problem solving or problem solving ap style

“Problem-Solving” Or “Problem Solving”? Learn If It Is Hyphenated

Is it problem-solving or problem solving? Hyphenation rules seem to be a little confusing when you’re first picking up a language. Don’t worry, though. They’re not nearly as complicated as the language may have led you to believe!

Problem-Solving Or Problem Solving – Hyphenated Or Not?

When we discuss the problem-solving hyphen rule, we learn that problem-solving is hyphenated when used to modify a noun or object in a sentence. We keep the two words separated when using them as their own noun and not modifying anything else in the sentence.

Examples Of When To Use “Problem-Solving”

Now that we’re into the whole debate of problem-solving vs problem solving, let’s look through some examples of how we can use “problem-solving” with a hyphen. As stated above, we use “problem-solving” when modifying a noun or object in a sentence. It’s the most common way to write “problem-solving.” Even the spelling without a hyphen is slowly being pushed out of common language use!

• This is a problem-solving class.
• I hold a problem-solving position at my workplace.
• My manager put me in charge of the problem-solving accounts.
• They say I have a problem-solving mind.
• We’re known as problem-solving children.

Examples Of When To Use “Problem Solving”

Though much less common to be seen written as a phrase noun, it is still worth mentioning. It’s grammatically correct to use “problem solving” at the end of a sentence or clause without a hyphen. However, as we stated above, many people are beginning to prefer the ease of sticking to the hyphenated spelling, meaning that it’s slowly phasing out of existence even in this form.

• I’m good at problem solving.
• This requires a lot of problem solving.
• We are all trained in problem solving.
• My job asks for problem solving.
• Did you say you were good at problem solving?

Is Problem-Solving Hyphenated AP Style?

Have you had a look through the rules in the AP stylebook before? Even if you haven’t, there’s a good explanation for hyphens there. As we stated above, we use hyphens when linking close words that modify a noun or object in a sentence. They’re used to help a reader better understand what is going on through the modification of the clause.

Should I Capitalize “Solving” In The Word “Problem-Solving”?

The question of “is problem-solving hyphenated” was answered, but now we’ve got a new question. What happens to capitalization rules when we add a hyphen to a title. It depends on your own title choices, so let’s look a little further into the three potential options. The first option capitalizes only the first word and any proper nouns in a title. In this case, neither word in “problem-solving” is capitalized.

The second option capitalizes all words except for short conjunctions, short prepositions, and articles. In this case, you will always capitalize “problem” but always leave “solving” uncapitalized. The final option capitalizes every single word in a title. No matter what, you’ll capitalize both words in “problem-solving” when using this style to write your titles.

Does The Rule Also Apply To “Problem Solver” Vs “Problem-Solver”?

The same rule does apply when we use “problem solver” instead of “problem solving.” However, it’s not often that we’ll see a “problem-solver” modifying a noun or object (unless it’s a problem-solver robot or something). So, it’s most likely you’ll write “problem solver.”

Alternatives To “Problem-Solving”

If you’re still struggling with the hyphen rule of whether it’s problem solving or problem-solving, there’s one last thing we can help you with. We can give you some alternatives that have the same meanings but don’t require a hyphen. This way, you can be safe in your own knowledge without having to worry about getting the rules wrong.

• interpretive

Quiz – Problem-Solving Or Problem Solving?

We’ll finish with a quiz to see how much you’ve learned from this article. The answers are all multiple choice, so you should have a blast with them! We’ll include the answers at the end to reference as well.

• I’ve been told that I’m good at (A. problem-solving / B. problem solving).
• I hold my (A. problem-solving / B. problem solving) skills close to my heart.
• We aren’t great at (A. problem-solving / B. problem solving).
• These are all the best (A. problem-solving / B. problem solving) subjects.
• Can we have a go at a (A. problem-solving / B. problem solving) puzzle?

Quiz Answers

Martin holds a Master’s degree in Finance and International Business. He has six years of experience in professional communication with clients, executives, and colleagues. Furthermore, he has teaching experience from Aarhus University. Martin has been featured as an expert in communication and teaching on Forbes and Shopify. Read more about Martin here .

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• Year round or Year-round? (Hyphen Rule Explained)

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5.7 Introduction to Thinking and Problem Solving

4 min read • december 22, 2022

Sadiyya Holsey

Haseung Jun

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So, we just went over memory, but how do we actually think and problem solve? 🤔

Problem Solving

There are two different ways by which you could solve problems:

An algorithm is a step by step method that guarantees to solve a particular problem.  

If you lost your phone📱, the algorithm might look like this: 

Remember where you put the phone last. If you don’t, go to the next step. ⤵️

Retrace your steps. If you can’t, go to the next step.⤵️

Call your phone to determine the location.

Algorithms are process oriented 🔄

Heuristics 

A heuristic is also known as a “rule of thumb.” Using heuristic is a quick way to solve a problem 💨 , but is usually less effective than using an algorithm (more error prone). Heuristics also involve using trial and error ❌

An example of a heuristic would be trying to find the x value that makes this equal true: 3x+6=24. You might plug in multiple x values until you determine the x value that works. 

Heuristics are the opposite of algorithms and are more result oriented. We use our mental set , schemas , prototypes , and concepts automatically when using heuristics .

How would you solve 3x624 using an algorithm?

Instead of using a heuristic and just plugging in answers till you find the right one, you could also solve this problem step by step using an algorithm . The steps may look like this:

Subtract 6 on both sides: 3x=18

Divide by 3 on both sides: x=6

You use a mixture of these two when taking a test and overall in everyday life activities🏃🍳.

Trial and Error

Trial and error is when you try to solve a problem multiple times using multiple methods. If you try to solve a problem one time using one method, the next time you solve it, you may use a different method. This process is repeated until a solution is reached.

Image Courtesy of Giphy .

How do we think?

A mental set is when individuals try to solve a problem the same way all the time because it has worked in the past. However, that doesn’t mean this problem solving method is applicable to the problem at hand or will work for other people. Having a mental set makes it harder to solve problems. Similarly, fixation is the inability to look at a problem with a different perspective.

Intuition is colloquially known as a “gut feeling.” It is sensing something without a direct reason and basically an automatic thought💾

When problem-solving and making difficult decisions, our brain intuits for us.

As we learn and grow, our intuition does, too. Our learned associations surface as this gut feeling that we have because of how we know the world works around us 🌎

Insight was discovered by Wolfgang Kohler. It occurs when an individual has an all-of the sudden understanding when solving a problem or learning something. It's that light bulb💡 moment!

Inductive Reasoning

Reasoning from something specific to something general, which puts your thought into concepts and groups.

Deductive Reasoning

Reasoning from something general to something specific. Think of mind-maps: you have one central idea in the middle (general) and then branch out into specific ideas.

These are usually more logical 🤔

Image Courtesy of Kristjan Pecanac .

Did you ever wonder how we get our creativity and to the extent to which it exists? Being creative is having the ability to produce ideas that are valuable. That's it; we're all creative in our own way.

There are five components of creativity 📸:

Expertise —The more knowledge we have, the more ideas we build. Knowledge is the foundation of every idea that comes about.

Generally, greater intelligence leads to a higher creativity 🎭 According to the threshold theory, a certain level of intelligence is necessary for creative work. However, it's not necessary sufficient, meaning other factors play in when it comes to creativity .

Imaginative thinking skills —In order to be creative, you must be open-minded and see things in different ways. These skills also include being able to make connections and recognize patterns in ideas.

A venturesome personality —Be willing to take risks, explore ideas, and try new things! 🧗

Intrinsic Motivation —This is to be driven by your interests and the will to explore for your own satisfaction.

A creative environment —All the above help fuel your creativity , but creativity can't exist without a supportive environment🌲

There are two different ways of thinking:

Convergent Thinking

This is the more logical way of thinking, in which we narrow the solutions to a problem till we find the best one. Convergent thinking is used in IQ and intelligence tests.

Divergent Thinking

The more creative way of thinking! You can think of this as brainstorming and diverging into different directions of thought. Rather than finding the best solution, divergent thinkers expand the number of solutions.

Divergent thinkers have a much easier time when problem solving since they have more of an open mind to trying different solutions.

Key Terms to Review ( 17 )

Creative Environment

Imaginative Thinking Skills

Intrinsic Motivation

Venturesome Personality

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Problem Solving or Problem-Solving?

The term problem-solving with a hyphen is an adjective that describes nouns in a sentence. E.g., “The problem-solving abilities of this program are excellent.” Furthermore, the term problem solving without a hyphen is a noun. E.g., “He is excellent at problem solving.”

The term problem solving can appear as two different word types, a noun or a compound adjective.

When problem-solving has a hyphen, it is an adjective describing a noun .

For example, in this sentence, problem-solving describes capabilities , which is a noun. Therefore, you need to include a hyphen.

• Her problem-solving capabilities are outstanding.

However, when you use problem solving as a noun , the term is usually not hyphenated because it doesn’t modify anything.

• She is an expert at all types of problem solving .

Furthermore, the rule of using a hyphen for the adjective but not for the noun is correct grammar according to AP Style and the Chicago Manual of Style .

Sometimes in English, the rules concerning punctuation, such as hyphens, are vague. For example, in the Cambridge and Oxford dictionaries, the noun problem-solving appears with a hyphen.

Essentially, you should always use a hyphen for the term problem-solving as an adjective. However, for problem solving as a noun, unless you follow a specific style guide like AP that does not use a hyphen, you can choose whether to hyphenate the noun. Just make sure that you are consistent with your choice.

In addition, Google Ngram shows that problem solving without a hyphen is slightly more popular in the US than with a hyphen. However, in the UK , the hyphenated version is slightly more frequent.

Now that you have learned the basics concerning the term problem solving, please keep reading the rest of the article to learn more about using the two variations of problem solving .

Problem Solving

The term problem solving as two words without a hyphen is the noun form.

In terms of use, it relates to resolving problems, which can relate to problems in a specific discipline or more general problems.

In the following examples, the term problem solving refers to the act of solving problems. Therefore, we do not need a hyphen.

• Her inability to perform problem solving on any task means that she loses her temper quickly.
• You should include problem solving as a skill on your resume.
• In team projects, effective problem solving can lead to innovative solutions and successful outcomes.
• Problem solving is not just about finding immediate answers but understanding the root cause of the issue.
• Her approach to problem solving often involves breaking down complex tasks into manageable steps.

Furthermore, you should use the above rule when following both AP Style or the Chicago Manual of Style.

Problem-Solving

The term problem-solving as one word with a hyphen is a compound word that appears before a noun. Hence, it modifies the noun .

For example, in the following sentence, the word problem-solving modifies the noun skills.

• His problem-solving skills are the best in the company.
• Her problem-solving approach is both methodical and creative, making her a valuable asset to the team.
• The workshop focuses on problem-solving techniques to address everyday challenges.
• Many employers prioritize candidates with strong problem-solving abilities in dynamic work environments.
• The game challenges children to use their problem-solving instincts to navigate various puzzles.

Furthermore, you will come across some sentences in which the noun problem solving has a hyphen.

As shown in these examples:

• You need to work on your problem-solving if you want to become an engineer.
• Improving your problem-solving is essential for success in mathematics.
• Mastering problem-solving will greatly benefit you in software development.

