– all angles 60°
Before they can solve problems, however, students must first know what type of visual representation to create and use for a given mathematics problem. Some students—specifically, high-achieving students, gifted students—do this automatically, whereas others need to be explicitly taught how. This is especially the case for students who struggle with mathematics and those with mathematics learning disabilities. Without explicit, systematic instruction on how to create and use visual representations, these students often create visual representations that are disorganized or contain incorrect or partial information. Consider the examples below.
Mrs. Aldridge ask her first-grade students to add 2 + 4 by drawing dots.
Notice that Talia gets the correct answer. However, because Colby draws his dots in haphazard fashion, he fails to count all of them and consequently arrives at the wrong solution.
Mr. Huang asks his students to solve the following word problem:
The flagpole needs to be replaced. The school would like to replace it with the same size pole. When Juan stands 11 feet from the base of the pole, the angle of elevation from Juan’s feet to the top of the pole is 70 degrees. How tall is the pole?
Compare the drawings below created by Brody and Zoe to represent this problem. Notice that Brody drew an accurate representation and applied the correct strategy. In contrast, Zoe drew a picture with partially correct information. The 11 is in the correct place, but the 70° is not. As a result of her inaccurate representation, Zoe is unable to move forward and solve the problem. However, given an accurate representation developed by someone else, Zoe is more likely to solve the problem correctly.
Some students will not be able to grasp mathematics skills and concepts using only the types of visual representations noted in the table above. Very young children and students who struggle with mathematics often require different types of visual representations known as manipulatives. These concrete, hands-on materials and objects—for example, an abacus or coins—help students to represent the mathematical idea they are trying to learn or the problem they are attempting to solve. Manipulatives can help students develop a conceptual understanding of mathematical topics. (For the purpose of this module, the term concrete objects refers to manipulatives and the term visual representations refers to schematic diagrams.)
It is important that the teacher make explicit the connection between the concrete object and the abstract concept being taught. The goal is for the student to eventually understand the concepts and procedures without the use of manipulatives. For secondary students who struggle with mathematics, teachers should show the abstract along with the concrete or visual representation and explicitly make the connection between them.
A move from concrete objects or visual representations to using abstract equations can be difficult for some students. One strategy teachers can use to help students systematically transition among concrete objects, visual representations, and abstract equations is the Concrete-Representational-Abstract (CRA) framework.
If you would like to learn more about this framework, click here.
CRA is effective across all age levels and can assist students in learning concepts, procedures, and applications. When implementing each component, teachers should use explicit, systematic instruction and continually monitor student work to assess their understanding, asking them questions about their thinking and providing clarification as needed. Concrete and representational activities must reflect the actual process of solving the problem so that students are able to generalize the process to solve an abstract equation. The illustration below highlights each of these components.
One promising practice for moving secondary students with mathematics difficulties or disabilities from the use of manipulatives and visual representations to the abstract equation quickly is the CRA-I strategy . In this modified version of CRA, the teacher simultaneously presents the content using concrete objects, visual representations of the concrete objects, and the abstract equation. Studies have shown that this framework is effective for teaching algebra to this population of students (Strickland & Maccini, 2012; Strickland & Maccini, 2013; Strickland, 2017).
Kim Paulsen discusses the benefits of manipulatives and a number of things to keep in mind when using them (time: 2:35).
Kim Paulsen, EdD Associate Professor, Special Education Vanderbilt University
View Transcript
Transcript: Kim Paulsen, EdD
Manipulatives are a great way of helping kids understand conceptually. The use of manipulatives really helps students see that conceptually, and it clicks a little more with them. Some of the things, though, that we need to remember when we’re using manipulatives is that it is important to give students a little bit of free time when you’re using a new manipulative so that they can just explore with them. We need to have specific rules for how to use manipulatives, that they aren’t toys, that they really are learning materials, and how students pick them up, how they put them away, the right time to use them, and making sure that they’re not distracters while we’re actually doing the presentation part of the lesson. One of the important things is that we don’t want students to memorize the algorithm or the procedures while they’re using the manipulatives. It really is just to help them understand conceptually. That doesn’t mean that kids are automatically going to understand conceptually or be able to make that bridge between using the concrete manipulatives into them being able to solve the problems. For some kids, it is difficult to use the manipulatives. That’s not how they learn, and so we don’t want to force kids to have to use manipulatives if it’s not something that is helpful for them. So we have to remember that manipulatives are one way to think about teaching math.
I think part of the reason that some teachers don’t use them is because it takes a lot of time, it takes a lot of organization, and they also feel that students get too reliant on using manipulatives. One way to think about using manipulatives is that you do it a couple of lessons when you’re teaching a new concept, and then take those away so that students are able to do just the computation part of it. It is true we can’t walk around life with manipulatives in our hands. And I think one of the other reasons that a lot of schools or teachers don’t use manipulatives is because they’re very expensive. And so it’s very helpful if all of the teachers in the school can pool resources and have a manipulative room where teachers can go check out manipulatives so that it’s not so expensive. Teachers have to know how to use them, and that takes a lot of practice.
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There’s a growing demand for business analytics and data expertise in the workforce. But you don’t need to be a professional analyst to benefit from data-related skills.
