Author Archives: Dr. Darrin

Review of “The Titanic: Lost…and Found”

This post is a review of the book The Titanic: Lost and Found (Step-Into-Reading, Step 4) by Judy Donnelly (pp. 48).

The Summary

This text covers the classic story of the sinking of the Titanic in the early part of the 20th century. Originally build as unsinkable the Titanic collided with an iceberg and sank on its first voyage from Europe to America.

The text describes the accommodations and size of the ship. Such amenities as a pool and dining halls are depicted. At the time, the Titanic was also the largest passenger ship ever built.

When the ship had its incident with the iceberg people were supposedly laughing and joking as they were called to the deck for evacuation. This is actually an emotionally poweful moment in the text that a small child will miss. The people actually believed the foolish claim that a ship was unsinkable. To make matters worse, there were not enough lifeboats as even the builders of the ship arrogantly believed this as well.

Adding to the discouragement was the fact that a nearby ship ignored the radio calls of the sinking Titanic because their radio was turned off. When the people finally began to realize the danger they were in fear quickly set in. For whatever reason, the musicians continue to play music to try and keep the people calm and even played a hymn right before the final sinking of the ship. A somewhat chilling ending.

The book then concludes with the people in the lifeboat being rescued, it mentions changes to laws to prevent this disaster from happening again, and the final section of the text shares the story of how the Titanic was found in the 1980’s by researchers.

The Good

This book is written in simple language for small children. It can be read by early primary students. This text also provides a good introduction into one of the great tragedies of modern western history.

The illustrations also help to describe what is happening in the text. Lastly, the text is not that long and probably can be read in a few days by a child alone.

The Bad

There is little to complain about with this text. It should be in any primary teacher’s library. The only problem may be that it is a paperback book so it will not last long enduring the wear and tear that comes from small children.

The Recommendation

There will be no regrets if you purchase this book for your classroom or home.

Advertisements

Absolute Value Equations & Inequalities

The absolute value of a number is its distance from 0.  For example, 5 and -5 both have an absolute value of 5 because both are 5 units from 0. The symbols used for absolute value are |  | with a number or variable placed inside the vertical bars. With this knowledge lets look at an example of an absolute value.

1.png

The answer is +5 because both 5 and -5 are 5 units from 0.

In this post, we will look at equations and inequalities that use absolute values.

Solving one Absolute Value Equations

It is also possible to have inequalities with absolute values. To solve these you want to isolate the absolute value and solve the positive and also the negative version of the answer. Lastly, you never manipulate anything inside the absolute value brackets. you only manipulate and simplify values outside of the brackets. Below is an example.

1.png

As you can see absolute value inequalities involves solving two equations. Below is an example involving multiplication.

1

Notice again how the values inside the absolute value were never changed. This is important when solving absolute value inequalities.

Solving Two Absolute Values Equations

Solving two absolute values is not that difficult. You simply make one of the absolute values negative for one equation and positive for another. Below is an example.

1.png

Absolute Value Inequalities

Absolute value inequalities require a slightly different approach. You can rewrite the inequality in double inequality form and solve appropriately when the inequality is “less than.” Below is an example.

1.png

You can see that we put the absolute value in the middle and simply solved for x. you can even write this using interval notation as shown below.

1

“Greater than” inequalities are solved the same as inequalities with equal signs. You use the “or” concept to solve both inequalities.

1.png

The interval notation is as follows

1

We use the union sign in the middle is used in place of the word “or”.

Conclusion 

This post provided a brief overview of how to deal with absolute values in both equations and inequalities.

Modifying Text and Creating Commands in LaTeX

In this post, we are going to explore to separate features available in LaTeX. These two features are modifying the text size and creating custom commands.

Modifying Text

You can change the size and shape of text using many different declarations/environments in LaTeX. Declarations and environments serve the same purpose the difference is in the readability of the code. In the example below, we use an environment to make the text bigger than normal. The code is first followed by the example

\documentclass{article}
\usepackage[english]{babel}
\usepackage{blindtext}
\begin{document}
\begin{huge}
\blindtext
\end{huge}
\end{document}

1.png

Here is what we did.

  1. We create a document with the class of article
  2. We used the “babel” and “blindtext” packages to create some filler text.
  3. Next, we began the document
  4. We create the environment “huge” for enlarging the text.
  5. We used the declaration  “\blindtext” to create the paragraph
  6. We closed the “huge” environment with the “end” declaration
  7. We end the document

If you ran this code you will notice the size of the text is larger than normal. Of course, you can bold and do many more complex things to the text simultaneously. Below is the same example but with the text bold and in italics

\documentclass{article}
\usepackage[english]{babel}
\usepackage{blindtext}
\begin{document}
\begin{huge}
\bfseries
\textit
\blindtext
\end{huge}
\end{document}

1.png

The code is mostly the same with the addition of “\bfseries” for bold and  “\texit” for italics.

Making Commands

It is also possible to make custom commands in LaTeX. This can save a lot of time for repetitive practices. In the example below, we create a command to automatically print the name of this blog’s web address.

\documentclass{article}
\newcommand{\ert}{\bfseries{educationalresearchtechniques}}
\begin{document}
The coolest blog on the web is \ert
\end{document}

1.png

In the code, we use the declaration “\newcommand” in the preamble. This declaration had the command “\ert” which is the shorthand for the code to the right which is “\bfseries{educationalresearchtechniques}. This code tells LaTeX to bold the contents inside the brackets.

The next step was to begin the document. Notice how we used the “\ert” declaration and the entire word educationalresearchtechniques was printed in bold in the actual pdf.

It is also possible to make commands that format text. Below is an example.

\documentclass{article}
\newcommand{\mod}[1]{\textbf{\textit{#1}}}
\begin{document}
The is an example of modified \mod{text}
\end{document}

1.png

What is new is in line 2. Here we use the “\newcommand” declaration again but this time we create a command call “\mode” and give it an argument of 1 (see [1]) this is more important when you have more than one argument. Next, we put in curly brackets what we want to be done to the text. Here we want the text to be bold “\textbf” and in italics “\textit”. Lastly, we set the definition {#1}. Definition works with arguments in that argument 1 uses definition 1, argument 2 uses definition 2, etc.  Having more than one argument and definition can be confusing for beginners so this will not be explored for now.

Conclusion

This post provided assistance in understanding LaTeX’s font size capabilities as well as ways to make new commands.

Review of “Peter the Great”

In this post, we will take a look at the book Peter the Great by Diane Stanley (32 pp).

The Summary
This book covers the life and death of Peter the Great (1672-1725) one of the most influential Tsars of Russia. The book begins by showing Peter as small boy play war games with his friends. What is unique is that Peter is not the leader, despite his status, but is rather one of the junior soldiers taking orders from the other boys. This points already to an everyman personality of Peter.

The story does not neglect that Peter was royalty and shows some of the luxuries Peter enjoyed such as dancing animals and his own horses. Peter even designed and sailed his own ships.

As a student, Peter was educated by Europeans. He saw how they lived compared to how people in his country lived and it planted a seed for reformation in Peter’s heart.

I his early twenties Peter travels to Europe. While there he absorbs as much culture about Europe as possible and focuses heavily on learning various trades such as shipbuilding. His status as a King made it difficult to learn trades as people found this strange of someone of his rank.

Upon returning home, Peter began immediately to reform Russia. Immediately the long beards that Russian men favored were removed at least among the elite. In addition, the long robes were shortened. Men and women were encouraged to mingle at social settings and arranged marriages were discouraged.

Peter also built schools and canals. His greatest achievement may have been the founding and building of St. Petersburg. Today St. Petersburg is one of the largest cities in Russia.

Of course, all of these reforms had drawbacks. The poor were tax practically to death. Everything was taxed from candle to beards. Peasant young men had to spend as much as 25 years in the military. Lastly, thousands perhaps tens of thousands lost there lives in wars and building projects push by Peter.

Peter died in 1725 of a fever. He was 53 years old at the time of his death.

The Good
This book was extremely interesting. It captures your attention by giving with the rich illustration that has a renaissance feel to it. The illustrations always depict Peter as a man of action. The text is well-written and simple enough for an upper elementary student to understand and appreciate by themselves.

The Bad
There is little to complain about. This text is well-balanced between picture and text. Younger students (below grade 4) may need help with the text. Otherwise, this book is great for all kids and provides some understanding of the history of Russia.

The Recommendation
This is an excellent book to add ou your library as teacher or parent. Younger kids can enjoy the pictures while older kids can enjoy the text. Even an adult can benefit from reading this book if they have not been exposed to Russian history.

Solving Compound Inequalities

Compound inequalities are two inequalities that are joined by the word “and” or the word “or”. Solving a compound inequality means finding all values that make the compound inequality true.

For compound inequalities join ed by the word “and” we look for solutions that are true for both inequalities. Fo compound inequalities joined by the word “or” we look for solutions that work for either inequality.

It is also possible to graph compound inequalities on a number line as well as indicate the final answer using interval notation. Below is a compound inequality with the line graph solution

1.png

Solving the answer is the same as a regular equation. Below is the number line for this answer.