This is often a stylistic choice . However, in writing that follows AP Style , you should not use a hyphen with the noun form of problem solving.

Problemsolving

The word problemsolving as a single word with no hyphen or space is incorrect , and you shouldn’t use it in this format.

There are two ways you can use the term problem solving .

The first is as a noun, in which case there is no hyphen, but there is a space.

• Correct: He is excellent at problem solving because he never gives up. (noun)
• Incorrect: He is excellent at problemsolving because he never gives up. (noun)
• Correct : She has a knack for problem solving , especially in high-pressure situations. (noun)
• Incorrect : She has a knack for problemsolving , especially in high-pressure situations. (noun)

Also, you can write problem-solving as an adjective, in which case you need a hyphen.

• Correct: He is excellent at problem solving because he never gives up. (adjective)
• Incorrect: He is excellent at problemsolving because he never gives up. (adjective)
• Correct : They often turn to her for advice on problem solving in challenging situations. (adjective)
• Incorrect : They often turn to her for advice on problemsolving in challenging situations. (adjective)

That’s all you need to know about the grammar rules concerning the words problem solving , problem-solving , and problemsolving . Rest assured that your sentences will be correct if you follow these!

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Proofed Guide to AP Style

• 36-minute read
• 12th October 2023

Please note that this guide is based on the AP Stylebook, last updated June 1, 2022.

This guide does not include everything contained in the stylebook. Rather, it aims to cover the most salient points and provide details of AP Style’s approach to key editorial issues.

Dictionaries and Other Authorities

The following should be used as additional authorities to AP Style:

Webster’s New World College Dictionary (use the first spelling listed unless AP Style specifies otherwise).

New York Stock Exchange , Nasdaq or the Securities and Exchange Commission for formal company names. Use Co., Inc., Ltd., Corp. after the name as appropriate.

National Geographic Atlas of the World for place names not in Webster’s.

Punctuation

It wouldn’t be practical to list all the punctuation rules here. Instead, we’ve included those AP Style punctuation rules that might go against common practice.

Note that AP Style values consistency, so you can often determine what to do in a given situation by following related rules.

Otherwise, use standard U.S. English practice unless the client has requested another dialect.

Apostrophes

• Always use ‘s if the word does not end in the letter s.
• Singular common nouns ending in s: add ‘s: the class’s inattention, the business’s opening day .
• Singular proper names ending in s: use only an apostrophe: Paris’ history, Achilles’ heel.
• Special expressions: words that end in an s sound and are followed by a word that begins with s: for appearance’ sake, for conscience’ sake, for goodness’ sake.
• Descriptive phrases: do not add an apostrophe to a word ending in s when it is used primarily in a descriptive sense: a teachers college, a writers guide . You’ll see this a lot in organization names.
• Inanimate objects: avoid excessive personalization of inanimate objects, and give preference to an “of” construction when it fits the makeup of the sentence: mathematics’ rules >> the rules of mathematics .
• Plurals of a single letter: include an apostrophe in single-letter plurals: he learned the three R’s and brought home a report card with four A’s and two B’s .
• Possessive of Jr./Sr.: John Smith Jr.’s house .

Brackets and Parentheses

• Avoid using square brackets.
• Use parentheses sparingly; consider rewording the sentence.

Bulleted Lists

• Use em dash + space to introduce bullets (bullet points are also acceptable).
• Capitalize the first word.
• Always end in a period.
• Watch out for parallelism.
• Introduce the list with a short phrase + colon.
• Capitalize the first word after a colon only if it’s a proper noun or the start of a full sentence.
• Use a colon to introduce a direct quotation over one sentence long, either within a paragraph or as a block quote.
• Use a colon in interviews/reporting of dialogue/Q&As: Anthony: I disagree .
• Only use a serial comma if it’s required to make the meaning clear.
• Use a comma to separate an introductory clause or phrase from the rest of the sentence. It can be omitted from short introductory phrases if no ambiguity results from doing so.
• Use a comma when giving a person’s name and town/country, or between a place name and the country where it is located: John Smith, Tennessee ; The Giant’s Causeway, County Antrim .
• Use a comma to separate similar/duplicated words: The question is, is this the right path to take?
• Use a comma separator in words <999 except in certain situations (e.g., 1460 kHz ; Room 1001 ).
• i.e./e.g. are always followed by a comma.
• Use the parenthetical spaced em dash (sparingly).
• Use a dash before an author’s or composer’s name at the end of a quotation: “I’ll probably always be interested in this planet — it’s my favorite.” — Sagan.
• Do not use en dashes (use a hyphen for ranges).
• Use a space before and after an ellipsis.
• In a quotation, if the words before an ellipsis form a complete sentence, place a period before the ellipsis ( . … ).
• Follow a similar approach if another type of punctuation is required before the ellipsis (e.g., ? … ).
• When material is deleted at the end of one paragraph and at the beginning of the one that follows, place an ellipsis in both locations.

Exclamation Points

• Use sparingly — only where it’s really warranted.
• Do not include a comma after an exclamation mark in dialogue: “Stop!” she shouted .
• Use hyphens to aid clarity. If it gets too confusing, then consider rewording the sentence.
• If you’re unsure whether a hyphen is required in a compound adjective, check its use in Webster’s New World College Dictionary .
• Don’t use hyphens in phrasal verbs, but do use them in compound verbs: he backed up the vehicle vs. they double-checked the results or she air-dried the strawberries .
• Words that are usually one-word compounds should be separated when a modifier is added. However, it’s difficult to think of an example of how this would work in practice: perhaps external-mail box ?
• Do not use a hyphen with dual heritages ( Italian American ) but do use it with “Anglo-” when the word that follows is a capital ( Anglo-Saxon but Anglophile ).
• Prefixes that usually require a hyphen: self-, all-, ex-, half-.
• Suffixes that usually require a hyphen: -free, -based, -elect.
• Use a hyphen to avoid duplicated vowels (except ee): anti-establishment, preempt .
• Other examples; state-of-the-art, arm-in-arm, non-life-threatening, 5- and 6-year-olds, 10-to-12-inch needle .
• Co-: Keep the hyphen when used to create a word that indicates occupation or status ( co-author, co-host, co-worker etc).
• Don’t use hyphens in e.g., fourth grade student , unless needed to avoid confusion.
• Use a period at the end of a rhetorical question if it’s more of a sentence than a question: Why not eat all the cake.
• Use a period in initials (no space between multiple initials): J.R.R. Tolkein . Avoid using single initials ( J. Smith ) unless it’s personal preference or the first name cannot be established.

Quotation Marks

• Use double quotation marks (nested single), except in headlines (use single).
• For quotations over two paragraphs: when the quoted material in the first paragraph is a full sentence, don’t include closed quotation marks at the end of the first paragraph: She said, “I think that’s a great idea. ¶ “Everyone should get a free cake.”
• Quotation marks are not used in Q&As.
• Use quotation marks to indicate irony.
• Use quotation marks to define unfamiliar terms upon first use.
• The dash, the semicolon, the colon, the question mark and the exclamation point go within the quotation marks when they apply to the quoted matter only.

See also Quotations in the Style section of this guide.

References and Third-Party Sources

AP Style doesn’t say much about references/citations. The following are some notes about the mention of creative works in text.

See also Third-party Sources in the Style section of this guide.

• Put all names of creative works in AP Style title case. Capitalize both parts of a phrasal verb and “to” in “to be.”
• Put quotation marks around: books, articles, songs, albums, paintings, movies (including “Star Wars”, “Star Wars” Day etc.), plays, poems, operas, radio/TV programs and episodes, lectures, speeches, event names, classical music (when using its popular name, e.g., Beethoven’s “Eroica” ).
• Don’t put quotation marks around: holy books, almanacs, directories, dictionaries, encyclopedias, handbooks, sculptures, software, video/online games, classical music (official/numbered name, e.g., Beethoven’s Symphony No. 3 in E flat major ).
• Classical music: put the key in lowercase ( C sharp minor ). Put the instruments in lowercase if not part of the work’s official name but added as a description: … for harpsichord.
• AP does not use italics.
• Translate foreign titles into English unless they’re better known by their original name. Exception: musical compositions — refer to the work in the language it was sung in, unless in a Slavic language (translate works in Slavic languages).
• Capitalize “the” in newspaper/magazine names if that is the way the newspaper is known: The Times; The Economist .
• Poetry: If giving in-line, separate each line with a forward slash with a space either side ( Despite the storms, / beauty arrives like ). Follow the author’s approach to capitalizing the first word of lines.

Spelling, Capitalization and Form

Here are some (perhaps) non-standard approaches to spelling, capitalization and form found in AP Style. Only those that differ from Webster’s Dictionary (or do not clearly appear in it) are included here.

(If we’ve missed any out, please let us know!)

Tip: To confirm whether the first word in a Webster’s entry is capitalized, scroll down to look at the “other word forms” in the dictionary entry.

Preferred Terms

Here is a list of terms for which AP Style has stated a preference. Note that:

• The preferred term is given first, with the non-preferred term given in parentheses.
• Non-preferred terms that should never be used (outside of direct quotes) are *asterisked*.
• Additional information is given where needed.
• Most non-preferred terms can be used in direct quotes (with the exception of offensive or vulgar terms).
• Where AP Style doesn’t include a preferred term, the column is left blank.
• Further information about preferred terms relating to age, gender, disability and race is given in Inclusive Language in the Usage section of this guide.

Abbreviations and Acronyms

• Avoid “alphabet soup,” i.e., when there are too many acronyms for the sake of having acronyms.
• DO NOT DEFINE ACRONYMS: The Prudential Regulation Authority released a statement … The PRA said that … not The Prudential Regulation Authority (PRA) released a statement … The PRA said that …
• Give an organization’s name in full on first use then use the acronym or shortened version thereafter. If it would not be clear what the acronym is referring to, don’t use it.
• Consider the audience when deciding how/whether to introduce or explain an acronym.
• Some acronyms are acceptable on first reference (then should be defined or explained in text). Some you need to use the full term and then the acronym thereafter. Finally, some (famous ones) use the acronym throughout, no definition required. These last are listed in Exceptions , below.
• Some common abbreviations: a.m./p.m.; No.; Corp.; Sept.; B.C.; Lt. Col.; Rep.; Sen., Dr.; Ave .
• Generally speaking, only use abbreviations when they accompany the noun they relate to (a numbered address in the case of Blvd., St. and Ave.): Room No. 23; 37 B.C; Lt. Col. Johnson; Washington Blvd.; 10 Castle St.; 20 Cornwall Ave . Don’t use No. in addresses, except for No. 10 Downing St .
• Don’t use periods in acronyms, but use them in most two-letter abbreviations: U.S., U.N., U.K., M.D. etc.
• Use the following abbreviations in all cases, no need to define:

911 (emergency number) app (application) AT&T (company name) ATM (automated teller machine c.o.d. (cash on delivery) CBD (cannabidiol) CD (compact disc) CIA (Central Intelligence Agency) DNA (deoxyribonucleic acid) dpa (Deutsche Presse-Agentur GmbH) DVD (digital versatile disc), E. coli (Escherichia coli) f.o.b. (free on board) FAQ (frequently asked questions) FBI (Federal Bureau of Investigation) FM (frequency modulation) GPA (grade point average) GPS (global positioning system) HDMI (high-definition multimedia interface) IBM (company name) Interpol (International Criminal Police Organization) IQ (intelligence quotient) IRS (Internal Revenue Service) IT (Information Technology; don’t spell out in technical articles) IV (intravenous line) JPG/JPEG (Joint Photographic Experts Group) LED (light-emitting diode) mpg (miles per gallon; use with a figure, e.g., 40 mpg) MRI (magnetic resonance imaging) MRSA (methicillin-resistant Staphylococcus aureus) NAACP (National Association for the Advancement of Colored People) NASA (National Aeronautics and Space Administration) NATO (North Atlantic Treaty Organization) NBC (National Broadcasting Company) OB-GYN (obstetrician gynecologist) OPEC (Organization of the Petroleum Exporting Countries) PC (personal computer) PDA (personal digital assistant) PDF (portable document format) PT (patrol torpedo) boat PTA (parent–teacher association) Q&A (questions and answers) R&B (rhythm and blues) radar (radio detection and ranging) ROM (read-only memory) ROTC (Reserve Officers’ Training Corps) rpm (revolutions per minute; use in auto magazines etc.), S&P 500 (Standard & Poors 500) SAT (Scholastic Aptitude Test) SST (supersonic transport) SWAT (special weapons and tactics) Tass (tactical air-to-surface system) THC (tetrahydrocannabinol) TNT (trinitrotoluene) TV (television) U.K. (United Kingdom) U.S. (United States) UFO (unidentified flying object) UHF (ultra-high frequency) UNESCO (United Nations Educational, Scientific and Cultural Organization) UNICEF (United Nations Children’s Fund) UPS (United Parcel Service) Inc. URL (uniform resource locator) USB (universal serial bus) USO (united service organizations) USS (United States ship) VHF (very high frequency) VIP (very important person) Wi-Fi (wireless fidelity) XML (extensible markup language) ZIP (zone improvement plan) code

• Always abbreviate Jr./Sr. after names.
• Notable exceptions to the rule that two-letter acronyms use periods: AP (Associated Press), GI (Army-related), ID (identification), EU (European Union).
• Capitalize only the first letter or abbreviations and acronyms longer than five letters unless Webster capitalizes them. (To be honest, Webster capitalizes most of the famous ones, like UNESCO).
• Unusually capitalized acronyms: dpa (Deutche Presse-Agentur GmbH), app (application), MiG (Russian fighter plane).
• Do not abbreviate: associate, association, assistant, government, governor general, hertz (but, for kilohertz/megahertz, kHz/MHz are acceptable on second use), horsepower, International Space Station (i.e., don’t use ISS), justice of the peace, U.S. Marines (don’t use USMC), professor, route, terrace.

Scientific Names

• Scientific names: Spell out genus/species on first use, then abbreviate the genus: Canis lupus; C. lupus .
• The genus is in uppercase, the species in lowercase.

Academic Matters

Academic Titles

• Avoid abbreviations after people’s names: John Smith , M.Sc. , who has a master’s in business administration .
• Use abbreviations after people’s names if doing otherwise would be very cumbersome (e.g., if listing lots of people or in a table).
• Use abbreviations only after a full name, never after just a surname.
• Use Dr. in front of someone’s name who has a medical qualification, but only on the first mention.
• Don’t use Dr. for non-medical qualifications: Jane Smith, who has a doctorate in Spanish literature, will be joining us later.
• Professor: Never abbreviate. Lowercase before a name (except Professor Emeritus).

Courses and Departments

• Capitalize course titles and use Arabic numbers after them: Biology 104 . Otherwise put into sentence case: he had an interest in biology .
• Academic departments: Use capitals only for proper nouns or when the department is given its full and proper name: the history department, the department of Spanish literature, University of Oxford English Department .
• Military academies: retain capitalization even if the “U.S.” is dropped: U.S. Air Force Academy/the Air Force Academy . Lowercase academy on its own.

Qualifications

• Bachelor’s, master’s degree — but Bachelor of Arts , Master of Science .
• GED is an adjective, not a noun: GED diploma , for example.

Capitalization

• Avoid unnecessary capitals.
• Generally, capitalize common nouns when they form part of a formal name or title ( the River Nile ); mentions not written in full (e.g., the river) would be made in lowercase.
• Generally, capitalize single nouns when they form part of a formal name/title/designation and lowercase plural ( Size 12 but sizes 12 and 14 , the Amazon and Nile rivers . The exception is formal titles + full names: Presidents Barack Obama and Donald Trump .
• Similarly: Chapter 3/next chapter; Channel 4/the other channel; the U.S. Census/census data; Captain Jones/the team captain etc.
• Capitalize words that are derived from a proper noun and still depend on it for their meaning: Christian, French, Marxist .
• Lowercase words that are derived from a proper noun but no longer depend on it for their meaning: french fries, venetian blind, epicurean .

The Arts and Architecture

• Buildings: Only capitalize “building” if it’s a part of the building’s name: the Shard building; the Empire State Building .
• Artistic/literary/dramatic works: Go in AP Style title case. See References in the Punctuation section.
• Lowercase art styles ( impressionism, modernism, cubism ).
• Capitalize artistic periods ( Renaissance, Gothic, Baroque ).
• Capitalize brand/product names used in common parlance: Mace , Frisbee , Breathalyzer etc. Lowercase those that have now become common terms (e.g., linoleum ). There are no instructions to say exactly how AP determines which is which.
• Use lowercase at all times for terms that are job descriptions rather than formal titles: the vice president of the company vs. Vice President Jones .
• Lowercase annual meeting .
• Capitalize the first letter of brands/products when they begin a sentence (e.g., iPhone / IPhone ).
• Capitalize Air Force, Army, Coast Guard, National Guard when referring to the U.S. versions, lowercase for other nations’ equivalents unless it’s part of their formal name.
• U.S. Civil War: Capitalize Union and Confederacy.
• North/South/East/West: Lowercase compass directions ( he went north ), capitalize compass directions when they relate to regions ( the travelers from the East were weary ; he had a Northern accent ).
• However, lowercase compass directions when describing an area of a specific country/state, unless it’s part of that country’s/state’s actual name ( northern Franc e; western Montana ; Northern Ireland ; West Virginia ; Southern California ).
• Earth: capitalize for the proper name of the planet, lowercase in all other instances.

Governance and Legislation

• Government: Always lowercase, never abbreviate: the U.S. government .
• Constitutions: U.S. Constitution/the Constitution ; e.g., French Constitution/the constitution (for constitutions of other countries); the organization’s constitution. Lowercase constitutional in all cases.
• House of Representatives: Capitalize, even when not given in full ( the House decided ). Same with U.S. Chamber of Commerce/the Chamber .
• Don’t capitalize “primary” in e.g., the New Hampshire primary .
• Tea party (lowercase for the movement generally). Capitalize when part of a group name.
• U.S. Courts: First reference e.g., U.S. Court of Appeals, 8th U.S. Circuit Court of Appeals or the U.S. Court of Appeals for the 8th Circuit all acceptable. Subsequent references: the Court of Appeals, the 2nd Circuit, the appeals court, the appellate court, the circuit court, the court . The district courts follow a similar approach to capitalization.
• Grand jury: always lowercase.
• U.S. Supreme Court/the Supreme Court.
• British Parliament: House of Commons, House of Lords on first instance, then Commons/Lords or the Commons/the Lords afterward.
• International Court of Justice on first instance, then international court/world court.
• In general, capitalize the proper name of non-U.S. legislative bodies (e.g., the Knesset ).
• United Nations: U.N., U.N. General Assembly, U.N. Secretariat, and U.N. Security Council, drop U.N. on second reference. Lowercase the assembly/the council.
• Words like nationalist, socialist etc are lowercase unless part of a party name.

Collective Nouns and Other Singular/Plural Issues

• Nouns that denote a unit take singular verbs and pronouns.
• Plural team/group names and teams/groups with no plural form take plural verbs: the Yankees are winning; the Beatles are famous; Orlando Magic are playing.
• Single team/group names take singular verbs: Queen was formed in 1970.
• Couple: in the sense of two people: the couple were married . In the sense of a unit: each couple was asked… .
• Each: takes a singular verb ( each of the options is … ).
• Emoji: the word serves as singular and plural.
• Group: takes the singular.
• Headquarters: can take singular or plural.
• Latin words: Latin-root words take the Latin ending (e.g., alumnus/i, medium/a ) unless they have taken on English endings by common usage (e.g., syllabuses ). Check Webster if unsure.
• Insignia: same for singular and plural.

AP Style datelines are a specific device used by journalists to indicate the location and date of a news story. They appear at the top of articles and take the form detailed below.

• CITY NAME, state abbreviation (or country name if outside the U.S.) date.
• Certain very well-known cities and regions don’t require a state or country.
• Never abbreviate: Alaska, Hawaii, Idaho, Iowa, Maine, Ohio, Texas or Utah .
• Never abbreviate : March, April, May, June, July .
• KANSAS CITY, Mo., May 2023.
• COLUMBUS, Ohio, July 4, 2023.
• LONDON, Sept. 2021.

Foreign Terms

• Use inverted commas to define foreign terms upon first use: “lunettes,” the French word for eye glasses .
• If a foreign word has been adopted into English, consider whether it is universally known by the intended audience. If it isn’t, then define as you would a foreign term.
• Include accent marks/diacritics when using a word in the original language, but remove them when the word is anglicized: Ou est le café? vs. Where is the cafe?
• Personal names: follow personal preferences, otherwise use the nearest phonetic equivalent in English.
• Lowercase particles (e.g., de, der, la, le, van, von ) in names unless personal preference says otherwise.
• Vodou/Voodoo: the religious/ritual practice in Haiti and Lousiana, respectively. Avoid using lowercase voodoo e.g., voodoo rituals .

Numbers, Dates, Currency, etc.

• Numbers: spell out millions, billions. In headlines, K, M, B (thousands, millions, billions) are permitted when accompanying a number. No space: 10K, 10M, 10B . Don’t use a hyphen when used adjectivally.
• Spell out 1-9. Use figures for 10 or above; preceding a unit; when referring to ages of people/animals/events/things; in tables; in statistics; in sequences ( Act 3; Size 12; Type 2 ); in mathematical usage; in military titles (unless they come after the name) and weapon names. 
• Spell out numbers at the start of a sentence. Years and number/letter combinations can be left as is.
• Spell out numbers in formal language, rhetorical quotations and figures of speech.
• Data is (general; data journalism); data are (scientific contexts).
• Fractions: two-thirds, seven-fifteenths Use decimals for figures over 1, e.g., 1.33, 20.6 .
• Minus signs: Use a hyphen as a minus sign. Spell out when relating to temperature ( minus 5 degrees Fahrenheit ).
• Plus signs: Use as part of a company/brand name. Otherwise spell out as “plus” (e.g., B-plus grade ).
• Percentages: Use numbers and %, with no space. Spell out zero percent. Spell out at the beginning of a sentence (or, preferably, reword). Spell out percentage point (NB: this is different to percentage).
• Ordinals: Don’t use superscript, e.g., 10th Ward . Don’t use ordinals in dates.
• Ranges: repeat the unit symbol, a hyphen or “to” are both acceptable e.g., 12%-14%, $3 million to $5 million .
• Ratios: the ratio was 3-to-1, a ratio of 3-to-1, a 3-1 ratio (“to” omitted when the number precedes “ratio”).
• Roman numerals: Use Roman numerals for wars and to establish personal sequence for people and animals: World War I, Native Dancer II, King George V . Also for certain legislative acts ( Title IX ).