Becoming skilled at common data visualization techniques can help you reap the rewards of data-driven decision-making , including increased confidence and potential cost savings. Learning how to effectively visualize data could be the first step toward using data analytics and data science to your advantage to add value to your organization.
Several data visualization techniques can help you become more effective in your role. Here are 17 essential data visualization techniques all professionals should know, as well as tips to help you effectively present your data.
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Data visualization is the process of creating graphical representations of information. This process helps the presenter communicate data in a way that’s easy for the viewer to interpret and draw conclusions.
There are many different techniques and tools you can leverage to visualize data, so you want to know which ones to use and when. Here are some of the most important data visualization techniques all professionals should know.
The type of data visualization technique you leverage will vary based on the type of data you’re working with, in addition to the story you’re telling with your data .
Here are some important data visualization techniques to know:
Pie charts are one of the most common and basic data visualization techniques, used across a wide range of applications. Pie charts are ideal for illustrating proportions, or part-to-whole comparisons.
Because pie charts are relatively simple and easy to read, they’re best suited for audiences who might be unfamiliar with the information or are only interested in the key takeaways. For viewers who require a more thorough explanation of the data, pie charts fall short in their ability to display complex information.
The classic bar chart , or bar graph, is another common and easy-to-use method of data visualization. In this type of visualization, one axis of the chart shows the categories being compared, and the other, a measured value. The length of the bar indicates how each group measures according to the value.
One drawback is that labeling and clarity can become problematic when there are too many categories included. Like pie charts, they can also be too simple for more complex data sets.
Unlike bar charts, histograms illustrate the distribution of data over a continuous interval or defined period. These visualizations are helpful in identifying where values are concentrated, as well as where there are gaps or unusual values.
Histograms are especially useful for showing the frequency of a particular occurrence. For instance, if you’d like to show how many clicks your website received each day over the last week, you can use a histogram. From this visualization, you can quickly determine which days your website saw the greatest and fewest number of clicks.
Gantt charts are particularly common in project management, as they’re useful in illustrating a project timeline or progression of tasks. In this type of chart, tasks to be performed are listed on the vertical axis and time intervals on the horizontal axis. Horizontal bars in the body of the chart represent the duration of each activity.
Utilizing Gantt charts to display timelines can be incredibly helpful, and enable team members to keep track of every aspect of a project. Even if you’re not a project management professional, familiarizing yourself with Gantt charts can help you stay organized.
A heat map is a type of visualization used to show differences in data through variations in color. These charts use color to communicate values in a way that makes it easy for the viewer to quickly identify trends. Having a clear legend is necessary in order for a user to successfully read and interpret a heatmap.
There are many possible applications of heat maps. For example, if you want to analyze which time of day a retail store makes the most sales, you can use a heat map that shows the day of the week on the vertical axis and time of day on the horizontal axis. Then, by shading in the matrix with colors that correspond to the number of sales at each time of day, you can identify trends in the data that allow you to determine the exact times your store experiences the most sales.
A box and whisker plot , or box plot, provides a visual summary of data through its quartiles. First, a box is drawn from the first quartile to the third of the data set. A line within the box represents the median. “Whiskers,” or lines, are then drawn extending from the box to the minimum (lower extreme) and maximum (upper extreme). Outliers are represented by individual points that are in-line with the whiskers.
This type of chart is helpful in quickly identifying whether or not the data is symmetrical or skewed, as well as providing a visual summary of the data set that can be easily interpreted.
A waterfall chart is a visual representation that illustrates how a value changes as it’s influenced by different factors, such as time. The main goal of this chart is to show the viewer how a value has grown or declined over a defined period. For example, waterfall charts are popular for showing spending or earnings over time.
An area chart , or area graph, is a variation on a basic line graph in which the area underneath the line is shaded to represent the total value of each data point. When several data series must be compared on the same graph, stacked area charts are used.
This method of data visualization is useful for showing changes in one or more quantities over time, as well as showing how each quantity combines to make up the whole. Stacked area charts are effective in showing part-to-whole comparisons.
Another technique commonly used to display data is a scatter plot . A scatter plot displays data for two variables as represented by points plotted against the horizontal and vertical axis. This type of data visualization is useful in illustrating the relationships that exist between variables and can be used to identify trends or correlations in data.
Scatter plots are most effective for fairly large data sets, since it’s often easier to identify trends when there are more data points present. Additionally, the closer the data points are grouped together, the stronger the correlation or trend tends to be.
Pictogram charts , or pictograph charts, are particularly useful for presenting simple data in a more visual and engaging way. These charts use icons to visualize data, with each icon representing a different value or category. For example, data about time might be represented by icons of clocks or watches. Each icon can correspond to either a single unit or a set number of units (for example, each icon represents 100 units).
In addition to making the data more engaging, pictogram charts are helpful in situations where language or cultural differences might be a barrier to the audience’s understanding of the data.
Timelines are the most effective way to visualize a sequence of events in chronological order. They’re typically linear, with key events outlined along the axis. Timelines are used to communicate time-related information and display historical data.
Timelines allow you to highlight the most important events that occurred, or need to occur in the future, and make it easy for the viewer to identify any patterns appearing within the selected time period. While timelines are often relatively simple linear visualizations, they can be made more visually appealing by adding images, colors, fonts, and decorative shapes.