1.png

The empty circle at -8 means that -8 is not part of the solution. This means all values less than -8 are acceptable answers. This is why the line moves from right to left. All values less than -8 until infinity are acceptable answers. Below is the interval notation.

1

The parentheses mean that the value next to it is not included as a solution. This corresponds to the empty circle over the -8 in the lin graph. If the value should be included such as with a less/greater than sign you would use a bracket.

Double Inequality

A double inequality is a more concise version of a compound inequality. The goal is to isolate the variable in the middle. Below is an example

1.png

This is not complex. We simply isolate x in the middle using appropriate steps. The number line and interval notation or as follows

1.png

[-4, 2/3)

This time there is a bracket next to -4 which means that -4 is also a potential solution. In addition, notice how the -4 has a filled circle on the number line. This is another indication that -4 is a solution.

Practical Application

You have signed up for internet access through your cell phone. Your bill is a flat $49.00 per month please $0.05 per minute for internet use. How many minutes can you use internet per month if you want to keep your bill somewhere between $54-$74 per month?

Below is the solution using a double inequality

1

The answer indicates that you can spend anywhere from 100 to 500 minutes on the internet through your phone per month to stay within the budget. You can make the number line and develop the interval notation yourself.

Conclusion

Compound inequalities are useful for not only as an intellectual exercise. They can also be used to determine practical solutions that include more than one specific answer.

Benefits of Coding in Schools

There is a push in education to have more students learn to code. In fact, some schools are considering having computer coding count as the foreign language requirement to graduate from high school. This in many ways almost singles that coding has “arrived” and is not a legitimate subject not just for the computer nerd but for everybody.

In this post, we will look at several benefits of learning to code while in school.

Problem-Solving

People often code to solve a problem. It can be something as small as making an entertaining game or to try and get the computer to do something.  Generally, the process of developing code leads to all kinds of small problems that have to be solved along the way. For example, you want the code to A but instead, it does B. This leads to all kinds of google searches and asking around to try and get the code to do what you want.

All this is happening in the context in which the student is truly motivated to learn. This is perhaps no better situation in which problem-solving skills are developed.

Attention to Detail

Coding involves the ability to see the smallest details. I cannot remember how many times my code would not run because I forgot a comma or a semicolon or perhaps I misspelled a variable. These problems are small but they must be noticed in order to get the code to run.

When students develop code they must write the code perfect (not necessarily efficiently) in order for it to work. This attention to the small things helps in developing students who are not careless.

Computational Thinking

Computational thinking is the skill of being able to explain systematically what you are doing. When developing code students must be able to capture every step needed to execute an action in their code. It is not possible to skip steps. Everything must be planned for in order to have success.

This type of thinking carries over into the real world when communicating with people. The computational thinking comes out when presenting information, teaching, etc. This logical thinking is a key skill in today’s world where miscommunication is becoming so common.

Employment Opportunities

Naturally, learning to code can lead to employment opportunities. There is a growing demand for people with coding skills. Some of the strongest demands are in fields such as Data Science in which people need a blend of coding and domain expertise to develop powerful insights. In other words, it is better to be well-rounded rather than a super coder for the average person as the domain knowledge is useful in interpreting whatever results the coding helped to produce.

Conclusion

There are naturally other benefits of coding as well. The purpose here was just to consider a few reasons. As a minimum, learning to code should be experienced by most students just as they are exposed to music appreciation, art appreciation, and other subjects for the sake of exposure.

Writing Discussion & Conclusions in Research

The Discussion & Conclusion section of a research article/thesis/dissertation is probably the trickiest part of a project to write. Unlike the other parts of a paper, the Discussion & Conclusions are hard to plan in advance as it depends on the results. In addition, since this is the end of a paper the writer is often excited and wants to finish it quickly, which can lead to superficial analysis.

This post will discuss common components of the Discussion & Conclusion section of a paper. Not all disciplines have all of these components nor do they use the same terms as the ones mentioned below.

Discussion

The discussion is often a summary of the findings of a paper. For a thesis/dissertation, you would provide the purpose of the study again but you probably would not need to share this in a short article. In addition, you also provide highlights of what you learn with interpretation. In the results section of a paper, you simply state the statistical results. In the discussion section, you can now explain what those results mean for the average person.

The ordering of the summary matters as well. Some recommend that you go from the most important finding to the least important. Personally, I prefer to share the findings by the order in which the research questions are presented. This maintains a cohesiveness across sections of a paper that a reader can appreciate. However, there is nothing superior to either approach. Just remember to connect the findings with the purpose of the study as this helps to connect the themes of the paper together.

What really makes this a discussion is to compare/contrast your results with the results of other studies and to explain why the results are similar and or different. You also can consider how your results extend the works of other writers. This takes a great deal of critical thinking and familiarity with the relevant literature.

Recommendation/Implications

The next component of this final section of the paper is either recommendations or implications but almost never both. Recommendations are practical ways to apply the results of this study through action. For example, if your study finds that sleeping 8 hours a night improves test scores then the recommendation would be that students should sleep 8 hours a night to improve their test scores. This is not an amazing insight but the recommendations must be grounded in the results and not just opinion.

Implications, on the other hand, explain why the results are important. Implications are often more theoretical in nature and lack the application of recommendations. Often implications are used when it is not possible to provide a strong recommendation.

The terms conclusion and implications are often used interchangeably in different disciplines and this is highly confusing. Therefore, keep in mind your own academic background when considering what these terms mean.

There is one type of recommendation that is almost always present in a study and that is recommendations for further study. This is self-explanatory but recommendations for further study are especially important if the results are preliminary in nature. A common way to recommend further studies is to deal with inconclusive results in the current study. In other words, if something weird happened in your current paper or if something surprised you this could be studied in the future. Another term for this is “suggestions for further research.”

Limitations

Limitations involve discussing some of the weaknesses of your paper. There is always some sort of weakness with a sampling method, statistical analysis, measurement, data collection etc. This section is an opportunity to confess these problems in a transparent matter that further researchers may want to control for.

Conclusion

Finally, the conclusion of the Discussion & Conclusion is where you try to summarize the results in a sentence or two and connect them with the purpose of the study. In other words, trying to shrink the study down to a one-liner. If this sounds repetitive it is and often the conclusion just repeats parts of the discussion.

Blog Conclusion

This post provides an overview of writing the final section of a research paper. The explanation here provides just one view on how to do this. Every discipline and every researcher has there own view on how to construct this section of a paper.

Time Mangagement and E-Learning

Time management is a critical component of having success when studying online. However, many people struggle with the discipline to manage their time so that they can complete an online learning experience. In this post, we will look at several strategies that can help a student to complete an online course.

Routine is King

Perhaps the single most valuable piece of advice that can be given is the benefit of routine. Routine is as simple as dedicating a certain time of each day to the class. Routine can also manifest itself through setting aside a designated place for studying as well.  The beauty of a schedule is that you have taken control of what you will do during a given day.

It is important that a student has a dedicated place and time for e-learning studies. Even though e-learning can happen anywhere few people can learn anywhere as they must be in a learning mindset first. Often, the mindset to learn is dependent on the environment which the student needs to control.

One Thing at a Time

Multi-tasking is tempting but does not work. This is an extra special problem for e-learning because a student is on their computer. Being on the computer can allow the student to check emails, chat on facebook, listen to music, while they are also supposed to be learning.

These other activities are only distractions when trying to learning content online. As such, it is important to close these other applications and turn off notifications from various websites when really trying to learn online.

Ask for Help

Elearning can be an isolating experience. A student is all alone trying to maneuver the complexities of a subject. At times, a student may even get stuck and not know what to do.

In such situations, it is important that a student reaches out to a peer or the teacher for support. The feedback that is received can make a difference in completing a course or not.

Something that was alluded to in this section is the benefit of taking an online course with a friend. Having a friend in a course can be a source of encouragement and also a way to share the burden of larger assignments.

Conclusion

The anytime anywhere freedom of e-learning is perhaps the greatest blessing and also the greatest curse of this platform. The flexibility gives people the impression they can study whenever. However, when there is no structure to the learning experience there is usually no progress made either. Therefore, each student must put in constraints in order to function at a high level academically when studying online.

Major Challenges of Teachers

This post will provide some examples of common problems teachers face. Although the post may seem overwhelmingly negative the purpose here is to provide insight into the actual realities of teaching rather than the romantic experience portrayed in many venues.

Adminstration

Administrators are in charge of the “big picture” of guiding a school towards particular goals that are often laid out by local laws and the results of the prior accreditation visit. This focus on large institutional goals can often cause the administrator to lose sight of the needs of the teachers (unless this was a recommendation from the last accreditation visit),

What results is a task-oriented leadership that is focused on attaining goals or at least showing progress towards goals. This can lead administrators to step on, overwork, and even mistreat teachers. It is hard to blame administrators because if they do not meet specific targets they could lose their own employment.

The constant meetings and incredulous policies that are derived to “help the students” can become exceedingly frustrating for any teacher. Rest assure that few administrators just randomly think up bad ideas. Often the inspiration is from a higher source that is abusing the local administrator.

Co-Workers

There is a surprising amount of petty bickering and fighting among teachers that can become Machevellini in nature. Gossiping backbiting and of course backstabbing all take place. A teacher A confides in teacher B there having problems handling their students and teacher B spreads this to everyone on-campus that teacher A is a terrible teacher who cannot handle her duties.