Dates/Times

• Minutes, seconds, hours: spell out in full.
• Times: 12 p.m.; 8 o’clock; 10 minutes; 10 seconds; 8 hours (but an eight-hour day ). Use noon and midnight in place of 12 p.m./a.m.
• Days: Don’t abbreviate days of the week unless in a table; when you do, use the three-letter form with no period ( Mon, Tue ).
• Months: When used as part of a date, abbreviate , Feb., Aug., Sept., Oct., Nov. and Dec . In tables, use three-letter forms for all months, no period.
• Years: 1920s, ‘95, the ‘80s, 2022-23 . It’s OK to start a sentence with a year. Don’t include years when referencing an occurrence in the same year as the story ( the pilot will air on 25 February ).
• Centuries: use numbers for 10 and over ( 10th century; seventh century ).
• Dates: Friday, Sept. 1, 2023; Sept. 1; Sept. 1, 2023; Sept. 2023.
• Biannual = twice a year; biennial = every two years; bimonthly = every two months; semimonthly = twice a month; biweekly = every two weeks; semiweekly = twice a week.
• Time zones: include the time zone if the item: involves TV/radio programs (always EDT/EST); has no dateline; is an advisory to editors. Do not convert clock times from other time zones in the U.S. to Eastern time. If there is high U.S. interest in the precise time something happened, add CDT, PST etc. to the local time zone so that readers can determine what time it happened in their equivalent local time. If the time is needed to make sense outside the U.S., provide a conversion to Eastern time in parentheses: … 9 a.m. (3 a.m. EDT) .
• Don’t use today/tomorrow/tonight etc. in news stories; use a day, date, etc. as appropriate. Avoid next/last (e.g., next Monday ). 

Temperatures

• 40 degrees Celsius / 40 C; 105 degrees Fahrenheit / 105 F; 10 kelvins / 10 K .
• Zero is always spelled out.
• US currency: use e.g., $5, $500, $50,000 (rather than five dollars, five-hundred dollars etc.). For numbers up to $1, use e.g., 5 cents, 97 cents , then e.g., $1.24 . For amounts over $1 million, use up to two decimal places as appropriate: $5.25 million; a $100 million budget .
• Currency conversions: give non-USD amounts in parentheses the first time a currency is mentioned: The company made a profit of $12.5 million (£10.3 million) last year . Use USD after that. Only do so for current amounts, not historical, and note that the conversion is at the current exchange rate, if necessary for clarity). 
• Non-US dollars: e.g., CA$1 million, HK$250,000 .
• Currencies are given in sentence case ( euro, dollar, pound, Canadian dollar , etc.).

Measurements

• Units: use the system (metric/imperial) most widely accepted in the location of the dateline. Do not use unit symbols except mm in the case of film widths and weapons ( a 9 mm pistol ).
• Avoid decimals unless a greater level of precision is necessary. Use up to two decimal places if necessary, except for blood alcohol level and baseball batting averages (these take 3 decimal points). 
• Measurements: 5 feet 3 inches tall; a 5-foot-3-inch man; 3,000 square feet, 4 miles . Use e.g., 5’3” only in very technical contexts.
• Tons: There are three types of ton — short ton (2,000 pounds); long/British ton (2,240 pounds); metric ton (1,000 kilograms).
• Two-by-four (piece of wood). Always spell out.

Miscellaneous

• Road numbering examples: S. Highway 1, Route 66, Route 3A, Interstate 40 ( 1-40 subsequently ).
• Votes: 7-3, but a four-vote margin .
• Serial numbers: Use figures/capital letters, no hyphens or spaces unless absolutely required. Social Security numbers are hyphenated: 123-45-6789 .
• Clothes sizes: size 10 pants; size 12 long; size 6 1/2 shoes; 16 1/2 inch neck, XL sweatshirt . 
• Tanks: M-60
• Telephone numbers: 123-456-7890 (national); 011-44-20-8535-1515 (international — 011 when calling from the U.S., country code, city code (minus the first zero) and telephone number); 800-111-1000 (toll-free numbers); 123-456-7890, ext. 415 .
• Radio/TV programs: Always use Eastern Time, and put EDT/EST after the time, as appropriate.

Inclusive Language

• Avoid tokenism and generalizations, recognize conscious and unconscious biases, avoid placing White/straight/non-disabled as the figurative norm. 
• Be conscious of who the story uses as expert witnesses, general witnesses, subjects of photos and videos. When covering issues related to marginal groups, home in on individual voices/stories focused on that group. 
• Recognize the difference between first- and second-hand experiences. 
• Be careful with the biases indicated by carelessly used language choices. 
• Make your content accessible (consider text, graphics, video). 
• Use Plain English.

Disabilities

• Check whether the individual/group prefers identity first (an autistic person) or person-first (a person with autism) language. Similarly: person with disabilities/disabled person.
• When preferences can’t be established, use a mixture of person-first and identity-first language.
• Generally, only mention disabilities and other conditions if it is directly relevant to the story.
• Avoid e.g., he is battling cancer >> he has cancer; a victim of heart disease >> she has heart disease.
• Do not use handi-capable, differently abled, physically challenged, handicapped, handicap.
• Avoid disorder, impairment, abnormality, special (unless part of a technical name for a condition).
• Refer to a disability only if relevant to the story and a medical diagnosis has been made or the person uses that term to describe themselves.
• Avoid writing that implies ableism: the belief that the abilities of people who aren’t disabled are superior.
• Avoid “inspiration porn” that implies that people with disabilities are objects of pity or wonder.
• Avoid using disability-related words casually or in unrelated situations (e.g., demented, psychotic, lame, blind, retarded, on the spectrum).
• Don’t use cliches (inspiring, brave etc.)
• Don’t use dehumanizing mass nouns (the disabled, the blind etc.)
• Don’t use normal/typical to describe someone who doesn’t have a disability (instead, use nondisabled, those without a disability). Able-bodied should be used only in instances where it has a specific meaning.
• Avoid the terms high/low functioning; instead, be specific about the condition.
• Mental illness: as with other disabilities/conditions, be specific; don’t say that someone was “mentally ill,” name their condition. Don’t use words such as demented, psychotic (including outside of a mental-health context). Don’t use terms lightly/casually, e.g., “I’m feeling very OCD.”
• Neurodiversity, -divergence, -diverse: Use these terms only in quotations.
• Wheelchair user (not wheelchair-bound or similar). Mention only if relevant to the story.
• Capitalize “deaf” when used to refer to the Deaf community (check the appropriateness of this term).
• Depending on individual preference, “dwarf” is acceptable (or person with dwarfism, little person).
• Autism: don’t use ASD, on the spectrum, Asperger’s syndrome (outside of direct quotes/due to personal preference). Don’t use as a noun (an autistic/autistics) unless it is personal preference.
• Lou Gehrig’s disease/ALS/amyotrophic lateral sclerosis in the U.S. Motor neuron (or neurone) disease outside of the U.S.
• Do not presume maleness when constructing a sentence.
• Do not use e.g., his/her, his or her .
• Use the gender-neutral “them” when necessary/personal preference/to hide someone’s identity.
• Where possible, reword sentences to avoid using “they” as an alternative to “his or her.”
• Be careful when using woman/women and female, as female is seen as purely describing sex, not gender, which can have an effect on representations of gender identity.
• Use terms that can apply to any gender in general parlance: businessperson, business owner, police officer, city leaders, confidant, workforce.
• However, avoid torturous constructions like snowperson .
• It’s alumnus/alumni (male); alumna/alumnae (female); alum/alums (neutral).
• Hair color: use the adjectives blond, brown (not blonde, brunette).
• See preferred terms for examples of gendered terms and their alternatives.
• Do not use the pronoun “her” in reference to nations, ships, storms or voice assistants.
• The terms “husband” and “wife” are acceptable in any legally recognized marriage. Spouse or partner can be used as gender-neutral options, if requested/preferred.
• Do not use lady/gentleman as a synonym for woman/man.
• Pregnant women/girls or women/girls seeking abortions are acceptable phrasings. Use pregnant people if needed to acknowledge transgender or non-binary pregnant people, but don’t use clinical terms like “people with uteruses.”
• When using “they” as a singular pronoun, explain if it isn’t clear from the context. The singular reflexive “themself” is acceptable.
• Be careful about using the term “boys” to refer to young Black men or children; use child/teen/youth as appropriate.
• Never use the N-word (including in this form) except when it is absolutely crucial to the story or an understanding of a news event.

Titles and Names

• As a general rule, capitalize titles/roles when applied to a name, not when they are used generally. I mam Shamsi Ali/the imam; Director Diane Carter/the director, Queen Anne/the queen , etc.
• Legislative titles can sometimes be omitted if the individual is well-known.
• Don’t use courtesy or honorary titles (as a general rule). Note that surgeons in the U.K. use Mr./Mrs./Ms. instead of Dr.
• Judge (law): use in front of the judge’s name upon first instance, but not thereafter — federal Judge John Smith; U.S. District Judge John Smith; Chief Judge John Smith . “Justice” is used instead in some jurisdictions.
• Applicable religious titles: see the guidance on Writing Explained .
• First lady/first gentleman is always in lowercase.
• Military titles: See the full guidance on the AP Stylebook blog .
• Governor: Abbreviate to Gov. in front of someone’s name. Do not abbreviate governor general.
• Representatives/senators: Rep./Reps., Sen./Sens . Add U.S. or state if necessary to avoid confusion. Only use such titles on first mention. Don’t use Congressman/Congresswoman before a name/as a title.
• See the Academic Matters section, above, for information about academic titles.
• Where things are unclear, the individual’s personal preference/usual habits always take precedence.
• In general, use only last names on second reference (use both names if necessary for clarity). Call children <15 yo by their first name on second reference, unless it’s a serious story such as a murder case. Use your judgment for 16/17 yos.
• Arabic: Two/three names on first mention/surname on second.
• Portuguese: (usually) given name, mother’s surname, father’s surname/father’s surname (but e.g., Canto e Castro if ‘e’ (and) is used).
• Russian: use the closest phonetic equivalent in English, if available; otherwise spell phonetically.
• Spanish: (usually) given name, father’s surname, mother’s surname/father’s surname
• China: Deng Xiaoping/Deng
• North Korea: Kim Jong Un/Kim
• South Korea: Sung Jinwoo/Sung . If given name is hyphenated, second part is lowercase – e.g., Hyo-ri . 
• Jesus: Jesus Christ or Christ is acceptable. Pronouns should be in lowercase.
• Muhammad: Use this spelling in relation to the Prophet Muhammad unless in a title/name of an organization. Prophet is lowercase when used on its own. 
• Match the headline tone to that of the story.
• Attribute carefully.
• Consider keywords and SEO.
• Update online headlines to reflect the latest news, as needed.
• Headlines go in sentence case (capitalize first word after a colon).
• Always capitalize the first letter of a headline (consider rewording if e.g., eBay is the first word).
• Avoid all but universally recognizable abbreviations and acronyms.
• Put no periods in US/UK/EU/UN when they appear in headlines, and avoid using them in other acronyms.
• Avoid using state abbreviations in headlines, but if you must, remove periods from those with two capital letters (others retain periods).
• Use numerals except in casual use or formal names ( thousands not 1000s; Big Ten; but Forbes 500 ).
• Spell out ordinals under 10, use numerals for over 10 (except eleventh hour ).
• Use single quote marks.
• Label opinion ( Opinion: Headline ), analysis ( Analysis: Headline ) and review ( Review: Headline ) pieces.
• Limit headlines to 100 characters.
• Write headlines for a global audience. Only use locators when it will increase readership, improve SEO, or is needed for clarity.
• Avoid Co. for company.
• Fed is acceptable for Federal Reserve.
• It’s possible to abbreviate millions/billions in headlines: $30M/$5B .
• Financial quarters: Q3 , not 3Q , for example.