A highlight table is a more engaging alternative to traditional tables. By highlighting cells in the table with color, you can make it easier for viewers to quickly spot trends and patterns in the data. These visualizations are useful for comparing categorical data.
Depending on the data visualization tool you’re using, you may be able to add conditional formatting rules to the table that automatically color cells that meet specified conditions. For instance, when using a highlight table to visualize a company’s sales data, you may color cells red if the sales data is below the goal, or green if sales were above the goal. Unlike a heat map, the colors in a highlight table are discrete and represent a single meaning or value.
A bullet graph is a variation of a bar graph that can act as an alternative to dashboard gauges to represent performance data. The main use for a bullet graph is to inform the viewer of how a business is performing in comparison to benchmarks that are in place for key business metrics.
In a bullet graph, the darker horizontal bar in the middle of the chart represents the actual value, while the vertical line represents a comparative value, or target. If the horizontal bar passes the vertical line, the target for that metric has been surpassed. Additionally, the segmented colored sections behind the horizontal bar represent range scores, such as “poor,” “fair,” or “good.”
A choropleth map uses color, shading, and other patterns to visualize numerical values across geographic regions. These visualizations use a progression of color (or shading) on a spectrum to distinguish high values from low.
Choropleth maps allow viewers to see how a variable changes from one region to the next. A potential downside to this type of visualization is that the exact numerical values aren’t easily accessible because the colors represent a range of values. Some data visualization tools, however, allow you to add interactivity to your map so the exact values are accessible.
A word cloud , or tag cloud, is a visual representation of text data in which the size of the word is proportional to its frequency. The more often a specific word appears in a dataset, the larger it appears in the visualization. In addition to size, words often appear bolder or follow a specific color scheme depending on their frequency.
Word clouds are often used on websites and blogs to identify significant keywords and compare differences in textual data between two sources. They are also useful when analyzing qualitative datasets, such as the specific words consumers used to describe a product.
Network diagrams are a type of data visualization that represent relationships between qualitative data points. These visualizations are composed of nodes and links, also called edges. Nodes are singular data points that are connected to other nodes through edges, which show the relationship between multiple nodes.
There are many use cases for network diagrams, including depicting social networks, highlighting the relationships between employees at an organization, or visualizing product sales across geographic regions.
A correlation matrix is a table that shows correlation coefficients between variables. Each cell represents the relationship between two variables, and a color scale is used to communicate whether the variables are correlated and to what extent.
Correlation matrices are useful to summarize and find patterns in large data sets. In business, a correlation matrix might be used to analyze how different data points about a specific product might be related, such as price, advertising spend, launch date, etc.
While the examples listed above are some of the most commonly used techniques, there are many other ways you can visualize data to become a more effective communicator. Some other data visualization options include:
Creating effective data visualizations requires more than just knowing how to choose the best technique for your needs. There are several considerations you should take into account to maximize your effectiveness when it comes to presenting data.
Related : What to Keep in Mind When Creating Data Visualizations in Excel
One of the most important steps is to evaluate your audience. For example, if you’re presenting financial data to a team that works in an unrelated department, you’ll want to choose a fairly simple illustration. On the other hand, if you’re presenting financial data to a team of finance experts, it’s likely you can safely include more complex information.
Another helpful tip is to avoid unnecessary distractions. Although visual elements like animation can be a great way to add interest, they can also distract from the key points the illustration is trying to convey and hinder the viewer’s ability to quickly understand the information.
Finally, be mindful of the colors you utilize, as well as your overall design. While it’s important that your graphs or charts are visually appealing, there are more practical reasons you might choose one color palette over another. For instance, using low contrast colors can make it difficult for your audience to discern differences between data points. Using colors that are too bold, however, can make the illustration overwhelming or distracting for the viewer.
Related : Bad Data Visualization: 5 Examples of Misleading Data
No matter your role or title within an organization, data visualization is a skill that’s important for all professionals. Being able to effectively present complex data through easy-to-understand visual representations is invaluable when it comes to communicating information with members both inside and outside your business.
There’s no shortage in how data visualization can be applied in the real world. Data is playing an increasingly important role in the marketplace today, and data literacy is the first step in understanding how analytics can be used in business.
Are you interested in improving your analytical skills? Learn more about Business Analytics , our eight-week online course that can help you use data to generate insights and tackle business decisions.
This post was updated on January 20, 2022. It was originally published on September 17, 2019.
Not long ago, the ability to create smart data visualizations (or dataviz) was a nice-to-have skill for design- and data-minded managers. But now it’s a must-have skill for all managers, because it’s often the only way to make sense of the work they do. Decision making increasingly relies on data, which arrives with such overwhelming velocity, and in such volume, that some level of abstraction is crucial. Thanks to the internet and a growing number of affordable tools, visualization is accessible for everyone—but that convenience can lead to charts that are merely adequate or even ineffective.
By answering just two questions, Berinato writes, you can set yourself up to succeed: Is the information conceptual or data-driven? and Am I declaring something or exploring something? He leads readers through a simple process of identifying which of the four types of visualization they might use to achieve their goals most effectively: idea illustration, idea generation, visual discovery, or everyday dataviz.
This article is adapted from the author’s just-published book, Good Charts: The HBR Guide to Making Smarter, More Persuasive Data Visualizations.