I’ve heard of teachers complaining that other teachers do not collaborate during lunch with them as though lunchtime is meant to be a meeting that has required attendance. In another setting, I’ve seen teachers slander another teacher in order to help a friend get the job. Petty jealousy can lead teachers to isolate themselves to avoid political attacks which makes it harder to support students.

Parents & Students

Perhaps the biggest problem facing teachers is not necessarily students but parents. If a child is out of line it should only take a simple phone call home to resolve the problem. However, this is almost never the case. Today many parents are indifferent to the behavior of their children. This leaves the teacher only to provide intervention towards a wayward student.

The other extreme is the parent who overly protects and defends everything their child does. This undercuts the teacher’s authority in the same way as a parent who does not provide any sort of behavioral support. The same parents are often quick to get the attention of the administration which is always.

Class Administration

There are a bevy of things that a teacher must do in their own classroom such as

  • Class preparation
  • Marking assignments
  • Decorating
  • Meetings
  • Communicating with parents/students
  • Professional development

This all requires serious time management. It is hard to stay on top of all of these expectations if you are laid back and easy going. It requires strict discipline in order to keep some sort of sanity.

Conclusion

Teaching is tough. However, it is not all bad. There are many rewarding moments in being a teacher. Yet to be successful a teacher must be aware of the common problems that will face so that they are able to weather them.

Gamification

The concept of gamification has really picked up steam over the past few years in education. Gamification is the use design elements from traditional games within a curriculum. Examples of gamification include the use of badges, leveling up, progress bars, hit points, etc. As students focus on acquiring the various rewards in the gamification experience they also learn the content. In other words, gamification is primarily behavioral in nature.

The idea of gamification is not completely new. Many adults can remember earning stickers and or points for excellent behavior as a child. Gamification, however, tends to be focused on the online/ technology context. Instead of earning stickers like students in the 1980’s and 90’s today’s students can earn badges online in their schools learning management system as an example.

It may be clear that there are some pros and cons to gamification and these will be addressed below.

Pros

  • Engagement-Nothing motivates a student like playing a game. The badges and points in gamification often heighten engagement at least temporarily for many students.
  • Feedback-Through leveling up and earning badges students are provided with instant feedback. If they are unsuccessful it is readily apparent and there is no need to wait for teacher feedback.
  • Technology exposure-Gamification is focused in a technology domain. As such, it is a great way to help students to develop their technology skills.

Cons

  • Attention span-Games are often fast-paced. However, the real world is not often moving at the same speed. This can lead kids to struggle with everyday tasks that have not been gamified.
  • Assessment-Often the game serves as a platform to master a skill. However, the nuanced nature of grading can be difficult to apply to a gamified learning experience. In addition, the game may not always transfer to actual real-world skill, which further impairs grading. The focus on gaming often makes the learning take a backseat.
  • Cost/logistics-The cost of using software and other materials can be high. Even if you use a free system, such as Moodle, there is the logistics of setting up the badge system in your online platform.
  • Work ethic-A critical skill that students need to acquire is how to do something they don’t like to do. Gamification can make almost anything fun. However, in the actual world, there are a little of boring things that people have to do. Students must develop the discipline to engage in an activity because it needs to be done rather than because it’s fun.

Conclusion

The appropriate use of gamification is dependent on the extent to which it is used. Having a progress bar in a course probably will not influence attention spans detrimentally. However, more complex gamification is probably where you will start to see problems as shared in the con section.

Therefore, an appropriate analogy would be to compare gamification to salt. A little bit of salt makes food taste better. However, too much salt can ruin the food and impact the health of the eater. As such, a little gamification can enrich a learning experience but heavy doses could harm learning and perhaps character development.

Shaping the Results of a Research Paper

Writing the results of a research paper is difficult. As a researcher, you have to try and figure out if you answered the question. In addition, you have to figure out what information is important enough to share. As such it is easy to get stuck at this stage of the research experience. Below are some ideas to help with speeding up this process.

Consider the Order of the Answers

This may seem obvious but probably the best advice I could give a student when writing their results section is to be sure to answer their questions in the order they presented them in the introduction of their study. This helps with cohesion and coherency. The reader is anticipating answers to these questions and they often subconsciously remember the order the questions came in.

If a student answers the questions out of order it can be jarring for the reader. When this happens the reader starts to double check what the questions were and they begin to second-guess their understanding of the paper which reflects poorly on the writer. An analogy would be that if you introduce three of your friends to your parents you might share each person’s name and then you might go back and share a little bit of personal information about each friend. When we do this we often go in order 1st 2nd 3rd friend and then going back and talking about the 1st friend. The same courtesy should apply when answering research questions in the results section. Whoever was first is shared first etc.

Consider how to Represent the Answers

Another aspect to consider is the presentation of the answers. Should everything be in text? What about the use of visuals and tables?  The answers depend on several factors

  • If you have a small amount of information to share writing in paragraphs is practical. Defining small depends on how much space you have to write as well but generally anything more than five ideas should be placed in a table and referred too.
  • Tables are for sharing large amounts of information. If an answer to a research question requires more than five different pieces of information a table may be best. You can extract really useful information from a table and place it directly in paragraphs while referring the reader to the table for even more information.
  • Visuals such as graphs and plots are not used as frequently in research papers as I would have thought. This may be because they take up so much space in articles that usually have page limits. In addition, readers of an academic journal are pretty good at visually results mentally based on numbers that can be placed in a table. Therefore, visuals are most appropriate for presentations and writing situations in which there are fewer constraints on the length of the document such as a thesis or dissertation.

Know when to Interpret

Sometimes I have had students try to explain the results while presenting them. I cannot say this is wrong, however, it can be confusing. The reason it is so confusing is that the student is trying to do two things at the same time which are present the results and interpret them. This would be ok in a presentation and even expected but when someone is reading a paper it is difficult to keep two separate threads of thought going at the same time.  Therefore, the meaning or interpretation of the results should be saved for the Discussion Conclusion section.

Conclusion

Presenting the results is in many ways the high point of a research experience. It is not easy to take numerical results and try to capture the useful information clearly. As such, the advice given here is intending to help support this experience

Purpose of a Quantitative Methodology

Students often struggle with shaping their methodology section in their paper. The problem is often that students do not see the connection between the different sections of a research paper. This inability to connect the dots leads to isolated thinking on the topic and inability to move forward.

The methodology section of a research paper plays a critical role. In brief, the purpose of a methodology is to explain to your readers how you will answer your research questions. In the strictest sense, this is important for reproducing a study. Therefore, what is really important when writing a methodology is the research questions of the study. The research questions determine the following of a methodology.

What this means is that a student must know what they want to know in order to explain how they will find the answers. Below is a description of these sections along with one section that is not often influenced by the research questions.

Sample & Setting

In the sample section of the methodology, it is common or the student to explain the setting of the study, provide some demographics, and explain the sampling method. In this section of the methodology, the goal is to describe what the reader needs to know about the participants in order to understand the context from which the results were derived.

Research Design & Scales

The research design explains specifically how the data was collected. There are several standard ways to do this in the social sciences such.

  • Survey design
  • experimental design
  • correlational design

Within this section, some academic disciplines also explain the scales or the tool used to measure the variable(s) of the study. Again, it is impossible to develop this section of the research questions are unclear or unknown.

Data Analysis

The data analysis section provides an explanation of how the answers were calculated in a study. Success in this section requires a knowledge of the various statistical tools that are available. However, understanding the research questions is key to articulating this section clearly.

Ethics

A final section in many methodologies is ethics. The ethical section is a place where the student can explain how the protected participant’s anonymity, made sure to get the permission and other aspects of morals. This section is required by most universities in order to gain permission to do research. However, it is often missing from journals.

Conclusion

The methodology is part of the larger picture of communicating one’s research. It is important that a research paper is not seen as isolated parts but rather as a whole. The reason for this position is that a paper cannot make sense on its own if any of these aspects are missing.

Solving Inequalities

Inequalities are equations that use symbols related to less than, greater than, etc. This allows for the solution to be a range of values rather than only one specific one as in many standard equations where you solve for x.

Unique Property of Inequalities

The rules for solving inequalities are mostly the same as for solving a regular equation with one exception. If you multiply or divide both sides of an inequality by a negative number you need to flip the inequality sign.  Below is an example of the sign flipping

1

If you look at the final answer you can see that the x must be greater than -2. This makes sense as -5 * -2 would come to 10 which is not less than 10. Naturally,  any number that is larger than -2 would only be worst. Below is a word problem that employs an inequality.

Single Inequality

You have $8,000 to buy math textbooks for your classroom. Each math book cost $127.06. What is the maximum number of math books you can buy?

In the problem above, the keyword is maximum. In other words, there is a range of potential answers from 1 book to whatever the max is. This indicates that this problem is an inequality. Therefore,

  • Let 127.06 be the price of a math book
  • x the number of math books we can buy
  • < use less than because we do not want to exceed our budget of 8000

Below is the solution to the problem.