In-Line Style Points

• Watch for sentences with more than one comma or clause. Consider splitting them up.
• Dashes and semicolons are often an indication that the sentence could be split up.
• Don’t use cliches, jargon, or bureaucratese.
• Define terms that could be unfamiliar to readers, or choose simpler terms.
• Use mostly active voice.
• Bracket additional information about a noun in commas: John Smith, of New York, ; located in Hartford, Connecticut, ; Kevin Jones, who has a Ph.D. in aeronautics, ”.
• Jargon: Avoid, unless writing for an audience that would be familiar with the term or phrase.
• Irrelevance: Avoid tautology and irrelevance: e.g., they went to a local hospital >> they went to a hospital .
• Euphemisms: avoid euphemisms wherever possible: died rather than passed away ; recreational cannabis (for example) rather than adult-use cannabis .
• Days of the week: Do not use “on” before a day of the week (e.g., the store will open Monday ).
• That, which: Follow Proofed’s approach to that/which.
• Versus: Spell out versus in longer sentences, vs. in short pithy expressions, v. in court cases.

Age-Related Nouns

• Embryo: up to 10 weeks of pregnancy.
• Fetus: 10 weeks of pregnancy — birth.
• Unborn baby: any time from gestation to birth (less clinical than fetus).
• Infant: up to 12 months old.
• Boy/girl: from 12 months old — 18 years. Use child/youth/teen if more appropriate.
• Youth: 13-18.
• Adult: over 18 years old.

Obscenities, Profanities, Slurs etc.

• Do not use except in direct speech.
• Try to describe them rather than use them directly (e.g., a racial/sexist slur ).
• For very offensive language, replace all but the first letter with hyphens, e.g., f—- . 
• An alternative is to replace the offensive word with a generic description in parentheses, e.g., (obscenity) .
• Never alter quotations even to correct minor grammatical errors or word usage.
• Casual minor slips can be removed using ellipses, but do so with caution.
• Do not use “sic”. Instead, paraphrase if possible or use the quotation exactly if the quotation is essential.
• Don’t try to replicate dialect with words such as gonna .
• When quoting spoken words, present them in the format that reflects AP style, e.g., $20, 1 Church St .. Don’t make any other changes for style, however.
• Use quotations if they are the best way to convey the text. Often, paraphrasing is preferable.
• Avoid fragmentary quotations. For cumbersome or awkward speech, leave quotation marks for sensitive or controversial statements that must be shown to come from the speaker.
• Describe emojis used in text: e.g., … following that with an emoji of a birthday cake .

Third-Party Sources

• Who does the source belong to? Avoid unverified sources.
• Clearly state the source.
• How accurate does the source appear to you?
• Any obvious signs of bias?
• Be mindful of Photoshop and deepfakes.
• Use common sense.

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About learnapphysics.

This site was developed to help AP Physics students of both B and C courses develop their skills in physics problem-solving, by:

• providing access to resources that students might not otherwise have, and
• offering lots of opportunities to practice solving AP-style problems.

This site is maintained by Richard White , an AP Physics teacher in southern California. I am happy to correspond with physics students and teachers—particularly if you notice an error in a problem!—but am unable to offer tutoring or individual problem-solving advice. I am a full-time instructor, and working with my own students keeps me very busy.

The expressions "AP" and "Advanced Placement Program" are registered trademarks of The College Board , which was not involved in the production of, and does not endorse, this site. Copyrighted materials from The College Board are not available via this website.

2.6 Problem-Solving Basics for One-Dimensional Kinematics

Learning objectives.

By the end of this section, you will be able to:

• Apply problem-solving steps and strategies to solve problems of one-dimensional kinematics.
• Apply strategies to determine whether or not the result of a problem is reasonable, and if not, determine the cause.

Problem-solving skills are obviously essential to success in a quantitative course in physics. More importantly, the ability to apply broad physical principles, usually represented by equations, to specific situations is a very powerful form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations, whereas a list of facts cannot be made long enough to contain every possible circumstance. Such analytical skills are useful both for solving problems in this text and for applying physics in everyday and professional life.

Problem-Solving Steps

While there is no simple step-by-step method that works for every problem, the following general procedures facilitate problem solving and make it more meaningful. A certain amount of creativity and insight is required as well.

Examine the situation to determine which physical principles are involved . It often helps to draw a simple sketch at the outset. You will also need to decide which direction is positive and note that on your sketch. Once you have identified the physical principles, it is much easier to find and apply the equations representing those principles. Although finding the correct equation is essential, keep in mind that equations represent physical principles, laws of nature, and relationships among physical quantities. Without a conceptual understanding of a problem, a numerical solution is meaningless.

Make a list of what is given or can be inferred from the problem as stated (identify the knowns) . Many problems are stated very succinctly and require some inspection to determine what is known. A sketch can also be very useful at this point. Formally identifying the knowns is of particular importance in applying physics to real-world situations. Remember, “stopped” means velocity is zero, and we often can take initial time and position as zero.

Identify exactly what needs to be determined in the problem (identify the unknowns) . In complex problems, especially, it is not always obvious what needs to be found or in what sequence. Making a list can help.

Find an equation or set of equations that can help you solve the problem . Your list of knowns and unknowns can help here. It is easiest if you can find equations that contain only one unknown—that is, all of the other variables are known, so you can easily solve for the unknown. If the equation contains more than one unknown, then an additional equation is needed to solve the problem. In some problems, several unknowns must be determined to get at the one needed most. In such problems it is especially important to keep physical principles in mind to avoid going astray in a sea of equations. You may have to use two (or more) different equations to get the final answer.

Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units . This step produces the numerical answer; it also provides a check on units that can help you find errors. If the units of the answer are incorrect, then an error has been made. However, be warned that correct units do not guarantee that the numerical part of the answer is also correct.

Check the answer to see if it is reasonable: Does it make sense? This final step is extremely important—the goal of physics is to accurately describe nature. To see if the answer is reasonable, check both its magnitude and its sign, in addition to its units. Your judgment will improve as you solve more and more physics problems, and it will become possible for you to make finer and finer judgments regarding whether nature is adequately described by the answer to a problem. This step brings the problem back to its conceptual meaning. If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to mechanically solve a problem.

When solving problems, we often perform these steps in different order, and we also tend to do several steps simultaneously. There is no rigid procedure that will work every time. Creativity and insight grow with experience, and the basics of problem solving become almost automatic. One way to get practice is to work out the text’s examples for yourself as you read. Another is to work as many end-of-section problems as possible, starting with the easiest to build confidence and progressing to the more difficult. Once you become involved in physics, you will see it all around you, and you can begin to apply it to situations you encounter outside the classroom, just as is done in many of the applications in this text.

Unreasonable Results

Physics must describe nature accurately. Some problems have results that are unreasonable because one premise is unreasonable or because certain premises are inconsistent with one another. The physical principle applied correctly then produces an unreasonable result. For example, if a person starting a foot race accelerates at 0 . 40 m/s 2 0 . 40 m/s 2 for 100 s, his final speed will be 40 m/s (about 150 km/h)—clearly unreasonable because the time of 100 s is an unreasonable premise. The physics is correct in a sense, but there is more to describing nature than just manipulating equations correctly. Checking the result of a problem to see if it is reasonable does more than help uncover errors in problem solving—it also builds intuition in judging whether nature is being accurately described.

Use the following strategies to determine whether an answer is reasonable and, if it is not, to determine what is the cause.

Solve the problem using strategies as outlined and in the format followed in the worked examples in the text . In the example given in the preceding paragraph, you would identify the givens as the acceleration and time and use the equation below to find the unknown final velocity. That is,

Check to see if the answer is reasonable . Is it too large or too small, or does it have the wrong sign, improper units, …? In this case, you may need to convert meters per second into a more familiar unit, such as miles per hour.

This velocity is about four times greater than a person can run—so it is too large.

If the answer is unreasonable, look for what specifically could cause the identified difficulty . In the example of the runner, there are only two assumptions that are suspect. The acceleration could be too great or the time too long. First look at the acceleration and think about what the number means. If someone accelerates at 0 . 40 m/s 2 0 . 40 m/s 2 , their velocity is increasing by 0.4 m/s each second. Does this seem reasonable? If so, the time must be too long. It is not possible for someone to accelerate at a constant rate of 0 . 40 m/s 2 0 . 40 m/s 2 for 100 s (almost two minutes).

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Mathematical Problem-Solving Style and Performance Of Students

• Luvie Jhun S. Gahi
• Ronald E. Almagro
• Richie Ryan C. Sudoy
• Dec 21, 2023

Mathematical Problem-Solving Style and Performance of Students

1 Luvie Jhun S. Gahi, 2 Ronald E. Almagro, 3 Richie Ryan C. Sudoy

1 Student, Master of Arts in Education Major in Mathematics, St. Mary’s College of Tagum,

2 Student, Master of Arts in Education Major in English, St. Mary’s College of Tagum,

3 Student, Doctor of Philosophy in Educational Management, Davao del Norte State College, Philippines

DOI: https://dx.doi.org/10.47772/IJRISS.2023.7011142

Received: 10 October 2023; Revised: 17 November 2023; Accepted: 20 November 2023; Published: 21 December 2023

This study aimed to determine whether the mathematical problem-solving style significantly affects the students’ performance in which a descriptive-correlational research design was used. Through stratified sampling, there were 291 first-year college respondents in the local college in Sto. Tomas, Davao del Norte who were chosen. This study used one adapted questionnaire and one researchers-made questionnaire with Mean, Pearson r, Standard Deviation, T-test, and Analysis of Variance used as the statistical tools. The students’ mathematics attained very good performance with the mathematical problem-solving style of students’ sensing, intuition, feeling, and thinking moderately observed. The findings revealed that the mathematical problem-solving style has a significant relationship with Students’ performance. However, there is a significant difference in the mathematical problem-solving style of students when grouped according to various programs. The result also revealed that there is no significant difference in the mathematical problem-solving style of students when grouped according to sex (male and female).  Students, instructors, college administrators, and Commission of Higher Education officials (CHED) are encouraged to value the importance of mathematical problem-solving style in the performance of the students. Instructors, college administrators, and CHED officials must establish programs that will enhance the mathematical problem-solving styles and performance of students. College instructors and administrators must work collaboratively to achieve better performance in the mathematics. Therefore, these should ensure that the necessary materials, resources, activities, and differentiated instruction are provided and used to meet the students’ needs to learn and to encourage in the problem-solving style.