Know what message you’re trying to communicate before you get down in the weeds.
Knowledge workers need greater visual literacy than they used to, because so much data—and so many ideas—are now presented graphically. But few of us have been taught data-visualization skills.
Inexpensive tools allow anyone to perform simple tasks such as importing spreadsheet data into a bar chart. But that means it’s easy to create terrible charts. Visualization can be so much more: It’s an agile, powerful way to explore ideas and communicate information.
Don’t jump straight to execution. Instead, first think about what you’re representing—ideas or data? Then consider your purpose: Do you want to inform, persuade, or explore? The answers will suggest what tools and resources you need.
Not long ago, the ability to create smart data visualizations, or dataviz, was a nice-to-have skill. For the most part, it benefited design- and data-minded managers who made a deliberate decision to invest in acquiring it. That’s changed. Now visual communication is a must-have skill for all managers, because more and more often, it’s the only way to make sense of the work they do.
Information visualization is not as easy as it might first appear, particularly when you are examining complex data sets. How do you deliver a “good” representation of the information that you bring out of the data that you are working with?
While this may be a subjective area of information visualization and, of course, there are exceptions to the guidelines (as with all areas of design – rules are for breaking if by breaking them you achieve your purpose) it’s best to begin with the four guidelines outlined by Edward Tufte.
Edward Tufte is, perhaps, the world’s leading authority on information design and data visualization . He is an American statistician and a Professor Emeritus at Yale University (for political sciences, computer sciences and statistics).
He has authored several books and papers on analytic design and is a strong proponent for the power of visualizing data. In particular his books, Visual Display of Quantitative Information, Envisioning Information, Visual Explanations and Beautiful Evidence are considered to be definitive works in the field of information visualization. The New York Times called him; “The Leonardo da Vinci of data.”
Within his works you can find four essential guidelines for visual information representation:
Visual integrity, maximizing the data-ink ratio, aesthetic elegance, tufte’s criteria for good visual information representation.
The purpose of “good’ representations is to deliver a visual representation of data to the user of that representation which is “most fit for purpose”. This will enable the user of the information to make the most out of the representation. There is no single hard and fast rule for creating good representations because the nature of the data, the users of that data, etc. are enormously varied.
Thus we find ourselves with a set of criteria which can be applied to most visual representations, as suggested by Tufte, to judge their fitness for purpose. It must be acknowledged, however, that these criteria can be bent or even broken if doing so serves a purpose for the user of the information representation.
There could be hours of debate as to what constitutes graphical excellence but Tufte offers that in data representations at least it should provide the user with; “the greatest number of ideas, in the shortest time, using the least amount of ink, in the smallest space.”
In short as with many other areas of user experience – the focus here is on usability ; it is completely possible to create beautiful graphical representations of data which fail to deliver on these premises. In fact, it might be said that this occurs so often that the power of data visualization is muted because people have come to expect such visualizations to be decorative rather than valuable.
The graphic above, relating to US employment statistics in March 2015, offers many ideas in a very small space and is easy to digest. We’d suggest it meets the criteria of “graphical excellence”.
This is a confusing term. When Tufte refers to “visual integrity” he is invoking an almost moral position in that the representation should neither distort the underlying data nor create a false impression or interpretation of that data.
In practice this means that numerical scales should be properly proportionate (and not fudged to exaggerate the fall or rise of a curve at a particular point, for example). That variations, when they occur, should relate to the data rather than to the artistic interpretation of that data. The dimensions used within an image should be limited to the dimensions within the data and should never exceed them and finally that the keys (or legends) should be undistorted and unambiguous.
This bar graph fails to give us enough information to be useful and thus fails in delivering “visual integrity”.
Tufte recommends that we pay attention to the way that a visualization is compiled; in that all superfluous elements (to the user) should be removed. He offers the idea that borders, backgrounds, use of 3D, etc. may do nothing but serve to distract the user from the information itself. He promotes that you give priority to the data and how it will be used and not to the visual appearance of that representation.
He also provides a mathematical formula for a data-ink ratio:
Data-Ink/Total Ink Used
This is simply a comparison of the ink needed to clearly and unambiguously present the data to the ink actually used (including aesthetic considerations). The closer the ratio is to 1 – the less distracting your representation is likely to be and thus the more useful it is likely to be for your user.
This image of business processes with an ERP environment is quite good at conveying which business functions are affected by the ERP processes but what purpose does the color scheme serve?
Tufte’s interpretation of aesthetic elegance is not based on the “physical beauty” of an information visualization but rather the simplicity of the design evoking the complexity of the data clearly.
He holds up Minard’s visualization (pictured below) of Napoleon’s March in the Russian Campaign as an example of aesthetic elegance.
Tufte’s guidelines are not prescriptive but rather designed to assist the information visualization professional in creating usable and useful information representations. At their core his rules can be boiled down to keeping things as simple and as honest as possible. The rest simply ensure that you adapt to complexity in the most creative and basic way possible.
UX designers will see clear links between their own design work on products and the design of information representations.
You can find all of Edward Tufte’s work via his website .
Find out more about Charles Joseph Minard and his map of Napoleon’s Russian Campaign.
You can also find an interesting analysis of Minard’s map here .
