its physical characteristics are all dependent on a specific allele combination. This is the difference between phenotype and genotype. Students will use Punnett Squares in this lesson to help determine all the possible allele combinations in a genetic cross and their probabilities.

Crossbreeding allows breeders to create better quality apples by incorporating traits from two parent plants into the seeds of a new generation of plants. Breeders must understand both genotypes and phenotypes to accomplish this task. Breeders must also decide which traits are desirable and should be selected. This is an intensive process that involves breeding successive generations of apples with the preferred traits in order to get the final product. There are several crop modification techniques breeders use to develop new plant/fruit varieties.

Connections to Agriculture

Apples are an important agricultural crop. There are about 7,500 apple producers in the United States. Washington, New York, and Michigan are the leaders in apple production. Growers produce a variety of different kinds of apples. Some apples are better for baking while others are typically consumed fresh. Apples are a good snack choice as they satiate hunger, contain no fat and relatively few calories while being high in fiber and vitamin C.

Apples are grown through a process called grafting rather than being grown from seed. This is done because most apple varieties are self-unfruitful, which means their blossoms must be fertilized with the pollen of a separate variety in order to produce fruit. The fruit has traits from the parent tree, but the seeds inside will be a cross of the two varieties. This mixture of genetic material in the seeds means the grower won&rsquot know what traits a tree grown from these seeds will have and what the resulting fruit will taste like.

To avoid this uncertainty apple growers do not grow new trees from seed. Instead, new apple trees are propagated through a process called grafting. In this process a special cut is made into the rootstock of a tree. Then, they graft or transplant a section of a stem with leaf buds called a scion from a variety that has desirable traits into the cut. In time the two pieces fuse together allowing for growth of the scion. Eventually, blossoms on the scion will be pollinated and will produce a consistent variety of fruit with the desired traits.

The goal of apple breeding is to continuously produce quality apples with desirable traits. Cross breeding and genetic engineering are two methods that have allowed breeders to produce better quality apples. See Crop Modification Techniques)

##### Interest Approach - Engagement

This lesson has been adapted for online instruction and can be found on the 6-8th grade eLearning site.

1. Ask students to think about their favorite apple. Ask them why that variety is their favorite. Ask them why they think a green Granny Smith apple is so tart/sour? This should lead to a discussion about various apple traits such as sweetness, tartness, flavors, crunchiness, color, etc.
2. Tell students that there are thousands of varieties of apples grown in the United States. Most of the varieties will not be familiar to them because they are only found in orchards grown for research, the development of new apple varieties, or hobby orchards. Challenge students to try to list the top 10 apple varieties in the United States. These varieties are more likely to be familiar to your students in addition to other local varieties.
3. Ask students if they know how these different apple varieties became available.
4. Ask your students to use their understanding of heredity and genetics to explain how apple varieties could be developed. Use student responses to transition to Activity 1.
##### Procedures

This lesson investigates the phenomenon of apple taste along with other observed apple characteristics. Natural phenomena are observable events that occur in the universe that we can use our science knowledge to explain or predict.

Phenomenon-Based Episode: What makes apple varieties different?
Disciplinary Core Ideas: Growth and Development of Organisms
National Agricultural Literacy Outcome Theme: Science, Technology, Engineering, and Math

1. How do apple characteristics differ?
• Planning and Carrying Out Investigations
1. How are new varieties of apples created?
• Asking Questions and Defining Problems
1. What makes every apple of a given variety taste and look the same?
• Constructing Explanations and Designing Solutions

Activity 1: Apple Genetics - Making them Different (Episode Questions 1 and 2)