Keywords: Student’s Profile, Mathematical Problem-Solving Style, and Performance of Students

INTRODUCTION

Good problem-solving abilities are required for all issues emerging from daily activity or to progress through the developmental stages. Effective problem-solving style has been linked to beneficial psychological outcomes such as competence, productivity, and optimism (Carver & Scheier, 1999; Chang & D’Zurilla, 1996; Elliott, et al., 1994). Additionally, according to the National Council of Teachers of Mathematics (NCTM), problem-solving ability is an essential component of all mathematics learning. The ability to solve problems can provide significant benefits in everyday life and the workplace. However, problem-solving is not only a goal of learning mathematics, but also a major method of learning mathematical concepts.

Likewise, the process of problem-solving begins with the observation of a gap, the application, and the complete evaluation of a theory to close that loophole. Styles of problem-solving are viewed as contrasting individuals’ unique characteristics with the behaviors that people prefer to draw and concentrate on their efforts to arrive at some comprehension or awareness, generate ideas, and make plans for the work(Sutherland, 2002).In the local college of Sto. Tomas, Davao del Norte, the varying levels of student performance in mathematics courses was used in this study. Surprisingly, there seemed to be a gap in the published research addressing this issue (Almagro etal., 2023)

The ability and style to solve problems increase the students’ comfort level when solving mathematical problems and practical difficulties. In turn, having the ability to solve problems has several advantages. For instance, problem-solving style is a feature of mathematical activity and a key method for developing mathematical understanding (NCTM, 2000). This statement implies that problem-solving style is an essential component of mathematics education. Furthermore, students learn to apply their mathematical skills in various ways; they gain a deeper understanding of mathematical concepts and gain firsthand experience as a mathematician by solving problems (Badger et al., 2012). Consequently, instruction should be advanced to enable the students to recognize and address the issues they encountered in real-life situations (Phonapichat et al., 2014). Nevertheless, several research findings suggest that children struggle to solve problems because of (Herawatty et al., 2018). Thus, learning mathematics should encourage students to solve problems confidently using mathematics. Learning mathematics in school should assist the students in understanding and applying mathematics to their problems that occur in their daily lives and in the workplace. The learning program must enable students to develop new mathematical knowledge through problem-solving style, solve mathematics and other problems, implement and adjust various problem-solving strategies, and monitor and reflect on the problem-solving style (NCTM, 2000).

The existing literature acknowledges challenges in mathematical problem-solving, but there is a significant gap in understanding the specific difficulties students face and the effectiveness of different problem-solving strategies. GanzonandEdig (2022), recognize the challenges, there is a need for in-depth investigations into the categories of difficulties encountered during the problem-solving process, low academic performance during in the pandemic. Additionally, the literature notes the importance of problem-solving models (Foshay & Kirkley, 2003; Almagro & Edig,2023), but a gap exists in understanding the comparative effectiveness of these motivated learning strategies. Moreover, within Realistic Mathematics Education (RME), recognized for its real-world emphasis, there is a need to examine the specific contextual factors that contribute to or hinder students’ success in mathematical problem-solving. Addressing these gaps will provide valuable insights for supporting students in developing effective problem-solving skills in mathematics.

The objective of this study is to investigate the relationship between mathematical problem-solving styles and the performance of students. Specifically, it aims to identify the various problem-solving styles employed by students, explore the challenges they face in mathematical problem-solving, and assess the impact of different problem-solving strategies on overall performance. The study seeks to contribute valuable insights that can inform educational practices and enhance students’ proficiency in mathematical problem-solving.

Statement of the Problem

The purpose of this study was to determine the relationship between mathematical problem-solving style and performance of first-year college students in Sto. Tomas College of Agriculture, Science and Technology (STCAST)in the academic year 2022-2023.

Specifically, these research questions sought to answer the following:

1. What is the measurable level of students’ performance in solving math problems? 2. What is the quantifiable level of students’ mathematical problem-solving style, considering the dimensions of sensing, intuition, feeling, and thinking? 3. Is there a significant and measurable relationship between students’ mathematical problem-solving style and their performance? 4. Can the mathematical problem-solving style of students be significantly differentiated when grouped according to sex and program, making it a specific and measurable analysis? 5. What specific and measurable instructional interventions can be proposed based on the study’s results, ensuring relevance and time-bound applicability?

The following hypotheses was tested at a 0.05 level of significance. Specifically, this was drawn to determine whether mathematical problem-solving style of students differ in terms of their sex and program.

• There is no significant relationship between the mathematical problem-solving style and performance of students.
• There is no significant difference on the mathematical problem-solving style of the students when classified according to sex and programs.

Theoretical Framework

The study is grounded in the original problem-solving style model, rooted in the concept of psychological functions as proposed by Jung (1923) and further developed by Moon (2008) and Taylor & Mackenny (2008). This model encompasses thinking, feeling, sensation, and intuition as the four psychological functions (Ghodrati et al., 2014). Building upon this foundation, the research draws attention to gender-specific problem-solving tendencies, with Burkey and Miller (2005) finding that women often employ intuition in work settings, contrasting the emphasis on rational problem-solving linked to masculinity by Wang, Heppner, and Berry (2007). Additionally, Conner (2000) identifies women as more intuitive global thinkers, emphasizing simultaneous, interconnected processing of information. The study aligns with the belief that students’ problem-solving methods significantly influence their academic achievement and success (Poshtiban, 2007; Morton, 2001).

Conceptual Framework

The study’s conceptual paradigm, which is shown in Figure 1, summarizes the variables which composed of mathematical problem-solving style and performance of students. On the one hand, the independent variable consists of mathematical problem-solving style which includes the indicators of sensing, intuitive, feeling, and thinking. On the other hand, the dependent variable consists of the performance of students in mathematics which composed of the moderating variable, i.e., the student respondent’s profile which are classified into sex and program. The researchers’ interpretation explains what the study wants to achieve as emphasized in the figure below. As such, the study aims to assess the Mathematical Problem-Solving Style of College students, and its relationship to their performance, and produce Students’ Instructional Intervention Plan that will improve their styles in solving mathematical problems.

 Figure 1. The Conceptual Paradigm of the Study

METHODOLOGY

This section covers the study’s numerous methodologies, which include the research design, respondents, research instrument, data gathering procedures, statistical treatment of data, and ethical considerations.

Research Design

This study employed descriptive and correlational study design. Descriptive research entails gatherings of quantitative data that may be tabulated along a scale in numerical forms, such as test scores. It entails collecting data by describing occurrences and then arranging, tabulating, displaying, and summarizing the data (Glass & Hopkins, 1984). The researcher will utilize this design to determine and describe the variables employed in this study. It utilized the mean test since this aimed to measure the level of performance of students.

Correlational study, meanwhile, tried to establish correlations between two or more variables. It looked to see if a rise or drop in one variable corresponded to an increase or decrease in another (Tan, 2014). This design will be utilized by the researcher to examine and determine the existing correlations between the variables in this research.

This study was concerned with data collection utilizing adopted research instrument and a pilot-tested researchers’ made examination to evaluate the hypotheses whether the mathematical-problem solving style influences the student performance. It will test the data using the proper statistical tools. Furthermore, the study’s major objective is to distinguish between the mathematical problem-solving styles of students when they are classified by sex and program. Thus, the study intends to look into the relationship between mathematical problem-solving style and performance of freshmen students in various programs at Sto. Tomas College of Agriculture, Sciences, and Technology.

Participants of the Study

The respondents of this research were the first-year college students enrolled in bachelor of Technical and Vocational Teacher Education (BTVTED), Bachelor of Science in Agricultural Business (BSAB), Bachelor of Science in Office Administration (BSOA), and Bachelor of Public Administration (BPA)programs for the school year 2022-2023. The respondents’ total population size of this study comprises of 1,188 students coming from four (4) programs in Sto. Tomas College of Agriculture, Sciences and Technology (STCAST). Specifically, BSOA department consists of 426 first-year students, BSAB department consists of 381 first-year students, BPA department consists of 217 first-year students, and BTVTED department consists of 164 first-year students. By using Qualtrics online sample size calculator, given the identified collective population size of 1,188 students, the ideal sample size of this quantitative study will consist of 291 students in total.

Moreover, this study utilized a stratified sampling technique to determine the sample size and determine the final total number of respondents. As a result, the BSOA program has an ideal sample of 176 students, BSAB program with 94 students, BPA with 53 students, and BTVTED with 40 students.

Materials/Research Instrument

One adapted research instrument and one researcher-made examination were used in this study. This was selected and modified to match the overall objectives of the study. These research instruments were validated by a panel of experts.

Problem-Solving Style Questionnaire (PSSQ). This instrument contains a 20-item survey questionnaire comprising the six (4) components problem-solving in mathematics such as Sensing (5 items), Intuitive (5 items), Feeling (5 items), and Thinking (5 items). This questionnaire was anchored on a 5-point Likert scale ranging from 5 as strongly agree to 1 as strongly disagree.

The following parameter limits, with its corresponding descriptions, were applied for the level of students’ mathematical problem-solving style.

The instrument for performance of students in Mathematics was a pilot-tested researcher-made questionnaire worth 40-item questionnaire. This instrument had been determined to possess good psychometric validity and reliability. The value of Cronbach’s a for the total scale is 0.747. All items of problem-solving skills are acceptable.

The percentage of the test score was computed by dividing the number of correct responses over the total highest possible score by multiplying it by 70 add 30. The highest probable score to achieve will be 40.

For the level of the mathematical problem-solving skills, the following parameter was used.

Data Gathering Procedure

The necessary data was gathered in a systematic procedure, which will involve the following.

Seeking permission to conduct the study. The researcher sought approval to conduct the research project. Primarily, the researcher will acquire a letter of recommendation from the College President of Santo Tomas College of Agriculture, Sciences, and Technology (STCAST). After acceptance, the researcher submitted a copy of the recommendation to the respective Department Heads of the following four (4) Degree Programs such as Bachelor of Science in Agriculture and Business (BSAB), Bachelor of Technical Vocational Teacher Education (BTVTED), Bachelor of Science in Office Administration (BSOA), and Bachelor of Public Administration (BPA) to finalize the approval to conduct the entire study.

General orientation and seeking of consent from research respondents . The study’s conduct was to regulate by ethical values, i.e., respect for individuals, beneficence, and justice, particularly in terms of data privacy and protection. Prior to data collection, the researcher will generate informed consent or assent forms and request them from respondents by e-mail. As evidence of their voluntary involvement in the full study, all forms will be delivered and signed electronically by those research respondents through e-mail message.

In addition, the researcher gave a brief 30-minute virtual presentation about the findings to the respondents. In accordance to this procedure, respondents who confirmed their voluntary involvement in the study were given a unique connection to Google Classroom developed by the researcher where the participants can partake in a brief virtual presentation. This was done specifically before conducting the survey. However, those respondents who were unable to attend the orientation due to unforeseen circumstances or personal reasons were educated about the research by phone call or chat through Facebook Messenger by the researcher. In addition, all respondents received a recorded video from the virtual presentation, which can be observed within the Google Classroom built by the researcher for the research.