1.png

You can buy up to 62 books and be less than or equal to 8000. We round down to 62 because we must stay under $8,000 in spending.

Below is another example but slightly more complex as it contains additional information.

Complex Single Inequality

You are planning a three-day camping trip for your students. Currently, there is $420 of money available. The students can earn $22.50 per hour through tutoring. The trip will cost $525 for transportation,  $390 for food, and $47.50 per night for the campground.  How many hours do the students need to tutor in order to have enough money for the trip?

This problem has three pieces of information on the left of the inequality

  • Transportation (525)
  • Food (390)
  • campground per night (47.5 * 3)

The information to the right is the following

  • The money available (420)
  • The earning rate per hour (22.50)
  • The variable for the hours to tutor (x)

We use the less than or equal to inequality <

Below is the solution

1.png

The students need to tutor for at least 28hours and 20 minutes in order to meet the expenses for the trip.

Conclusion

Inequalities are another useful tool taught in algebra. The applications are limitless. The key to appreciating inequalities is being able to determine when they can be used to solve real-world problems.

Tips for Writing a Quantitative Review of Literature

Writing a review of literature can be challenging for students. The purpose here is to try and synthesize a huge amount of information and to try and communicate it clearly to someone who has not read what you have read.

From my experience working with students, I have developed several tips that help them to make faster decisions and to develop their writing as well.

Remember the  Purpose

Often a student will collect as many articles as possible and try to throw them all together to make a review of the literature. This naturally leads to problems of the paper sounded like a shopping list of various articles. Neither interesting nor coherent.

Instead, when writing a review of literature a student should keep in mind the question

What do my readers need to know in order to understand my study?

This is a foundational principle when writing. Readers don’t need to know everything only what they need to know to appreciate the study they are ready. An extension of this is that different readers need to know different things. As such, there is always a contextual element to framing a review of the literature.

Consider the Format

When working with a student, I always recommend the following format to get there writing started.

For each major variable in your study do the following…

  1. Define it
  2. Provide examples or explain theories about it
  3. Go through relevant studies thematically

Definition

There first thing that needs to be done is to provide a definition of the construct. This is important because many constructs are defined many different ways. This can lead to confusion if the reader is thinking one definition and the writer is thinking another.

Examples and Theories

Step 2 is more complex. After a definition is provided the student can either provide an example of what this looks like in the real world and or provide more information in regards to theories related to the construct.

Sometimes examples are useful. For example, if writing a paper on addiction it would be useful to not only define it but also to provide examples of the symptoms of addiction. The examples help the reader to see what used to be an abstract definition in the real world.

Theories are important for providing a deeper explanation of a construct. Theories tend to be highly abstract and often do not help a reader to understand the construct better. One benefit of theories is that they provide a historical background of where the construct came from and can be used to develop the significance of the study as the student tries to find some sort of gap to explore in their own paper.

Often it can be beneficial to include both examples and theories as this demonstrates stronger expertise in the subject matter. In theses and dissertations, both are expected whenever possible. However, for articles space limitations and knowing the audience affects the inclusion of both.

Relevant Studies

The relevant studies section is similar breaking news on CNN. The relevant studies should generally be newer. In the social sciences, we are often encouraged to look at literature from the last five years, perhaps ten years in some cases. Generally, readers want to know what has happened recently as experience experts are familiar with older papers. This rule does not apply as strictly to theses and dissertations.

Once recent literature has been found the student needs to organize it thematically. The reason for a thematic organization is that the theme serves as the main idea of the section and the studies themselves serve as the supporting details. This structure is surprisingly clear for many readers as the predictable nature allows the reader to focus on content rather than on trying to figure out what the author is tiring to say. Below is an example

There are several challenges with using technology in class(ref, 2003; ref 2010). For example, Doe (2009) found that technology can be unpredictable in the classroom. James (2010) found that like of training can lead some teachers to resent having to use new technology

The main idea here is “challenges with technology.” The supporting details are Doe (2009) and James (2010). This concept of themes is much more complex than this and can include several paragraphs and or pages.

Conclusion

This process really cuts down on the confusion of students writing. For stronger students, they can be free to do what they want. However, many students require structure and guidance when the first begin writing research papers

Uniform Motion Equations

A uniform motion equation involves trying to make calculations when an object(s) is moving at a constant rate. The formula for this type of equation is below.

rate * time = distance

Generally, you want to make a table that includes all of the known information. This allows you to determine what the unknown information is that needs to be solved. Below is a table that you can use.

Rate            * Time            = Distance

Let’s go through some examples

Example 1

Dan and William are riding bicycles. Dan’s speed is 4 kph faster than William’s speed. It takes William 1.5 hours to reach the beach while it takes Dan 1 hour. Find the speed of both bicyclists.

Here is what we know

  • Dan is 4 kph faster than William
  • It takes Dan 1 hour to get to the beach
  • William is 4 kph slower than Dan
  • It takes William 1.5 hours to get to the beach

We will now setup our table

Rate            * Time            = Distance
 Dan  r + 4  1  1(r + 4)
 William  r  1.5  1.5r

We will now solve this equation by placing Dan’s information on one side of the equation and William’s information on the other side of the equation. Below is the solution

1.png

We now know what r is so we need to plug this into the table to get the answers

Rate            * Time            = Distance
 Dan  8 + 4 = 12  1  1(8 + 4) = 12
 William  8  1.5  1.5(8) = 12

The speed of Dan was 12kph while the speed of William was 8kph. This first example was two people traveling the same distance. The next example will be two people travel a different distance.

Example 2

Jenny is traveling to meet her brother. She travels from Saraburi to Chang Mai while her brother travels from Chang Mai to Saraburi. They meet in Bangkok. The distance from Saraburi to Chang Mai is 620km. It takes Jenny 2 hours to get to Bangkok while it takes the brother 7.5 hours to get there. Jenny’s brother’s average speed is 30kph faster than hers. Find the average speed for both people.

The table below captures all of our information

Rate            * Time            = Distance
 Jenny r  2  2r
 Brother  r + 30  7.5  7.5(r + 30)
 620

To solve this problem we combine the information about Jenny and her brother and set this information to equal 620 which is the total distance. Below is the solved equation.

1.png

We can now place this information in our table.

Rate            * Time            = Distance
 Jenny 41.57  2  2(41.57) = 83.14
 Brother  41.57 + 30 = 71.57  7.5  7.5(41.57 + 30) = 536.78
 620

Jenny average speed was 41.57kph while her brother’s speed was 71.57kph. If you add up the distance traveled it will sum to 620.

Our final example will look at determining the time travel when we know the rate of the two objects.

Example 3

A husband and wife both leave their home. The wife travels east and the husband travels west. Wife travels 80kph while the husband travels 100kph. How long will they travel before they are 360km apart?

Below is what we know

Rate            * Time            = Distance
Husband 100  t 100t
Wife 80 t  80t
 360

To solve this we combine the wife and husband information on one side of the equation and put the total distance traveled on the other side. The solution is below.

1.png

We place our answer inside our table

Rate            * Time            = Distance
Husband 100  2 100(2) = 200
Wife 80 2  80(2) = 160
 360

It takes two hours for the wife and husband to be 360km apart.

Conclusion

Understanding uniform equations involve determining first what you know and then determining what the problem wants you to figure out. If you follow this simple process and are able to identify when an equation involves a uniform application it should not be difficult to find the solution.

Basics of LateX

In this post, we will explore more concepts about Latex the typesetting language.

Optional Commands

Optional commands appear in brackets [   ]  when you are using Latex. In the example below, we will set the font size to 20pt in the preamble of the document. The code is as follows.

\documentclass[4paper,12pt]{article} \begin{document} 
     Behold the power of \LaTeX
\end{document}

Here is what it looks like

Screenshot 2018-02-05 14:01:57.png

Inside the brackets, we set the paper size to A4 and the font size to 12pt. Many if not most commands have optional commands that can be used to customize the behavior of the document.

Comments

Like most coding languages Latex allows you to make comments. To do this you need to place a % sign in front of your comment. As shown below

\documentclass[4paper,12pt]{article} \begin{document} 
   Behold the power of \LaTeX
   %This will not print
\end{document}

Screenshot 2018-02-05 14:01:57

Everything after the % did not print. To stop this action simply press enter to move to the next line and you can continue with your document.

Fun with Fonts

There are many different ways to set the fonts. Generally, you can use the \text**{  } code. Where the asterisks are is where you can specify the behavior you want of the text. Below is a simple example of the use of several different formats to the font.

\documentclass[4paper,12pt]{article} 
\begin{document} 
   You can \textit{italicized} 
   Text can be \textsl{slanted} 
   Off course, you can \textbf{bold} text 
   You can also make text in \textsc{small caps} 
   It is also possible to use several commands at the same \textit{\textbf{time}} 
   Behold the power of \LaTeX  
\end{document}

Screenshot 2018-02-05 14:01:57.png

Notice how you put the command in front of the word that you want to format. This might seem cumbersome. However, once you get comfortable with this it is much faster to format documents then the point and click style of Word.

Environments

If you want a certain effect to last for awhile you can use an environment. An environment is a space you declare in your document in which a center behavior takes place. Generally, environments are used to improve the readability of your code. Below is an example.