1. Give each student one copy of the Apple Genetics activity sheet. Divide the class into small groups of students (2-4).
2. Give each group of students the following supplies:
• 1 paper plate (this will be the cutting board as well as an area to keep the apples)
• 1 Braeburn Apple
• 1 Royal Gala Apple (Note: DO NOT hand out the Jazz apple yet).
• 1 knife (or pre-slice apples)
3. Have students draw a line down the center of their paper plate and label each side with "Gala" or "Braeburn." The apples will look similar, so it will be important to avoid confusing the two apples.
4. Have students complete "Part 1" and "Part 2" of the worksheet and then stop.
5. Project the Apple Genetics PowerPoint slides for students to see. Using slide 2, hold a brief class discussion about the traits they have observed in the apples so far. Draw on the student's prior knowledge of heredity and genetics to conclude that each trait is an expression of its genotype.
6. Use slide 3 of the PowerPoint to review vocabulary if needed. Make sure students are familiar with the terms.
7. Have students complete "Part 3" of the worksheet to review the possible genotypes of the Gala and Braeburn apples. These genotypes can be found on the worksheet and slide 4-5 of the PowerPoint.
8. Once students have finished their Punnet squares, give each group of students a Jazz apple. Students will follow the same procedure and complete "Part 4" and "Part 5" of the worksheet.
9. Facilitate a class discussion about the 3 varieties of apple (slide 6). Reveal to the students that the Jazz apple is a cross between the Gala and Braeburn apple. Using slide 7, share a few more facts about the Jazz Apple.
10. Talk about the concept of crossbreeding and how it is used to produce better quality organisms (slide 8).
11. Explain that the Honeycrisp apple (slide 9) was also developed by crossbreeding, and is a competitor of the Jazz apple.
12. Summarize with students by connecting what they know about genetics with what they have learned about apples:
• Genes determine genetic traits found in apples such as color, taste, and texture.
• To develop a new, improved variety of apple, apple breeders cross pollinate apple varieties. This form of sexual reproduction results in an offspring (seed) that is genetically different from the parent trees.
• Scientists use a knowledge of genetics and heredity to cross breed apples and produce new varieties of apples. The Jazz and Honeycrisp apples are examples.

Three Dimensional Learning Proficiency: Crosscutting Concepts
Students engage in scientific investigation as they investigate and build models and theories about the natural world.

Stability and Change: For both designed and natural systems, conditions that affect stability and factors that control states of change are critical elements to consider and understand.

Activity 2: Apple Genetics - Keeping Them the Same (Episode Question 3)

1. Ask students if they have ever eaten Jelly Belly jelly beans. Have they ever eaten or heard of the Jelly Belly jelly beans that have "bad" flavors like toothpaste, stinkbug, or stinky socks? (Perhaps in the game Beanboozled.) While this may be a fun game or practical joke, have a discussion with your students about what they (as consumers) want in their food. Conclude that every time they purchase milk, meat, bread, vegetables. or an apple, they want it to taste consistently the same without surprises.
2. Students have just learned how new varieties of apples are created. Ask, "How do apple farmers all across the nation grow specific varieties of apple that all taste and look the same? For example, how does a Granny Smith always taste like a Granny Smith and a Gala always taste like a Gala?" Does a [Granny Smith] grown in one region of the country taste the same as a [Granny Smith] grown in another region of the country?
3. To discover the answer, show Apple - How Does it Grow?

• Grafting, a form of asexual propagation is used by apple farmers to produce the apples we eat. It produces apples consistent to consumer expectations for each variety of apple by eliminating the genetic variability of sexual propagation methods.

Concept Elaboration and Evaluation:

After completing these activities, have students create a Venn Diagram to list both the similarities and differences found in sexual and asexual propagation methods. Discuss the benefits and drawbacks of each.

Phenomena Episode Extensions:

Effective phenomena-based instruction continues to evolve as students learn. New questions should arise throughout the learning process. The following questions may arise providing opportunity for other episodes in this storyline:

• Why can other fruits and vegetables be propagated with sexual reproduction (seeds) and produce a consistent crop, but apples cannot?
• What makes an apple (such as the Honeycrisp) crunchy?
• How was the Opal apple selectively bred to not brown after it is cut?
• How was the Arctic® apple genetically engineered to be non-browning?
##### Enriching Activities

Show the 4-minute video clip, Have We Engineered The Perfect Apple? to see the science behind the taste of the Honeycrisp apple.

If cut apples are in the room at the end of the lesson, ask students if they see any browning occurring. Discuss what causes this. Teach students about Arctic apples, a genetically modified apple which does not brown. Compare and contrast to the Opal apple, an apple variety selectively bred to be non-browning.

Listen to the NPR podcast "The Miracle Apple."

Guide students through an simulation activity to Make a New Apple Cultivar.

##### Sources

Activity 1 was originally written in the lesson "Apple Genetics" written by Kevin Atterberg (Culler Middle School, Lincoln NE), Erin Ingram, and Molly Brandt (University of Nebraska-Lincoln, IANR Science Literacy Initiative). The lesson was updated in 2018 to follow a phenomena-based format.

Phenomenon chart adapted from work by Susan German.
German, S. (2017, December). Creating conceptual storylines. Science Scope, 41(4), 26-28.
German, S. (2018, January). The steps of a conceptual storyline. Science Scope, 41(5), 32-34.

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