Hero Image: Author/Copyright holder: Kitware Inc. Copyright terms and licence: CC BY-ND 2.0
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Humans make decisions about food every day. The visual system provides important information that forms a basis for these food decisions. Although previous research has focused on visual object and category representations in the brain, it is still unclear how visually presented food is encoded by the brain. Here, we investigate the time-course of food representations in the brain. We used time-resolved multivariate analyses of electroencephalography (EEG) data, obtained from human participants (both sexes), to determine which food features are represented in the brain and whether focused attention is needed for this. We recorded EEG while participants engaged in two different tasks. In one task, the stimuli were task relevant, whereas in the other task, the stimuli were not task relevant. Our findings indicate that the brain can differentiate between food and nonfood items from ∼112 ms after the stimulus onset. The neural signal at later latencies contained information about food naturalness, how much the food was transformed, as well as the perceived caloric content. This information was present regardless of the task. Information about whether food is immediately ready to eat, however, was only present when the food was task relevant and presented at a slow presentation rate. Furthermore, the recorded brain activity correlated with the behavioral responses in an odd-item-out task. The fast representation of these food features, along with the finding that this information is used to guide food categorization decision-making, suggests that these features are important dimensions along which the representation of foods is organized.
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If you’re struggling to get enough protein this visual guide will help you select which foods to eat each day.
Protein is an essential part of our diets. It's a crucial element in what helps our bodies function properly. Protein helps regulate hormones, transports molecules, acts as an enzyme for chemical reactions and more.
Everyone has different dietary requirements, but for the average person, 100 grams of protein daily is ideal. Keep in mind that if you're active, you may need more protein in your diet.
This visual guide shows what 100 grams of protein look like whether you follow a vegan, vegetarian or omnivore diet. Use it to put your daily protein needs into perspective.
The grams were calculated by taking the information from the nutrition facts label on packaged items and weighing them when necessary. The gram amounts listed in this guide are specific to the products used for this experiment, so your numbers may vary if you look at a different brand of bread or yogurt.
If you don't have any dietary restrictions, eating 100 grams of protein per day should be pretty easy. Here's one way to do it:
Everything pictured above comes to 103 grams, which puts you slightly over the 100-gram goal.
As you can see, getting 100 grams of protein from animal products doesn't take much. This photo shows:
This amounts to a perfect 100. If you ate all of this in a day, plus bread and other nonanimal products, you would easily surpass 100 grams of protein in a day.
For vegetarians, 100 grams of protein might look like:
This actually comes out to 99 grams of protein, which is pretty close and still a great number to hit for a day.
What you see isn't totally what you get with this photo. In the photo, you see:
This amounts to 79 grams of protein. If we double up on the mixed nuts, chia seeds and hemp seeds, this brings us to 93 grams of protein. You could add an extra tablespoon of peanut butter or eat a full cup of oats, instead of half a cup, to come closer to that 100-gram goal.
Also, this plate doesn't include any high-protein vegan meat substitutes, such as tofu, tempeh or plant-based meats like the Impossible Burger . Those food sources can make it much easier to get 100 grams of protein than someone who eats a vegan diet .
Transfer learning, the re-application of previously learned higher-level regularities to novel input, is a key challenge in cognition. While previous empirical studies investigated human transfer learning in supervised or reinforcement learning for explicit knowledge, it is unknown whether such transfer occurs during naturally more common implicit and unsupervised learning and if so, how it is related to memory consolidation. We compared the transfer of newly acquired explicit and implicit abstract knowledge during unsupervised learning by extending a visual statistical learning paradigm to a transfer learning context. We found transfer during unsupervised learning but with important differences depending on the explicitness/implicitness of the acquired knowledge. Observers acquiring explicit knowledge during initial learning could transfer the learned structures immediately. In contrast, observers with the same amount but implicit knowledge showed the opposite effect, a structural interference during transfer. However, with sleep between the learning phases, implicit observers switched their behaviour and showed the same pattern of transfer as explicit observers did while still remaining implicit. This effect was specific to sleep and not found after non-sleep consolidation. Our results highlight similarities and differences between explicit and implicit learning while acquiring generalizable higher-level knowledge and relying on consolidation for restructuring internal representations.
The authors have declared no competing interest.
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As knowledge can be condensed in different non-verbal ways of representation, the integration of graphic and visual representations and design in research output helps to expand insight and understanding. Layers of visual charts, maps, diagrams not only aim at synergizing the complexity of a topic with visual simplicity, but also to guide a personal search for and insights into knowledge. However, from research over graphic representation to interpretation and understanding implies a move that is scientific, epistemic, artistic and, last but not least, ethical. This article will consider these four aspects from both the side of the researcher and the receiver/interpreter from three different perspectives. The first perspective will consider the importance of visual representations in science and its recent developments. As a second perspective, we will analyse the discussion concerning the use of diagrams in the philosophy of mathematics. A third perspective will be from an artistic perspective on diagrams, where the visual tells us (sometimes) more than the verbal.
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This is the school typically associated with the mathematician David Hilbert. Although he himself saw formalism as a particular strategy to solve certain specific mathematical questions such as the consistency of arithmetic, nevertheless in the hands mainly of the French Bourbaki group it became an overall philosophy and the famous expression that mathematics is a game of meaningless signs was born. See (Detlefsen, 2005 ).