\documentclass[4paper,12pt]{article} 
    \begin{document} 
        \begin{bfseries} 
            Everything is bold here 
        \end{bfseries} 
        \begin{itshape} 
            Everything is bold here 
        \end{itshape} 
        Behold the power of \LaTeX  
   \end{document}

Screenshot 2018-02-05 14:01:57

An environment always begins with the \begin command and ends with the \end command. In the curly braces, you type whatever is required for your formatting goals. There are scores of commands you can place inside the curly braces.

Conclusion

There is so much more to learn but this is just a beginning. One of the main benefits of learning Latex is the fixed nature of the formatting and the speed at which you can produce content once you are familiar with how to use this language.

Common Problems with Research for Students

I have worked with supporting undergrad and graduate students with research projects for several years. This post is what I consider to be the top reasons why students and even the occasional faculty member struggles to conduct research. The reasons are as follows

  1. They don’t read
  2. No clue what  a problem
  3. No questions
  4. No clue how to measure
  5. No clue how to analyze
  6. No clue how to report

Lack of Reading

The first obstacle to conducting research is that students frequently do not read enough to conceptualize how research is done. Reading not just anything bust specifically research allows a student to synthesize the vocabulary and format of research writing. You cannot do research unless you first read research. This axiom applies to all genres of writing.

A common complaint is the difficulty with understanding research articles. For whatever reason, the academic community has chosen to write research articles in an exceedingly dense and unclear manner. This is not going to change because one graduate student cannot understand what the experts are saying. Therefore, the only solution to understand research English is exposure to this form of communication.

Determining the Problem

If a student actually reads they often go to the extreme of trying to conduct Nobel Prize type research. In other words, their expectations are overinflated given what they know. What this means is that the problem they want to study is infeasible given the skillset they currently possess.

The opposite extreme is to find such a minute problem that nobody cares about it. Again, reading will help in avoiding this two pitfalls.

Another problem is not knowing exactly how to articulate a problem. A student will come to me with excellent examples of a problem but they never abstract or take a step away from the examples of the problem to develop a researchable problem. There can be no progress without a clearly defined research problem.

Lack the Ability to Ask Questions about the Problem

If a student actually has a problem they never think of questions that they want to answer about the problem. Another extreme is they ask questions they cannot answer. Without question, you can never better understand your problem. Bad questions or no questions means no answers.

Generally, there are three types of quantitative research questions while qualitative is more flexible. If a student does not know this they have no clue how to even begin to explore their problem.

Issues with Measurement

Let’s say a student does know what their questions are, the next mystery for many is measuring the variables if the study is quantitative. This is were applying statistical knowledge rather than simply taking quizzes and test comes to play. The typical student does not understand often how to operationalize their variables and determine what type of variables they will include in their study. If you don’t know how you will measure your variables you cannot answer any questions about your problem.

Lost at the Analysis Stage

The measurement affects the analysis. I cannot tell you how many times a student or even a colleague wanted me to analyze their data without telling me what the research questions were. How can you find answers without questions? The type of measurement affects the potential ways of analyzing data. How you summary categorical data is different from continuous data. Lacking this knowledge leads to inaction.

No Plan for the Write-Up

If a student makes it to this stage, firstly congratulations are in order, however, many students have no idea what to report or how. This is because students lose track of the purpose of their study which was to answer their research questions about the problem. Therefore, in the write-up, you present the answers systematically. First, you answer question 1, then 2, etc.

f necessary you include visuals of the answers. Again Visuals are determined by the type of variable as well as the type of question. A top reason for article rejection is an unclear write-up. Therefore, great care is needed in order for this process to be successful.

Conclusion

Whenever I deal with research students I often walk through these six concepts. Most students never make it past the second or third concept. Perhaps the results will differ for others.

Successful research writing requires the ability to see the big picture and connection the various section of a paper so that the present a cohesive whole. Too many students focus on the little details and forget the purpose of their study. Losing the main idea makes the details worthless.

If I left out any common problems with research please add them in the comments section.

Algebraic Mixture Problems

There are many examples in the world in which you want to know the quantity of several different items that make up a whole. When such a situation arise it is an example of mixture problem.

In this post, we will look at several examples of mixture problems. First, we need to look at the general equation for a mixture problem.

number * value = total value

The problems we will tackle will all involve some variation of the equation above. Below is our first example

Example 1

There are times when you want to figure out how many coins are needed to equal a certain dollar amount such as in the problem below

Tom has $6.04 of pennies and nickels. The number of nickels is 4 more and 6 times the number of pennies. How many nickels and pennies does Tom have?

To have success with this problem we need to convert the information into a table to see what we know. The table is below.

Type Number * Value = Total Value
Pennies x .01 .01x
Nickels 6x+4 .05 .05(6x+4)

total 6.04

We can now solve our equation.

1.png

We know that there are 18.83 pennies. To determine the number of nickels we put 18.83 into x and get the following.

1

Almost 117 nickels

You can check if this works for yourself.

Example 2 

For those of us who love to cook, mixture equations can be used for this as well below is an example.

Tom is mixing nuts and cranberries to make 20 pounds of trail mix. Nuts cost $8.00 per pound and cranberries cost $3.00 per pound. If Tom wants to his trail mix to cost $5.50 per pound how many pounds of raisins and cranberries should he use? 

Our information is in the table below. What is new is subtracting the number of pounds from x. Doing so will help us to determine the number of pounds of cranberries.

Type Number of Pounds* Price Per Pound = Total Value
nuts x 8 8x
Cranberries 20-x 3 3(20-x)
Trail Mix 20 5.5 20(5.50)

We can now solve our equation with the information in the table above.

1.png

Once you solve for x you simply place this value into the equation. When you do this you see that we need ten pounds of nuts and berries to reach our target cost.

Conclusion

This post provided to practical examples of using algebra realistically. It is important to realize that understanding these basics concepts can be useful beyond the classroom.

Intro to LaTex

History

LaTex is an open-sourced typesetting document developed about 30 years ago by Leslie Lamport and based on the Tex typesetting of Donald Knuth. It is commonly used in the domains of physics and math for producing mathematical equations and other technical documents. Below is a simple example of an equation developed using LaTex

1

LaTex is a document markup language, which means that you indicate the commands and then it is processed to produce the desired effect. This is in contrast to Microsoft Word which utilizes a WYSIWYG (What you see is what you get) approach.

Benefits

Using LaTex provides several benefits. Cross-referencing is easily accomplish especially with the help of BibTex. It is also multi-lingual and able to make glossaries, indexes, and figures/tables with ease. In addition, LaTex is highly portable and opening a file on any computer is not a problem. Sometimes moving to another computer using Microsoft Word can cause issues with formatting.

Another benefit is psychology, using LaTex allows the author to focus on content and not appearance when writing. It is easy to get distracted when using Word to try and make something work through the point and click mechanism we are so used to when writing.

Cons

It takes extensive time to use LaTex. It looks similar to coding which is intimidating for many. However, once a certain mastery is achieved. Producing documents can be faster as everything is text-based and not point click based using a mouse.

Using LaTex

To use LaTex you need to install TexLive and TexWorks. TextLive is a LaTex distribution and TexWorks is one of many LaTex editors. The editor allows you to manipulate the LaTex code that you generate.

Once you have installed both programs you can type the following into TexWorks. Make sure the typeset is set to pdfLaTex. This allows the output to be a pdf file.

\documentclass{article}
\begin{document}
This is an example of what LaTex does
\end{document}

1.png

What happened is as follows

  1. We entered the command \documentclass{article}. All commands begin with a slash followed by the name. The curly braces are required arguments. In this case, we are using the article template which is one of many templates available in LaTex.
  2. The next command is \begin and this command indicates the beginning of the actual text of the document. Everything above the \begin command is part of what we call the preamble.
  3. Next is the actual text that we want to appear in the pdf.
  4. Lastly, we have the \end command which tells LaTex that the document is finished. Everything between \begin and \end command is part of the environment.

Conclusion

There is so much more that can be accomplished with this typesetting software. The possibilities will be explored in the near future.

Solving Equations with Fractions or Decimals

This post will provide an explanation of how to solve equations that include fractions or decimals. The processes are similar in that both involve determining the least common denominator.

Solving Equations with Fractions

The key step to solving equations with fractions is to make sure that the denominators of all the fractions are the same. This can be done by finding the least common denominator. The least common denominator (LCD) is the smallest multiple of the denominators. For example, if we look at the multiples of 4 and 6 we see the following.

1

You can see clearly that the number 12 is the first multiple that 4 and 6 have in common. You can find the LCD by making factor trees but that is beyond the scope of this post. The primary reason we would need the LCD is when we are adding fractions in an equation. If we are multiplying we could simply multiply straight across.

Below is an equation that has fractions. We will find the LCD

1.png

Here is an explanation of each step

  • A. This is the original problem. We first need to find the LCD
  • B. We then multiply each fraction by the LCD
  • C. This is the equation we solve for
  • D. We get the variable alone by subtracting 20 from each side
  • E. We have our new simplified equation
  • F. We further isolate the variable by dividing by 3 on both sides.
  • G. This is our answer

Solving Equations with Decimals

The process for solving equations with decimals is almost the same as for fractions. The LCD of all decimals is 100. Therefore, one common way to deal with decimals is to multiply all decimals by 100 and the continue to solve the equation.