Administration and retrieval of the questionnaire . The study took place in February of the school year 2022-2023. In order to carry out the research, the researcher will first develop Google Forms that will be utilized to collect responses from respondents based on the survey questions from the questionnaires. The quantitative data for this study was collected online using Google Forms. The researcher managed all direct contact and administration of surveys to respondents.

All surveys were allotted within one 90-minute session commencing with the Mathematical Problem-Style Questionnaire (MPSQ) and researcher-made examination. To protect the data, respondents was required to take the survey at a location and on a technological device (e.g., laptop, cellphone, or tablet) where only they have access to those offered online survey surveys via Google Forms. This was explicitly stated in their Informed Consent Form (ICF). In addition, the data questionnaire was returned to the researcher on time. Furthermore, the researcher handled personal communication and questionnaire administration. Finally, questionnaires were administered when the respondents’ 90-minute session has expired.

Checking, collating, and processing of data. The researcher gathered, validated, and quantified the respondents’ scores collected in an Excel file throughout this step. Following the tabulation, the data were submitted to an expert or certified statistician for data analysis. The researcher analyzed the results based on the data analysis for specific discoveries, discussions, and conclusions. This was accomplished mostly through data table and graphical presentations. Furthermore, descriptive statements were used to further explain and easily grasp the findings in relation to the study’s variables.

Statistical Tool for Data Analysis

The study’s findings were examined and comprehended properly using statistical methods such as Mean, Standard Deviation, Pearson r, T-test, and Analysis of Variance (ANOVA).

Mean . This method of analysis was used to measure the level of performance of students and their mathematical problem-solving style. Specifically, this was addressed in the first and second research questions.

Standard Deviation . A standard deviation is a statistical measure of the dispersion of a dataset in reference to its mean. This kind of analysis was used to determine how widely scattered the data is or how close the scores are to the mean. This was specifically answer the first and second research questions.

Pearson r. This statistical analysis was utilized to establish the existence of a significant relationship between mathematical problem-solving style and the performance of students. This will be utilized to specifically address the third research question.

T-test. This statistical analysis was utilized to determine if there was significant difference in the mathematical problem-solving style of students when classified into sex. This will specifically answer the fourth research question.

Analysis of Variance . This statistical analysis was utilized to determine if there was a significant difference in the mathematical problem-solving style of students when classified into four (4) different programs.

RESULTS AND DISCUSSION

In this chapter, the researchers present the results and discussions from the data gathered. In particular, this shows the data in tables and its corresponding descriptive interpretations.

Level of Mathematical Problem-Solving Style of Students in terms of Sensing

Table 1 presents the level of Mathematical Problem-Solving Style of Students in terms of Sensing. The item “As a student, I like to solve math problems and I am comfortable to trying to learn new skills.” has the highest mean of 3.69 with a descriptive equivalent of high. This is followed by the item “Before I put energy into solving math problems, I want to know first the benefits I can get from it.”, with a mean of 3.62 and high descriptive equivalent. On the contrary, the item “I tend to focus on immediate problems and let others worry about the distant future.” with the lowest mean of 3.20 and descriptive equivalent of moderate.

Table 1 Level of Mathematical Problem-Solving Style of Students in terms of Sensing

Furthermore, it has a category mean of 3.42 with descriptive equivalent of high. This indicates that the mathematical problem-solving style of students in terms of sensing is observed. Moreover, it has an Standard Deviation (SD) of 0.97.

The dispersion of the mathematical problem-solving style of students in terms of sensing based on the answers of the students revealed that the SD is 0.97. This indicates that the measures of variability of sensing as a mathematical problem-solving style of students are near the mean.

The result shows that students are interested in solving math problems and they are comfortable in trying new skills. It is also much observed that before solving math problems, students want to identify first the benefits they can get from it. Furthermore, students focus on immediate problems. Moreover, Vicente et al. (2002) observed that individuals with a sensation-type problem-solving style tend to focus on details and gather specific, factual data from their environment using their five senses. This approach involves a preference for concrete, practical, and tangible information, rather than abstract or theoretical concepts. These individuals tend to use a step-by-step approach to problem-solving, relying on established rules and procedures, and often prefer to work with real-world problems that have clear and immediate applications. Similarly, the study of Hsieh and Lin (2006) investigated the connection between mathematical problem-solving style and sensory preference among high school students. The study established that students with a sensing preference tended to use a more practical, sequential, and concrete problem-solving approach.

Level of Mathematical Problem-Solving Style of Students in terms of Intuitive

Table 2 presents the level of Mathematical Problem-Solving Style of Students in terms of Intuitive. The item “ As a student, I solve math problems accurately by knowing all the details of the problem.” has the highest mean of 3.68 with a descriptive equivalent of high. This is followed by the item “As a student, I enjoy solving mathematical problems.” with a mean of 3.36 and a descriptive equivalent of moderate. On the contrary, the item “ As a student, I solve mathematical problems quickly without wasting a lot of time on details.” has a mean of 3.05 with a descriptive equivalent of moderate.

Table 2 Level of Mathematical Problem-Solving Style of Students in terms of Intuitive

Furthermore, it has a category mean of 3.26 with a descriptive equivalent of moderate. It means that the mathematical problem-solving style of students in terms of intuitive is moderately observed. Moreover, the standard deviation of 1.01 in the category mean indicates that the measures of the variability of the mathematical problem-solving style of students in terms of intuition are near the mean.

It is observed that the students solve math problems accurately by knowing all the details of the problem. Additionally, it is moderately observed that students enjoy solving mathematical problems quickly and without wasting a lot of time on details. Hafriani (2018) suggests that students rely heavily on their intuition when it comes to solving mathematical problems. Students who use intuitive thinking to solve mathematical problems exhibit several characteristics: directness, self-evidence, intrinsic certainty, perseverance, coercion, extrapolation, globality, and implicitness.

Similarly, in the study conducted by Wuryanieet al., (2020) they found that students tend to rely on intuition when solving problems, exhibiting traits such as directness, self-evidence, extrapolation, intrinsic certainty, coercion, and decisiveness.

Level of Mathematical Problem-Solving Style of Students in terms of Feeling

Table 3 presents the level of Mathematical Problem-Solving Style of students in terms of feeling. The item “I want to solve math problems within a group and not individually.” has the highest mean of 3.73 with a descriptive equivalent of high. This is followed by the item “As a student, I can tell how others feel about solving math problems.” with a mean of 3.60 and a descriptive equivalent of high. On the other hand, the item “I try to please others and need occasional praise for myself.” has the lowest mean of 3.05 with a descriptive equivalent of moderate.

Table 3 Level of Mathematical Problem-Solving Style of Students in terms of Feeling

Moreover, it has a category mean of 3.48 with a descriptive equivalent of high. It implies that the mathematical problem-solving style of students in terms of feeling is observed. Consequently, the standard deviation of 0.98 in the category mean indicates that the measures of variability of the mathematical problem-solving style of students in terms of feeling are close to the mean.

Based on the results, it is observed that students want to solve math problems within a group and not individually. In addition, it is also observed that students can tell how others feel about solving math problems. It is moderately observed that in solving math problems, students try to please others and need occasional praise for their selves. A study by Goez et al. (2005) found that students must gain information and abilities related to feelings. Moreover, Altun (2003) examined the students who tend to rely on their feelings when solving problems prioritize their emotional and personal approaches in the problem-solving process. Along with this, Ahmed et al. (2014) stated that the mathematical problem-solving style has a favorable effect on the student’s attention, motivation to learn, choice of learning tools, self-regulation of learning, and academic performance.

Level of Mathematical Problem-Solving Style of students in terms of Thinking

Table 4 presents the level of mathematical problem-solving style of students in terms of thinking. The item “ As a student, I don’t let mathematics word problems discourage me, no matter how difficult they are.” has the highest mean of 3.65 with a descriptive equivalent of high. This is followed by the item “I solve math problems by analyzing all the facts and putting them in systematic order.” with a mean of 3.56 with a descriptive equivalent of high. On other hand, the item “ When I have a math problem to be solved, I solve it, even if others’ feelings might get hurt in the process.” has the lowest mean of 3.08 with a descriptive equivalent of moderate.

Furthermore, it has a category mean of 3.36 with a descriptive equivalent of moderate. This implies that the mathematical problem-solving style of students in terms of intuitive is moderately observed. Consequently, the standard deviation of 1.01 in the category means indicates that the measures of variability of the mathematical problem-solving style of students in terms of intuition are close to the mean.

Table 4 Level of Mathematical Problem-Solving Style of Students in terms of Thinking

Based on the results, it is observed that students approach mathematics word problems in a thoughtful and analytical manner. They are not easily discouraged by the difficulty of the problems and instead persevere until they find a solution. Students employ a systematic approach to problem-solving, carefully considering all the relevant facts and putting them in a logical order. This approach aligns with the findings of Khan et al. (2016), who emphasize the importance of analysis and research in effective problem-solving. Additionally, students demonstrate a commitment to objectivity and impartiality, seeking solutions that are grounded in evidence and reason. This aligns with the notion that mathematical problem-solving requires a high level of reflective thinking, as highlighted by Kneeland (2001) and Macaso and Dagohoy (2022). Moreover, the results of this study suggest that students are generally well-equipped to tackle mathematics word problems. They possess the necessary skills and mindset to approach these problems in a thoughtful, analytical, and objective manner. Thinking based on a mental process is an essential component of problem-solving solving, and problem-solving abilities are dependent on the correct application of thinking and solution processes. Furthermore, high-level thinking skills are involved in the intricate process of problem-solving (Gürsan & Yazgan, 2020).

Summary of the Level of Mathematical-Problem Solving Style of Students

Table summarizes the level of mathematical problem-solving style of students. Among the four indicators, “feeling” acquire the highest mean of 3.48 with descriptive equivalent of high. “Sensing” developed a mean of 3.42 with a descriptive equivalent of high. They have an SD of 0.98 and 0.97, respectively. It is followed by “thinking” with a mean of 3.36 with a descriptive equivalent of moderate and an SD of 1.01. On other hand, “intuitive” got the lowest mean of 3.26 with a descriptive equivalent of moderate and an SD of 1.01.

Table 5 Summary of the Level of Mathematical-Problem Solving Style of Students

Furthermore, it has an overall mean of 3.38 with a descriptive equivalent of moderate. This means that the mathematical problem-solving style of students is moderately observed. The findings suggest that there is a high degree of homogeneity in the mathematical problem-solving styles of the students, as evidenced by the small standard deviation of 0.99 in the overall mean. This indicates that the measures of variability in the students’ responses are clustered closely around the mean. Such a narrow range of variability in the problem-solving styles implies that the students have similar levels of proficiency in this variable.

Particularly, the results suggest that “feeling and sensing” as students’ mathematical problem-solving style is observed. This means that students want to solve math problems within a group and they like to solve math problems and are comfortable trying to learn new things. Moreover, “thinking and intuitive” as students’ mathematical problem-solving style is moderately observed. This indicates that students solve math problems by analyzing all the facts and knowing all the details of the problem.