This seemingly simple graph consisting of 10 vertices and 15 edges is nevertheless of supreme importance in graph theory because of the impressive list of properties it possesses. Starikova ( 2017 ) presents a nice and thorough analysis of the graph (in order to discuss its aesthetic qualities). We just mention that the graph has 120 symmetries.
To be found at http://mathworld.wolfram.com/PetersenGraph.html , consulted Sunday, 17 September 2017.
A famous example is a proof of Augustin Cauchy wherein he made the mistake of inverting the quantifiers. A statement of the form ‘For all x, there is a y such that …’ was interpreted as ‘There is a y, such that for all x …’, which is a stronger statement. It is interesting to mention that this case was already (partially) studied by Imre Lakatos, see (Lakatos, 1976 , Appendix 1), who is often seen as the founding father of the study of mathematical practices.
That being said, the interest in the topic is growing. We just mention (Giaquinto, ), (Manders, ), (Giardino, ) and (Carter, 2010 ) as initiators. Of special interest is the connection that is being made between the philosophical approach and the opportunities offered by cognitive science to study the multiple ways that diagrams can be used an interpreted, see (Mumma & Hamami, 2013 ).
It is interesting that, under the same topic, David Bridges (this volume) develops a similar point of view on arts-based research for education. While Bridges questions the ambiguity of the potential and use of artistic means and expressions as research, we rather consider artistic expressions as enriching methods for knowledge construction, opening new insights by their complexity and layeredness.
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Thanks to Joachim Frans (2017) who directed my attention to the work of Nelsen (1993, 2000) in his inspiring Ph.D. thesis on ‘Mathematical explanation’.
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Kathleen Coessens
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Karen François
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Jean Paul Van Bendegem
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Coessens, K., François, K., Van Bendegem, J.P. (2021). Understanding Without Words: Visual Representations in Math, Science and Art. In: Smeyers, P., Depaepe, M. (eds) Production, Presentation, and Acceleration of Educational Research: Could Less be More?. Educational Research, vol 11. Springer, Singapore. https://doi.org/10.1007/978-981-16-3017-0_9
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Hair shedding scores: more than heat stress.
Jamie Courter State Beef Genetics Extension Specialist
Traditionally, when the topic of hair shedding arises, it is in the context of mitigating summer heat stress among cattle in the southeastern U.S. while grazing toxic fescue. This is not in error. It is estimated that cattle suffering from fescue toxicosis and heat stress alone cost the beef industry over a billion dollars a year. This makes hair shedding an economically relevant trait in cattle.
As research into hair shedding continues, more information about its importance becomes known. Most recently, a relationship between hair shedding in cattle and their ability to sense change in daylight has been discovered. This suggests that cattle who shed their winter coats earlier are more able to adapt to their environment making hair shedding an indicator trait in cattle regardless of location.
This MU Extension guide is meant to 1) provide background information on the economic importance of hair shedding scores, 2) introduce the hair shedding (1-5) scoring system, 3) discuss how to implement hair shedding into a selection program, and 4) promote hair shedding as a management tool.
Anytime a trait can directly be linked to profitability, it is characterized as economically relevant. In the case of hair shedding, cows who shed their winter coat earlier tend to wean heavier calves (Genetic correlation = -0.19). Figure 1 shows the average weaning weight of calves born to dams who began shedding their winter coats between March and July. The average weight of a calf born to a dam that shed in March was 57.2 lbs heavier than those that shed in July.
Most research that investigates the relationship between hair shedding and profitability target weaning weight because it encompasses many different production issues created by heat stress. To start, cows who undergo heat stress in the summer are less likely to get pregnant early in the breeding season. This could be due to low body condition scoring (BCS) because of heat stress while grazing summer pasture before fall breeding, or directly due to heat stress during spring breeding. Regardless, cows bred later in the season also calve later and therefore wean lighter calves.
Secondly, cows who undergo stress after calving may see an impact on milk production, which also impacts weaning weight.
Because increases in temperature happen at the same time hours of daylight are increasing, it is difficult to identify whether animals start shedding due to changes in temperature or daylight. Recent research conducted at the University of Missouri investigated the relationship between the DNA of the animal and temperature 30 days prior to the collection of a hair shedding score. This analysis only identified 17 interactions between DNA and temperature that influenced hair shedding. In a second analysis, researchers instead investigated the interaction between the DNA of the animal and the average length of daylight 30 days before the hair shedding score was observed. This time, there were 1,040 significant DNA-by-daylength associations identified. This supports the idea that cattle shed their winter coats in response to increasing amounts of daylight instead of a drop in temperature. This association is important because it promotes hair shedding as an indicator of an animal’s ability to sense and respond to their environment.
What/How: Hair shedding scores represent a visual appraisal of the extent of hair shedding and are reported on a 1 to 5 scale ( Figure 2 ) in which:
Half scores, such as 3.5, are not reported. In general, cattle tend to shed hair from the front to the back and from their topline to their belly ( Figure 3 ), but there is individual animal variation in this pattern. Typically, animals begin shedding around their neck, followed by their topline. The last spots to shed are an animal’s lower quarter above its hock and its underline.