The primary benefit of multiply by 100 is to remove the decimals because sometimes we make mistakes with where to place decimals. Below is an example of an equation with decimals.

1.pngHere is what we did

  • A. Initial equation
  • B. we distribute the 0.10
  • C. Revised equation
  • D. We multiply everything by 100.
  • E. Revised equation
  • F. Subtract 20 from both sides to isolate the variable
  • G. Revised equation
  • H. Divide both sides by 30 to isolate the variable
  • I. Final answer

Conclusion

Understand the process of solving equations with fractions or decimals is not to complicated. However, this information is much more valuable when dealing with more complex mathematical ideas.

Linear Equations

In this post, we will look at several types of equations that you would encounter when learning algebra. Algebra is a foundational subject to know when conducting most quantitative research.

Equations

An equation is a statement that balances two expressions. Often equations include a variable or an unknown value. By solving for the unknown value you are able to balance the equation.

There are many different types of equations such as

(1) Linear equation

1.png
(2) Quadratic equation

1
(3) Polynomial equation

See number 1 or 2. The rule for polynomial equation is that the exponent must be positive
(4) Trigonometric equation

1
(5) Radical equation

1
(6) Exponential equation
1

This post will focus on linear equations.

Linear Equation

A linear equation is an equation that if it is graph will render a straight line. It is common to have to solve for the variable in a linear equation by isolating as in the example below.

1

There are also several terms related to equations and the include the following

  • Conditional equation: An equation that is true for only one value of the variable. The example above is a conditional equation.
  • Identity: An equation that is true for any value of a variable. Below is an example
    1

Any value of x will work with an identity equation.

  • A contradiction is an equation that is false for all values. Below is an example

1.png

No value of x will work with the equation above.

Conclusion

This post provided an overview of the types of equations commonly encountered in algebra.

Real Number Properties

In this post, we will examine several properties of math. The word property in this context means characteristic or trait. In other words, we are going to look at characteristics of math.

Communicative Property

The communicative property applies both to addition and multiplication. To express this simply the communicative property states that the order the of the numbers does not matter when we add or multiply. Examine the example below

1.png

Whether 8 =+ 9 or 9+ 8 it doesn’t matter as the answer is the same. However, this does not work for subtraction or division because the order of the numbers is critical. Consider the following

1

The point is order does not matter for addition and multiplication but the order does matter for subtraction and division.

Associative Property

The associative property has to do with the grouping of numbers. Often, numbers are grouped in equations with parentheses or brackets. When this is done for an addition and multiplication problem there is no change to the results. You can see this in the example below.

1

However, the associative property does not work with subtraction and division as the order of the numbers affects the final values as shown below.

1

Identity Property

The identity property of addition states that any number added to zero does not change. The identity property of multiplication states that any number multiplied by 1 does not change.

Inverse Property

There are also inverse properties for addition and nultiplication. The inverse property of addition states that adding the opposite value to a real number will result in zero. Or in other words

1

The inverse property of multiplication states any number multiplied by its reciprocal will equal 1. Or as shown below

1.png

Distributive Property

The distributive property is used to get rid of parentheses in order to simplify expressions. This is hard to explain but easy to figure out with an example as shown below.

1.png

In the example, you can see that we distributed the 6 by multiplying it with the values within the parentheses. By doing this we were able to remove the parentheses.

Conclusion

Understanding these properties are useful when it is necessary to do more complex calculations. When you know how the numbers should behave it is easier to identify when they do not behave appropriately.

Teaching Math

Probably one of the most dreaded subjects in school is math. Many students fear this subject and perhaps rightfully so. This post will provide some basic tips on how to help students to understand what is happening in math class.

Chunk the Material

Many math textbooks, especially at the college level, are huge. By huge we are talking over 1000 pages. That is a tremendous amount of content to cover in a single semester even if the majority of the pages are practice problems.

To overcome this, many have chapters that are broken down into 5 sub-sections such as 1.1, 1.2, etc. This means that in a given class period, students should be exposed to 2 or 3 new concepts. Depending on their background this might be too many for a student, especially if they are not a math major.

Therefore, a math teacher must provide new concepts only after previous concepts are mastered. This means that the syllabus needs to flexible and the focus is on the growth of students rather than covering all of the material.

Verbal Walk Through

When teaching math to a class, normally a teacher will provide an example of how to do a problem. The verbal walkthrough is when the teacher completes another example of the problem and the students tell the teacher what to do verbally. This helps to solidify the problem-solving process in the students’ minds.

A useful technique in relation to the verbal walkthrough is to intentional make mistakes when the students are coaching you. This requires the students to think about what is corrected and to be able to explain what was wrong with what the teacher did. The wisest approach is to make mistakes that have been experienced in the past as these are the ones that are likely to be repeated.

The verbal walkthrough works with all students of all ages. It can be more chaotic with younger children but this is a classic approach to teaching the step-by-step process of learning math calculations.

Practice Practice Practice

Daily practice is needed when learning mathematical concepts. Students should be learning new material while reviewing old material. The old material is reviewed until it becomes automatic.

This requires the teacher to determine the most appropriate mix of new and old. Normally, math has a cumulative effect in that new material builds on old. This means that students are usually required to use old skills to achieve new skills. The challenge is in making sure the old skills are at a certain minimum level that they can be used to acquire new skills.

Conclusion

Math is tough but if a student can learn it math can become a highly practical tool in everyday life. The job of the teacher is to develop a context in which math goes from mysterious to useful.

Algebraic Expressions and Equations

This post will focus mainly on expressions and their role in algebra. Expressions play a critical role in mathematics and we all have had to try and understand what they are as well as what they mean.

Expression Defined

To understand what an expression is you first need to know what operation symbols are. Operation symbols tell you to do something to numbers or variables. Examples include the plus, minus, multiply, divide, etc.

An expression is a number, variable, or a combination of numbers[s] and variable[s] that use operation symbols. Below is an example

1.png

Expressions consist of two terms and these are variables and constants. A variable is a letter that represents a number that can change. In our example above, the letter a is a variable.

A constant is a number whose value remains the same. In our example above, the numbers 2 and 4 are constants.

Expression vs Equation

An equation is when two expressions connected by an equal sign as shown below.

1

In the example above we have to expressions. To the left of the equal sign is 2a * 4 and to the right of the equal sign is 16. Remember that an expression can be numbers and or variables so 16 is an expression because it is a number.

Simplify an Expression

Simplifying an expression involves completing as much math as possible to reduce the complexity of an expression. Below is an example.

1

In order to complete this expression  above you need to know the order of operations which is explained below

Parentheses
Exponents
Multiplication Division
Adition Subtraction

In the example above, we begin with multiplication of 8 and 4 before we do the addition of adding 2. It’s important to remember that for multiplication/division or addition/subtraction that you move from left to right when dealing with these operation symbols in an expression. It is also important to know that subtraction and division are not associative (or commutative) that is: (1 – 2) – 3 != 1 – (2 – 3).

Evaluating an Expression

Evaluating an expression is finding the value of an expression when the variable is replaced with a specific number.

1.png

Combining Like Terms

A common skill in algebra is the ability to combine like terms. A term is a constant or constant with one or more variables. Terms can include a constant such as 7 or a number and variable product such as 7a. The constant that multiplies the variable is called a coefficient. For example, 7a, 7 is the constant and the coefficient while a is the variable.

Combining like terms involves combining constant are variables that have the same characteristics for example

  • 3 and 2 are like terms because they are both constants
  • 2x and 3x are like terms because they are both constants with the same variable

Below is an example of combining like terms

1

In this example, we first placed like terms next to each other. This makes it easier to add them together. The rest is basic math.

Conclusion

Hopefully, the concept of expressions makes more sense. This is a foundational concept in mathematics that if you do not understand. It is difficult to go forward in the study of math.

Basic Algebraic Concepts

This post will provide insights into some basic algebraic concepts. Such information is actually useful for people who are doing research but may not have the foundational mathematical experience.

Multiple

A multiple is a product of  and a counting number of n. In the preceding sentence, we actually have two unknown values which are.

  • n
  • Counting number

The can be any value, while the counting number usually starts at 1 and continues by increasing by 1 each time until you want it to stop. This is how this would look if we used the term n,  counting number, and multiple of n. 

n * counting number = multiple of n

For example, if we say that = 2 and the counting numbers are 1,2,3,4,5. We get the following multiples of 2.

1

You can see that the never changes and remains constant as the value 2. The counting number starts at 1 and increases each time. Lastly, the multiple is the product of n and the counting number.

Let’s take one example from above

2 * 3 = 6

Here are some conclusions we can make from this simple equation

  • 6 is a multiple of 2. In other words, if I multiply 2 by a certain counting number I can get the whole number of 6.
  • 6 is divisible by 2. This means that if I divide 2 into six I will get a whole number counting number which in this case is 3.

Divisibility Rules

There are also several divisibility rules in math. They can be used as shortcuts to determine if a number is divisible by another without having to do any calculation.