This finding is supported by a recent study by TIMSS and PISA, which found that students can use their mathematical understanding and knowledge to solve problems (IEA, 2016). PISA assesses students’ ability to use their knowledge and skills in recognizing, analyzing, and solving problems in a variety of situations(OECD, 2019). Moreover, the result confirms the findings of Schoenfeld (2013), who mentioned that the mental state of students is an essential aspect of learning mathematics. The belief system of the student regarding himself, mathematics, and problem-solving determines student progress in problem-solving. This is also agreed by Hendriana et al., (2017) who mentioned that one of the fundamental mathematical abilities that students who study mathematics must develop is the ability to solve problems.

Level of Performance of Students in Solving Math Problems

Table 6 depicts the level of performance of students in solving math problems. The level of performance of students in terms of answering the 40-item Mathematics test has a mean of 60.84 with a descriptive equivalent of above average. This indicates that the performance of students in mathematics is very good.  Furthermore, it has an SD of 10.48. This demonstrates whether one’s scores on mathematical problem-solving style were extremely high or extremely low. This suggests that students’ capacity to solve mathematical problems is more likely to deviate from the mean.

The result shows that the students were able to select and correctly identify the appropriate answer to the given questions. They could choose the correct equation from the problem and accurately determine the answer in the problem scenario. Additionally, they were able to verify their responses by selecting the correct answer to the given problem.

Table 6 Level of Performance of Students in Solving Math Problems

As cited by Heris and Sumarmo (2014), problem-solving style are basic mathematical skills students need to learn. Fernandez et al. (2017) also added that problem-solving is a significant part of learning, making it of particular importance for the study of mathematics. Furthermore, another component of learning mathematics is problem-solving. This means that students need to become proficient in a variety of problem-solving style in mathematics in order to improve their creativity, reasoning, critique, and systematic thinking (NCTM, 2000). Therefore, completing mathematical problems is a crucial component of the learning objectives that must be encountered (Surya et al., 2017).

Significance of the Relationship Between the Variables

Table 7 presents the relationship between Mathematical Problem-Solving Style and the Performance of Students in Mathematics.

The correlation of Mathematical Problem-Solving Style has a significant relationship with the Performance of Students in Mathematics (p<0.05) with a coefficient determination of 0.733. Specifically, there is a strong positive correlation between the variables, and the p-value of the two variables is less than the 0.05 level of significance, which indicates that there is a significant relationship between the mathematical problem-solving style and the performance of students in mathematics (r=0.733,p=0.000. Thus, the null hypothesis is rejected.

Since the result confirmed that the mathematical problem-solving style and performance of students have a very high relationship, this means that the mathematical problem-solving style of the students significantly affects their performance in math. It can be seen from the aforementioned discussion that the mathematical problem-solving style of students is an important factor that affects their performance.

Table 7 Significance of the Relationship Between the Variables

It can be deduced that if students’ mathematical problem-solving styles will be developed, then it enhances their performance in school. This is supported by the study conducted by Suratno et al., (2020) which found that there is a significant positive correlation between students’ problem-solving styles and academic performance of students. Additionally, Mustafić et al. (2017) established that a student would perform exceptionally well in any science subject if they propose a high level of self-concept toward problem-solving skills.

The significant difference in the Mathematical Problem-solving Style of students when grouped according to sex

To determine if there was a significant difference in the mathematical problem-solving style of students when grouped according to sex (male and female), a t-test was used. Table 8 shows the result.

Using the t-test, the obtained t-value is -2.211 and the resulting p-value is 0.9860. This result indicates that the difference in mathematical problem-solving style between males and females is not statistically significant at the 0.05 level. Therefore, the researchers fail to reject the null hypothesis that there is no significant difference in mathematical problem-solving style between male and female students.

Table 8 The significant difference in mathematical-problem solving style of students when grouped according to sex

This means that students’ mathematical problem-solving style when grouped according to sex (male and female) does not vary. This finding is consistent with previous research conducted by Rusdiet al., (2020) who mentioned that men and women have similar mathematical problem-solving styles. Male and female students may understand the information, describe their knowledge, and ask relevant questions. They can use notations, symbols, and mathematical models to describe the problem and solution properly.  Moreover, Goos et al. (2017) studies examined the impact of gender differences on students’ mathematical learning outcomes have yielded inconsistent results. While some studies have demonstrated differences between genders, indicating that either men or women perform better, other studies have found no significant gender differences.

The significant difference in mathematical problem-solving style of students when grouped according to Program

Table 9 presents the significant difference in the mathematical problem-solving style of students when grouped according to four different programs: BTVTED, BSAB, BSOA, and BPA.

The result shows that there is a positive significant difference in the mathematical problem-solving style of students when grouped according to different programs since the f-value is 12.46 and the p-value is 0.000 which is lesser than 0.05 alpha level of significance. It indicates the mathematical problem-solving style of students from BTVTED, BSAB, BSOA, and BPA is significantly different. Therefore, the null hypothesis is rejected.

Table 9 The significant difference in mathematical problem-solving style of students when grouped according to Program.

The result means that students coming from four programs have different problem-solving styles in mathematics. This finding is concurrent to the first model of problem-solving styles based on the concept of psychological functions (Jung, 1923; Moon, 2008; Taylor & Mackenny, 2008). This model consists of four psychological functions as thinking, feeling, sensation, and intuition (Ghodrati et al., 2014). As such, Hacısalihlioğlu et al. (2003)stated that achieving success in problem-solving is linked to possessing abilities such as critical thinking, decision-making, reflective thinking, inquiring, analyzing, and synthesizing. This is further confirmed by Wen-Chun et al. (2015) that students use different problem-solving styles in solving math problems.

SUMMARY, CONCLUSION, AND RECOMMENDATIONS

This chapter presents the summary of the major findings of the study, the conclusion, and the proposed recommendations for possible implementations.

Summary of Findings

The major findings of the study are the following:

• For the level of performance of students in solving math problem, the level of performance of students in terms of answering the 40-item Mathematics test has a mean of 60.84 with a descriptive equivalent of high. This indicates that the performance of students in mathematics is very good. Furthermore, it has an SD of 10.48. This demonstrates whether one’s scores on mathematical problem-solving style were extremely high or extremely low.
• For the level of mathematical problem-solving style, “feeling” got the highest mean of 3.48 with an SD of 0.98. This is followed by “sensing” with a mean of 3.42 and an SD of 0.97. Both indicators got a similar descriptive equivalent of high. On other hand, “intuitive” got the lowest mean of 3.26 and an SD of 1.01 with a descriptive equivalent of moderate. Furthermore, it has an overall mean of 3.38 and an SD of 0.99 with a descriptive equivalent of moderate.
• The statistical analysis shows that there is a strong positive correlation (r=0.733) between the mathematical problem-solving style and students’ performance. The p-value of the two variables is less than 0.05, indicating that the correlation is statistically significant. This means that there is a significant relationship between the students’ mathematical problem-solving style and their academic performance. As a result, the null hypothesis is rejected.
• The statistical analysis shows that there is no significant difference (t=211, p=0.9860) in the mathematical problem-solving style of students when grouped according to sex (male and female). The result indicates that the difference in mathematical problem-solving style between males and females is not statistically significant at the 0.05 level. Therefore, the researchers fail to reject the null hypothesis that there is no significant difference in the mathematical problem-solving style of students when grouped according to sex (male and female). Meanwhile, there is a significant difference (f=12.46, p=0.000) in the mathematical problem-solving style of students when grouped in accordance to different programs since the f-value is 12.46 and the p-value is 0.000 which is lesser than 0.05 alpha level of significance. This indicates the mathematical problem-solving style of students from BTVTED, BSAB, BSOA, and BPA is significantly different. Therefore, the null hypothesis is rejected.
• Based on the results of the study, here are the instructional intervention plan that can be proposed:
• For thinking and intuitive problem solvers – Provide explicit instruction on how to justify mathematical solutions using evidence and logical reasoning. Encourage intuitive problem-solvers to verbalize their thought process and explain how they arrived at their solution. Moreover, provide opportunities for students to develop their critical thinking skills through activities such as puzzles, brain teasers, and logic games. Encourage students to explain their reasoning and thought processes when solving problems.
• For sensing and feeling problem solvers – Provide opportunities for collaborative problem-solvers to work in pairs or small groups to solve problems. Encourage the students to take turns explaining their thought process and to ask questions to deepen their understanding. Provide opportunities for independent problem-solving and self-directed learning while allowing for collaborative work. Furthermore, provide support and guidance as needed to help students build their problem-solving skills. Gradually remove support as students become more confident and independent.

The findings from the study led the researcher to draw the following conclusions:

• Performance of students in solving math problems is very good.
• Mathematical problem-solving style of students is moderately observed.
• There is a significant relationship between mathematical problem-solving style and the performance of students.
• There is no significant difference in the mathematical problem-solving style of students when grouped according to sex. However, students’ mathematical problem-solving style is significantly different when grouped according to different programs.
• Understanding students’ different mathematical problem-solving styles can help instructors tailor their instruction to meet the needs of different learners and create a more inclusive classroom environment.

Recommendations

Based on the findings, analysis, and conclusion drawn in this study, the following recommendations were summarized:

• Students are encouraged to learn effectively and independently in solving mathematical tasks. They may discover that strengthening their different mathematical problem-solving style would boost their performance in math. This can be achieved by helping them discover their different problem-solving styles and identifying strategies that can assist them in solving mathematical problems. By doing so, they can improve their performance in mathematics and unleash their full potential in this subject area.
• Instructors, college administrators, and local college officials are urged to develop enrichment activities to help their students in developing their mathematical problem-solving styles. This is especially true when it comes to encouraging students to take the initiative, establish their own mathematical problem-solving strategy, set learning goals, and assess their abilities to specify the sources they need to learn, particularly in mathematics. Teachers may create engaging instructional intervention programs to deepen students’ interests in problem-solving in Mathematics. Furthermore, they can engage in and be creative with technology, mentoring, and coaching students who are having difficulty completing mathematical problems.
• The current study’s findings emphasize the relevance of problem-solving style in mathematics and provide ways to apply and improve it. The establishment of a curriculum and instructional strategy for applying it to students is required so that teachers and students can continue and maximize their mathematical problem-solving style. To ensure that these tactics are embedded in students, ongoing efforts will be required. To maximize students’ problem-solving style in Mathematics, STCAST instructors and administrators should work collaboratively to achieve the objective. They should ensure that the necessary materials, resources, activities, and differentiated instruction are available and used to meet students’ needs to be motivated in learning.
• Future research for developing other intervention programs needed to identify the factors that might improve the mathematical problem-solving style to enhance students’ performance.

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Advances in Systematic Creativity pp 71–85 Cite as

The Adaptive Problem Sensing and Solving (APSS) Model and Its Use for Efficient TRIZ Tool Selection

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In increasingly complex environments, TRIZ offers a versatile set of tools and processes for problem solving. However, when it comes to defining which TRIZ tool(s) should be used in a given problem situation, many TRIZ users appear to be uncertain and seem to follow personal preferences rather than a structured process. The Adaptive Problem Sensing and Solving (APSS) Model provides a general process that is applied for TRIZ tool selection here. Based on the Cynefin framework, which allows sensing the characteristics of a problem situation, the authors suggest identifying which of the domains are actually available and provide strategies on how to extend the number of available domains. The user can set up the problem-solving process such that the solutions will more likely satisfy the respective expectations on the solution.

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