When: It is only necessary to collect hair shedding scores once in late spring or early summer. The date to evaluate cattle for shedding progress will vary by geographic location and environmental conditions. The goal should be to score cattle when there is the most variation in hair shedding within a herd. In other words, a few animals with a hair shedding score of 1 or 5 with a majority receiving a hair shedding score of 3. Mid-May has been identified as an ideal hair shedding evaluation period for cattle in the Southeastern U.S. As a rule of thumb, the hotter and more humid the climate the earlier in the spring scores should be collected.
Who: If using hair shedding score as a selection method or reporting scores to a breed association, all cows in the herd should be observed. While it is recommended to score all animals in a herd on the same day, it is important to keep in mind that males tend to begin shedding a few weeks prior to females and therefore should likely be scored separately.
Where: Being a subjective observation of the amount of hair an animal has shed, these scores are easy to collect. This can be done as cattle pass through a chute during routine handling timeframes or while out on pasture.
As a moderately heritable trait (h2 = 0.35 to 0.42), producers can expect to create positive genetic change in their herds by simply adding hair shedding scores as a selection criterion when making selection decisions. To do this, producers would need to assess the hair shedding score of the whole herd on the same day, consider culling older animals with higher scores (more hair), and selecting the replacement heifers who shed earlier in the season in addition to other components of interest.
In addition to phenotypic selection, some bulls will also have an expected progeny difference (EPD) for hair shedding available for use. If available, selecting bulls with a lower hair shedding EPD will result in daughters born who shed earlier in the season, on average.
When using hair shedding as a selection criterion, it is important to also consider the age and nutrient requirements of the female. Yearlings and first-calf heifers tend to have higher hair shedding scores compared to older, established cows ( Figure 4 ). This does not mean younger females are necessarily worse shedders than their dams. Younger cows are, by default, the most nutritionally stressed as they are growing and raising a calf while also growing themselves. Therefore, it may be more beneficial to rank and select younger females within their age group instead of comparing them to older herd mates.
When evaluating the effect of age on hair shedding score, the average score for each age group tends to decrease as age increases ( Figure 4 ). This could reflect the impact of late shedding on production. Cows who shed their winter coats later in the summer may have fallen out of the herd due to weaning lighter calves, failure to conceive, or low body condition.
It is anticipated that hair shedding scores could be used in conjunction with body condition scores to assess the current nutritional stress of the herd. Genetic associations were also discovered between hair shedding and functions related to metabolism. Therefore, hair shedding may also pose as an indication of an animal’s overall nutritional plane, thus helping to inform management decisions. It is no coincidence that younger females shed their coats later than their older herd mates. Similarly, older cows who may have had a ‘hard winter’ would shed later in the year. The repeatability of hair shedding is only 45%, which indicates over half (55%) of the variance in hair shedding is due to year-to-year differences in management and environment of the cow. Understanding that later hair shedding (higher scores) indicates increased nutritional demands could be used to identify animals who would benefit from additional supplemental feed heading into spring and summer.
Although hair shedding has traditionally been associated with heat stress and fescue toxicosis, recent research shows this quick and easy phenotypic assessment of cattle could be a trait of even more economic importance. Producers wishing to select females based on hair shedding scores can do so based on a simple 1 to 5 scoring system. With its moderate heritability, combining this score with a hair shedding EPD or score on bulls would result in positive genetic progress over time.
More detailed information on the scoring system and some frequently asked questions can be found in publication G2041, How to Use the Hair Shedding Scale .
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Visual Representation refers to the principles by which markings on a surface are made and interpreted. Designers use representations like typography and illustrations to communicate information, emotions and concepts. Color, imagery, typography and layout are crucial in this communication. Alan Blackwell, cognition scientist and professor ...
Definition. The concept of "representation" captures the signs that stand in for and take the place of something else [ 5 ]. Visual representation, in particular, refers to the special case when these signs are visual (as opposed to textual, mathematical, etc.). On the other hand, there is no limit on what may be (visually) represented ...
The use of visual representations (i.e., photographs, diagrams, models) has been part of science, and their use makes it possible for scientists to interact with and represent complex phenomena, not observable in other ways. Despite a wealth of research in science education on visual representations, the emphasis of such research has mainly been on the conceptual understanding when using ...
Despite the notable number of publications on the benefits of using visual representations in a variety of fields (Meyer, Höllerer, Jancsary, & Van Leeuwen, 2013), few studies have systematically investigated the possible pitfalls that exist when creating or interpreting visual representations.Some information visualization researchers, however, have raised the issue and called to action ...
Information visualization is the process of representing data in a visual and meaningful way so that a user can better understand it. Dashboards and scatter plots are common examples of information visualization. Via its depicting an overview and showing relevant connections, information visualization allows users to draw insights from abstract ...
Consequently, a visual representation is an event, process, state, or object that carries meaning and that is perceived through the visual sensory channel. Of course, this is a broad definition. It includes writing, too, because writing is perceived visually and refers to a given meaning.
Visual representations not only promote generative learning (Fiorella & Mayer, 2015) but also, in our understanding, active learning, when it manages to mobilize people to think about what they want to learn, for example, about living beings or to plan a way of approaching a problem, such as the design of a prototype to move fragile objects ...