A number is divisible by

  • 2 when the last digit of the number 0, 2, 4, 6, 8
    • Example 14, 20, 26,
  • 3 when the sum of the digits is divisible by 3
    • Example 27 is divisible by 3 because 2 + 7 = 9 and 9 is divisible by 3
  • 5 when the number’s last digit is 0 or 5
    • Example 10, 20, 25
  • 6 when the number is divisible by 2 and 3
    • Example 24 is divisible by 6 because it is divisible by 2 because the last digit is for and it is divisible by 3 because 2 + 4 = 6 and six is divisible by 3
  • 10 when the number ends with 0
    • Example 20, 30 , 40, 100

Factors

Factors are two or more numbers that when multiplied produce a number. For example

1.png

The numbers 7 and 6 are factors of 42. In other words, 7 and 6 are divisible by 42. A number that has only itself and one as factors is known as a prime number. Examples include 2, 3, 5, 7, 11, 13. A number that has many factors is called a composite number and includes such examples as 4, 8, 10, 12, 14.

An important concept in basic algebra is understanding how to find the prime numbers of a composite number. This is known as prime factorization and is done through the development of a factor tree. A factor tree breaks down a composite number into the various factors of it. These factors are further broken down into their factors until you reach the bottom of a tree that only contains prime numbers. Below is an example

 

1

You can see in the tree above that the prime factors of 12 are 2 and 3. If we take all of the prime factors and multiply them together we will get the answer 12.

1.png

Conclusion

Understanding these basic terms can only help someone who maybe jumped straight into statistics in grad school without have the prior thorough experience in basic algebra.

APA Tables in R

Anybody who has ever had to do any writing for academic purposes or in industry has had to deal with APA formatting. The rules and expectations seem to be endless and always changing. If you are able to maneuver the endless list of rules you still have to determine what to report and how when writing an article.

There is a package in R that can at least take away the mystery of how to report ANOVA, correlation, and regression tables. This package is called “apaTables”. In this post, we will look at how to use this package for making tables that are formatted according to APA.

We are going to create examples of ANOVA, correlation, and regression tables using the ‘mtcars’ dataset. Below is the initial code that we need to begin.

library(apaTables)
data("mtcars")

ANOVA

We will begin with the results of ANOVA. In order for this to be successful, you have to use the “lm” function to create the model. If you are familiar with ANOVA and regression this should not be surprising as they both find the same answer using different approaches. After the “lm” function you must use the “filename” argument and give the output a name in quotations. This file will be saved in your R working directory. You can also provide other information such as the table number and confidence level if you desire.

There will be two outputs in our code. The output to the console is in R. A second output will be in a word doc. Below is the code.

apa.aov.table(lm(mpg~cyl,mtcars),filename = "Example1.doc",table.number = 1)
## 
## 
## Table 1 
## 
## ANOVA results using mpg as the dependent variable
##  
## 
##    Predictor      SS df      MS      F    p partial_eta2
##  (Intercept) 3429.84  1 3429.84 333.71 .000             
##          cyl  817.71  1  817.71  79.56 .000          .73
##        Error  308.33 30   10.28                         
##  CI_90_partial_eta2
##                    
##          [.56, .80]
##                    
## 
## Note: Values in square brackets indicate the bounds of the 90% confidence interval for partial eta-squared

Here is the word doc output

1.png

Perhaps you are beginning to see the beauty of using this package and its functions. The “apa.aov.table”” function provides a nice table that requires no formatting by the researcher.

You can even make a table of the means and standard deviations of ANOVA. This is similar to what you would get if you used the “aggregate” function. Below is the code.

apa.1way.table(cyl, mpg,mtcars,filename = "Example2.doc",table.number = 2)
## 
## 
## Table 2 
## 
## Descriptive statistics for mpg as a function of cyl.  
## 
##  cyl     M   SD
##    4 26.66 4.51
##    6 19.74 1.45
##    8 15.10 2.56
## 
## Note. M and SD represent mean and standard deviation, respectively.
## 

Here is what it looks like in word

1.png

Correlation 

We will now look at an example of a correlation table. The function for this is “apa.cor.table”. This function works best with only a few variables. Otherwise, the table becomes bigger than a single sheet of paper. In addition, you probably will want to suppress the confidence intervals to save space. There are other arguments that you can explore on your own. Below is the code

apa.cor.table(mtcars,filename = "Example3.doc",table.number = 3,show.conf.interval = F)
## 
## 
## Table 3 
## 
## Means, standard deviations, and correlations
##  
## 
##   Variable M      SD     1      2      3      4      5      6      7     
##   1. mpg   20.09  6.03                                                   
##                                                                          
##   2. cyl   6.19   1.79   -.85**                                          
##                                                                          
##   3. disp  230.72 123.94 -.85** .90**                                    
##                                                                          
##   4. hp    146.69 68.56  -.78** .83**  .79**                             
##                                                                          
##   5. drat  3.60   0.53   .68**  -.70** -.71** -.45**                     
##                                                                          
##   6. wt    3.22   0.98   -.87** .78**  .89**  .66**  -.71**              
##                                                                          
##   7. qsec  17.85  1.79   .42*   -.59** -.43*  -.71** .09    -.17         
##                                                                          
##   8. vs    0.44   0.50   .66**  -.81** -.71** -.72** .44*   -.55** .74** 
##                                                                          
##   9. am    0.41   0.50   .60**  -.52** -.59** -.24   .71**  -.69** -.23  
##                                                                          
##   10. gear 3.69   0.74   .48**  -.49** -.56** -.13   .70**  -.58** -.21  
##                                                                          
##   11. carb 2.81   1.62   -.55** .53**  .39*   .75**  -.09   .43*   -.66**
##                                                                          
##   8      9     10 
##                   
##                   
##                   
##                   
##                   
##                   
##                   
##                   
##                   
##                   
##                   
##                   
##                   
##                   
##                   
##                   
##   .17             
##                   
##   .21    .79**    
##                   
##   -.57** .06   .27
##                   
## 
## Note. * indicates p < .05; ** indicates p < .01.
## M and SD are used to represent mean and standard deviation, respectively.
## 

Here is the word doc results

1.png

If you run this code at home and open the word doc in Word you will not see variables 9 and 10 because the table is too big by itself for a single page. I hade to resize it manually. One way to get around this is to delate the M and SD column and place those as rows below the table.

Regression

Our final example will be a regression table. The code is as follows

apa.reg.table(lm(mpg~disp,mtcars),filename = "Example4",table.number = 4)
## 
## 
## Table 4 
## 
## Regression results using mpg as the criterion
##  
## 
##    Predictor       b       b_95%_CI  beta    beta_95%_CI sr2 sr2_95%_CI
##  (Intercept) 29.60** [27.09, 32.11]                                    
##         disp -0.04** [-0.05, -0.03] -0.85 [-1.05, -0.65] .72 [.51, .81]
##                                                                        
##                                                                        
##                                                                        
##       r             Fit
##                        
##  -.85**                
##             R2 = .718**
##         95% CI[.51,.81]
##                        
## 
## Note. * indicates p < .05; ** indicates p < .01.
## A significant b-weight indicates the beta-weight and semi-partial correlation are also significant.
## b represents unstandardized regression weights; beta indicates the standardized regression weights; 
## sr2 represents the semi-partial correlation squared; r represents the zero-order correlation.
## Square brackets are used to enclose the lower and upper limits of a confidence interval.
## 

Here are the results in word

1.png

You can also make regression tables that have multiple blocks or models. Below is an example

apa.reg.table(lm(mpg~disp,mtcars),lm(mpg~disp+hp,mtcars),filename = "Example5",table.number = 5)
## 
## 
## Table 5 
## 
## Regression results using mpg as the criterion
##  
## 
##    Predictor       b       b_95%_CI  beta    beta_95%_CI sr2  sr2_95%_CI
##  (Intercept) 29.60** [27.09, 32.11]                                     
##         disp -0.04** [-0.05, -0.03] -0.85 [-1.05, -0.65] .72  [.51, .81]
##                                                                         
##                                                                         
##                                                                         
##  (Intercept) 30.74** [28.01, 33.46]                                     
##         disp -0.03** [-0.05, -0.02] -0.62 [-0.94, -0.31] .15  [.00, .29]
##           hp   -0.02  [-0.05, 0.00] -0.28  [-0.59, 0.03] .03 [-.03, .09]
##                                                                         
##                                                                         
##                                                                         
##       r             Fit        Difference
##                                          
##  -.85**                                  
##             R2 = .718**                  
##         95% CI[.51,.81]                  
##                                          
##                                          
##  -.85**                                  
##  -.78**                                  
##             R2 = .748**    Delta R2 = .03
##         95% CI[.54,.83] 95% CI[-.03, .09]
##                                          
## 
## Note. * indicates p < .05; ** indicates p < .01.
## A significant b-weight indicates the beta-weight and semi-partial correlation are also significant.
## b represents unstandardized regression weights; beta indicates the standardized regression weights; 
## sr2 represents the semi-partial correlation squared; r represents the zero-order correlation.
## Square brackets are used to enclose the lower and upper limits of a confidence interval.
## 

Here is the word doc version

1.png

Conculsion 

This is a real time saver for those of us who need to write and share statistical information.