The Role of EFs in Making Sense of Visual Representations. Cognitive scientists broadly agree that the complexity and amount of information to be processed in visual representations can be cognitively demanding, particularly when the relationships between representations are meaningful and will lead the learner to build a broader understanding of the concept being represented (Phillips et al ...
A good visual representation is ideally a general-purpose representation that can be used profitably not just in visual recognition tasks, but in a broader range of downstream tasks.
The prostriata has a distinctive representation from V1 over a wide visual field (up to 60 deg), and responds more to fast motion (570 deg per sec) than the moderate-speed motion (38 deg per sec) 29.
Visual representation is one of the most efficient decision making techniques. Visual representations illuminate the links and connections, presenting a fuller picture. It's like having a compass in your decision-making journey, guiding you toward the correct answer. 3. Professional Development.
A good pictorial representation need not simulate visual experience any more than a good painting of a unicorn need resemble an actual unicorn. When designing user interfaces, all of these techniques are available for use, and new styles of pictorial rendering are constantly being introduced. 5.4.1 Summary
Visual representations can also be simple illustrations that young children add to stories to enhance a story they are telling together (Gelmini-Hornsby et al., 2011). By externalizing one's understanding in a visual representation, not only is a peer's understanding improved, ...
Data and information visualization ( data viz/vis or info viz/vis) [2] is the practice of designing and creating easy-to-communicate and easy-to-understand graphic or visual representations of a large amount [3] of complex quantitative and qualitative data and information with the help of static, dynamic or interactive visual items.
Such diverse representations of the visual world likely triggered the language models' misconceptions. While the models struggled to perceive these abstract depictions, they demonstrated the creativity to draw the same concepts differently each time. When the researchers queried LLMs to draw concepts like strawberries and arcades multiple ...
When representation is not enough. However, representation simply is not enough—especially when it is one-dimensional, superficial, or not actually representative. Some scholars describe how ...
Page 5: Visual Representations. Yet another evidence-based strategy to help students learn abstract mathematics concepts and solve problems is the use of visual representations. More than simply a picture or detailed illustration, a visual representation—often referred to as a schematic representation or schematic diagram— is an accurate ...
This type of chart is helpful in quickly identifying whether or not the data is symmetrical or skewed, as well as providing a visual summary of the data set that can be easily interpreted. 7. Waterfall Chart. A waterfall chart is a visual representation that illustrates how a value changes as it's influenced by different factors, such as time ...
Visu al information plays a fundamental role in our understanding, more than any other form of information (Colin, 2012). Colin (2012: 2) defines. visualisation as "a graphica l representation ...
Data visualization is the graphical representation of information and data. By using v isual elements like charts, graphs, and maps, data visualization tools provide an accessible way to see and understand trends, outliers, and patterns in data. Additionally, it provides an excellent way for employees or business owners to present data to non ...
Summary. Not long ago, the ability to create smart data visualizations (or dataviz) was a nice-to-have skill for design- and data-minded managers. But now it's a must-have skill for all managers ...
Tufte's Criteria for Good Visual Information Representation. The purpose of "good' representations is to deliver a visual representation of data to the user of that representation which is "most fit for purpose". This will enable the user of the information to make the most out of the representation. There is no single hard and fast ...
Visual representation techniques in CAS aim to display all available data to the surgeon in order to facilitate pre and intraoperative decision-making. At the most basic level, visualization techniques are applied to medical images to show the patient's anatomy, the relevant pathology, and the embedding of the latter in the former. ...
However, multiple visual representations are not always more effective for promoting learning (Rau, Aleven, & Rummel, 2015). It is critical to remember that—even if a visual representation is more concrete than a symbolic representation—the visual representation
The visual system provides important information that forms a basis for these food decisions. Although previous research has focused on visual object and category representations in the brain, it is still unclear how visually presented food is encoded by the brain. Here, we investigate the time-course of food representations in the brain.
In an influential article, Jones et al. (1995) provide evidence that auditory distraction by changing relative to repetitive auditory distracters (the changing-state effect) did not differ between a visual-verbal and visual-spatial serial recall task, providing evidence for an amodal mechanism for the representation of serial order in short-term memory that transcends modalities.
Two slices (2 ounces) of turkey bacon (10 grams) 3 ounces of turkey breast (24 grams) One can of tuna (27 grams) This amounts to a perfect 100. If you ate all of this in a day, plus bread and ...
Transfer learning, the re-application of previously learned higher-level regularities to novel input, is a key challenge in cognition. While previous empirical studies investigated human transfer learning in supervised or reinforcement learning for explicit knowledge, it is unknown whether such transfer occurs during naturally more common implicit and unsupervised learning and if so, how it is ...
The first one considered the importance of visual representations in science and its recent debate in education. It was already shown by philosophers of the Wiener Kreis that visual representation could serve for a better understanding and dissemination of knowledge to the broader public. As knowledge can be condensed in different non-verbal ...
Hair Shedding Scores What/How: Hair shedding scores represent a visual appraisal of the extent of hair shedding and are reported on a 1 to 5 scale (Figure 2) in which: 1: Cattle have shed 100% of their winter coat. All that remains is a shorter, smooth, summer coat. 2: Cattle have shed 75% of their winter coat, with a small amount of hair left ...