Reading Comprehension Strategies

Students frequently struggle with understanding what they read. There can be many reasons for this such as vocabulary issues, to struggles with just sounding out the text. Another common problem, frequently seen among native speakers of a language, is the students just read without taking a moment to think about what they read. This lack of reflection and intellectual wrestling with the text can make so that the student knows they read something but knows nothing about what they read.

In this post, we will look at several common strategies to support reading comprehension. These strategies include the following…

Walking a Student Through the Text

As students get older, there is a tendency for many teachers to ignore the need to guide students through a reading before the students read it. One way to improve reading comprehension is to go through the assigned reading and give an idea to the students of what to expect from the text.

Doing this provides a framework within the student’s mind in which they can add the details to as they do the reading. When walking through a text with students the teacher can provide insights into important ideas, explain complex words, explain visuals, and give general ideas as to what is important.

Ask Questions

Asking question either before or after a reading is another great way to support students understanding. Prior questions give an idea of what the students should be expected to know after reading. On the other hand, questions after the reading should aim to help students to coalesce the ideals they were exposed to in the reading.

The type of questions is endless. The questions can be based on Bloom’s taxonomy in order to stimulate various thinking skills. Another skill is probing and soliciting responses from students through encouraging and asking reasonable follow-up questions.

Develop Relevance

Connecting what a student knows what they do not know is known as relevance.If a teacher can stretch a student from what they know and use it to understand what is new it will dramatically improve comprehension.

This is trickier than it sounds. It requires the teacher to have a firm grasp of the subject as well as the habits and knowledge of the students. Therefore, patience is required.

Conclusion

Reading is a skill that can improve a great deal through practice. However, mastery will require the knowledge and application of strategies. Without this next level of training, a student will often become more and more frustrated with reading challenging text.

Criticism of Grades

Grading has recently been under attack with people bringing strong criticism against the practice. Some schools have even stopped using grades altogether. In this post, we will look at problems with grading as well as alternatives.

It Depends on the Subject

The weakness of grading is often seen much more clearly in subjects that have more of a subjective nature to them from the Social sciences and humanities such as English, History, or Music. Subjects from the hard sciences such as biology, math, and engineering are more objective in nature. If a student states that 2 + 2 = 5 there is little left to persuasion or critical thinking to influence the grade.

However, when it comes to judging thinking or musical performance it is much more difficult to assess this without bringing the subjectivity of opinion. This is not bad as a teacher should be an expert in their domain but it still brings an arbitrary unpredictability to the system of grading that is difficult to avoid.

Returning to the math problem, if a student stats 2 +2 =  4 this answer is always right whether the teacher likes the student or not. However, an excellent historical essay on slavery can be graded poorly if the history teacher has issues with the thesis of the student. To assess the essay requires subjective though into the quality of the student’s writing and subjectivity means that the assessment cannot be objective.

Obsession of Students

Many students become obsess and almost worship the grades they receive. This often means that the focus becomes more about getting an ‘A’ than on actually learning. This means that the students take no-risk in their learning and conform strictly to the directions of the teacher. Mindless conformity is not a sign of future success.

There are many comments on the internet about the differences between ‘A’ and ‘C’ students. How ‘A’ students are conformist and ‘C’ students are innovators. The point is that the better the academic performance of a student the better they are at obeying orders and not necessarily on thinking independently.

Alternatives to Grades

There are several alternatives to grading. One of the most common is Pass/fail. Either the student passes the course or they do not. This is common at the tertiary level especially in highly subjective courses such as writing a thesis or dissertation. In such cases, the student meets the “mysterious” standard or they do not.

Another alternative is has been the explosion in the use of gamification. As the student acquires the badges, hit points, etc. it is evidence of learning. Of course, this idea is applied primarily at the K-12 level but it the concept of gamification seems to be used in almost all of the game apps available on cellphones as well as many websites.

Lastly, observation is another alternative. In this approach, the teacher makes weekly observations of each student. These observations are then used to provide feedback for the students. Although time-consuming this is a way to support students without grades.

Conclusion

As long as there is education there must be some sort of way to determine if students are meeting expectations. Grades are the current standard. As with any system, grades have their strengths and weaknesses. With this in mind, it is the responsibility of teachers to always search for ways to improve how students are assessed.

Passive vs Active Learning

Passive and active learning are two extremes in the world of teaching. Traditionally, learning has been mostly passive in nature. However, in the last 2-3 decades, there has been a push, particularly in the United States to encourage active learning in the classroom.

This post will define passive and active learning and provide examples of each.

Passive Learning

Passive learning is defined from the perspective of the student and means learning in which the students do little to nothing to acquire the knowledge. The most common form of passive learning is direct instruction aka lecture-style teaching.

With passive learning, the student is viewed as an empty receptacle of knowledge that the teacher must fill with his knowledge. Freire called this banking education as the student serves as an account in which the teacher or banker places the knowledge or money.

There is a heavy emphasis on memorizing and recalling information. The objective is the preservation of knowledge and the students should take notes and be ready to repeat or at least paraphrase what the teacher said. The teacher is the all-wise sage on the stage.

Even though it sounds as though passive learning is always bad there are times when it is beneficial. When people have no prior knowledge of a subject passive learning can provide a foundation for future active learning activities. In addition, if it is necessary to provide a large amount of information direct instruction can help in achieving this.

Active Learning

Active learning is learning in which the students must do something in order to learn. Common examples of this include project-based learning, flipped classroom, and any form of discussion in the classroom.

Active learning is derived from the philosophy of constructivism. Constructivism is the belief that students used their current knowledge to build new understanding. For example, with project-based learning students must take what they know in order to complete the unknown of the project.

For the flipped classroom, students review the lecture style information before class. During class, the students participate in activities in which the use what they learned outside of class. This in turn “flips” the learning experience. Out of class is the passive part while in class is the active part.

There is a reduction or total absence of lecturing in an active learning classroom. Rather students interact with each and the teacher to develop their understanding of the content. This transactional nature of learning is another characteristic of active learning.

There are some challenges with active learning. Since it is constructivist in nature it can be difficult to assess what the students learned. This is due in part to the subjective nature of constructivism. If everybody constructs their own understanding everybody understands differently which makes it difficult to have one objective assessment.

Furthermore, active learning is time-consuming in terms of preparation and the learning experience. Developing activities and leading a discussion forces the class to move slower. If the demands of the course require large amounts of content this can be challenging for many teachers.

Conclusion

There is room in the world of education for passive and active learning strategies. The main goal should be to find a balance between these two extremes as over reliance on either one will probably be a disadvantage to students.

Teacher Burnout

Teacher burnout is a common problem within education. The statistics vary but you can safely say about 1/3 of teachers suffer from some form of burnout at one point or another during their career. This post will define burnout, explain some of the causes, the stages of burnout, as well as ways to deal with burnout.

Definition

Essentially, teacher burnout is an experience of a person who is overwhelmed by the stress of teaching. The most common victims of this are young teachers as well as female teachers.

Young teachers are often at higher risk because they have not developed coping mechanisms for the rigors of teaching. Women are also more often to fall victim to teacher burnout because of the added burning of maintaining the home as well as difficulties with distancing themselves emotionally from their profession as a teacher.

Causes

Teacher burnout is generally caused by stress. Below are several forms of stress that can plague the teaching profession.

  • Workload-This is especially true for those who can never say “no.” Committees, field trips, student activities, grading, lesson plans, accreditation. All of these important tasks can overwhelm a person
  • Student behavioral problems-Classroom management is always a challenge as families continue to collapse.
  • Issues with leadership
  • Boredom-This stressor is more common with experienced teachers who have taught the same content for years. There are only so many ways to teach content that are appealing to the teacher before there is some repetition. Boredom can also be especially challenging for a teacher who values learning more than personal relationships with students.

Stages of Burnout

The stages of teacher burnout follow the same progression as burnout in other social work like professions. Below are four stages as developed by McMullen

  1. Closed off- The burnout victim stops socializing and is rigid against feedback. Signs include self-neglect.
  2. Irritable-The victim temper shortens. In addition, he begins to complain about everything. Problems are observed everywhere whether they are legitimate or not.
  3. Paranoia-The teacher is worried about everything. Depression is common at this point as well as a loss of motivation.
  4. Exhaustion-THe teacher is emotionally drained. They no longer “care” as they see no way to improve the situation. Compassion fatigue sets in which means that there is no more emotional support to give to students.

Dealing with Burnout

Perhaps the most important step coping with burnout is to prioritize. It is necessary for a sake of sanity to say no to various request at times. Personal time away from any job is critical to being able to return refreshed. Therefore, teaching cannot be the sole driving force of the typical person’s life but should be balanced with other activities and even downtime.

It may also be necessary to consider changing professions. If you are not able to give your best in the classroom perhaps there are other opportunities available. It is impractical to think that someone who becomes a teacher must stay a teacher their entire life as though there is no other way to use the skills developed in the classroom in other professions.

Conclusion

Burnout is a problem but it is not unique to education. What really matters is that people take control and responsibility of their time and not chase every problem that comes into their life. Doing so will help in coping with the rigors of the teaching profession.